Class 10 — Maths

Find the HCF and LCM of 96 and 404 by prime factorisation.

Prove that √5 is an irrational number.

Find the HCF of 510 and 92 and verify HCF × LCM = product of numbers.

Without long division, state whether 13/3125 has terminating or non-terminating decimal expansion.

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

Show that 3 + 2√5 is irrational.

Three bells toll at intervals of 9, 12, 15 minutes. If they toll together at 8 a.m., when will they toll together next?

Express 156 as a product of its prime factors.

Find the zeroes of x^2 + 7x + 10 and verify the relationship between zeroes and coefficients.

Find a quadratic polynomial whose zeroes are 3 and -2.

If α and β are zeroes of x^2 - 5x + 6, find α^2 + β^2.

Find the zeroes of 6x^2 - 7x - 3 by factorisation.

If the sum of zeroes of kx^2 + 2x + 3k = 0 equals their product, find k.

Find the zeroes of x^2 - 2√2 x - 16 by factorisation.

Form a quadratic polynomial whose sum of zeroes is 4 and product is 3.

Draw the graph of y = x^2 - 4 and find the number of zeroes.

Solve graphically: 2x + y = 6 and 2x - y = 2.

Solve by substitution: x + y = 14 and x - y = 4.

Solve by elimination: 3x + 4y = 10 and 2x - 2y = 2.

The sum of two numbers is 60 and their difference is 8. Find the numbers.

For what value of k will kx + 2y = 5 and 3x + y = 1 have a unique solution?

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

A fraction becomes 1/3 when 1 is subtracted from the numerator and 1/4 when 8 is added to the denominator. Find the fraction.

Father's age is three times the age of his son. After 5 years, father's age will be 2.5 times son's age. Find their present ages.

Solve by factorisation: x^2 - 7x + 10 = 0.

Find the roots of 2x^2 - 5x + 3 = 0 using the quadratic formula.

Find the nature of the roots of x^2 + 4x + 5 = 0.

If the roots of (b-c)x^2 + (c-a)x + (a-b) = 0 are equal, prove that 2b = a + c.

The sum of squares of two consecutive natural numbers is 365. Find the numbers.

A train travels 480 km at a uniform speed. If speed had been 8 km/h more, it would have taken 3 hours less. Find the speed.

The product of Aman's age 5 years ago and his age 8 years later is 30. Find his present age.

Find the value of k so that 2x^2 + kx + 3 = 0 has equal roots.

Find the 15th term of the AP: 7, 13, 19, 25, …

Which term of the AP: 21, 18, 15, … is -81?

Find the sum of the first 25 terms of the AP: 5, 12, 19, 26, …

The sum of the first n terms is S_n = 3n^2 + 5n. Find the nth term.

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

How many terms of AP: -6, -11/2, -5, … are needed for a sum of -25?

The 17th term of an AP exceeds its 10th term by 7. Find the common difference.

A man saves Rs. 200 in the first month and increases savings by Rs. 50 each month. Find total savings after 1 year.

In △ABC, DE ∥ BC. If AD = 3 cm, DB = 5 cm and AE = 4.5 cm, find EC.

State and prove the Basic Proportionality Theorem.

Prove that the ratio of areas of two similar triangles equals the square of the ratio of their corresponding sides.

Prove the Pythagoras theorem.

A ladder 25 m long reaches a window 24 m above the ground. Find the distance of the foot from the wall.

In equilateral △ABC, BC is trisected at D. Prove that 9 AD^2 = 7 AB^2.

If corresponding sides of two similar triangles are in ratio 4:7, find the ratio of their areas.

△ABC is an isosceles right triangle right-angled at C. Prove that AB^2 = 2 AC^2.

Find the distance between the points (3, 4) and (-2, 1).

Find the mid-point of the segment joining (4, -3) and (-2, 7).

Find the coordinates of the point dividing the segment joining (-1, 3) and (4, -7) in the ratio 3:2.

Show whether the points (1, 7), (4, 2) and (-1, -1) are collinear.

Find the value of k if the distance between (k, -1) and (3, 2) is 5.

Find the area of the triangle with vertices (2, 3), (-1, 0) and (2, -4).

A(1,2), B(4,y), C(x,6) and D(3,5) are vertices of a parallelogram. Find x and y.

Find the ratio in which the y-axis divides the segment joining (5, -6) and (-1, -4).

If tan A = 4/3, find sin A and cos A.

Evaluate: sin^2 60° + cos^2 60° - tan 45°.

Prove that (1 + sin A)/(1 - sin A) = (sec A + tan A)^2.

If sin A = 3/5, find cos A and tan A.

Evaluate: (sin 30° + cos 60°)/tan 45°.

Express sin 72° in terms of trigonometric ratios of an angle less than 45°.

Prove: tan A/(1-cot A) + cot A/(1-tan A) = 1 + sec A · cosec A.

If cos A = 7/25, find the value of tan A + cot A.

A tower stands on the ground. From a point 30 m from its foot, the angle of elevation of the top is 30°. Find the height of the tower.

From the top of a 75 m high building, the angle of depression of a car is 30°. Find the distance of the car from the building.

The shadow of a tower is 40 m longer when sun's altitude is 30° than when it is 60°. Find the height of the tower.

The angle of elevation of the top of a pole from a point 12 m from its foot is 60°. Find the height of the pole.

From the top of a 100 m tower, two cars are seen with angles of depression 30° and 45°. Find the distance between the cars.

A ladder leans against a wall at 60° to the ground. If the foot is 2.5 m from the wall, find the length of the ladder.

A man on a cliff observes a boat with angles of depression changing from 30° to 60° in 3 minutes. Find the time taken to reach the shore.

Angles of elevation of the top of a tower from two points 4 m and 9 m from the base are complementary. Find the height of the tower.

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

Two tangents PT and PS are drawn from P to a circle with centre O. If ∠TPS = 60°, find ∠TOS.

The length of a tangent from a point 5 cm from the centre is 4 cm. Find the radius.

Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

Two concentric circles have radii 7 cm and 25 cm. Find the length of the chord of the larger circle that touches the smaller circle.

A quadrilateral ABCD circumscribes a circle. Prove that AB + CD = AD + BC.

Prove that the perpendicular at the point of contact to the tangent passes through the centre.

Two tangents from an external point are inclined at 80° to each other. Find the angle subtended at the centre by the radii to the points of contact.

Find the area of a circle whose circumference is 44 cm.

Find the area of a sector of radius 14 cm and central angle 90°.

Find the length of an arc of radius 21 cm that subtends 60° at the centre.

A chord of a circle of radius 10 cm subtends a right angle at the centre. Find the area of the minor segment. (π = 3.14)

A horse is tied at a corner of a square field of side 15 m by a 5 m rope. Find the area the horse can graze.

The minute hand of a clock is 14 cm long. Find the area swept in 10 minutes.

Wheel diameter is 80 cm. How many revolutions does it make in 10 minutes at 66 km/h?

Find the area of the shaded region: a square of side 14 cm with a circle of radius 7 cm inside it.

Find the total surface area of a hemisphere of radius 7 cm.

A solid is a cone mounted on a hemisphere of the same radius 3.5 cm; cone height = 7 cm. Find the total volume.

A sphere of radius 6 cm is melted and recast into three balls. If two radii are 3 cm and 4 cm, find the third radius.

A cylindrical bucket (h = 32 cm, r = 18 cm) is filled with sand forming a conical heap of radius 36 cm. Find the height of the heap.

A cone of radius 3.5 cm is mounted on a hemisphere of the same radius; total height = 15.5 cm. Find the total surface area.

A frustum bucket has radii 28 cm and 21 cm and height 24 cm. Find the volume and surface area.

A frustum of a cone has circular ends of radii 6 cm and 14 cm and slant height 10 cm. Find the curved surface area.

A hemispherical bowl of internal radius 9 cm contains liquid filled into bottles of radius 1.5 cm and height 4 cm. How many bottles are needed?

Find the mean of: classes 0-10 (f=5), 10-20 (8), 20-30 (12), 30-40 (10), 40-50 (5).

Find the median of: classes 100-120 (12), 120-140 (14), 140-160 (8), 160-180 (6), 180-200 (10).

Find the mode of: classes 10-20 (5), 20-30 (8), 30-40 (15), 40-50 (12), 50-60 (10).

The mean of 50 numbers is 30. If each number is increased by 5, find the new mean.

Draw a less than ogive for: classes 0-10 (5), 10-20 (10), 20-30 (15), 30-40 (12), 40-50 (8).

The median of the following data is 525. Find x and y if total frequency is 100. (Standard NCERT question)

Find the mean by step deviation method: 0-50(4), 50-100(8), 100-150(10), 150-200(12), 200-250(10), 250-300(4), 300-350(2).

State the empirical relationship between mean, median and mode and apply it to a given problem.

Two dice are thrown together. Find the probability that the sum is 8.

A card is drawn from a deck of 52. Find P(king), P(red queen) and P(face card).

A bag contains 5 red, 4 green and 3 blue balls. A ball is drawn at random. Find P(not blue).

Probability of guessing the correct answer is x/12. If P(wrong) = 2/3, find x.

Cards numbered 1 to 20 are in a box. Find P(multiple of 3) and P(prime number).

Two coins are tossed simultaneously. Find P(at most one head).

From 52 cards, all kings, queens and aces are removed. A card is drawn. Find P(red card).

A die is thrown twice. Find P(5 will not come up either time).