Class 10 MathematicsImportant Questions for Board Exams 2025-26

Prove that √2 is irrational.

Find the HCF of 96 and 404 using Euclid's algorithm.

Find the LCM of 12, 15, and 21 using prime factorisation.

Explain why 0.14114111411114... is irrational.

The HCF of two numbers is 9 and their LCM is 2016. If one number is 54, find the other.

Find the zeros of the polynomial 4x² + 5x − 6 and verify the relationship between the zeros and coefficients.

If α and β are zeros of x² − 4x + 3, find the value of α² + β².

Divide 3x³ + x² + 2x + 5 by 1 + 2x + x² and verify the division algorithm.

If one zero of the polynomial (a² + 9)x² + 13x + 6a is the reciprocal of the other, find a.

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

Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu. Find 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

Solve the following pair by cross-multiplication: x/a + y/b = a + b; x/a² + y/b² = 2

Find the values of k for which the quadratic equation kx(x − 2) + 6 = 0 has two equal 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. Find the speed of the train.

Find the roots of 4x² + 4√3x + 3 = 0.

Is it possible to design a rectangular mango grove whose length is twice its breadth and area is 800 m²?

The sum of the 4th and 8th terms of an AP is 24 and the sum of the 6th and 10th terms is 44. Find the first three terms.

How many three-digit numbers are divisible by 7?

In an AP, if the 12th term is −13 and the sum of the first 4 terms is 24, find the sum of the first 10 terms.

Find the sum of all two-digit odd positive integers.

Prove the Basic Proportionality Theorem.

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

In triangle ABC, DE is parallel to BC. If AD/DB = 3/5 and AC = 5.6 cm, find AE.

Prove Pythagoras theorem.

A vertical pole 6 m high casts a shadow of 4 m. Find the height of a nearby tower which casts a shadow of 28 m.

Find the ratio in which the point P(−6, a) divides the join of A(−3, 10) and B(6, −8).

Find the coordinates of the points which divide the line segment AB into four equal parts where A(−4, 0) and B(0, 6).

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

Find the area of the rhombus whose vertices are (3, 0), (4, 5), (−1, 4) and (−2, −1).

Prove: (sin θ − cosec θ)(cos θ − sec θ) = 1/(tan θ + cot θ)

If tan(A + B) = √3 and tan(A − B) = 1/√3, find A and B.

Evaluate: (sin 30° + tan 45° − cosec 60°) / (sec 30° + cos 60° + cot 45°)

If sin A = 3/4, find the value of all other trigonometric ratios.

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 was 60°. Find the height of the tower.

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°. Find the height of the tower.

Two poles of equal heights are standing opposite each other on either side of the road. From a point between them on the road, the angles of elevation are 60° and 30°. Find the height of each pole.

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

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

Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.

PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents to the circle at P and Q intersect at T. Find TP.

Find the area of the shaded region where ABCD is a square of side 14 cm and four circles of diameter 7 cm are drawn inside it.

Find the area of the sector of a circle of radius 7 cm with central angle 60°.

A brooch is made with silver wire in the form of a circle of diameter 35 mm with 5 diameters as spokes. Find the total length of silver wire required.

A solid is in the form of a cylinder with hemispherical ends. The total height of the solid is 19 cm and the diameter is 7 cm. Find its total surface area.

A metallic sphere of radius 4.2 cm is melted and recast into small spheres of radius 0.6 cm. Find the number of small spheres.

A container shaped like a right circular cylinder with diameter 12 cm and height 15 cm is full of ice cream. The ice cream is to be filled in cones of height 12 cm and diameter 6 cm, having a hemispherical shape at the top. Find the number of cones.

The following distribution shows the daily pocket allowance of children of a locality. The mean daily pocket allowance is Rs 18. Find the missing frequency.

Find the mode of the following frequency distribution: [table with class intervals]

The median of the following data is 525. Find the values of x and y if the total frequency is 100.

Draw the ogive for the given distribution and find the median.

A bag contains 3 red balls and 5 black balls. A ball is drawn at random. What is the probability that it is red?

A card is drawn from a well-shuffled deck of 52 playing cards. Find the probability that the card drawn is (i) a face card (ii) a red king.

Two dice are thrown simultaneously. What is the probability that: (i) 5 will not come up on either of them; (ii) 5 will come up on at least one.

A game consists of tossing a coin 3 times. Find the probability of getting exactly 2 heads.