Previous CBSE board exam questions — Mathematics (Class 10)

Previous-year CBSE board exam questions in Mathematics, with model answers. 966 question(s), grouped by chapter and skill.

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Chapter 1: Real Numbers (70)

• Assertion (A): $(\sqrt{3} + \sqrt{5})$ is an irrational number. Reason (R): Sum of any two irrational numbers…
• Prove that $2 + 3\sqrt{5}$ is an irrational number given that $\sqrt{5}$ is irrational.
• If the HCF of 210 and 55 is expressed as $210 \times 5 + 55m$, then find the value of $m$.
• Find the greatest number less than 10,000 which is exactly divisible by 48, 60 and 65.
• If $\text{HCF}(98, 28) = m$ and $\text{LCM}(98, 28) = n$, then the value of $n - 7m$ is:
• If $(-1)^n + (-1)^8 = 0$, then $n$ is:
• Which of the following is a rational number between $\sqrt{3}$ and $\sqrt{5}$?
• Prove that $\sqrt{5}$ is an irrational number.
• The least number which is a perfect square and is divisible by each of 16, 20 and 50, is :
• The sum of the exponents of prime factors in the prime factorisation of 4004 is :
• The HCF of 40, 110 and 360 is :
• Prove that $\sqrt{3}$ is an irrational number.
• If $x$ is the LCM of 4, 6, 8 and $y$ is the LCM of 3, 5, 7 and $p$ is the LCM of $x$ and $y$, then which of t…
• Find the smallest number which is divisible by both 644 and 462.
• Two numbers are in the ratio $4:5$ and their HCF is 11. Find the LCM of these numbers.
• Prove that $4 + 3\sqrt{2}$ is an irrational number given that $\sqrt{2}$ is an irrational number.
• If $\text{HCF}(2520, 6600) = 40$ and $\text{LCM}(2520, 6600) = 252 \times k$, then the value of $k$ is
• Can the number $(15)^n$, $n$ being a natural number, end with the digit 0? Give reasons.
• If $a = 2^2 \times 3^x$, $b = 2^2 \times 3 \times 5$, $c = 2^2 \times 3 \times 7$ and $\text{LCM}(a, b, c) =…
• Given $\text{HCF}(2520, 6600) = 40$, $\text{LCM}(2520, 6600) = 252 \times k$, then the value of $k$ is :
• A pair of irrational numbers whose product is a rational number is :
• Now, Mukta asked some questions as given below to the students :
• The HCF of two numbers 65 and 104 is 13. If LCM of 65 and 104 is $40x$, then the value of $x$ is :
• Three bells toll at intervals of 9, 12 and 15 minutes respectively. If they start tolling together, after wha…
• Prove that $\dfrac{2 - \sqrt{3}}{5}$ is an irrational number, given that $\sqrt{3}$ is an irrational number.
• The greatest number which divides 281 and 1249, leaving remainder 5 and 7 respectively, is :
• The LCM of three numbers 28, 44, 132 is :
• If the product of two co-prime numbers is 553, then their HCF is :
• Prove that $\left(\sqrt{2} + \dfrac{\sqrt{3}}{2}\right)$ is an irrational number, given that $\sqrt{6}$ is an…
• If $p$ and $q$ are natural numbers and $p$ is a multiple of $q$, then what is the HCF of $p$ and $q$?
• Prove that $2 + \sqrt{3}$ is an irrational number, given that $\sqrt{3}$ is an irrational number.
• Find by prime factorisation the LCM of the numbers 18180 and 7575. Also, find the HCF of the two numbers.
• Three bells ring at intervals of 6, 12 and 18 minutes. If all the three bells rang at 6 a.m., when will they…
• The ratio of HCF to LCM of the least composite number and the least prime number is :
• Two numbers are in the ratio $2 : 3$ and their LCM is 180. What is the HCF of these numbers ?
• Assertion (A) : The number $5^n$ cannot end with the digit 0, where $n$ is a natural number. Reason (R) : Pri…
• Using prime factorisation, find HCF and LCM of 96 and 120.
• The traffic lights at three different road crossings change after every 48 seconds, 72 seconds and 108 second…
• The HCF of 960 and 432 is :
• For any natural number $n$, $6^n$ ends with the digit :
• The LCM of 960 and 240 is :
• The natural number 1 is :
• For any natural number n, $5^n$ ends with the digit :
• $(3 \times 11 \times 13 + 3)$ is :
• Find the length of the plank that can be used to measure the lengths 4 m 20 cm and 5 m 4 cm exactly, in the l…
• If $x = ab^3$ and $y = a^3b$, where $a$ and $b$ are prime numbers, then $[\text{HCF}(x,y) - \text{LCM}(x,y)]$…
• $\left(1+\sqrt{3}\right)^2 - \left(1-\sqrt{3}\right)^2$ is :
• In question number 19 and 20, a statement of Assertion (A) is followed by a statement of Reason (R). Choose t…
• Prove that $\sqrt{2}$ is an irrational number.
• Let $x$ and $y$ be two distinct prime numbers and $p = x^2 y^3$, $q = xy^4$, $r = x^5 y^2$. Find the HCF and…
• $\left(\sqrt{3}+2\right)^2 + \left(\sqrt{3}-2\right)^2$ is a/an
• Let $x = a^2 b^3 c^n$ and $y = a^3 b^m c^2$, where a, b, c are prime numbers. If LCM of $x$ and $y$ is $a^3 b…
• For any prime number p, if p divides $a^2$, where a is any real number then p also divides
• State true or false for each of the following statements and justify in each case :
• $\sqrt{0.4}$ is a/an
• Which of the following cannot be the unit digit of $8^n$, where n is a natural number ?
• Directions: A statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.…
• Let p, q and r be three distinct prime numbers. Check whether $p \cdot q \cdot r + q$ is a composite number o…
• Assertion (A): The perimeter of $\triangle ABC$ is a rational number. Reason (R): The sum of the squares of t…
• Find the greatest number which divides 85 and 72 leaving remainders 1 and 2 respectively.
• If $p^2 = \frac{32}{50}$, then p is a/an
• Show that $6^n$ can not end with digit 0 for any natural number 'n'.
• Find the HCF and LCM of 72 and 120.
• Find the HCF and LCM of 26, 65 and 117, using prime factorisation.
• Directions: Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select th…
• Prove that $2 - 5\sqrt{3}$ is an irrational number given that $\sqrt{3}$ is irrational.
• The dimensions of a window are 156 cm $\times$ 216 cm. Arjun wants to put grill on the window creating comple…
• If two positive integers $p$ and $q$ can be expressed as $p = 18\,a^2b^4$ and $q = 20\,a^3b^2$, where $a$ and…
• Prove that $5 - 2\sqrt{3}$ is an irrational number. It is given that $\sqrt{3}$ is an irrational number.
• In a teachers' workshop, the number of teachers teaching French, Hindi and English are 48, 80 and 144 respect…

Chapter 2: Polynomials (54)

• Observe the graph of polynomial $p(x)$. Number of zeroes of $p(x)$ is
• $\alpha$ and $\beta$ are the zeroes of the polynomial $5x^2 - 16x - 10$. Find the value of $\dfrac{\alpha}{\b…
• If $\alpha$ and $\beta$ are the zeroes of polynomial $3x^2 + 6x + k$ such that $\alpha + \beta + \alpha\beta…
• Two polynomials are shown in the graph below. The number of distinct zeroes of both the polynomials is:
• Find the zeroes of the polynomial $p(x) = x^2 + \dfrac{1}{2}x - 1$.
• If $4$ is a zero of the polynomial $p(x) = x^2 - x - (2 + 2k)$, then the value of $k$ is :
• If $p$ and $q$ are zeroes of the polynomial $p(y) = 21y^2 - y - 2$, then find the value of $(1 - p) \cdot (1…
• If $\alpha$ and $\beta$ are the zeroes of the polynomial $p(x) = x^2 - ax - b$, then the value of $(\alpha +…
• Zeroes of the polynomial $p(x) = x^2 - 3x + 4$ are:
• If $\alpha$ and $\beta$ are the zeroes of the polynomial $p(y) = y^2 - 4\sqrt{3}y + 3$, then find the value o…
• If $\alpha$ and $\beta$ are zeroes of the polynomial $5x^2 + 3x - 7$, the value of $\frac{1}{\alpha} + \frac{…
• A ball is thrown in the air so that $t$ seconds after it is thrown, its height $h$ metre above its starting p…
• The zeroes of the quadratic polynomial $2x^2 - 3x - 9$ are :
• Find the zeroes of the quadratic polynomial $x^2 - 15$ and verify the relationship between the zeroes and the…
• If a polynomial $p(x)$ is given by $p(x) = x^2 - 5x + 6$, then the value of $p(1) + p(4)$ is :
• Find the zeroes of the polynomial $4x^2 + 4x - 3$ and verify the relationship between zeroes and coefficients…
• If $\alpha$ and $\beta$ are the zeroes of the polynomial $x^2 + x - 2$, then find the value of $\dfrac{\alpha…
• The ratio of the sum and product of the roots of the quadratic equation $5x^2 - 6x + 21 = 0$ is :
• If $\alpha$ and $\beta$ are the zeroes of the polynomial $p(x) = kx^2 - 30x + 45k$ and $\alpha + \beta = \alp…
• Assertion (A) : Degree of a zero polynomial is not defined. Reason (R) : Degree of a non-zero constant polyno…
• If $\alpha$, $\beta$ are zeroes of the polynomial $p(x) = 5x^2 - 6x + 1$, then find the value of $\alpha + \b…
• If one zero of the polynomial $6x^2 + 37x - (k - 2)$ is reciprocal of the other, then what is the value of $k…
• The zeroes of the polynomial $p(x) = x^2 + 4x + 3$ are given by:
• If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $p(x) = x^2 - ax - b$, then the value of $…
• Assertion (A): The polynomial $p(x) = x^2 + 3x + 3$ has two real zeroes. Reason (R): A quadratic polynomial c…
• If $\alpha$, $\beta$ are zeroes of the polynomial $x^2 - 1$, then the value of $(\alpha + \beta)$ is :
• If $\alpha$, $\beta$ are the zeroes of the polynomial $p(x) = 4x^2 - 3x - 7$, then $\left(\dfrac{1}{\alpha} +…
• If one zero of the polynomial $p(x) = 6x^2 + 37x - (k - 2)$ is reciprocal of the other, then find the value o…
• The number of polynomials having zeroes 3 and 5 is :
• Based on the above, answer the following questions :
• The graph of $y = f(x)$ is given. The number of zeroes of $f(x)$ is :
• Directions: Two statements are given, one labelled as Assertion (A) and the other labelled as Reason (R). Sel…
• If $\alpha, \beta$ are the zeroes of the polynomial $p(x) = x^2 - 3x - 1$, then find the value of $\frac{1}{\…
• The graph of $y = f(x)$ is given. The number of distinct zeroes of $y = f(x)$ is :
• If $\alpha$ and $\beta$ are two zeroes of a polynomial $f(x) = px^2 - 2x + 3p$ and $\alpha + \beta = \alpha\b…
• If $\alpha, \beta$ are the zeroes of the quadratic polynomial $px^2 + qx + r$, then find the value of $\alpha…
• The sum and product of zeroes of a quadratic polynomial p(x) are $\frac{-1}{3}$ and 2 respectively. The polyn…
• Based on the figure given above, answer the following questions:
• Which of the following statements is true for a polynomial $p(x)$ of degree 3?
• $\alpha$ and $\beta$ are zeroes of a quadratic polynomial $px^2 + qx + 1$. Form a quadratic polynomial whose…
• Obtain the zeroes of the polynomial $7x^2 + 18x - 9$. Hence, write a polynomial each of whose zeroes is twice…
• If the zeroes of the polynomial $ax^2 + bx + \frac{2a}{b}$ are reciprocal of each other, then the value of b…
• Find the zeroes of the polynomial $p(x) = 3x^2 - 4x - 4$. Hence, write a polynomial whose each of the zeroes…
• The graph of $y = p(x)$ is given, for a polynomial $p(x)$. The number of zeroes of $p(x)$ from the graph is
• If $\alpha$, $\beta$ are the zeroes of a polynomial $p(x) = x^2 + x - 1$, then $\dfrac{1}{\alpha} + \dfrac{1}…
• Which of the following is a quadratic polynomial having zeroes $\dfrac{-2}{3}$ and $\dfrac{2}{3}$ ?
• The number of quadratic polynomials having zeroes $-5$ and $-3$ is
• If $\alpha$ and $\beta$ are the zeroes of the polynomial $x^2 - 1$, then the value of $(\alpha + \beta)$ is
• If the zeroes of the quadratic polynomial $x^2 + (a + 1)x + b$ are 2 and $-3$, then
• If the zeroes of a polynomial $p(x)$ are $-3$ and $8$, then $p(x)$ equals
• Observe the diagram and based on the above information, answer the following questions:
• If the sum of zeroes of the polynomial $p(x) = 2x^2 - k\sqrt{2}\,x + 1$ is $\sqrt{2}$, then value of $k$ is :
• The zeroes of a polynomial $x^2 + px + q$ are twice the zeroes of the polynomial $4x^2 - 5x - 6$. The value o…
• In Q. No. 19 and 20 a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct…

Chapter 3: Pair of Linear Equations in Two Variables (61)

• The value of $k$ for which the system of linear equations $\dfrac{x}{2} - \dfrac{y}{3} = 5$ and $2x + ky = 7$…
• In a class test, Veer scored 6 more than twice as many marks as Kevin scored. If one of them had scored 4 mor…
• Solve the linear equations $3x + y = 14$ and $y = 2$ graphically.
• If $x = 1$ and $y = 2$ is a solution of the pair of linear equations $2x - 3y + a = 0$ and $2x + 3y - b = 0$,…
• Vijay invested certain amounts of money in two schemes $A$ and $B$, which offer interest at the rate of 8% pe…
• The line represented by the equation $x - y = 0$ is :
• Assertion (A) : The pair of linear equations $px + 3y + 59 = 0$ and $2x + 6y + 118 = 0$ will have infinitely…
• Based on the above information, answer the following questions :
• The value of $k$ for which the system of equations $6x + y = 3k$ and $36x + 6y = 3$ have infinitely many solu…
• The value of $k$ for which the system of equations $3x - y + 8 = 0$ and $6x - ky + 16 = 0$ has infinitely man…
• Using graphical method, solve the following system of equations: $$3x + y + 4 = 0 \quad \text{and} \quad 3x -…
• Tara scored 40 marks in a test, getting 3 marks for each right answer and losing 1 mark for each wrong answer…
• The pair of linear equations $x + 2y + 5 = 0$ and $-3x = 6y - 1$ has
• If $2x + y = 13$ and $4x - y = 17$, find the value of $(x - y)$.
• Sum of two numbers is 105 and their difference is 45. Find the numbers.
• Solve the following system of linear equations graphically : $$x - y + 1 = 0$$ $$x + y = 5$$
• If $ax + by = a^2 - b^2$ and $bx + ay = 0$, then the value of $x + y$ is :
• Two lines are given to be parallel. The equation of one of these lines is $5x - 3y = 2$. The equation of the…
• Rehana went to a bank to withdraw ₹2,000. She asked the cashier to give her ₹50 and ₹100 notes only. Rehana g…
• Which out of the following type of straight lines will be represented by the system of equations $3x + 4y = 5…
• Based on the above, answer the following questions :
• If the pair of equations $3x - y + 8 = 0$ and $6x - ry + 16 = 0$ represent coincident lines, then the value o…
• The pair of equations $x = a$ and $y = b$ graphically represents lines which are:
• If the system of linear equations $2x + 3y = 7$ and $2ax + (a + b)y = 28$ has infinitely many solutions, then…
• If $217x + 131y = 913$ and $131x + 217y = 827$, then solve the equations for the values of $x$ and $y$.
• The pair of linear equations $2x = 5y + 6$ and $15y = 6x - 18$ represents two lines which are :
• Based on the above information, answer the following questions :
• The pair of equations $ax + 2y = 9$ and $3x + by = 18$ represent parallel lines, where $a$, $b$ are integers,…
• Two people are 16 km apart on a straight road. They start walking at the same time. If they walk towards each…
• If a pair of linear equations in two variables is represented by two coincident lines, then the pair of equat…
• Determine graphically, the coordinates of vertices of a triangle whose equations are $2x - 3y + 6 = 0$; $2x +…
• If the pair of linear equations : $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ is consistent and depen…
• Aarush bought 2 pencils and 3 chocolates for ₹ 11 and Tanish bought 1 pencil and 2 chocolates for ₹ 7 from th…
• Equation of another line parallel to the line represented by $2x - 6y = 7$ is :
• Solve the following system of equations graphically : $x - 2y = 3$, $3x - 8y = 7$.
• Five years ago, Adil was thrice as old as Bharat. Ten years later Adil shall be twice as old as Bharat. To kn…
• The value of '$p$' for which the equations $px + 3y = p - 3$, $12x + py = p$ has infinitely many solutions is…
• Solve the following system of equations algebraically : $30x + 44y = 10$; $40x + 55y = 13$
• The monthly incomes of two persons are in the ratio 9 : 7 and their monthly expenditures are in the ratio 4 :…
• The cost of 2 kg apples and 1 kg of grapes on a day was found to be ₹ 320. The cost of 4 kg apples and 2 kg g…
• Solve for $x$ and y : $\sqrt{2}\,x + \sqrt{3}\,y = 5$ and $\sqrt{3}\,x - \sqrt{8}\,y = -\sqrt{6}$
• Solve the following system of equations graphically : $2x - y - 2 = 0$, $-4x + y + 4 = 0$. Also, find the abs…
• A 2-digit number is seven times the sum of its digits and two (2) more than 5 times the product of its digits…
• The system of equations $2x + 1 = 0$ and $3y - 5 = 0$ has
• Solve the following pair of equations algebraically: $101x + 102y = 304$ $102x + 101y = 305$
• In a pair of supplementary angles, the greater angle exceeds the smaller by 50°. Express the given situation…
• Check whether the following pair of equations is consistent or not. If consistent, solve graphically $x + 3y…
• The value of k for which the pair of equations $kx = y + 2$ and $6x = 2y + 3$ has infinitely many solutions,
• Solve the pair of equations $x = 3$ and $y = -4$ graphically.
• Using graphical method, find whether following system of linear equations is consistent or not : $x = 0$ and…
• Half of the difference between two numbers is 2. The sum of the greater number and twice the smaller number i…
• The point of intersection of the line represented by $3x - y = 3$ and y-axis is given by
• 3 chairs and 1 table cost ₹ 900; whereas 5 chairs and 3 tables cost ₹ 2,100. If the cost of 1 chair is ₹ $x$…
• Based on the above information, answer the following questions:
• A 2-digit number is such that the product of its digits is 24. If 18 is subtracted from the number, the digit…
• Directions: Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select th…
• Solve the system of linear equations: $x = 4$ and $3x - 2y = 6$ graphically.
• Venkat can row a boat in still water at the speed of 12 km/h. He ferries tourists 15 km upstream and 18 km do…
• In the given figure, graphs of two linear equations are shown. The pair of these linear equations is :
• Solve the following system of linear equations $7x - 2y = 5$ and $8x + 7y = 15$ and verify your answer.
• Three years ago, Rashmi was thrice as old as Nazma. Ten years later, Rashmi will be twice as old as Nazma. Ho…

Chapter 4: Quadratic Equations (69)

• If the quadratic equation $9x^2 + 8kx + 16 = 0$ has real and equal roots, then the value of $k$ is
• Verify that roots of the quadratic equation $(p - q)x^2 + (q - r)x + (r - p) = 0$ are equal when $q + r = 2p$.
• A person on tour has ₹5,400 for his expenses. If he extends his tour by 5 days, he has to cut down his daily…
• The total cost of certain piece of cloth was ₹2,100. During special sale time, the shopkeeper offered 2 m ext…
• The perimeter of a right triangle is 60 cm and its hypotenuse is 25 cm. Find the lengths of other two sides o…
• A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would hav…
• The quadratic equation whose roots are $7$ and $\dfrac{1}{7}$ is :
• The sum of the areas of two squares is 52 cm² and difference of their perimeters is 8 cm. Find the lengths of…
• The time taken by a person to travel an upward distance of 150 km was $2\dfrac{1}{2}$ hours more than the tim…
• If $\dfrac{x+1}{x-1} + \dfrac{x-2}{x+2} = 0$, then the values of $x$ are:
• Find the value of $k$ for which the quadratic equation $4x^2 + kx + 1 = 0$ has real and equal roots.
• Based on the information given above, answer the following questions:
• The quadratic equation $x^2 + x + 1 = 0$ has ______ roots.
• In an A.P., the sum of three consecutive terms is 24 and the sum of their squares is 194. Find the numbers.
• In a 2-digit number, the digit at the unit's place is 5 less than the digit at the ten's place. The product o…
• In a flight of 2800 km, an aircraft was slowed down due to bad weather. Its average speed is reduced by 100 k…
• The denominator of a fraction is one more than twice the numerator. If the sum of the fraction and its recipr…
• If the discriminant of the quadratic equation $3x^2 - 2x + c = 0$ is 16, then the value of $c$ is :
• A train travels a distance of 90 km at a constant speed. Had the speed been 15 km/h more, it would have taken…
• Find the value of $'c'$ for which the quadratic equation $(c + 1)x^2 - 6(c + 1)x + 3(c + 9) = 0$; $c \neq -1$…
• The ratio of the sum and product of the roots of the quadratic equation $5x^2 - 6x + 21 = 0$ is :
• Find the value of '$k$' for which the quadratic equation $(k + 1)x^2 - 6(k + 1)x + 3(k + 9) = 0$, $k \neq -1$…
• The age of a man is twice the square of the age of his son. Eight years hence, the age of the man will be 4 y…
• Which of the following quadratic equations has sum of its roots as 4?
• The roots of the equation $x^2 + 3x - 10 = 0$ are :
• Find the sum and product of the roots of the quadratic equation $2x^2 - 9x + 4 = 0$.
• Find the discriminant of the quadratic equation $4x^2 - 5 = 0$ and hence comment on the nature of roots of th…
• Find the value of $p$ for which the quadratic equation $px(x - 2) + 6 = 0$ has two equal real roots.
• A quadratic equation whose roots are $(2 + \sqrt{3})$ and $(2 - \sqrt{3})$ is :
• A train travels at a certain average speed for a distance of 54 km and then travels a distance of 63 km at an…
• Two pipes together can fill a tank in $\dfrac{15}{8}$ hours. The pipe with larger diameter takes 2 hours less…
• Find the value of $m$ for which the quadratic equation $(m-1)x^2 + 2(m-1)x + 1 = 0$ has two real and equal ro…
• Solve the following quadratic equation for $x$: $\sqrt{3}\,x^2 + 10x + 7\sqrt{3} = 0$
• The product of Rehan's age (in years) 5 years ago and his age 7 years from now, is one more than twice his pr…
• Solve the quadratic equation for $x$: $$x^2 - 2ax - (4b^2 - a^2) = 0$$
• If the quadratic equation $(1 + a^2)x^2 + 2abx + (b^2 - c^2) = 0$ has equal and real roots, then prove that:…
• Based on the above information, answer the following questions:
• A faster train takes one hour less than a slower train for a journey of 200 km. If the speed of the slower tr…
• The sum of the areas of two squares is 640 m$^2$. If the difference in their perimeters is 64 m, find the sid…
• A person on a tour has ₹ 4,200 for expenses. If he extends his tour for 3 days, he has to cut down his daily…
• The area of a right-angled triangle is 600 $cm^2$. If the base of the triangle exceeds the altitude by 10 cm,…
• The roots of the quadratic equation $(x-1)^2 = 16$ are :
• If the roots of the quadratic equation $\sqrt{3}x^2 - kx + 2\sqrt{3} = 0$ are real and equal, then the value(…
• Find two consecutive negative integers, sum of whose squares is 481.
• The value of '$a$' for which $ax^2 + x + a = 0$ has equal and positive roots is :
• The sides of a right triangle are such that the longest side is 4 m more than the shortest side and the third…
• Express the equation $\dfrac{x-2}{x-3} + \dfrac{x-4}{x-5} = \dfrac{10}{3}$; $(x \neq 3, 5)$ as a quadratic eq…
• Which of the following equations is a quadratic equation ?
• If $x^2 + bx + b = 0$ has two real and distinct roots, then the value of b can be
• Obtain the zeroes of the polynomial $7x^2 + 18x - 9$. Hence, write a polynomial each of whose zeroes is twice…
• Find the value(s) of p for which the quadratic equation given as $(p + 4) x^2 - (p + 1) x + 1 = 0$ has real a…
• Which of the following quadratic equations has real and equal roots ?
• Find the smallest value of p for which the quadratic equation $x^2 - 2(p + 1)x + p^2 = 0$ has real roots. Hen…
• The least positive value of k, for which the quadratic equation $2x^2 + kx - 4 = 0$ has rational roots, is
• Based on the above information, answer the following questions.
• If the quadratic equation $ax^2 + bx + c = 0$ has two real and equal roots, then 'c' is equal to
• In question numbers 19 and 20, a statement of Assertion (A) is followed by a statement of Reason (R). Choose…
• The sum of two numbers is 15. If the sum of their reciprocals is $\frac{3}{10}$, find the two numbers.
• If $\alpha$ and $\beta$ are roots of the quadratic equation $x^2 - 7x + 10 = 0$, find the quadratic equation…
• Solve the quadratic equation : $x^2 + 2\sqrt{2}x - 6 = 0$ for $x$.
• The difference of the squares of two numbers is 180. The square of the smaller number is 8 times the greater…
• If the sum of the roots of the quadratic equation $ky^2 - 11y + (k - 23) = 0$ is $\frac{13}{21}$ more than th…
• If $x = -2$ is the common solution of quadratic equations $ax^2 + x - 3a = 0$ and $x^2 + bx + b = 0$, then fi…
• Find the value of 'p' for which the quadratic equation $p(x - 4)(x - 2) + (x - 1)^2 = 0$ has real and equal r…
• Had Aarush scored 8 more marks in a Mathematics test, out of 35 marks, 7 times these marks would have been 4…
• The value of $k$ for which the equation $kx^2 - 6x - 4 = 0$ has real and equal roots, is
• The value of $p$ for which roots of the quadratic equation $x^2 - px + 6 = 0$ are rational, is
• If the roots of equation $ax^2 + bx + c = 0$, $a \neq 0$ are real and equal, then which of the following rela…
• Based on the above information, answer the following questions :

Chapter 5: Arithmetic Progressions (65)

• The $n$th term of an A.P. is $\sqrt{2}\,n + 1$. Its common difference is
• Based on the above, answer the following questions:
• If the sum of first $m$ terms of an AP is $2m^2 + 3m$, then its second term is:
• Based on the information given above, answer the following questions:
• The 10th term of the AP $-5, \dfrac{-5}{2}, 0, \dfrac{5}{2}, \ldots$ is :
• Assertion (A) : Common difference of the AP : $5, 1, -3, \ldots$ is $4$. Reason (R) : Common difference of th…
• A sum of $\rupee 2{,}000$ is invested at 7% per annum simple interest. Calculate the interests at the end of…
• The sum of the third term and the seventh term of an AP is 6 and their product is 8. Find the sum of the firs…
• The minimum age of children eligible to participate in a painting competition is 8 years. It is observed that…
• The common difference of an A.P. in which $a_{15} - a_{11} = 48$, is
• Which term of the A.P. $-29, -26, -23, \ldots, 61$ is $16$?
• If the sum of first $m$ terms of an A.P. is same as sum of its first $n$ terms ($m \neq n$), then show that t…
• In an A.P., the sum of three consecutive terms is 24 and the sum of their squares is 194. Find the numbers.
• The common difference of the A.P. $\dfrac{1}{2x}, \dfrac{1-4x}{2x}, \dfrac{1-8x}{2x}, \ldots$ is :
• If the sum of first 7 terms of an A.P. is 49 and that of first 17 terms is 289, find the sum of its first 20…
• The ratio of the 10th term to its 30th term of an A.P. is 1 : 3 and the sum of its first six terms is 42. Fin…
• Three numbers in A.P. have the sum 30. What is its middle term ?
• Based on the above, answer the following questions to help the players in spotting the clues :
• The next (4th) term of the A.P. $\sqrt{18}$, $\sqrt{50}$, $\sqrt{98}$, … is :
• The number of terms in the A.P. 3, 6, 9, 12, …, 111 is :
• If the sum of the first 14 terms of an A.P. is 1050 and the first term is 10, then find the 20th term and the…
• The first term of an A.P. is 5, the last term is 45 and the sum of all the terms is 400. Find the number of t…
• How many terms of the arithmetic progression 45, 39, 33, ........ must be taken so that their sum is 180? Exp…
• The next term of the A.P. : $6$, $\sqrt{24}$, $\sqrt{54}$ is :
• Assertion (A) : $a$, $b$, $c$ are in A.P. if and only if $2b = a + c$. Reason (R) : The sum of first $n$ odd…
• How many terms are there in an A.P. whose first and fifth terms are $-14$ and $2$, respectively and the last…
• Which term of the A.P. : $65, 61, 57, 53, \ldots$ is the first negative term ?
• The common difference of the A.P. whose $n$th term is given by $a_n = 3n + 7$, is :
• The 11th term from the end of the A.P. : $10, 7, 4, \ldots, -62$ is :
• If $p$th term of an A.P. is $q$ and $q$th term is $p$, then prove that its $n$th term is $(p + q - n)$.
• Find the sum of first 30 terms of AP : $30, 24, 18, \ldots$
• In an AP if $S_n = n(4n + 1)$, then find the AP.
• Observe the pattern and answer the following questions using Arithmetic Progression:
• Find the sum of first 20 terms of an AP in which $d = 5$ and $a_{20} = 135$.
• For what value of $n$, are the $n$th terms of the APs: $9, 7, 5, \ldots$ and $15, 12, 9, \ldots$ the same?
• The common difference of the AP : $\sqrt{2}, 2\sqrt{2}, 3\sqrt{2}, 4\sqrt{2}, .....$ is :
• Based on the above information, answer the following questions :
• Which of the following sequence is *not* an A.P. ?
• Assertion (A) : The mean of first 'n' natural numbers is $\frac{n-1}{2}$. Reason (R): The sum of first 'n' na…
• Based on the information given above, answer the following questions :
• The number of multiples of 4 lying between 12 and 250 is :
• The $n^{th}$ term of the A.P. $\frac{-1}{3}, \frac{2}{3}, \frac{5}{3}, \frac{8}{3}, \dots$ is :
• In an A.P., the first term is 32 and the last term is $-10$. If the common difference is $-2$, then find the…
• Find the sum of the first 28 terms of an A.P. whose $n^{th}$ term is given by $a_n = 3n - 2$.
• Based on above information, answer the following questions using Arithmetic Progression :
• Directions: A statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.…
• Based on given information, answer the following questions using Arithmetic Progression.
• Based on given information, answer the following questions, using concept of Arithmetic Progression.
• If $p - 1$, $p + 1$ and $2p + 3$ are in A.P., then the value of p is
• The ratio of the 11th term to 17th term of an A.P. is 3 : 4. Find the ratio of 5th term to 21st term of the s…
• 250 logs are stacked in the following manner : 22 logs in the bottom row, 21 in the next row, 20 in the row n…
• If the sum of the first n terms of an A.P be $3n^2 + n$ and its common difference is 6, then its first term is
• The ratio of the 11th term to the 18th term of an A.P. is 2 : 3. Find the ratio of the 5th term to the 21st t…
• If the sum of first 6 terms of an A.P. is 36 and that of the first 16 terms is 256, find the sum of first 10…
• Which term of the A.P. $-\frac{11}{2}, -3, -\frac{1}{2}, \dots$ is $\frac{49}{2}$ ?
• Find $a$ and $b$ so that the numbers $a, 7, b, 23$ are in A.P.
• Find the sum of first 20 terms of an A.P. whose $n^{th}$ term is given as $a_n = 5 - 2n$.
• How many natural numbers are there between 1 and 1000 which are divisible by 5 but not by 2 ?
• If the last term of an A.P. of 30 terms is 119 and the 8th term from the end (towards the first term) is 91,…
• If $a_n$ represents $n$th term of the A.P. $-\frac{15}{4}, -\frac{10}{4}, -\frac{5}{4}, \ldots$ then value of…
• In an A.P., 15th term exceeds the 8th term by 21. If sum of first 10 terms is 55, then form the A.P.
• The sum of first $n$ terms of an A.P. is $2n^2 + 13n$. Find its $n$th term and hence $10$th term.
• In an A.P., if the first term $a = 7$, $n$th term $a_n = 84$ and the sum of first $n$ terms $s_n = \frac{2093…
• The sum of first and eighth terms of an A.P. is 32 and their product is 60. Find the first term and common di…
• In an A.P. of 40 terms, the sum of first 9 terms is 153 and the sum of last 6 terms is 687. Determine the fir…

Chapter 6: Triangles (93)

• It is given that $\triangle ABC \sim \triangle EDF$. Which of the following is not true?
• In the given figure, $DE \parallel BC$. If $\dfrac{AD}{DB} = \dfrac{1}{3}$ and $AC = 6$ cm, then length $AE$…
• D is a point on the side BC of $\triangle ABC$ such that $\angle CAB = \angle CDA$. Show that $CA^2 = CB \tim…
• Based on the above information, answer the following questions:
• In triangles $ABC$ and $DEF$, $\angle B = \angle E$, $\angle F = \angle C$ and $AB = 3\,DE$. Then, the two tr…
• If $\triangle ABC \sim \triangle PQR$ in which $AB = 6$ cm, $BC = 4$ cm, $AC = 8$ cm and $PR = 6$ cm, then fi…
• In the given figure, $\angle 1 = \angle 2$, show that $\triangle PQS \sim \triangle TQR$.
• The diagonal $BD$ of a parallelogram $ABCD$ intersects the line segment $AE$ at the point $F$, where $E$ is a…
• In $\triangle ABC$, if $AD \perp BC$ and $AD^2 = BD \cdot DC$, then prove that $\angle BAC = 90°$.
• In the given figure, $PQ \| BC$. If $\dfrac{AP}{PB} = \dfrac{2}{3}$ and $AC = 20\cdot4$ cm, then the length o…
• In the given figure, D is a point on the side BC of $\triangle ABC$ such that $\angle ADC = \angle BAC$. Show…
• In the given figure, $OA \cdot OB = OC \cdot OD$. Show that $\angle A = \angle C$ and $\angle B = \angle D$.
• Prove that a line drawn parallel to one side of a triangle to intersect the other two sides in distinct point…
• If in two triangles DEF and PQR, $\angle D = \angle Q$ and $\angle R = \angle E$, then which of the following…
• The measurements of $\triangle LMN$ and $\triangle ABC$ are shown in the figure given below. The length of si…
• In the given figure, in $\triangle ABC$, $AD \perp BC$ and $\angle BAC = 90^\circ$. If $BC = 16$ cm and $DC =…
• The perimeter of an isosceles triangle is 32 cm. If each equal side is $\dfrac{6}{8}$ th of the base, find th…
• In the given figure, PA, QB and RC are perpendicular to AC. If $PA = x$ units, $QB = y$ units and $RC = z$ un…
• Sides AB and BC and median AD of triangle ABC are respectively proportional to sides PQ and QR and median PM…
• In the given figure $\triangle ABC$ is shown. DE is parallel to BC. If $AD = 5$ cm, $DB = 2.5$ cm and $BC = 1…
• The perimeters of two similar triangles ABC and PQR are 56 cm and 48 cm respectively. $\frac{PQ}{AB}$ is equa…
• AB and CD are two chords of a circle intersecting at P. Choose the correct statement from the following:
• If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, th…
• Sides AB and AC and median AD to $\triangle ABC$ are respectively proportional to sides PQ and PR and median…
• If the diagonals of a quadrilateral divide each other proportionally, then it is a :
• In $\triangle ABC$, $DE \parallel BC$ (as shown in the figure). If $AD = 2$ cm, $BD = 3$ cm, $BC = 7.5$ cm, t…
• In the given figure, $\dfrac{EA}{EC} = \dfrac{EB}{ED}$, prove that $\triangle EAB \sim \triangle ECD$.
• State and prove Basic Proportionality theorem.
• In $\triangle ABC$, $DE \parallel BC$ (as shown in the figure). If $AD = 4$ cm, $AB = 9$ cm and $AC = 13.5$ c…
• In the given figure, $\triangle AHK \sim \triangle ABC$. If $AK = 8$ cm, $BC = 3.2$ cm and $HK = 6.4$ cm, the…
• $E$ is a point on the side $AD$ produced of a parallelogram $ABCD$ and $BE$ intersects $CD$ at $F$. Show that…
• Sides $AB$, $BC$ and the median $AD$ of $\triangle ABC$ are respectively proportional to sides $PQ$, $QR$ and…
• In the given figure, in $\triangle ABC$, $DE \parallel BC$. If $AD = 2 \cdot 4$ cm, $DB = 4$ cm and $AE = 2$…
• If a vertical pole of length 7·5 m casts a shadow 5 m long on the ground and at the same time, a tower casts…
• Assertion (A) : ABCD is a trapezium with $DC \parallel AB$. E and F are points on AD and BC respectively, suc…
• In the given figure, $\triangle FEC \cong \triangle GDB$ and $\angle 1 = \angle 2$. Prove that $\triangle ADE…
• Sides AB and AC and median AD of a $\triangle ABC$ are respectively proportional to sides PQ and PR and media…
• If $\triangle ABC \sim \triangle PQR$, $\angle A = 32°$ and $\angle R = 65°$, then the measure of $\angle B$…
• In the given figure, $DE \parallel BC$. If $AD = 2$ units, $DB = AE = 3$ units and $EC = x$ units, then the v…
• In the given figure, $AB \parallel PQ$. If $AB = 6$ cm, $PQ = 2$ cm and $OB = 3$ cm, then the length of $OP$…
• Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median P…
• Through the mid-point M of the side CD of a parallelogram ABCD, the line BM is drawn intersecting AC in L and…
• In the given figure, $\triangle ABC \sim \triangle QPR$. If $AC = 6$ cm, $BC = 5$ cm, $QR = 3$ cm and $PR = x…
• In $\triangle ABC$, $PQ \parallel BC$. If $PB = 6$ cm, $AP = 4$ cm, $AQ = 8$ cm, find the length of $AC$.
• D is a point on the side BC of a triangle ABC such that $\angle ADC = \angle BAC$, prove that $CA^2 = CB \cdo…
• If AD and PM are medians of triangles ABC and PQR, respectively where $\triangle ABC \sim \triangle PQR$, pro…
• In the given figure, $DE \parallel BC$. The value of $x$ is :
• In triangles $ABC$ and $DEF$, $\dfrac{AB}{DE} = \dfrac{BC}{FD}$. Which of the following makes the two triangl…
• In the given figure, $CD$ is the perpendicular bisector of $AB$. $EF$ is perpendicular to $CD$. $AE$ intersec…
• In the given figure, $ABCD$ is a parallelogram. $BE$ bisects $CD$ at $M$ and intersects $AC$ at $L$. Prove th…
• In Figure 1, a triangle ABC with $\angle B = 90^\circ$ is shown. Taking AB as diameter, a circle has been dra…
• Draw a line segment AB of length 8 cm and locate a point P on AB such that $AP : PB = 1 : 5$.
• In Figure 3, two circles with centres at O and O′ of radii 2r and r respectively, touch each other internally…
• If $\triangle ABC$ and $\triangle DEF$ are similar such that $2\,AB = DE$ and $BC = 8$ cm, then EF is equal t…
• In $\triangle ABC$, DE $\parallel$ BC. If AD = $x$, DB = $x - 2$, AE = $x + 2$ and EC = $x - 1$, then find th…
• In the figure given above, $\triangle ABC \sim \triangle XYZ$, then find the values of $x$ and $y$.
• In the given figure, CM and RN are respectively the medians of $\triangle ABC$ and $\triangle PQR$. If $\tria…
• In triangles ABC and PQR, $\angle A = \angle Q$ and $\angle B = \angle R$, then AB : AC is equal to :
• In the given figure, $\triangle AHK \sim \triangle ABC$. If AK = 10 cm, BC = 3·5 cm and HK = 7 cm, find the l…
• In the given figure, XY || QR, $\frac{PQ}{XQ} = \frac{7}{3}$ and PR = 6·3 cm. Find the length of YR.
• Prove that if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct…
• As shown in the given figure, a girl of height 90 cm is walking away from the base of a lamp post at a speed…
• In $\triangle DEF$, AB $\parallel$ EF. The value of x is :
• It is given that $\triangle ABC \sim \triangle QRP$ such that AB = 9 cm, BC = 5 cm and PR = 2 cm. Length of s…
• The diagonals of a quadrilateral ABCD intersect each other at the point O such that $\frac{AO}{OC} = \frac{BO…
• $\triangle ABC$ and $\triangle PQR$ are shown in the adjoining figures. The measure of $\angle C$ is :
• E and F are points on the sides AB and AC respectively of a $\triangle ABC$ such that $\frac{AE}{EB} = \frac{…
• In parallelogram ABCD, side AD is produced to a point E and BE intersects CD at F. Prove that $\triangle ABE…
• The corresponding sides of $\triangle ABC$ and $\triangle PQR$ are in the ratio 3 : 5. AD$\perp$BC and PS$\pe…
• State basic proportionality theorem. Use it to prove the following : If three parallel lines $l$, $m$, $n$ ar…
• In the adjoining figure, ABCD is a trapezium in which $XY \parallel AB \parallel CD$. If $AX = \frac{2}{3} AD…
• AD and PS are medians of triangles ABC and PQR respectively such that ΔABD ~ ΔPQS. Prove that ΔABC ~ ΔPQR.
• If a line is drawn parallel to one side of a triangle intersecting the other two sides in distinct points the…
• In the adjoining figure, PQ $\parallel$ XY $\parallel$ BC, AP = 2 cm, PX = 1.5 cm and BX = 4 cm. If QY = 0.75…
• Given $\triangle ABC \sim \triangle PQR$, $\angle A = 30°$ and $\angle Q = 90°$. The value of $(\angle R + \a…
• P is a point on the side BC of $\triangle ABC$ such that $\angle APC = \angle BAC$. Prove that $AC^2 = BC \cd…
• There is a circular park of diameter 65 m as shown in the following figure, where AB is a diameter. An entry…
• If a line drawn parallel to one side of triangle intersecting the other two sides in distinct points divides…
• In the adjoining figure, $\triangle CAB$ is a right triangle, right angled at A and $AD \perp BC$. Prove that…
• In the given figure, XZ is parallel to BC. AZ = 3 cm, ZC = 2 cm, BM = 3 cm and MC = 5 cm. Find the length of…
• If a line is drawn parallel to one side of a triangle to intersect the other two sides at distinct points, pr…
• If $\triangle PQR \sim \triangle ABC$; PQ = 6 cm, AB = 8 cm and the perimeter of $\triangle ABC$ is 36 cm, th…
• In the given figure, DE||BC. If AD = 3 cm, AB = 7 cm and EC = 3 cm, then the length of AE is
• In the given figure, ABC is a triangle in which DE||BC. If AD = $x$, DB = $x - 2$, AE = $x + 2$ and EC = $x -…
• Diagonals AC and BD of trapezium ABCD with AB||DC intersect each other at point O. Show that $\frac{OA}{OC} =…
• In the given figure, E is a point on the side CB produced of an isosceles triangle ABC with AB = AC. If AD $\…
• Devansh proved that $\triangle ABC \sim \triangle PQR$ using SAS similarity criteria. If he found $\angle C =…
• In the given figure, $PQ \parallel YZ$ such that $XP : PY = 2 : 3$. If $PQ = 5$ cm, then $YZ$ equals
• In the given figure, $AB \parallel DE$ and $AC \parallel DF$. Show that $\triangle ABC \sim \triangle DEF$. I…
• $D$ is the mid-point of side $BC$ of $\triangle ABC$. $CE$ and $BF$ intersect at $O$, a point on $AD$. $AD$ i…
• Through the mid-point $Q$ of side $CD$ of a parallelogram $ABCD$, the line $AR$ is drawn which intersects $BD…
• In the given figure, ABCD is a quadrilateral. Diagonal BD bisects $\angle B$ and $\angle D$ both. Prove that…
• In the given figure PA, QB and RC are each perpendicular to AC. If $AP = x$, $BQ = y$ and $CR = z$, then prov…

Chapter 7: Coordinate Geometry (79)

• In the given figure, a circle is centred at $(1, 2)$. The diameter of the circle is
• The distance between the points $(a\cos\theta + b\sin\theta,\ 0)$ and $(0,\ a\sin\theta - b\cos\theta)$ is
• Based on the above, answer the following questions:
• The mid-point of the line segment joining the points $P(-4, 5)$ and $Q(4, 6)$ lies on:
• The coordinates of the centre of a circle are $(2a, a - 7)$. Find the value(s) of $a$ if the circle passes th…
• Find the ratio in which the $y$-axis divides the line segment joining the points $(5, -6)$ and $(-1, -4)$. Al…
• The coordinates of the end points of a diameter of a circle are $(5, -2)$ and $(-5, 2)$. The length of the ra…
• The points $(-5, 0)$, $(5, 0)$ and $(0, 4)$ are the vertices of a triangle which is a/an :
• The equation of a line parallel to the x-axis and at a distance of 3 units below x-axis is :
• At point A on the diameter AB of a circle of radius 10 cm, tangent XAY is drawn to the circle. Find the lengt…
• If the mid-point of the line segment joining the points $A(3, 4)$ and $B(k, 6)$ is $P(x, y)$ and $x + y - 10…
• The line represented by $\dfrac{x}{4} + \dfrac{y}{6} = 1$ intersects x-axis and y-axis respectively at P and…
• Two of the vertices of $\triangle PQR$ are $P(-1, 5)$ and $Q(5, 2)$. The coordinates of a point which divides…
• AOBC is a rectangle whose three vertices are $A(0, 2)$, $O(0, 0)$ and $B(4, 0)$. The square of the length of…
• If the mid-point of the line segment joining the points $A(3, 4)$ and $B(k, 6)$ is $P(x, y)$ and $x + y - 10…
• Find the coordinates of the points which divide the line segment joining $A(-2, 2)$ and $B(2, 8)$ into four e…
• Point P divides the line segment joining the points $A(4, -5)$ and $B(1, 2)$ in the ratio $5:2$. Co-ordinates…
• XOYZ is a rectangle with vertices $X(-3, 0)$, $O(0, 0)$, $Y(0, 4)$ and $Z(x, y)$. The length of its each diag…
• Find the type of triangle ABC formed whose vertices are $A(1, 0)$, $B(-5, 0)$ and $C(-2, 5)$.
• In what ratio is the line segment joining the points $(3, -5)$ and $(-1, 6)$ divided by the line $y = x$?
• $A(3, 0)$, $B(6, 4)$ and $C(-1, 3)$ are vertices of a triangle ABC. Find length of its median BE.
• Directions: Two statements are given, one labelled as Assertion (A) and the other labelled as Reason (R). Sel…
• Find a relation between $x$ and $y$ such that the point $P(x, y)$ is equidistant from the points $A(7, 1)$ an…
• Points $A(-1, y)$ and $B(5, 7)$ lie on a circle with centre $O(2, -3y)$ such that $AB$ is a diameter of the c…
• Find the ratio in which the line segment joining the points $(5, 3)$ and $(-1, 6)$ is divided by Y-axis.
• $P(-2, 5)$ and $Q(3, 2)$ are two points. Find the coordinates of the point $R$ on line segment $PQ$ such that…
• The fourth vertex $D$ of a parallelogram $ABCD$ whose three vertices are $A(-2, 3)$, $B(6, 7)$ and $C(8, 3)$…
• Assertion (A) : Mid-point of a line segment divides the line segment in the ratio 1 : 1. Reason (R) : The rat…
• Based on the above, answer the following questions :
• Find the ratio in which the point $P(-4, 6)$ divides the line segment joining the points $A(-6, 10)$ and $B(3…
• Prove that the points $(3, 0)$, $(6, 4)$ and $(-1, 3)$ are the vertices of an isosceles triangle.
• Based on the above, answer the following questions :
• The coordinates of the vertex A of a rectangle ABCD whose three vertices are given as B(0, 0), C(3, 0) and D(…
• The area of the triangle formed by the line $\frac{x}{a} + \frac{y}{b} = 1$ with the coordinate axes is:
• Show that the points $(-2, 3)$, $(8, 3)$ and $(6, 7)$ are the vertices of a right-angled triangle.
• If $Q(0, 1)$ is equidistant from $P(5, -3)$ and $R(x, 6)$, find the values of $x$.
• The distance of the point $(-1, 7)$ from the $x$-axis is :
• The distance of the point $(-6, 8)$ from origin is :
• Based on the above information, answer the following questions :
• The distance between the points $P\left(\dfrac{11}{3}, 5\right)$ and $Q\left(\dfrac{2}{3}, 5\right)$ is :
• The line segment joining the points $A(4, -5)$ and $B(4, 5)$ is divided by the point $P$ such that $AP : AB =…
• Point $P(x, y)$ is equidistant from points $A(5, 1)$ and $B(1, 5)$. Prove that $x = y$.
• Find the ratio in which $y$-axis divides the line segment joining the points $(5, -6)$ and $(-1, -4)$.
• The mid-point of the line segment joining the points $(5, -4)$ and $(6, 4)$ lies on :
• The coordinates of the centre of a circle are $(x - 7, 2x)$. Find the value(s) of '$x$', if the circle passes…
• Find the ratio in which the $x$-axis divides the line segment joining the points $(-6, 5)$ and $(-4, -1)$. Al…
• The distance of the point A(4a, 3a) from x-axis is :
• If the points A(4, 5), B(m, 6), C(4, 3) and D(1, n) taken in this order are the vertices of a parallelogram A…
• Find the coordinates of the points of trisection of the line segment joining the points A(-1, 4) and B(-3, -2…
• In the given figure, $\triangle ABC$ is an equilateral triangle. AD is a median of the triangle joining the p…
• Diagonals AC and BD of square ABCD intersect at P. Coordinates of points B and D are $(9, -2)$ and $(1, 6)$ r…
• Find the coordinates of a point on the line $x + y = 5$ which is equidistant from $(6, 4)$ and $(5, 2)$.
• A point P(x, 7) divides a line segment joining the points A($-5$, 4) and B(7, 9) in a certain ratio. Find the…
• In $\triangle ABC$, AD is a median. X is a point on AD such that AX : XD = 2 : 3. BX is extended so that it i…
• The distance of a point A from $x$-axis is 3 units. Which of the following cannot be coordinates of the point…
• Find the coordinates of the point C which lies on the line AB produced such that AC = 2BC, where coordinates…
• P $(x, y)$, Q $(-2, -3)$ and R $(2, 3)$ are the vertices of a right triangle PQR right angled at P. Find the…
• In the figure given below, points P, Q, R divides the line segment AB in four equal parts. The point Q divide…
• The coordinates of the end points of the line segment AB are A(–2, –2) and B(2, –4). P is the point on AB suc…
• Find a relation between $x$ and y such that P($x$, y) is equidistant from the points A(3, 5) and B(7, 1). Hen…
• The distance of the point $A(-3, -4)$ from $x$-axis is
• Prove that abscissa of a point P which is equidistant from points with coordinates A(7, 1) and B(3, 5) is 2 m…
• If the points A(6, 1), B(p, 2), C(9, 4) and D(7, q) are the vertices of a parallelogram ABCD, then find the v…
• In what ratio, does x-axis divide the line segment joining the points A(3, 6) and B(-12, -3) ?
• The distance between the points $(0, 2\sqrt{5})$ and $(-2\sqrt{5}, 0)$ is
• In Question 19 and 20, an Assertion (A) statement is followed by a statement of Reason (R). Select the correc…
• If (-5, 3) and (5, 3) are two vertices of an equilateral triangle, then find co-ordinates of the third vertex…
• The distance of the point $(-6, 8)$ from x-axis is
• A line intersects y-axis and x-axis at point P and Q, respectively. If R(2, 5) is the mid-point of line segme…
• Find the ratio in which the line segment joining the points A(6, 3) and B($-2$, $-5$) is divided by x-axis.
• The line segment joining the points $P(-4, -2)$ and $Q(10, 4)$ is divided by $y$-axis in the ratio
• Using distance formula, prove that the points $A(2, 3)$, $B(-7, 0)$ and $C(-1, 2)$ are collinear.
• A circle centred at $(2, 1)$ passes through the points $A(5, 6)$ and $B(-3, K)$. Find the value(s) of $K$. He…
• Prove that the point $P$ dividing the line segment joining the points $A(-1, 7)$ and $B(4, -3)$ in the ratio…
• AD is a median of $\triangle ABC$ with vertices A(5, $-6$), B(6, 4) and C(0, 0). Length AD is equal to :
• If the distance between the points $(3, -5)$ and $(x, -5)$ is 15 units, then the values of $x$ are :
• The centre of a circle is at $(2, 0)$. If one end of a diameter is at $(6, 0)$, then the other end is at :
• Find the ratio in which the point $\left(\frac{8}{5}, y\right)$ divides the line segment joining the points $…
• ABCD is a rectangle formed by the points A $(-1, -1)$, B $(-1, 6)$, C $(3, 6)$ and D $(3, -1)$. P, Q, R and S…

Chapter 8: Introduction to Trigonometry (121)

• When $\sin A = \dfrac{1}{3}$, the value of $\cot A$ is
• $\dfrac{1 - \tan^2 A}{1 - \cot^2 A}$ equals to:
• The distance between the points $(a\cos\theta + b\sin\theta,\ 0)$ and $(0,\ a\sin\theta - b\cos\theta)$ is
• Prove that: $\sqrt{\dfrac{1 + \sin A}{1 - \sin A}} = \sec A + \tan A$
• Evaluate: $\dfrac{3\cos^2 30^\circ - 6\operatorname{cosec}^2 30^\circ}{\tan^2 60^\circ}$
• Prove that: $\dfrac{\sin\theta - \cos\theta + 1}{\sin\theta + \cos\theta - 1} = \dfrac{1}{\sec\theta - \tan\t…
• If $\theta$ is an acute angle and $7 + 4\sin\theta = 9$, then the value of $\theta$ is:
• The value of $\tan^2 30° - \frac{1}{\cos^2 60°}$ is:
• If $x\cos 60° + y\cos 0° + \sin 30° - \cot 45° = 5$, then find the value of $x + 2y$.
• Evaluate: $\dfrac{\sin^2 30° + \cos^2 60°}{1 + \tan^2 45°}$
• Prove that: $\dfrac{\sin\theta}{1 + \cos\theta} + \dfrac{1 + \cos\theta}{\sin\theta} = \dfrac{2}{\sin\theta}$
• Based on the above given information, answer the following questions:
• If $7\cos^2\theta + 3\sin^2\theta = 4$, then the value of $\theta$ is :
• If $\sin 30^\circ \cdot \tan 45^\circ = \dfrac{\sqrt{k}}{2}$, then the value of $k$ is :
• If $\tan A = \dfrac{\sqrt{3} - 1}{\sqrt{3} + 1}$, where $A$ is an acute angle, then find the value of $\sin A…
• Prove that : $\dfrac{\cosec\theta}{\cosec\theta - 1} + \dfrac{\cosec\theta}{\cosec\theta + 1} = 2\sec^2\theta$
• The angles of depression of the top and the bottom of an 8 m tall building from the top of another multistori…
• If $\tan 3\theta = \cot \theta$, then $\theta$ equals:
• $(\cot\theta + \tan\theta)$ equals:
• Assertion (A): A ladder leaning against a wall, stands at a horizontal distance of 6 m from the wall. If the…
• If $4k = \tan^2 60^\circ - 2\cosec^2 30^\circ - 2\tan^2 30^\circ$, then find the value of $k$.
• Prove that: $\dfrac{\sin A - 2\sin^3 A}{2\cos^3 A - \cos A} + \dfrac{\cos A}{\sin A} = 2\cosec A$
• Prove that: $\dfrac{\cosec A}{\cosec A - 1} + \dfrac{\cosec A}{\cosec A + 1} = 2\sec^2 A$
• Based on the information given above, answer the following questions:
• If $\sin\theta = \cos\theta$, $(0° < \theta < 90°)$, then value of $(\sec\theta \cdot \sin\theta)$ is
• Directions: In Question 19 and 20, Assertion (A) and Reason (R) are given. Select the correct option. **Asser…
• Evaluate: $2\sin^2 30° \cdot \sec 60° + \tan^2 60°$.
• If $2\sin(A + B) = \sqrt{3}$ and $\cos(A - B) = 1$, then find the measures of angles A and B. $0 \leq A, B, (…
• From the top of a 45 m high light house, the angles of depression of two ships, on the opposite side of it, a…
• From a point on the ground, which is 30 m away from the foot of a vertical tower, the angle of elevation of t…
• If $\cos\theta = \dfrac{\sqrt{3}}{2}$ and $\sin\phi = \dfrac{1}{2}$, then $\tan(\theta + \phi)$ is :
• Evaluate : $\dfrac{\cos 45° + \sin 60°}{\sec 30° + \cosec 30°}$
• Prove that $\dfrac{\sin\theta - 2\sin^3\theta}{2\cos^3\theta - \cos\theta} = \tan\theta$.
• From a point on the ground, the angles of elevation of the bottom and the top of a transmission tower fixed a…
• If $\sin\theta = 1$, then the value of $\dfrac{1}{2}\sin\left(\dfrac{\theta}{2}\right)$ is :
• At some time of the day, the length of the shadow of a tower is equal to its height. Then, the Sun's altitude…
• Evaluate : $$\frac{5\cos^2 60° + 4\sec^2 30° - \tan^2 45°}{\sin^2 30° + \sin^2 60°}$$
• If $\sin(A - B) = \dfrac{1}{2}$, $\cos(A + B) = \dfrac{1}{2}$; $0 < A + B \leq 90°$, $A > B$; find $\angle A$…
• If $\dfrac{x}{3} = 2\sin A$, $\dfrac{y}{3} = 2\cos A$, then the value of $x^2 + y^2$ is :
• If $4\sec\theta - 5 = 0$, then the value of $\cot\theta$ is :
• If two tangents inclined at an angle of 60° are drawn to a circle of radius 3 cm, then find the length of eac…
• Evaluate : $$\frac{2\tan 30° \cdot \sec 60° \cdot \tan 45°}{1 - \sin^2 60°}$$
• Prove that : $$\frac{\tan A}{1 - \cot A} + \frac{\cot A}{1 - \tan A} = 1 + \sec A \cosec A$$
• If $4\cot^2 45° - \sec^2 60° + \sin^2 60° + p = \frac{3}{4}$, then find the value of $p$.
• If $\cos A + \cos^2 A = 1$, then find the value of $\sin^2 A + \sin^4 A$.
• The angle of elevation of the top of a tower from a point on the ground which is 30 m away from the foot of t…
• As observed from the top of a 75 m high lighthouse from the sea-level, the angles of depression of two ships…
• From a point on the ground, the angle of elevation of the bottom and top of a transmission tower fixed at the…
• $\sec\theta$ when expressed in terms of $\cot\theta$, is equal to :
• Evaluate $\dfrac{5\cos^2 60^\circ + 4\sec^2 30^\circ - \tan^2 45^\circ}{\sin^2 30^\circ + \cos^2 30^\circ}$
• If A and B are acute angles such that $\sin(A - B) = 0$ and $2\cos(A + B) - 1 = 0$, then find angles A and B.
• Prove that $\dfrac{\sin A - 2\sin^3 A}{2\cos^3 A - \cos A} = \tan A$
• Prove that $\sec A (1 - \sin A)(\sec A + \tan A) = 1$.
• If $\tan \theta = \dfrac{5}{12}$, then the value of $\dfrac{\sin \theta - \cos \theta}{\sin \theta + \cos \th…
• In the given figure, $AB$ is a tangent to the circle centered at $O$. If $OA = 6$ cm and $\angle OAB = 30°$,…
• $\dfrac{2\tan 30°}{1 - \tan^2 30°}$ is equal to :
• If $a\cos\theta + b\sin\theta = m$ and $a\sin\theta - b\cos\theta = n$, then prove that $a^2 + b^2 = m^2 + n^…
• Prove that : $$\sqrt{\dfrac{\sec A - 1}{\sec A + 1}} + \sqrt{\dfrac{\sec A + 1}{\sec A - 1}} = 2\cosec A$$
• One observer estimates the angle of elevation to the basket of a hot air balloon to be 60°, while another obs…
• The tops of two poles of heights 20 m and 28 m are connected with a wire. The wire is inclined to the horizon…
• Given that $\sin\theta = \frac{a}{b}$, then $\cos\theta$ is equal to :
• If $\cos A = \frac{1}{2}$, then the value of $\sin^2 A + 2\cos^2 A$ is :
• If $\tan\theta = \frac{24}{7}$, then find the value of $\sin\theta + \cos\theta$.
• If $\cot\theta = \frac{7}{8}$, then find the value of $\frac{(1+\sin\theta)(1-\sin\theta)}{(1+\cos\theta)(1-\…
• If $x = h + a\cos\theta$, $y = k + b\sin\theta$, then prove that : $\left(\frac{x-h}{a}\right)^2 + \left(\fra…
• Prove that : $\frac{\tan A}{1 + \sec A} - \frac{\tan A}{1 - \sec A} = 2\,\mathrm{cosec}\,A$
• If $\cos A = \frac{4}{5}$, then the value of $\tan A$ is :
• If $2 \sin A = 1$, then the value of $\tan A + \cot A$ is :
• From a point on the ground, which is 60 m away from the foot of a vertical tower, the angle of elevation of t…
• If $\tan\theta + \frac{1}{\tan\theta} = 2$, find the value of $\tan^2\theta + \frac{1}{\tan^2\theta}$.
• Prove that : $\sqrt{\frac{1 - \sin\theta}{1 + \sin\theta}} = \sec\theta - \tan\theta$
• Prove that : $\frac{\sec^3\theta}{\sec^2\theta - 1} + \frac{\csc^3\theta}{\csc^2\theta - 1} = \sec\theta \cdo…
• If $\frac{\sec\alpha}{\csc\beta} = p$ and $\frac{\tan\alpha}{\csc\beta} = q$, then prove that $(p^2 - q^2)\se…
• In the given figure, PQ and PR are tangents to a circle with centre O and radius 3 cm. If $\angle QPR = 60°$,…
• Given that $\sin 2\alpha = \frac{\sqrt{3}}{2}$, the value of $\sin 3\alpha$ is :
• The value of $\left(\frac{1}{2}\tan^2 45° - \cos^2 60°\right)$ is :
• Directions: Select the correct answer from the codes (A), (B), (C) and (D). Assertion (A): $\tan 2\theta$ is…
• If $\sin\theta + \cos\theta = \sqrt{3}$, then prove that $\tan\theta + \cot\theta = 1$.
• Prove that: $(\sin A + \sec A)^2 + (\cos A + \csc A)^2 = (1 + \sec A\,\csc A)^2$.
• In the given figure, chord AB subtends an angle of $120°$ at the centre of the circle with radius 7 cm. Find…
• A boy standing on a horizontal plane is flying a kite with a string of length 60 m, at an angle of elevation…
• $\tan 2A = 3 \tan A$ is true, when the measure of $\angle A$ is :
• Find the value of $x$ for which $(\sin A + \text{cosec } A)^2 + (\cos A + \sec A)^2 = x + \tan^2 A + \cot^2 A$
• Evaluate the following : $\dfrac{3\sin 30° - 4\sin^3 30°}{2\sin^2 50° + 2\cos^2 50°}$
• Prove that $\dfrac{\cos A + \sin A - 1}{\cos A - \sin A + 1} = \text{cosec } A - \cot A$
• If $\cot\theta + \cos\theta = p$ and $\cot\theta - \cos\theta = q$, prove that $p^2 - q^2 = 4\sqrt{pq}$
• $\frac{1 - \tan^2 30°}{1 + \tan^2 30°}$ is equal to
• Directions: A statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.…
• It is given that $\sin(A - B) = \sin A \cos B - \cos A \sin B$. Use it to find the value of $\sin 15°$.
• If $\sin A = y$, then express $\cos A$ and $\tan A$ in terms of y.
• Prove the following trigonometric identity : $\frac{1 + \operatorname{cosec} A}{\operatorname{cosec} A} = \fr…
• Let 2A + B and A + 2B be acute angles such that $\sin(2A + B) = \frac{\sqrt{3}}{2}$ and $\tan(A + 2B) = 1$. F…
• If $x\left(\frac{2\tan 30°}{1 + \tan^2 30°}\right) = y\left(\frac{2\tan 30°}{1 - \tan^2 30°}\right)$, then $x…
• In a right triangle ABC, right-angled at A, if $\sin B = \frac{1}{4}$, then the value of $\sec B$ is
• If $a\sec\theta + b\tan\theta = m$ and $b\sec\theta + a\tan\theta = n$, prove that $a^2 + n^2 = b^2 + m^2$
• Use the identity: $\sin^2 A + \cos^2 A = 1$ to prove that $\tan^2 A + 1 = \sec^2 A$. Hence, find the value of…
• Prove that: $\frac{\cos\theta - 2\cos^3\theta}{\sin\theta - 2\sin^3\theta} + \cot\theta = 0$.
• Given that $\sin\theta + \cos\theta = x$, prove that $\sin^4\theta + \cos^4\theta = \frac{2 - (x^2 - 1)^2}{2}…
• In the adjoining figure, TP and TQ are tangents drawn to a circle with centre O. If $\angle OPQ = 15°$ and $\…
• Based on given information, answer the following questions.
• If $2\tan A = 3$, then the value of $\dfrac{4\sin A + 3\cos A}{4\sin A - 3\cos A}$ is
• $\left[\dfrac{3}{4}\tan^2 30° - \sec^2 45° + \sin^2 60°\right]$ is equal to
• If $\sin\theta + \cos\theta = \sqrt{3}$, then find the value of $\sin\theta \cdot \cos\theta$.
• If $\sin\alpha = \dfrac{1}{\sqrt{2}}$ and $\cot\beta = \sqrt{3}$, then find the value of $\csc\alpha + \csc\b…
• The angle of elevation of the top of a tower 24 m high from the foot of another tower in the same plane is 60…
• A chord of a circle of radius 14 cm subtends an angle of 60° at the centre. Find the area of the correspondin…
• $\frac{\cos^2\theta}{\sin^2\theta} - \frac{1}{\sin^2\theta}$, in simplified form, is :
• If $\theta$ is an acute angle of a right angled triangle, then which of the following equation is not true?
• A statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option. Statement A…
• Prove that $\frac{1 + \sec A}{\sec A} = \frac{\sin^2 A}{1 - \cos A}$.
• From the top of an 8 m high building, the angle of elevation of the top of a cable tower is 60° and the angle…
• For an acute angle $\theta$, if $\sin\theta = \frac{1}{9}$, then value of $\frac{9\csc\theta + 1}{9\csc\theta…
• Simplest form of $\dfrac{\sec A}{\sqrt{\sec^2 A - 1}}$ is
• Vertices of a right triangle $ABC$ with $\angle B = 90°$ are $A(3, 4)$, $B(1, 1)$ and $C(-8, 7)$. Find the va…
• Evaluate: $\dfrac{\sin^3 60° - \tan 30°}{\cos^2 45°}$
• For acute angles $A$ and $B$, if $A + 2B$ and $2A + B$ are acute, $\tan(A + 2B) = \sqrt{3}$ and $\sin(2A + B)…
• If $\sec\theta - \tan\theta = m$, then the value of $\sec\theta + \tan\theta$ is :
• If $\cos(\alpha + \beta) = 0$, then value of $\cos\left(\frac{\alpha + \beta}{2}\right)$ is equal to :
• Evaluate : $2\sqrt{2}\,\cos 45^\circ \sin 30^\circ + 2\sqrt{3}\,\cos 30^\circ$
• If $A = 60^\circ$ and $B = 30^\circ$, verify that : $\sin(A + B) = \sin A \cos B + \cos A \sin B$
• Prove that : $\dfrac{\tan\theta}{1 - \cot\theta} + \dfrac{\cot\theta}{1 - \tan\theta} = 1 + \sec\theta\,\math…

Chapter 9: Some Applications of Trigonometry (50)

• A kite is flying at a height of 60 m above the ground level. Ravi, standing at the roof of the house is holdi…
• A kite is flying at a height of 150 m from the ground. It is attached to a string inclined at an angle of $30…
• Based on the above given information, answer the following questions:
• Two ships are sailing in the sea on either side of a lighthouse. The angles of depression to the two ships as…
• The angles of depression of the top and the bottom of an 8 m tall building from the top of another multistori…
• Assertion (A): A ladder leaning against a wall, stands at a horizontal distance of 6 m from the wall. If the…
• Based on the information given above, answer the following questions:
• From the top of a 45 m high light house, the angles of depression of two ships, on the opposite side of it, a…
• From a point on the ground, which is 30 m away from the foot of a vertical tower, the angle of elevation of t…
• From a point on the ground, the angles of elevation of the bottom and the top of a transmission tower fixed a…
• At some time of the day, the length of the shadow of a tower is equal to its height. Then, the Sun's altitude…
• Two pillars of equal lengths stand on either side of a road which is 100 m wide, exactly opposite to each oth…
• From a point on a bridge across the river, the angles of depressions of the banks on opposite sides of the ri…
• The length of the shadow of a tower on the plane ground is $\sqrt{3}$ times the height of the tower. Find the…
• The angle of elevation of the top of a tower from a point on the ground which is 30 m away from the foot of t…
• As observed from the top of a 75 m high lighthouse from the sea-level, the angles of depression of two ships…
• From a point on the ground, the angle of elevation of the bottom and top of a transmission tower fixed at the…
• If a pole 6 m high casts a shadow $2\sqrt{3}$ m long on the ground, then the sun's elevation is :
• A straight highway leads to the foot of a tower. A man standing on the top of the 75 m high tower observes tw…
• From the top of a 7 m high building, the angle of elevation of the top of a cable tower is $60^\circ$ and the…
• One observer estimates the angle of elevation to the basket of a hot air balloon to be 60°, while another obs…
• The angle of elevation of the top of a building from the foot of the tower is $30^\circ$ and the angle of ele…
• From a point on a bridge across a river, the angles of depression of the banks on opposite sides of the river…
• Observe the picture and answer the following questions:
• The tops of two poles of heights 20 m and 28 m are connected with a wire. The wire is inclined to the horizon…
• A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an…
• A car is moving away from the base of a 30 m high tower. The angle of elevation of the top of the tower from…
• Based on the above information, answer the following questions :
• From a point on the ground, which is 60 m away from the foot of a vertical tower, the angle of elevation of t…
• Based on the information given above, answer the following questions :
• If the length of the shadow of a tower is $\sqrt{3}$ times that of its height, then altitude of the Sun is :
• A boy standing on a horizontal plane is flying a kite with a string of length 60 m, at an angle of elevation…
• A 30 m long rope is tightly stretched and tied from the top of pole to the ground. If the rope makes an angle…
• A 1.5 m tall boy is walking away from the base of a lamp post which is 12 m high, at the speed of 2.5 m/sec.…
• Based on above information, answer the following questions :
• An observer 1.8 m tall stands away from a chimney at a distance of 38.2 m along the ground. The angle of elev…
• Based on given information, answer the questions given in part (i) and (ii).
• A peacock sitting on the top of a tree of height 10 m observes a snake moving on the ground. If the snake is…
• Based on given information, answer the following questions.
• The angle of elevation of the top of a tower 24 m high from the foot of another tower in the same plane is 60…
• A spherical balloon of radius r subtends an angle of 60° at the eye of an observer. If the angle of elevation…
• Find the length of the shadow on the ground of a pole of height 18 m when angle of elevation $\theta$ of the…
• Based on the above information, answer the following questions:
• In Fig. 3, AB is tower of height 50 m. A man standing on its top, observes two cars on the opposite sides of…
• Case Study - 1 : Kite Festival. In Fig. 5, the angles of elevation of two kites (Points A and B) from the han…
• An aeroplane when flying at a height of 3125 m from the ground passes vertically below another plane at an in…
• From the top of an 8 m high building, the angle of elevation of the top of a cable tower is 60° and the angle…
• A wire is attached from a point $A$ on the ground to the top of a pole $BC$, making an angle of elevation as…
• Based on the above information, answer the following questions:
• A pole 6m high is fixed on the top of a tower. The angle of elevation of the top of the pole observed from a…

Chapter 10: Circles (98)

• If PQ and PR are tangents to the circle with centre O and radius 4 cm such that $\angle QPR = 90^\circ$, then…
• PQ is tangent to a circle with centre O. If $\angle POR = 65^\circ$, then $m\widehat{PTR}$ is
• In the given figure, TP and TQ are tangents to a circle with centre M, touching another circle with centre N…
• The tangents drawn at the extremities of the diameter of a circle are always:
• In the given figure, $PA$ is a tangent from an external point $P$ to a circle with centre $O$. If $\angle POB…
• A person is standing at $P$ outside a circular ground at a distance of 26 m from the centre of the ground. He…
• In the given figure, $O$ is the centre of the circle and $BCD$ is tangent to it at $C$. Prove that $\angle BA…
• Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the cent…
• In the given figure, RS is the tangent to the circle at the point L and MN is the diameter. If $\angle NML =…
• At point A on the diameter AB of a circle of radius 10 cm, tangent XAY is drawn to the circle. Find the lengt…
• Prove that the parallelogram circumscribing a circle is a rhombus.
• Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to th…
• If tangents PA and PB drawn from an external point P to the circle with centre O are inclined to each other a…
• Assertion (A): If two tangents are drawn to a circle from an external point, then they subtend equal angles a…
• Prove that the tangents drawn at the ends of a diameter of a circle are parallel.
• In the given figure, PC is a tangent to the circle at C. AOB is the diameter which when extended meets the ta…
• AB and CD are two chords of a circle intersecting at P. Choose the correct statement from the following:
• In the given figure, tangents PA and PB to the circle centred at O, from point P are perpendicular to each ot…
• In the given figure, AT is tangent to a circle centred at O. If $\angle CAT = 40°$, then $\angle CBA$ is equa…
• In the given figure, AB and CD are tangents to a circle centred at O. Is $\angle BAC = \angle DCA$? Justify y…
• In the given figure, PQ is tangent to a circle centred at O and $\angle BAQ = 30°$; show that $BP = BQ$.
• In the given figure, AB, BC, CD and DA are tangents to the circle with centre O forming a quadrilateral ABCD.…
• Maximum number of common tangents that can be drawn to two circles intersecting at two distinct points is :
• In the given figure, if PT is a tangent to a circle with centre O and $\angle TPO = 35°$, then the measure of…
• In the given figure, O is the centre of the circle. MN is the chord and the tangent ML at point M makes an an…
• Prove that the tangents drawn at the end points of a chord of a circle makes equal angles with the chord.
• In the given figure, $AB$ and $AC$ are tangents to the circle. If $\angle ABC = 42°$, then the measure of $\a…
• In the given figure, $QR$ is a common tangent to the two given circles touching externally at $A$. The tangen…
• In the given figure, $O$ is the centre of the circle. If $\angle AOB = 145°$, then find the value of $x$.
• In the given figure, RJ and RL are two tangents to the circle. If $\angle RJL = 42°$, then the measure of $\a…
• If two tangents inclined at an angle of 60° are drawn to a circle of radius 3 cm, then find the length of eac…
• In the given figure, the quadrilateral PQRS circumscribes a circle. Here PA + CS is equal to:
• Assertion (A): A tangent to a circle is perpendicular to the radius through the point of contact. Reason (R):…
• In the given figure, O is the centre of the circle. AB and AC are tangents drawn to the circle from point A.…
• In the given figure, O is the centre of the circle and QPR is a tangent to it at P. Prove that $\angle QAP +…
• In the given figure, PQ is a tangent to the circle with centre O. If $\angle OPQ = x$, $\angle POQ = y$, then…
• In the given figure, TA is a tangent to the circle with centre O such that $OT = 4$ cm, $\angle OTA = 30^\cir…
• Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which to…
• In the given figure, $AB = BC = 10$ cm. If $AC = 7$ cm, then the length of $BP$ is :
• In the given figure, $AB$ is a tangent to the circle centered at $O$. If $OA = 6$ cm and $\angle OAB = 30°$,…
• In the given figure, $AC$ and $AB$ are tangents to a circle centered at $O$. If $\angle COD = 120°$, then $\a…
• Questions number 19 and 20 are Assertion and Reason based questions carrying 1 mark each. Select the correct…
• In the given figure, $PT$ is a tangent to the circle centered at $O$. $OC$ is perpendicular to chord $AB$. Pr…
• A triangle $ABC$ is drawn to circumscribe a circle of radius 4 cm such that the segments $BD$ and $DC$ are of…
• Two circles with centres $O$ and $O'$ of radii 6 cm and 8 cm, respectively intersect at two points $P$ and $Q…
• Two concentric circles are of radii $4$ cm and $3$ cm. Find the length of the chord of the larger circle whic…
• Draw a circle of radius $3$ cm. Take two points P and Q on one of its extended diameter each at a distance of…
• In Figure 1, a triangle ABC with $\angle B = 90^\circ$ is shown. Taking AB as diameter, a circle has been dra…
• In Figure 2, PQ and PR are tangents to the circle centred at O. If $\angle OPR = 45°$, then prove that ORPQ i…
• Draw a circle of radius 3 cm. From a point P lying outside the circle at a distance of 6 cm from its centre,…
• In Figure 3, two circles with centres at O and O′ of radii 2r and r respectively, touch each other internally…
• In Figure 4, O is centre of a circle of radius 5 cm. PA and BC are tangents to the circle at A and B respecti…
• If TP and TQ are two tangents to a circle with centre O from an external point T so that $\angle POQ = 120°$,…
• In the given figure, PA is a tangent from an external point P to a circle with centre O. If $\angle POB = 125…
• Two concentric circles are of radii 5 cm and 4 cm. Find the length of the chord of the larger circle which to…
• In the given figure, $\triangle ABC$ is a right triangle in which $\angle B = 90°$, AB = 4 cm and BC = 3 cm.…
• In the given figure, if a circle touches the side QR of $\triangle PQR$ at S and extended sides PQ and PR at…
• In the given figure, PA and PB are tangents to a circle centred at O. If $\angle OAB = 15°$, then $\angle APB…
• In the given figure, PA and PB are tangents to a circle centred at O. If $\angle AOB = 130°$, then $\angle AP…
• In the given figure, O is the centre of the circle. PQ and PR are tangents. Show that the quadrilateral PQOR…
• Prove that the lengths of tangents drawn from an external point to a circle are equal.
• Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that $\angle PTQ =…
• In the given figure, PQ and PR are tangents to a circle with centre O and radius 3 cm. If $\angle QPR = 60°$,…
• In the given figure, PT is a tangent to the circle with centre O and radius r. If $\angle POT = 45°$, then th…
• Directions: Select the correct answer from the codes (A), (B), (C) and (D). Assertion (A): Radius is the smal…
• Two tangents PA and PB are drawn to a circle with centre O from an external point P. Prove that $\angle APB =…
• In the given figure, PA is the tangent to the circle with centre O such that OA = 10 cm, AB = 8 cm and AB $\p…
• In the adjoining figure, AB is the chord of the larger circle touching the smaller circle. The centre of both…
• A parallelogram having one of its sides 5 cm circumscribes a circle. The perimeter of parallelogram is :
• A statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option. Assertion (…
• Rectangle ABCD circumscribes the circle of radius 10 cm. Prove that ABCD is a square. Hence, find the perimet…
• In the adjoining figure, PA and PB are tangents to a circle with centre O. The measure of angle APB is
• In the adjoining figure, the sum of radii of two concentric circles is 16 cm. The length of chord AB which to…
• In the adjoining figure, XY and X'Y' are parallel tangents to a circle with centre O. Another tangent AB touc…
• In the adjoining figure, PA and PB are tangents to a circle with centre O such that $\angle P = 90°$. If $AB…
• For a circle with centre O and radius 5 cm, which of the following statements is true ? P : Distance between…
• In the adjoining figure, TS is a tangent to a circle with centre O. The value of $2x°$ is
• In the adjoining figure, TP and TQ are tangents drawn to a circle with centre O. If $\angle OPQ = 15°$ and $\…
• In the given figure, PQ is tangent to the circle centred at O. If $\angle AOB = 95°$, then the measure of $\a…
• In the given figure, a circle is inscribed in a quadrilateral ABCD in which $\angle B = 90°$. If AD = 17 cm,…
• Based on above information, answer the following questions.
• In the given figure, PA and PB are tangents from external point P to a circle with centre C and Q is any poin…
• In the given figure, PA is a tangent to the circle drawn from the external point P and PBC is the secant to t…
• A circle touches the side BC of a $\triangle ABC$ at a point P and touches AB and AC when produced at Q and R…
• In Fig. 1, AB is diameter of a circle centered at O. BC is tangent to the circle at B. If OP bisects the chor…
• In Fig. 2, XAY is a tangent to the circle centered at O. If $\angle ABO = 40°$, then find $m\angle BAY$ and $…
• Draw two concentric circles of radii 2 cm and 5 cm. From a point on the outer circle, construct a pair of tan…
• In Fig. 4, PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q meet at a point T…
• In Fig. 1, there are two concentric circles with centre O. If ARC and AQB are tangents to the smaller circle…
• Construct a pair of tangents to a circle of radius 4 cm which are inclined to each other at an angle of 60°.
• In Fig.-2, if a circle touches the side QR of $\triangle PQR$ at S and extended sides PQ and PR at M and N, r…
• In Fig. 3, a triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC i…
• In the given figure, $PQ$ is tangent to the circle with centre $O$. $S$ is a point on the circle such that $\…
• $PQ$ and $PR$ are two tangents to a circle with centre $O$ and radius 5 cm. $AB$ is another tangent to the ci…
• In Q. No. 19 and 20 a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct…
• In the given figure, AB is a diameter of the circle with centre O. AQ, BP and PQ are tangents to the circle.…
• A circle with centre O and radius 8 cm is inscribed in a quadrilateral ABCD in which P, Q, R, S are the point…
• Based on the above information, answer the following questions :

Chapter 11: Areas Related to Circles (46)

• A circle is divided into 16 identical sectors. If radius of the circle is 7 cm, area of each sector is
• Arc PQ subtends an angle $\theta$ at the centre of the circle with radius 6.3 cm. If $PQ = 11$ cm, then the v…
• Chord AB of a circle with centre O and radius 21 mm subtends an angle of $120^\circ$ at the centre. Find the…
• If a sector of a circle has an area of 40 sq. units and a central angle of $72°$, the radius of the circle is:
• A piece of wire 20 cm long is bent into the form of an arc of a circle of radius $\dfrac{60}{\pi}$ cm. The an…
• Based on the above given information, answer the following questions:
• If a large circular pizza is divided into 5 equal sectors, then the central angle of each sector will be :
• In the given figure, the shape of the top of a table is that of a sector of a circle with centre O and $\angl…
• In the given figure, three sectors of a circle of radius 5 cm, making angles $35^\circ$, $50^\circ$ and $95^\…
• The length of the hour hand of a clock is 10 cm. Find the area of the minor sector swept by the hour hand of…
• If the area of a sector of a circle of radius 36 cm is 54$\pi$ cm$^2$, then the length of the corresponding a…
• A chord of a circle of radius 10 cm subtends a right angle at the centre of the circle. Find the area of the…
• The perimeter of a certain sector of a circle of radius 5.6 m is 20.0 m. Find the area of the sector.
• Perimeter of a sector of a circle whose central angle is 90° and radius 7 cm is :
• Based on the above, answer the following questions :
• If an arc subtends an angle of 90° at the centre of a circle, then the ratio of its length to the circumferen…
• The area of the sector of a circle of radius 12 cm is $60\pi$ cm². The central angle of this sector is :
• Assertion (A) : If the circumference of a circle is 176 cm, then its radius is 28 cm. Reason (R) : Circumfere…
• The minute hand of a clock is 14 cm long. Find the area on the face of the clock described by the minute hand…
• Find the length of the arc of a circle which subtends an angle of 60° at the centre of the circle of radius 4…
• The perimeter of the sector of a circle of radius 21 cm which subtends an angle of 60° at the centre of circl…
• The length of an arc of a circle with radius 12 cm is $10\pi$ cm. The angle subtended by the arc at the centr…
• A chord of a circle of radius 10 cm subtends a right angle at its centre. The length of the chord (in cm) is :
• An arc of a circle of radius 10 cm subtends a right angle at the centre of the circle. Find the area of the c…
• What is the length of the arc of the sector of a circle with radius 14 cm and of central angle 90°?
• The hour-hand of a clock is 6 cm long. The angle swept by it between 7:20 a.m. and 7:55 a.m. is:
• A car has two wipers which do not overlap. Each wiper has a blade of length 21 cm sweeping through an angle o…
• Based on the above, answer the following questions:
• What is the area of a semi-circle of diameter $d$ ?
• Based on the above information, answer the following questions :
• A horse is tied to a peg at one corner of a square shaped grass field of side 15 m by means of a 5 m long rop…
• The length of the arc of the sector of a circle with radius 21 cm and of central angle $60°$, is :
• The hour hand of a clock is 7 cm long. The angle swept by it between 7:00 a.m. and 8:10 a.m. is :
• Based on the above information, answer the following questions :
• Area of a segment of a circle of radius 'r' and central angle 60° is :
• Find the area of the sector of a circle of radius 42 cm and of central angle 30°. Also, find the area of the…
• In the given figure, chord AB subtends an angle of $120°$ at the centre of the circle with radius 7 cm. Find…
• Based on above information, answer the following questions :
• Based on given information, answer the following questions :
• Based on given information, answer the following questions.
• A chord of a circle of radius 14 cm subtends an angle of 60° at the centre. Find the area of the correspondin…
• The circumferences of two circles are in the ratio $4 : 5$. What is the ratio of their radii?
• In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. Find the area of the sector forme…
• An arc of length 2.2 cm subtends an angle $\theta$ at the centre of the circle with radius 2.8 cm. The value…
• A circle of diameter 20 cm is equally divided into five sectors. Find the area and perimeter of one of the se…
• An arc of a circle of radius 21 cm subtends an angle of $60^\circ$ at the centre. Find : (i) the length of th…

Chapter 12: Surface Areas and Volumes (59)

• Three tennis balls are just packed in a cylindrical jar. If radius of each ball is $r$, volume of air inside…
• An ice-cream cone of radius $r$ and height $h$ is completely filled by two spherical scoops of ice-cream. If…
• To protect plants from heat, a shed of iron rods covered with green cloth is made. The lower part of the shed…
• The internal and external radii of a hollow hemisphere are $5\sqrt{2}$ cm and 10 cm respectively. A cone of h…
• Assertion (A): If we join two hemispheres of same radius along their bases, then we get a sphere. Reason (R):…
• A room is in the form of a cylinder surmounted by a hemispherical dome. The base radius of the hemisphere is…
• Based on the above information, answer the following questions :
• If the volumes of two cubes are in the ratio $8:125$, then the ratio of their surface areas is:
• A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of its top, which is open, is…
• **Assertion (A):** Two cubes each of edge length 10 cm are joined together. The total surface area of newly f…
• The word 'circus' has the same root as 'circle'. In a closed circular area, various entertainment acts includ…
• A solid iron pole consists of a solid cylinder of height 200 cm and base diameter 28 cm, which is surmounted…
• A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends. The length o…
• The ratio of total surface area of a solid hemisphere to the square of its radius is :
• Based on the above given information, answer the following questions :
• A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in the fig…
• The total surface area of a solid hemisphere whose diameter is $d$ is:
• Based on the above, answer the following questions:
• A student was asked to make a model shaped like a cylinder with two cones attached to its ends by using a thi…
• Water in a river which is 3 m deep and 40 m wide is flowing at the rate of 2 km/h. How much water will fall i…
• Based on the above, answer the following questions :
• A solid metallic sphere of radius $10.5$ cm is melted and recast into a number of smaller cones, each of radi…
• From a solid cylinder of height $30$ cm and radius $7$ cm, a conical cavity of height $24$ cm and same radius…
• Water in a canal, $8$ m wide and $6$ m deep, is flowing with a speed of $12$ km/hour. How much area will it i…
• 150 spherical marbles, each of diameter 1.4 cm, are dropped in a cylindrical vessel of diameter 7 cm containi…
• Three cubes of side 6 cm each, are joined as shown in Figure 1. Find the total surface area of the resulting…
• Based on the above information, answer the following questions:
• The total surface area of a solid hemisphere of diameter '$2d$' is :
• A solid is in the form of a cylinder with hemispherical ends. The total height of the solid is 20 cm and the…
• A hemispherical bowl is made of steel of thickness 1 cm. The outer radius of the bowl is 6 cm. The volume of…
• Assertion (A) : The surface area of the cuboid formed by joining two cubes of sides 4 cm each, end-to-end, is…
• Based on the information given above, answer the following questions :
• A cone of maximum size is carved out from a solid cube of edge length $l$. The volume of the cone is :
• A toy is in the form of a cone mounted on a hemisphere of radius 7 cm. The total height of the toy is 31 cm.…
• Observe the figure and answer the following questions:
• On the top face of the wooden cube of side 7 cm, hemispherical depressions of radius 0.35 cm are to be formed…
• A wooden cubical die is formed by forming hemispherical depressions on each face of the cube such that face 1…
• A cone of height 12 cm and slant height 13 cm is surmounted on a hemisphere having radius equal to that of co…
• From one of the faces of a solid wooden cube of side 14 cm, maximum number of hemispheres of diameter 1.4 cm…
• From a solid cylinder of height 24 cm and radius 5 cm, two cones of height 12 cm and radius 5 cm are hollowed…
• If a cone of greatest possible volume is hollowed out from a solid wooden cylinder, then the ratio of the vol…
• From one face of a solid cube of side 14 cm, the largest possible cone is carved out. Find the volume and sur…
• Curved surface area of a cylinder of height 5 cm is 94.2 cm². Radius of the cylinder is (Take $\pi = 3.14$)
• The curved surface area of a cone having height 24 cm and radius 7 cm, is
• A room is in the form of cylinder surmounted by a hemi-spherical dome. The base radius of hemisphere is one-h…
• An empty cone is of radius 3 cm and height 12 cm. Ice-cream is filled in it so that lower part of the cone wh…
• The volume of a right circular cone whose area of the base is 156 cm$^2$ and the vertical height is 8 cm, is
• A solid is in the shape of a right-circular cone surmounted on a hemisphere, the radius of each of them being…
• A solid piece of metal in the form of a cuboid of dimensions $11\,\text{cm} \times 7\,\text{cm} \times 7\,\te…
• Case Study - 2 : The tent is in the shape of a cylinder surmounted by a conical top. If the height and diamet…
• The curved surface area of a right circular cylinder is 176 sq cm and its volume is 1232 cu cm. What is the h…
• The largest sphere is carved out of a solid cube of side 21 cm. Find the volume of the sphere.
• Maths teacher who was accompanying the students asked following questions :
• A conical cavity of maximum volume is carved out from a wooden solid hemisphere of radius 10 cm. Curved surfa…
• A camping tent in hemispherical shape of radius 1.4 m, has a door opening of area 0.50 m$^2$. Outer surface a…
• Based on the above information, answer the following questions:
• A solid sphere is cut into two hemispheres. The ratio of the surface areas of sphere to that of two hemispher…
• The volume of the largest right circular cone that can be carved out from a solid cube of edge 2 cm is :
• The difference between the outer and inner radii of a hollow right circular cylinder of length 14 cm is 1 cm.…

Chapter 13: Statistics (61)

• Mean and Median of a frequency distribution are 43 and 40 respectively. The value of mode is
• The median of the following data is 50 and sum of all frequencies is 90. Find the values of $p$ and $q$.
• Find mean and mode of the following distribution:
• Mode and Mean of a data are $15x$ and $18x$, respectively. Then the median of the data is:
• Find the missing frequency $f$ in the following table, if the mean of the given data is 18. Hence find the mo…
• Find the Mean and Mode of the following frequency distribution :
• If the mean of 2, 9, $x+6$, $2x+3$, 5, 10, 5 is 7, then the value of $x$ is:
• Based on the information given above, answer the following questions:
• If value of each observation in a data is increased by 2, then median of the new data
• After an examination, a teacher wants to know the marks obtained by maximum number of the students in her cla…
• In a test, the marks obtained by 100 students (out of 50) are given below: | Marks obtained | 0–10 | 10–20 |…
• The mean of five observations is 15. If the mean of first three observations is 14 and that of the last three…
• From the above answer the following questions :
• If the difference of mode and median of a data is 24, then the difference of its median and mean is :
• The following table shows the ages of the patients admitted in a hospital during a year. Find the mode and me…
• For the data 2, 9, $x + 6$, $2x + 3$, 5, 10, 5; if the mean is 7, then the value of $x$ is :
• Based on the above, answer the following questions :
• A student noted the number of cars passing through a spot on a road for 100 periods each of 3 minutes and sum…
• The empirical relation between the mode, median and mean of a distribution is :
• The monthly expenditure on milk in 200 families of a Housing Society is given below : Find the value of $x$ a…
• If the mean and the median of a data are 12 and 15 respectively, then its mode is :
• If every term of the statistical data consisting of $n$ terms is decreased by 2, then the mean of the data :
• Find the mean of the following frequency distribution :
• Find the mode of the following frequency distribution:
• For what value of $x$, is the median of the following frequency distribution $34.5$?
• Following is the daily expenditure on lunch by 30 employees of a company. Find the mean daily expenditure of…
• Find the mode of the given frequency distribution:
• The weights (in kg) of 50 wild animals of a National Park were recorded and the following data was obtained.…
• For the following frequency distribution, find the median:
• If the mean and mode of a data are 12 and 21 respectively, then its median is :
• The mean of the following frequency distribution is 35. Find the values of $x$ and $y$, if the sum of frequen…
• The mean and median of a frequency distribution are 43 and 43·4 respectively. The mode of the distribution is…
• Assertion (A) : The mean of first 'n' natural numbers is $\frac{n-1}{2}$. Reason (R): The sum of first 'n' na…
• An SBI health insurance agent found the following data for distribution of ages of 100 policy holders. The he…
• The median and mode of a distribution are 25·2 and 26·1 respectively. The mean of the distribution is :
• The median of the following data is 137. Find the values of x and y, given that total of frequencies is 68.
• The cumulative frequency for calculating median is obtained by adding the frequencies of all the :
• If mean and median of given set of observations are 10 and 11 respectively, then the value of mode is :
• The following table shows the number of patients of different age group who were discharged from the hospital…
• If $x$ median + y mean = z mode; is the empirical relationship between mean, median and mode, then the value…
• Following data shows the marks obtained by 100 students in a class test : The median will be the average of w…
• Medical check-up was carried out for 35 students of a class and their weights were recorded as follows : Find…
• If the mode of some observations is 10 and sum of mean and median is 25, then the mean and median respectivel…
• If the maximum number of students has obtained 52 marks out of 80, then
• Following distribution shows the marks of 230 students in a particular subject. If the median marks are 46, t…
• The distribution below gives the marks obtained by 80 students on a test. The modal class of this distributio…
• If the value of each observation of a statistical data is increased by 3, then the mean of the data
• Based on the above information, answer the following questions.
• 250 apples of a box were weighed and the distribution of masses of the apples is given in the following table:
• If mode of the following frequency distribution is 55, then find the value of $x$.
• The mean of the following frequency distribution is 25. Find the value of f.
• Find the mean of the following data using assumed mean method :
• Heights of 50 students of class X of a school are recorded and following data is obtained : Find the median h…
• The mode of a grouped frequency distribution is 75 and the modal class is 65-80. The frequency of the class p…
• Based on the above, find the following :
• While calculating mean of a grouped frequency distribution, step deviation method was used $\left(\frac{x - a…
• The mean of the following frequency distribution is 28. If sum of all frequencies is 100, then find the value…
• Find median and mode of the following distribution:
• For some data $x_1, x_2, \ldots\, x_n$ with respective frequencies $f_1, f_2, \ldots\, f_n$, the value of $\s…
• The middle most observation of every data arranged in order is called :
• Based on the above information, answer the following :

Chapter 14: Probability (69)

• Two different dice are rolled together. The probability that both the obtained numbers are less than 4, is
• Assertion (A): If probability of happening of an event is $0.2p$, $p > 0$, then $p$ can't be more than $5$. R…
• A bag contains 30 balls out of which '$m$' number of balls are blue in colour.
• A card is selected at random from a deck of 52 playing cards. The probability of it being a red face card is:
• Assertion (A): The probability of selecting a number at random from the numbers 1 to 20 is 1. Reason (R): For…
• Two dice are thrown at the same time. Determine the probability that the difference of the numbers on the two…
• The probability of drawing an even prime number out of numbers from 1 to 30 is :
• In a cricket match, a batsman hits the boundary 7 times out of the 42 balls he plays. The probability of his…
• Answer the following questions if Rahul draws a card today :
• A die is thrown once. The probability of getting a number which is not a factor of 36, is:
• The probability of guessing the correct answer of a certain test question is $\dfrac{x}{12}$. If the probabil…
• Three unbiased coins are tossed simultaneously. Find the probability of getting:
• Two dice are rolled together. The probability of getting the sum of the two numbers to be more than 10, is
• A box contains cards numbered 6 to 55. A card is drawn at random from the box. The probability that the drawn…
• In a survey on holidays, 120 people were asked to state which type of transport they used on their last holid…
• Two dice are thrown together. The probability that they show different numbers is :
• The probability of guessing the correct answer to a certain test question is $\dfrac{x}{6}$. If the probabili…
• If a digit is chosen at random from the digits 1, 2, 3, 4, 5, 6, 7, 8, 9; then the probability that this digi…
• Assertion (A): In a cricket match, a batsman hits a boundary 9 times out of 45 balls he plays. The probabilit…
• One card is drawn at random from a well shuffled deck of 52 cards. Find the probability that the card drawn
• Two dice are tossed simultaneously. The probability of getting odd numbers on both the dice is :
• For an event $E$, if $P(E) + P(\bar{E}) = q$, then the value of $q^2 - 4$ is :
• Three coins are tossed simultaneously. What is the probability of getting
• A box contains 90 discs which are numbered 1 to 90. If one disc is drawn at random from the box, find the pro…
• One ticket is drawn at random from a bag containing tickets numbered 1 to 40. The probability that the select…
• A carton consists of 60 shirts of which 48 are good, 8 have major defects and 4 have minor defects. Nigam, a…
• Three unbiased coins are tossed simultaneously. Find the probability of getting :
• A bag contains 100 cards numbered 1 to 100. A card is drawn at random from the bag. What is the probability t…
• If three coins are tossed simultaneously, what is the probability of getting at most one tail?
• One school is chosen at random. Then:
• Two dice are thrown together. The probability of getting the difference of numbers on their upper faces equal…
• A card is drawn at random from a well-shuffled pack of 52 cards. The probability that the card drawn is not a…
• Assertion (A) : The probability that a leap year has 53 Sundays is $\dfrac{2}{7}$. Reason (R) : The probabili…
• If a fair coin is tossed twice, find the probability of getting 'atmost one head'.
• Two coins are tossed together. The probability of getting at least one tail is :
• Which of the following numbers cannot be the probability of happening of an event ?
• Based on the above, answer the following questions :
• A die is thrown once. Probability of getting a number other than 3 is :
• Directions: Two statements are given, one labelled as Assertion (A) and the other labelled as Reason (R). Sel…
• Two dice of different colours are thrown at the same time. Write down all the possible outcomes. What is the…
• The probability for a randomly selected number out of 1, 2, 3, 4, ..., 25 to be a composite number is :
• Two dice are thrown at the same time. Determine the probability that the (i) sum of the numbers on the two di…
• Two dice are rolled together. The probability that the sum of the numbers obtained is divisible by 6, is :
• Assuming that the dropped cards were a queen of hearts, a ten of spades and an ace of clubs, answer the follo…
• The number of red balls in a bag is 10 more than the number of black balls. If the probability of drawing a r…
• A pair of dice is thrown. The probability that sum of numbers appearing on top faces is at most 10 is :
• Two friends Anil and Ashraf were born in the December month in the year 2010. Find the probability that :
• A bag contains red balls and black balls in the ratio 3 : 7. A ball is drawn at random. The probability that…
• The probability of getting a composite number greater than 3 on throwing a die is
• While shuffling a pack of 52 cards, one card was accidently dropped. Find the probability that the dropped ca…
• Two coins are tossed simultaneously. The probability of getting atleast one head is
• Directions: A statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.…
• The number of red balls in a bag is three more than the number of black balls. If the probability of drawing…
• Probability of happening of an event is denoted by p and probability of non-happening of the event is denoted…
• A girl calculates that the probability of her winning the first prize in a lottery is 0.08. If 6000 tickets a…
• In a group of 20 people, 5 can't swim. If one person is selected at random, then the probability that he/she…
• A bag contains 4 red, 3 blue and 2 yellow balls. One ball is drawn at random from the bag. Find the probabili…
• A bag contains 5 pink, 8 blue and 7 yellow balls. One ball is drawn at random from the bag. What is the proba…
• A card is drawn at random from a well shuffled deck of 52 playing cards. The probability of getting a face ca…
• Based on the above information, answer the following questions:
• Which of the following can not be the probability of an event?
• A card is drawn at random from a well shuffled deck of 52 playing cards. The probability that it is either a…
• Meena calculates that the probability of her winning the first prize in a lottery is 0.08. If total 800 ticke…
• A bag contains 25 balls. Some of them are yellow and others are green. One ball is drawn at random. If probab…
• If the probability of a player winning a game is 0.79, then the probability of his losing the same game is :
• From the data 1, 4, 7, 9, 16, 21, 25, if all the even numbers are removed, then the probability of getting at…
• Two dice are rolled together. The probability of getting sum of numbers on the two dice as 2, 3 or 5, is :
• In a pack of 52 playing cards one card is lost. From the remaining cards, a card is drawn at random. Find the…
• Based on the above information, answer the following :

Chapter A1: Proofs in Mathematics (15)

• Prove that $\sqrt{5}$ is an irrational number.
• Which of the following statements is incorrect ?
• Prove that $\sqrt{3}$ is an irrational number.
• Prove that $4 + 3\sqrt{2}$ is an irrational number given that $\sqrt{2}$ is an irrational number.
• Prove that $\dfrac{2 - \sqrt{3}}{5}$ is an irrational number, given that $\sqrt{3}$ is an irrational number.
• Prove that $\left(\sqrt{2} + \dfrac{\sqrt{3}}{2}\right)$ is an irrational number, given that $\sqrt{6}$ is an…
• Prove that $2 + \sqrt{3}$ is an irrational number, given that $\sqrt{3}$ is an irrational number.
• Questions number 19 and 20 are Assertion and Reason based questions carrying 1 mark each. Select the correct…
• The natural number 2 is :
• Which of the following statements is true ?
• Prove that $\sqrt{2}$ is an irrational number.
• Which of the following statements is false ?
• Prove that $2 - 5\sqrt{3}$ is an irrational number given that $\sqrt{3}$ is irrational.
• Prove that $5 - 2\sqrt{3}$ is an irrational number. It is given that $\sqrt{3}$ is an irrational number.
• Show that the number $5 \times 11 \times 17 + 3 \times 11$ is a composite number.