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

X is a point on the side BC of ΔABC. XM and XN are drawn parallel to AB and AC respectively meeting AB in N and AC in M. MN produced meets CB produced at T. Prove that TX^{2} = TB × TC

In Figure, ABC is a triangle in which ∠B = 90º, BC = 48 cm and AB = 14 cm. A circle is inscribed in the triangle, whose centre is O. Find radius r of in-circle.

Find the linear relation between x and y such that P(x, y) is equidistant from the points A(1, 4) and B(–1, 2).

A, B, C are interior angles of ΔABC. Prove that cosec (A+B)/2 = sec C/2

A right circular cylinder and a cone have equal bases and equal heights. If their curved surface areas are in the ratio 8 : 5, show that the ratio between radius of their bases to their height is 3 : 4.

Seven times a two digit number is equal to four times the number obtained by reversing the order of its digits. If the difference of the digits is 3, determine the number.

Show that exactly one of the numbers n, n + 2 or n + 4 is divisible by 3.

If two positive integers p and q are written as p = a^{2}b^{3} and q = a^{3}b; a, b are prime numbers, then verify:

LCM (p, q) × HCF (p, q) = pq

Use Euclid’s algorithm to find the HCF of 4052 and 12576.

Using Euclid’s division algorithm find the HCF of the numbers 867 and 255.

Divide 27 into two parts such that the sum of their reciprocals is 3/20.

The sum of first n terms of an AP is given by S_{n} = 2n^{2} + 3n. Find the sixteenth term of the AP.

An AP consists of 37 terms. The sum of the three middle most terms is 225 and the sum of the last three terms is 429. Find the AP.

In an A.P if sum of its first n terms is 3n^{2} + 5n and its k^{th} term is 164, find the value of k.

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

If the area of triangle with vertices (x, 3), (4, 4) and (3, 5) is 4 square units, find x.

In figure, AB is a chord of length 8 cm of a circle of radius 5 cm. The tangents to the circle at A and B intersect at P. Find the length of AP.

The side of a square is 10 cm. Find the area between inscribed and circumscribed circles of the square.

The short and long hands of a clock are 4 cm and 6 cm long respectively. Find the sum of distances travelled by their tips in 48 hours.

If sin (A + 2B) = √3/2 and cos (A + 4B) = 0, A > B, and A + 4B ≤ 90º, then find A and B.

The following frequency distribution shows the distance (in metres) thrown by 68 students in a Javelin throw competition.

Draw a less than type Ogive for the given data and find the median distance thrown using this curve.

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

By changing the following frequency distribution ‘to less than type’ distribution, draw its ogive.

Find the value(s) of k for which the pair of linear equations kx + y = k^{2} and x + ky = 1 have infinitely many solutions.

For what values of m and n the following system of linear equations has infinitely many solutions.

3x + 4y = 12

(m + n)x + 2(m – n)y = 5m – 1

Find all the zeroes of the polynomial 3x^{4} + 6x^{3} - 2x^{2} - 10x - 5 if two of its zeroes are √(5/3) and – √(5/3)

Obtain all zeroes of 3x^{4} – 15x^{3} + 13x^{2} + 25x – 30, if two of its zeroes are √5/3 and – √5/3

A train travelling at a uniform speed for 360 km would have taken 48 minutes less to travel the same distance if its speed were 5 km/hour more. Find the original speed of the train.

Check whether the equation 5x^{2} – 6x – 2 = 0 has real roots and if it has, find them by the method of completing the square. Also verify that roots obtained satisfy the given equation.

A faster train takes one hour less than a slower train for a journey of 200 km. If the speed of slower train is 10 km/hr less than that of faster train, find the speeds of two trains.

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

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

If sin θ + cos θ = √2, then evaluate: tan θ + cot θ

The angle of elevation of the top of a hill at the foot of a tower is 60º and the angle of depression from the top of tower to the foot of hill is 30º. If tower is 50 metre high, find the height of the hill.

A cone of maximum size is carved out from a cube of edge 14 cm. Find the surface area of the remaining solid after the cone is carved out.

A man donates 10 aluminum buckets to an orphanage. A bucket made of aluminum is of height 20 cm and has its upper and lowest ends of radius 36 cm and 21 cm respectively. Find the cost of preparing 10 buckets if the cost of aluminum sheet is Rs. 42 per 100 cm^{2}.

Find the mean and mode for the following data:

A box contains cards numbered from 1 to 20. A card is drawn at random from the box. Find the probability that number on the drawn card is

- a prime number
- a composite number
- a number divisible by 3

A box contains cards numbered 11 to 123. A card is drawn at random from the box. Find the probability that the number on the drawn card is

- a square number
- a multiple of 7

A box contains 12 balls of which some are red in colour. If 6 more red balls are put in the box and a ball is drawn at random, the probability of drawing a red ball doubles than what it was before. Find the number of red balls in the bag.

The King, Queen and Jack of clubs are removed from a pack of 52 cards and then the remaining cards are well shuffled. A card is selected from the remaining cards. Find the probability of getting a card

- of spade
- of black king
- of club
- of jacks

In Figure, ABCD is a rectangle. Find the values of x and y.

Find the ratio in which P(4, m) divides the line segment joining the points A(2, 3) and B(6, –3). Hence find m.

Two different dice are tossed together. Find the probability:

- of getting a doublet
- of getting a sum 10, of the numbers on the two dice

An integer is chosen at random between 1 and 100. Find the probability that it is:

- divisible by 8
- not divisible by 8

Find all zeroes of the polynomial (2x^{4} – 9x^{3} + 5x^{2} + 3x – 1) if two of its zeroes are (2 + √3) and (2 – √3).