CBSE Class 10 Mathematics (Standard) — Short Answer Questions

Prove that $\sqrt{3}$ is an irrational number.

Find the ratio in which the x-axis divides the line segment joining the points $(-6, 5)$ and $(-4, -1)$. Also, find the point of intersection.

If $x = h + a\cos\theta, y = k + b\sin\theta$, then prove that : $\left(\frac{x-h}{a}\right)^2 + \left(\frac{y-k}{b}\right)^2 = 1$

OR Prove that : $\frac{\tan A}{1 + \sec A} - \frac{\tan A}{1 - \sec A} = 2\csc A$

In the given figure, Δ ABC is a right triangle in which $\angle B = 90^\circ$, AB = 4 cm and BC = 3 cm. Find the radius of the circle inscribed in the triangle ABC.

OR In the given figure, if a circle touches the side QR of Δ PQR at S and extended sides PQ and PR at M and N respectively, then prove that : $PM = \frac{1}{2}(PQ + QR + PR)$

A solid is in the form of a cylinder with hemispherical ends. The total height of the solid is 20 cm and the diameter of the cylinder is 7 cm. Find the total volume of the solid. (Use $\pi = \frac{22}{7}$)

Two dice of different colours are thrown at the same time. Write down all the possible outcomes. What is the probability that : (i) same number appears on both the dice ? (ii) different number appears on both the dice ?

What is the total distance the competitor has to run ?

OR If average speed of competitor is 5 m/s, then find the average time taken by competitor to put all the potatoes in the bucket.

(iii) (a) What is the total length of silver wire required ?

OR (iii) (b) What is the area of each sector of the brooch ?