Question 2

Prove that 3 + 2√5 is irrational.

Answer

Let us assume 3 + 2√5 is a rational number.

Therefore, 3 + 2√5 = p/q where p and q are co primes and q ≠ 0.

3 + 2√5 = ab

On solving, 2√5 =(a/b) - 3

√5 =1/2 (a/b - 3)

Since a, b are integers and 1/2 (a/b-3 ) is also a rational number.

But we know √5 is an irrational number.

Thus our assumption is wrong. 3 + 2√5 is not a rational number.

Hence proved.

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

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

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

- Q:-
Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:

(i) 2x

^{2 }– 3x + 5 = 0 (iii) 2x^{2}– 6x + 3 = 0 - 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 following pair of linear equations by the substitution method.

- Q:-
If tangents PA and PB from a point P to a circle with centre O are inclined to each other at angle of 80°, then ∠ POA is equal to

(A) 50° (B) 60°

(C) 70° (D) 80° - Q:-
Is it possible to design a rectangular park of perimeter 80 m and area 400 m

^{2}? If so, find its length and breadth. - Q:-
Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

- Q:-
A lot consists of 144 ball pens of which 20 are defective and the others are good. Nuri will buy a pen if it is good, but will not buy if it is defective. The shopkeeper draws one pen at random and gives it to her. What is the probability that

(i) She will buy it ?

(ii) She will not buy it ? - Q:-
In Fig. 10.13, XY and X′Y′ are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X′Y′ at B. Prove that ∠ AOB = 90°.

- Q:-
A die is thrown once. Find the probability of getting

(i) a prime number; (ii) a number lying between 2 and 6; (iii) an odd number. - Q:-
Prove that the parallelogram circumscribing a circle is a rhombus.

- 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. - Q:-
Find the roots of the following quadratic equations, if they exist, by the method of

completing the square:

(i) 2x^{2 }– 7x + 3 = 0 (ii) 2x^{2 }+ x – 4 = 0 (iv) 2x^{2}+ x + 4 = 0

- NCERT Chapter