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### Quadratic Equations is an important topic in CAT Algebra. The best way to master a topic is to practice problems and we bring to you this set of 5 quadratic equation problems that you can use for the same.

Question 1: Let a, b, c be real numbers a≠0. If p is a root of a2y2+by + c = 0, q is the root of a2y2+by + c = 0 and 0 <p< q, then the equation a2y2+by + c = 0 has a root r that always satisfies

(a) r = (p+q)/2
(b) r = p + (q/2)
(c) r = a
(d) p <r <q

Explanation: (d) ƒ(y) must have a root lying in the open interval (p, q).

Therefore p<r<q

Question 2: Let p, q be the roots of the equation (x-a)(x-b) = c, c is not equal to 0 . Then the roots of the equation (x-p)(x-q) + c = 0 are
(a) a, c
(b) b, c
(c) a, b
(d) a + c, b + c

Explanation: (c)

Given p, q are the roots of (x-a)(x-b) = c

(x-a)(x-b)-c = (x-p)(x-q)

(x-a)(x-b)= (x-p)(x-q)+c

a , b, are the roots of equation (x-p)(x-q)+c = 0

Question 3: If the roots of the equation x2-2bx+b2+b-3 = 0  are real and less than 3, then
(a) – 4 <b< 2
(b) 2<b< 3
(c) 0 <b< 4
(d) b> 3

Explanation: (a)   Question 4:

If p and q (p<q) are the roots of the equation x2+bx+c = 0 ,  where c< 0 <b.
(a) 0<p<q
(b) p< 0 <q< |p|
(c) p < q< 0
(d) p < 0 |p| <q

Explanation: (b)

Given , c<0<b

Since ce, p+q = -b …………………………………..(1)

and pq = c

From equation 2, c<0

pq<0

=> Either p is –ve , q is +ve

Or, p is +ve , q is –ve .

From Eq. 1, b>0

=> -b <0

=> p+q <0 therfore the sum is negative

Now since sum is negative and one of p and q is negative and also p < q, we can conclude that p is negative and q is positive but |p| > q

Hence we have p< 0 <q< |p|

Question 5: For all ‘x’, x2+2px+(10 – 3p)>0, then the interval in which ‘p’ lies is
(a) p< –5
(b) –5 <p < 2
(c) p > 5
(d) 2 <p< 5 