Find two consecutive positive integers, sum of whose squares is 365.

Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.

Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers.
   (i) 26 and 91     (ii) 510 and 92     (iii) 336 and 54

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.

(i) x² – 2x – 8 (ii) 4s² – 4s + 1 (iii) 6x² – 3 – 7x (iv) 4u² + 8u (v) t² – 15 (vi) 3x² – x – 4

Prove that 3 + 2√5 is irrational.

A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was Rs 90, find the number of articles produced and the cost of each article.

The graphs of y = p(x) are given in Fig. 2.10 below, for some polynomials p(x). Find the number of zeroes of p(x), in each case.

Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial:
(i) t² – 3, 2t⁴ + 3t³ – 2t² – 9t – 12

(ii) x² + 3x + 1, 3x⁴ + 5x³ – 7x² + 2x + 2

(iii) x³ – 3x + 1, x⁵ – 4x³ + x² + 3x + 1

Find two numbers whose sum is 27 and product is 182.

Given that HCF (306, 657) = 9, find LCM (306, 657).

Sum of the areas of two squares is 468 m². If the difference of their perimeters is 24 m, ind the sides of the two squares.

A box contains 5 red marbles, 8 white marbles and 4 green marbles. One marble is taken out of the box at random. What is the probability that the marble taken out will be (i) red ? (ii) white ? (iii) not green?

A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle 30° with it. The distance between the foot of the tree to the point where the top touches the ground is 8 m. Find the height of the tree.

A jar contains 24 marbles, some are green and others are blue. If a marble is drawn at random from the jar, the probability that it is green is 2/3. 

Find the number of blue balls in the jar.

Prove that √5 is irrational.

Prove that 3 + 2√5 is irrational.

Two water taps together can fill a tank in  hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

Fill in the blanks :
(i) A tangent to a circle intersects it in                  point (s).
(ii) A line intersecting a circle in two points is called a                  .
(iii) A circle can have                       parallel tangents at the most.
(iv) The common point of a tangent to a circle and the circle is called                   .

Which of the following cannot be the probability of an event?

 (B) –1.5 (C) 15% (D) 0.7