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

^{2} = TB × TC

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

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

^{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.

_{n} = 2n^{2} + 3n. Find the sixteenth term of the AP.

^{2} + 5n and its k^{th} term is 164, find the value of k.

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

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.

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

^{2} and x + ky = 1 have infinitely many solutions.

3x + 4y = 12

(m + n)x + 2(m – n)y = 5m – 1

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

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

^{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.

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

^{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

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.

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

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