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CAT Exam  >  CAT Test  >  Level-wise Tests  >  Test Level 1: Quadratic Equations & Linear Equations - 2 - CAT MCQ

Test Level 1: Quadratic Equations & Linear - 2 Solved MCQs CAT


MCQ Practice Test & Solutions: Test Level 1: Quadratic Equations & Linear Equations - 2 (15 Questions)

You can prepare effectively for CAT Level-wise Tests for CAT with this dedicated MCQ Practice Test (available with solutions) on the important topic of "Test Level 1: Quadratic Equations & Linear Equations - 2". These 15 questions have been designed by the experts with the latest curriculum of CAT 2026, to help you master the concept.

Test Highlights:

  • - Format: Multiple Choice Questions (MCQ)
  • - Duration: 30 minutes
  • - Number of Questions: 15

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Test Level 1: Quadratic Equations & Linear Equations - 2 - Question 1
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If the difference between the roots of the equation x2 - 5x + 2k = 1 is 3, then find the value of k.

Detailed Solution: Question 1

The given equation can be written as x2 - 5x + (2k - 1) = 0.
If p and q are the roots, it is given that p - q = 3.
⇒ (p + q)2 - 4pq = 9. (∵ (p - q)2 = (p + q)2 - 4pq)
⇒ 25 - 4(2k - 1) = 9 (∵ p + q = 5, pq = 2k - 1)
∴ k = 2.5

Test Level 1: Quadratic Equations & Linear Equations - 2 - Question 2
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If x = 3 + √8, then find the value of x2 + (1/x2).

Detailed Solution: Question 2

Given: x = 3 + √8
1/x = 1/(3 + √8) = (3 - √8)/[32 - (√8)2] = 3 - √8/(9 - 8) = 3 - √8
x + 1/x = 3 +√8 + 3 - √8 = 6
Squaring both sides, we get:
(x + 1/x)2 = 62
x2 + 1/x2 + 2 = 36
x2 + 1/x2 = 36 - 2 = 34

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Test Level 1: Quadratic Equations & Linear Equations - 2 - Question 3
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If the polynomial ax4 + bx3 + cx2 + dx + e has the property that the product of its roots is 1/3rd of the sum taking two roots at a time, then find the relationship between e and c.

Detailed Solution: Question 3

Product of all the roots = e/a
Sum of the roots taking two at a time = c/a

Test Level 1: Quadratic Equations & Linear Equations - 2 - Question 4
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If (c - 3)2 = c2 + kc + 9, then the value of k is:

Detailed Solution: Question 4

Given, (c - 3)2 = c2 + kc + 9
We know the formula: (a - b)2 = a2 - 2ab + b2
Applying the above formula, we get
c2 - 6c + 9 = c2 + kc + 9
Comparing both sides, we get
k = -6

Test Level 1: Quadratic Equations & Linear Equations - 2 - Question 5
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Find x + y + z if 5x + 9y + 13z = 50 and 24x + 20y + 16z = 66.  

Detailed Solution: Question 5

5x + 9y + 13z = 50 and 24x + 20y + 16z = 66
Adding both, we get
29x + 29y + 29z = 116
Thus, x + y + z = 4

Test Level 1: Quadratic Equations & Linear Equations - 2 - Question 6
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Find the nature of roots of the following equations:
(i) x2 + 9x + 27 = 0
(ii) 6x2 - 13x - 5 = 0

Detailed Solution: Question 6

(i) x2 + 9x + 27 = 0
D = 92 - 4(27)
D < 0; therefore, roots are imaginary.
(ii) 6x2 - 13x - 5 = 0
D = (-13)2 - 4(-5)(6)
D > 0; therefore, roots are real.
So, option (3) is the correct answer.

Test Level 1: Quadratic Equations & Linear Equations - 2 - Question 7
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If the roots of the equation 3x2 + 9x + k = 0 are in the ratio of 1 : 2, then what is the value of k?

Detailed Solution: Question 7

Since the roots are in the ratio of 1 : 2, they can be assumed as α and 2α.
Sum of the roots

Product of the roots = α (2α) = k/3 ⇒ k = 6

Test Level 1: Quadratic Equations & Linear Equations - 2 - Question 8
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Find the value of (x – a)(x – b)(x – c) … (x – z).

Detailed Solution: Question 8

The 24th term in the series will be (x – x).
Now, (x – x) = 0
So, (x – a)(x – b) … (x – z) = 0.

Test Level 1: Quadratic Equations & Linear Equations - 2 - Question 9
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After 15 years, Ralph's age will be five times his age 5 years back. What is the present age of Ralph?

Detailed Solution: Question 9

Let Ralph's present age be x years.
After 15 years, his age will be (x + 15) years. His age 5 years back was (x - 5) years.
According to the condition: x + 15 = 5(x - 5)
On solving, we get x = 10.
Therefore, Ralph's present age = 10 years

Test Level 1: Quadratic Equations & Linear Equations - 2 - Question 10
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a, b and c are three integers and they are not equal to each other. If a + b + c = 1, provided one of them is equal to zero, then the value of  Here, x will be equal to

Detailed Solution: Question 10

Since one of a, b, c is equal to zero, we can put the values of a, b and c as a = 2, b = -1, c = 0 in the following equation

On solving we get x = 2/3.

Test Level 1: Quadratic Equations & Linear Equations - 2 - Question 11
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If x, y, z are distinct positive real numbers, then  would be

Detailed Solution: Question 11

Since, x, y, z are distinct positive real numbers, use AM > GM

Hence, option (3) is correct.

Test Level 1: Quadratic Equations & Linear Equations - 2 - Question 12
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Find the maximum value of the expression

Detailed Solution: Question 12

For the given expression to be a maximum, the denominator should be minimized. (Since, the function in the denominator has imaginary roots and is always positive). x2 + 5x + 10 will be minimized at x = – 2.5 and its minimum values at x = –2.5 is 3.75. Hence, required answer = 1/3.75 = 4/15.

Test Level 1: Quadratic Equations & Linear Equations - 2 - Question 13
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Find the minimum value of the expression (p +1/p); p > 0.

Detailed Solution: Question 13

The minimum value of ( p + 1/p) is at p = 1. The value is 2.

Test Level 1: Quadratic Equations & Linear Equations - 2 - Question 14
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If the roots of equation x2 + bx + c = 0 differ by 2, then which of the following is true?

Detailed Solution: Question 14

To solve this take any expression whose roots differ by 2.
Thus, (x – 3) (x – 5) = 0
⇒ x2 – 8x + 15 = 0
In this case, a = 1, b = –8 and c = 15.
We can see that b2 = 4(c + 1).

Test Level 1: Quadratic Equations & Linear Equations - 2 - Question 15
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If f(x) = (x + 2) and g(x) = (4x + 5), and h(x) is defined as h(x) = f(x) · g(x), then sum of roots of h (x) will be

Detailed Solution: Question 15

h (x) = 4x2 + 13x + 10.
Sum of roots -13/4.

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About this CAT Practice Test

This test contains 15 MCQs on Test Level 1: Quadratic Equations & Linear Equations - 2 with detailed solutions for each question. It is part of the Level-wise Tests for CAT course on EduRev, designed to align with the current CAT syllabus. Each solution explains not just the correct answer but also gives you an understanding of the topic — not just to attempt the question but also help you prepare for the exam with practice.

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