Question 1

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

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:-
Complete the following statements:

(i) Probability of an event E + Probability of the event ‘not E’ =

(ii) The probability of an event that cannot happen is

(iii) The probability of an event that is certain to happen is

(iv) The sum of the probabilities of all the elementary events of an experiment is(v) The probability of an event is greater than or equal to

- 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 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:-
A die is thrown twice. What is the probability that

(i) 5 will not come up either time? (ii) 5 will come up at least once?

[Hint : Throwing a die twice and throwing two dice simultaneously are treated as the same experiment] - Q:-
Solve the following pair of linear equations by the substitution method.

- Q:-
A kite is flying at a height of 60 m above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is 60°. Find the length of the string, assuming that there is no slack in the string.

- Q:-
Find the values of k for each of the following quadratic equations, so that they have two equal roots.

(i) 2x^{2 }+ kx + 3 = 0 (ii) kx (x – 2) + 6 = 0 - Q:-
Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.

- Q:-
Draw the graphs of the equations x – y + 1 = 0 and 3x + 2y – 12 = 0. Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis, and shade the triangular region.

- Q:-
A statue, 1.6 m tall, stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is 60° and from the same point the angle of elevation of the top of the pedestal is 45°. Find the height of the pedestal.

- NCERT Chapter