Question 7

The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find the two numbers.

Answer

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

(i) x

^{2}– 2x – 8 (ii) 4s^{2}– 4s + 1 (iii) 6x^{2}– 3 – 7x (iv) 4u^{2}+ 8u (v) t^{2 }– 15 (vi) 3x^{2 }– x – 4 - Q:-
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.

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

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

- Q:-
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.

- Q:-
Refer to Example 13. (i) Complete the following table:

(ii) A student argues that ‘there are 11 possible outcomes 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12. Therefore, each of them has a probability

Do you agree with this argument? Justify your answer.

- Q:-
Find two numbers whose sum is 27 and product is 182.

- Q:-
Is the following situation possible? If so, determine their present ages.

The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48. - Q:-
Prove that 3 + 2√5 is irrational.

- Q:-
Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial:

(i) t^{2}– 3, 2t^{4}+ 3t^{3}– 2t^{2}– 9t – 12(ii) x

^{2}+ 3x + 1, 3x^{4}+ 5x^{3}– 7x^{2}+ 2x + 2(iii) x

^{3}– 3x + 1, x^{5}– 4x^{3}+ x^{2}+ 3x + 1

- Q:-
If the polynomial x

^{4 }– 6x^{3 }+ 16x^{2 }– 25x + 10 is divided by another polynomial x^{2 }– 2x + k, the remainder comes out to be x + a, find k and a. - Q:-
A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. Length PQ is:

- Q:-
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 for the same journey. Find the speed of the train.

- Q:-
Prove that the parallelogram circumscribing a circle is a rhombus.

- Q:-
Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

- Q:-
Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial:

(i) t^{2}– 3, 2t^{4}+ 3t^{3}– 2t^{2}– 9t – 12(ii) x

^{2}+ 3x + 1, 3x^{4}+ 5x^{3}– 7x^{2}+ 2x + 2(iii) x

^{3}– 3x + 1, x^{5}– 4x^{3}+ x^{2}+ 3x + 1 - Q:-
Solve the problems given in Example 1.

- Q:-
The sum of the reciprocals of Rehman’s ages, (in years) 3 years ago and 5 years from now is 1/3. Find his present age.

- Q:-
A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at random from the box, find the probability that it bears (i) a two-digit number (ii) a perfect square number (iii) a number divisible by 5.

- Q:-
Sum of the areas of two squares is 468 m

^{2}. If the difference of their perimeters is 24 m, ind the sides of the two squares.

- NCERT Chapter