1)  Construct a quadratic equation with roots 7 and −3.

Solution

2)  A quadratic polynomial has one of its zeros 1 +√5 and it satisfies p(1) = 2. Find the quadratic polynomial.

3)  If α and β are the roots of the quadratic equation x2 + √2x+3 = 0, form a quadratic polynomial with zeroes 1/α, 1/β .

4)  If one root of k(x − 1)2 = 5x − 7 is double the other root, show that k = 2 or −25.

5)  If the difference of the roots of the equation 2x2 − (a+1)x + a −1 = 0 is equal to their product, then prove that a = 2.

6)  Find the condition that one of the roots of ax2 + bx + c may be

(i) negative of the other,

(ii) thrice the other,

(iii) reciprocal of the other.  Solution

7)  If the equations x2 −ax+b = 0 and x2 −ex+f = 0 have one root in common and if the second equation has equal roots, then prove that ae = 2(b + f).  Solution

8)  Discuss the nature of roots of

(i) −x2 + 3x+1 = 0,

(ii) 4x2 − x − 2 = 0,

(iii) 9x2 + 5x = 0.    Solution

9)  Without sketching the graphs, find whether the graphs of the following functions will intersect the x-axis and if so in how many points.

(i) y = x2 + x + 2

(ii) y = x2 − 3x − 7

(iii) y = x2 + 6x + 9.     Solution

10)  Write f(x) = x2 + 5x + 4 in completed square form.

Solution

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