CBSE Class 10 Mathematics
Previous Year Questions

The HCF of 96 and 404 is 4. Find their LCM.

Without actual division, state whether the rational number 13/3125 has a terminating or non-terminating repeating decimal expansion.

Find the HCF and LCM of 336 and 54 using prime factorisation. Verify that HCF × LCM = product of the two numbers.

Find the HCF of 1848 and 3058 using Euclid's division algorithm.

Prove that √5 is irrational.

Prove that 3 + 2√5 is irrational, given that √5 is irrational.

Three bells ring at intervals of 6 minutes, 12 minutes and 18 minutes. If they all ring together at 6:00 a.m., at what time will they ring together again?

Find the largest number which divides 245 and 1029 leaving remainder 5 in each case.

Show that any positive odd integer is of the form 6q + 1, 6q + 3, or 6q + 5, where q is some integer.

Check whether 6ⁿ can end with the digit 0, for any natural number n. Give reasons.

Given that √2 is irrational, prove that (5 + 3√2) is an irrational number.

An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?

If α and β are the zeroes of the polynomial p(x) = x² − 5x + 6, find the value of α + β and αβ.

Find a quadratic polynomial, the sum and product of whose zeroes are −3 and 2 respectively.

Find the zeroes of the quadratic polynomial x² + 7x + 10 and verify the relationship between the zeroes and the coefficients.

If α and β are zeroes of the polynomial f(x) = x² − p(x + 1) − c, show that (α + 1)(β + 1) = 1 − c.

On dividing x³ − 3x² + x + 2 by a polynomial g(x), the quotient and remainder were (x − 2) and (−2x + 4) respectively. Find g(x).

If the sum of the zeroes of the quadratic polynomial kx² + 2x + 3k is equal to the product of its zeroes, find the value of k.

Find the zeroes of the polynomial 4s² − 4s + 1 and verify the relationship between the zeroes and the coefficients.

Find a quadratic polynomial whose zeroes are 2 + √3 and 2 − √3.

What number should be added to the polynomial x² − 5x + 4 so that 3 is a zero of the resulting polynomial?

If one zero of the polynomial 2x² + 3x + λ is 1/2, find the value of λ and the other zero.

Solve: 2x + 3y = 11 and 2x − 4y = −24. Hence find the value of m for which y = mx + 3.

Solve graphically: 2x − y = 4 and x + y = −1. Find the vertices of the triangle formed by these two lines and the y-axis.

A fraction becomes 9/11 if 2 is added to both numerator and denominator. If 3 is added to both numerator and denominator, the fraction becomes 5/6. Find the fraction.

Five years hence, the age of Jacob will be three times that of his son. Five years ago, Jacob's age was seven times that of his son. What are their present ages?

For what value of k will the following pair of linear equations have no solution? 3x + y = 1; (2k − 1)x + (k − 1)y = 2k + 1.

A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it goes 40 km upstream and 55 km downstream. Determine the speed of the stream and the speed of the boat in still water.

Solve for x and y: 2/x + 3/y = 13 and 5/x − 4/y = −2.

The sum of a two-digit number and the number obtained by reversing the order of its digits is 99. If the digits differ by 3, find the number.

For what values of k will the pair kx + 2y = 3 and 3x + 6y = 10 have (i) a unique solution, (ii) no solution?

Places A and B are 100 km apart on a highway. A car starts from A and another from B at the same time. If they travel in the same direction, they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?

Solve by cross multiplication method: 2x + y = 5 and 3x + 2y = 8.

Meena went to a bank to withdraw &rupee;2000. She asked the cashier to give her &rupee;50 and &rupee;100 notes only. She got 25 notes in all. Find how many notes of each denomination she received.

Solve the quadratic equation 2x² + x − 4 = 0 by the method of completing the square.

Find the discriminant of x² + 5x + 5 = 0 and hence find the nature of its roots.

Find the value of k for which the equation x² − kx + 9 = 0 has equal roots, and find the roots.

A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.

Two water taps together fill a tank in 9⅜ hours. The larger tap takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

If the roots of the equation (c² − ab)x² − 2(a² − bc)x + (b² − ac) = 0 in x are equal, then show that either a = 0 or a³ + b³ + c³ = 3abc.

The diagonal of a rectangular field is 60 metres more than its shorter side. If the longer side is 30 metres more than the shorter side, find the sides of the field.

Find the roots of the quadratic equation: ¼x² − ½x − 1 = 0 using the quadratic formula.

Is the following situation possible? If so, find their present ages. The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48.

A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was ₹90, find the number of articles produced and the cost of each article.

For what value of p does the quadratic equation 2x² + px + 8 = 0 have real roots?

Sum of the areas of two squares is 468 m². If the difference of their perimeters is 24 m, find the sides of the two squares.

The sum of the first 20 terms of an AP is 400 and the sum of the first 40 terms is 1600. Find the sum of the first 60 terms.

How many terms of the AP: 9, 17, 25, … must be taken to give a sum of 636?

Find the 31st term of an AP whose 11th term is 38 and 16th term is 73.

If the 7th term of an AP is 1/9 and the 9th term is 1/7, find the 63rd term.

The first term of an AP is 5, the last term is 45, and the sum is 400. Find the number of terms and the common difference.

Find the sum of all two-digit natural numbers which are divisible by 4.

The ratio of the 11th term to the 18th term of an AP is 2:3. Find the ratio of the 5th term to the 21st term, and also find the ratio of the sum of the first 5 terms to the sum of the first 21 terms.

In an AP, the sum of the first n terms is 3n²/2 + 5n/2. Find its 25th term.

Find the middle term(s) of the AP: 7, 13, 19, … 241.

A sum of ₹700 is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is ₹20 less than its preceding prize, find the value of each prize.

Which term of the AP: 3, 15, 27, 39, … will be 132 more than its 54th term?

Find the sum of all odd integers between 2 and 100 which are divisible by 3.

State and prove the Basic Proportionality Theorem (Thales Theorem).

In the figure, ▵ABC and ▵DBC are two triangles on the same base BC. If AD intersects BC at O, show that: ar(▵ABC) / ar(▵DBC) = AO / DO.

In ▵ABC, DE ∥ BC, AD = 3 cm, DB = 2 cm and AE = 6 cm. Find AC.

Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.

Prove that, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (Pythagoras Theorem).

In an equilateral triangle ABC, D is a point on side BC such that BD = (1/3)BC. Prove that 9AD² = 7AB².

If the areas of two similar triangles are 81 cm² and 49 cm², and the altitude of the bigger triangle is 4.5 cm, find the corresponding altitude of the smaller triangle.

O is any point inside a rectangle ABCD. Prove that OB² + OD² = OA² + OC².

In ▵ABC, ∠B = 90°. If AB = 7 cm and BC = 24 cm, find AC.

Diagonals of a trapezium ABCD with AB ∥ DC intersect each other at point O. If AB = 2CD, find the ratio of the areas of ▵AOB and ▵COD.

▵ABC is right-angled at C. If p is the length of the perpendicular from C to AB, and a, b, c are the sides opposite to angles A, B, C respectively, prove that 1/p² = 1/a² + 1/b².

Two triangles BAC and BDC are right-angled at A and D respectively and are on the same side of BC. If AC and BD intersect at P, prove that: AP × PC = DP × PB.

Find the ratio in which the line segment joining the points A(1, −5) and B(−4, 5) is divided by the x-axis. Also find the coordinates of the point of division.

If A(4, −8), B(3, 6) and C(5, −4) are the vertices of a ▵ABC, D is the mid-point of BC and P is a point on AD joined such that AP/PD = 2, find the coordinates of P.

Find the area of the triangle whose vertices are (−5, −1), (3, −5) and (5, 2).

If the points (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find x and y.

If the point C(−1, 2) divides internally the line segment joining A(2, 5) and B(x, y) in the ratio 3:4, find the coordinates of B.

The vertices of a ▵ABC are A(4, 6), B(1, 5) and C(7, 2). A line is drawn to intersect sides AB and AC at D and E respectively such that AD/AB = AE/AC = 1/4. Calculate the area of ▵ADE and compare it with the area of ▵ABC.

Show that the points A(3, 5), B(6, 0), C(1, −3) and D(−2, 2) are the vertices of a square ABCD.

Find the value of k, if the points A(2, 3), B(4, k) and C(6, −3) are collinear.

In what ratio does the point (−4, 6) divide the line segment joining the points A(−6, 10) and B(3, −8)?

If two adjacent vertices of a parallelogram are (3, 2) and (−1, 0) and the diagonals bisect each other at (2, −5), find the other two vertices.

In a right triangle ABC, right-angled at B, if tan A = 1, verify that 2 sin A cos A = 1.

Evaluate: (sin² 30° + 4 cot² 45° − sec² 60°) / (cosec² 30° + cos 60°).

Prove that: (tan A − sin A) / (tan A + sin A) = (sec A − 1) / (sec A + 1).

If sin θ + cos θ = √3, then prove that tan θ + cot θ = 1.

Prove the identity: (cos A − sin A + 1) / (cos A + sin A − 1) = cosec A + cot A, using the identity cosec² A = 1 + cot² A.

If tan 2A = cot(A − 18°), where 2A is an acute angle, find the value of A.

Prove that: (sin A + cosec A)² + (cos A + sec A)² = 7 + tan² A + cot² A.

If cos θ = 7/25, find the value of all other trigonometric ratios of θ.

Without using trigonometric tables, evaluate: (sin 65° / cos 25°) + (cos 32° / sin 58°) − sin 28° sec 62° + cosec² 30°.

Show that: tan⁴ θ + tan² θ = sec⁴ θ − sec² θ.

If sec θ = 5/4, evaluate (sin θ − 2 cos θ) / (tan θ − cot θ).

Prove that: (1 + tan² A) / (1 + cot² A) = (1 − tan A)² / (1 − cot A)² = tan² A.

A 1.2 m tall girl spots a balloon moving with the wind in a horizontal line at a height of 88.2 m from the ground. The angle of elevation of the balloon from the eyes of the girl at any instant is 60°. After some time, the angle of elevation reduces to 30°. Find the distance travelled by the balloon during the interval.

The shadow of a tower standing on level ground is found to be 40 m longer when the Sun's altitude is 30° than when it is 60°. Find the height of the tower.

From a point on the ground, the angles of elevation of the bottom and top of a transmission tower fixed at the top of a 20 m tall building are 45° and 60° respectively. Find the height of the tower.

Two poles of equal heights are standing opposite to each other on either side of a road, which is 80 m wide. From a point between them on the road, the angles of elevation of the tops are 60° and 30°. Find the height of the poles and the distances of the point from the poles.

A statue 1.6 m tall stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is 60° and from the same point, the angle of elevation of the top of the pedestal is 45°. Find the height of the pedestal.

The angle of elevation of the top of a building from the foot of a tower is 30° and the angle of elevation of the top of the tower from the foot of the building is 60°. If the tower is 50 m high, find the height of the building.

From the top of a 7 m high building, the angle of elevation of the top of a cable tower is 60° and the angle of depression of its foot is 45°. Determine the height of the tower.

A vertical tower stands on a horizontal plane and is surmounted by a flagstaff of height 7 m. At a point on the plane, the angle of elevation of the bottom of the flagstaff is 30° and of the top of the flagstaff is 45°. Find the height of the tower.

As observed from the top of a 75 m high lighthouse from the sea-level, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships. (Use √3 = 1.732)

The angle of elevation of the top of a hill from the foot of a tower is 60° and the angle of elevation of the top of the tower from the foot of the hill is 30°. If the tower is 50 m high, what is the height of the hill? Find the distance between the hill and the tower.

Prove that the lengths of tangents drawn from an external point to a circle are equal.

Two concentric circles of radii 5 cm and 3 cm are drawn. Find the length of the chord of the larger circle which is tangent to the smaller circle.

In the figure, PQ is a tangent to the outer circle and a chord of the inner circle. The two circles are concentric. Show that PQ is bisected at the point of tangency.

A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = AD + BC.

If tangents PA and PB are drawn from a point P to a circle with centre O such that ∠APB = 80°, find ∠OAB.

Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.

From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. Find the radius of the circle.

PQ is a tangent drawn from an external point P to a circle with centre O. QOR is a diameter of the circle. If ∠POR = 120°, find ∠OPQ.

In the figure, XY and X’Y’ are two parallel tangents to a circle with centre O and another tangent AB with point of tangency C intersects XY at A and X’Y’ at B. Prove that ∠AOB = 90°.

In the figure, O is the centre of a circle of radius 5 cm. T is a point such that OT = 13 cm. TP and TQ are tangents. Find the length of segment PQ if PT = 12 cm. Also find the area of ▵OPQ.

Questions for this chapter will be added soon.

The radii of two circles are 8 cm and 6 cm respectively. Find the radius of the circle having its area equal to the sum of the areas of the two circles.

In the figure, ABCD is a square of side 14 cm with four congruent circles each of radius 7/2 cm. Find the area of the shaded region.

Find the area of the shaded region in the figure, if ABCD is a rectangle with AB = 8 cm, BC = 20 cm and O is the centre of two semicircles. (Use π = 3.14)

A chord of a circle of radius 15 cm subtends an angle of 60° at the centre. Find the area of the corresponding minor and major segments of the circle. (Use π = 3.14 and √3 = 1.732)

The minute hand of a clock is 14 cm long. Find the area swept by the minute hand between 9:00 a.m. and 9:35 a.m.

Find the area of the sector of a circle with radius 4 cm and angle 30°. Also find the area of the corresponding major sector. (Take π = 3.14)

In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. Find: (i) the length of the arc, (ii) area of the sector formed by the arc, (iii) area of the segment formed by the corresponding chord.

Three horses are tethered with 7 m long ropes at the three corners of a triangular field having sides 20 m, 34 m and 42 m. Find the area of the field which cannot be grazed by the horses. (Use π = 22/7)

Find the area of the shaded design in the figure, where ABCD is a square of side 10 cm and semicircles are drawn with each side of the square as diameter. (Use π = 3.14)

Two circles touch externally. The sum of their areas is 130π cm² and the distance between their centres is 14 cm. Find the radii of the circles.

A sphere of diameter 6 cm is dropped in a right circular cylindrical vessel partly filled with water. The diameter of the cylindrical vessel is 12 cm. If the sphere is completely submerged in water, by how much will the level of water rise in the cylindrical vessel?

A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of same radius. The total height of the toy is 15.5 cm. Find the total surface area of the toy. (Use π = 22/7)

A copper rod of diameter 1 cm and length 8 cm is drawn into a wire of length 18 m of uniform thickness. Find the thickness of the wire.

A solid iron pole consists of a cylinder of height 220 cm and base radius 24 cm, surmounted by a cone of height 42 cm. Find the mass of the pole, given that 1 cm³ of iron has approximately 8 g mass. (Use π = 3.14)

A bucket open at the top is in the form of a frustum of a cone with a capacity of 12308.8 cm³. The radii of the top and bottom circular ends are 20 cm and 12 cm respectively. Find the height of the bucket and the area of the metal sheet used in making the bucket. (Use π = 3.14)

A solid consisting of a right cone standing on a hemisphere is placed upright in a right circular cylinder full of water and touches the bottom. Find the volume of water left in the cylinder, if the radius of the cylinder is 60 cm and its height is 180 cm, the radius of the hemisphere is 60 cm, and height of cone is 120 cm.

A hemispherical tank full of water is emptied by a pipe at the rate of 25/7 litres per second. How much time will it take to empty the full tank if the tank has a radius of 3.5 m? (Take π = 22/7)

Water in a canal, 6 m wide and 1.5 m deep, is flowing with a speed of 10 km/h. How much area will it irrigate in 30 minutes if 8 cm of standing water is required?

Metallic spheres of radii 6 cm, 8 cm and 10 cm respectively are melted to form a single solid sphere. Find the radius of the resulting sphere.

A conical vessel, with base radius 5 cm and height 24 cm, is full of water. This water is emptied into a cylindrical vessel of base radius 10 cm. Find the height to which the water will rise in the cylindrical vessel.

A medicine capsule is in the shape of a cylinder with two hemispheres stuck at each end. The length of the entire capsule is 14 mm and the diameter is 5 mm. Find its surface area.

A 20 m deep well with diameter 7 m is dug and the earth from digging is evenly spread out to form a platform 22 m × 14 m. Find the height of the platform. (Use π = 22/7)

The following distribution shows the daily pocket allowance of children of a locality. The mean pocket allowance is ₹18. Find the missing frequency f. Classes: 11–13 (7), 13–15 (6), 15–17 (9), 17–19 (13), 19–21 (f), 21–23 (5), 23–25 (4).

Find the median of the following frequency distribution: Classes: 0–10 (5), 10–20 (8), 20–30 (20), 30–40 (15), 40–50 (7), 50–60 (5).

The following frequency distribution gives the monthly consumption of electricity of 68 consumers of a locality. Find the median, mean and mode of the data and compare them. Classes: 65–85 (4), 85–105 (5), 105–125 (13), 125–145 (20), 145–165 (14), 165–185 (8), 185–205 (4).

The median of the following data is 525. Find x and y if total frequency is 100. Classes: 0–100 (2), 100–200 (5), 200–300 (x), 300–400 (12), 400–500 (17), 500–600 (20), 600–700 (y), 700–800 (9), 800–900 (7), 900–1000 (4).

The following data gives the distribution of total household expenditure (in ₹) of manual workers in a city. Find the average (mean) expenditure of a worker per month. Use the step-deviation method. Classes: 1000–1500 (24), 1500–2000 (40), 2000–2500 (33), 2500–3000 (28), 3000–3500 (30), 3500–4000 (22), 4000–4500 (16), 4500–5000 (7).

Find the mode of the following frequency distribution: Classes: 0–20 (6), 20–40 (8), 40–60 (10), 60–80 (12), 80–100 (6), 100–120 (5), 120–140 (3).

The following table gives production yield per hectare of wheat of 100 farms of a village. Draw a ‘more than’ ogive for this data and find the median from the graph. Classes: 50–55 (2), 55–60 (8), 60–65 (12), 65–70 (24), 70–75 (38), 75–80 (16).

100 surnames were randomly picked from a local telephone directory and a frequency distribution of the number of letters in the English alphabet in the surnames was found as follows. Determine the median number of letters in the surnames. Also find the modal size of the surnames. Classes: 1–4 (6), 4–7 (30), 7–10 (40), 10–13 (16), 13–16 (4), 16–19 (4).

If the mean of the following distribution is 54, find the value of p. Classes: 0–20 (7), 20–40 (p), 40–60 (10), 60–80 (9), 80–100 (13).

Given that the mean of five numbers is 27. If one of the numbers is excluded, the mean of the remaining numbers becomes 25. Find the excluded number.

Two dice are thrown at the same time and the product of numbers appearing on them is noted. Find the probability that the product is a prime number.

A box contains 90 discs numbered from 1 to 90. One disc is drawn at random. Find the probability that it bears: (i) a two-digit number, (ii) a perfect square number, (iii) a number divisible by 5.

Cards marked with numbers 13, 14, 15, …, 60 are placed in a box and mixed thoroughly. One card is drawn at random. Find the probability that the number on the card drawn is: (i) divisible by 5, (ii) a number which is a perfect square.

From a deck of 52 playing cards, Jacks, Queens, Kings and Aces are removed. From the remaining cards, a card is drawn at random. Find the probability of getting: (i) a black face card, (ii) a red card.

A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball at random from the bag is three times that of a red ball, find the number of blue balls.

Two customers Shyam and Ekta are visiting a particular shop in the same week (Tuesday to Saturday). Each is equally likely to visit the shop on any one day. What is the probability that both will visit the shop on: (i) the same day, (ii) consecutive days, (iii) different days?

A die is thrown twice. What is the probability that: (i) 5 will not come up either time, (ii) 5 will come up at least once?

A game consists of tossing a one-rupee coin 3 times and noting its outcome each time. Ramesh wins if all the tosses give the same result (i.e., three heads or three tails) and loses otherwise. Calculate the probability that Ramesh will lose the game.

One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting: (i) a non-face card, (ii) a black king, (iii) a spade card.

In a lottery there are 10 prizes and 25 blanks. What is the probability of getting a prize? Also find the probability of not getting a prize.

A bag contains 6 red balls and 4 blue balls. A ball is drawn at random. What is the probability that the ball drawn is (i) red, (ii) not red?

A number x is selected from the numbers 1, 2, 3 and then a second number y is randomly selected from the numbers 1, 4, 9. What is the probability that the product xy of the two numbers will be less than 9?