CAT 2022 Question Paper | Quant Slot 3

CAT 2022 Quant was dominated by Arithmetic followed by Algebra. In Arithmetic, the questions were dominated by topics like Speed-time-distance, Mixture and Alligations. This year, there was a surprise. The questions from Geometry were relatively on the lower side as compared to the previous years. There were 8 TITA Qs this year. Overall this section was at a medium level of difficulty.

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

Video Explanation

Explanatory Answer

\( 2 x ^ { 2 } + k x + 5 = 0 \)
Since this Quadratic equation has no real roots, \( k ^ { 2 } < 4 ( 2 ) ( 5 ) \)
\( k ^ { 2 } < 40 \)
Since k is an integer, the possible values of k are (-6, -5, …, 5, 6)
\( x ^ { 2 } + ( k - 5 ) x + 1 = 0 \)
Since this Quadratic equation has real & distinct roots, \( ( k - 5 ) ^ { 2 } > 4 ( 1 ) ( 1 ) \)
\( ( k - 5 ) ^ { 2 } > 4 \)
Since k - 5 is an integer, the possible values of k - 5 are (..., -4, -3, 3, 4, …)
the possible values of k are (..., 1, 2, 8, 9, …)
Putting both the inferences together, the possible values of k are (-6, -5, -4, -3, -2, -1, 0, 1, 2)
k can take 9 values.

The answer is '9'

Choice C is the correct answer.

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