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CAT 2022 Slot 3 — QA Question 9

Discriminant and Roots of Quadratic EquationEasy

Suppose k is any integer such that the equation 2x2 + kx + 5 = 0 has no real roots and the equation x2 + (k - 5)x + 1 = 0 has two distinct real roots for x. Then, the number of possible values of k is

Answer & solution

  • A

    8

  • B

    7

  • 9

  • D

    13

Solution

Easy

Two discriminant conditions on the same integer kk. "No real roots" gives D<0D<0; "two distinct real roots" gives D>0D>0. Find the integer values of kk satisfying both and count them.

1

First condition (2x2+kx+5=02x^2+kx+5=0 has no real roots, so D<0D<0):

\begin{aligned} &k^2-4\cdot2\cdot5<0\ \Rightarrow\ k^2<40\\ &-\sqrt{40}
2

Second condition (x2+(k−5)x+1=0x^2+(k-5)x+1=0 has two distinct roots, so D>0D>0):

(k−5)2−4>0 ⇒ k2−10k+21>0(k−3)(k−7)>0 ⇒ k<3  or  k>7...(2)\begin{aligned} &(k-5)^2-4>0\ \Rightarrow\ k^2-10k+21>0\\ &(k-3)(k-7)>0\ \Rightarrow\ k<3\ \text{ or }\ k>7\quad\text{...(2)} \end{aligned}
3

Intersect (1) and (2). From (1), k∈[−6,6]k\in[-6,6]; combine with k<3k<3 or k>7k>7. No value in [−6,6][-6,6] exceeds 77, so we need −6≤k<3-6\le k<3:

k∈{−6,−5,−4,−3,−2,−1,0,1,2}⇒9 valuesk\in\{-6,-5,-4,-3,-2,-1,0,1,2\}\Rightarrow 9\text{ values}
Number of possible k=9\text{Number of possible }k=\mathbf{9}
CAT 2022 Slot 3 QA Q9: Suppose k is any integer such that the equation 2x 2 + kx + 5 = 0 has no real roots and the equation x 2 + (k — Solution | TheCATExam