The CAT QA section requires speed and accuracy, along with a thorough understanding of the Quadratic Equation. This article provides a set of MCQs on Quadratic Equation to help you understand the topic and improve your problem-solving skills with the help of detailed solutions by ensuring conceptual clarity, which will help you in the CAT 2025 exam preparation
Whether you're revising the basics or testing your knowledge, these MCQs will serve as a valuable practice resource.
The CAT 2025 exam is expected to follow a similar trend to the CAT 2024, with 22 questions from the QA section out of a total of 68 questions.
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CAT MCQs on Quadratic Equation
1. Let a, b, x, y be real numbers such that \(a^2+b^2=25,x^2+y^2=169,\) and \(ax+by=65.\) If \(k= ay-bx,\) then
A
k=0
B
\(0
C
\(k=\frac{5}{13}\)
D
\(k>\frac{5}{13}\)
2. If x is a real number, then \(\sqrt{log_e\frac{4x-x^2}{3}}\) is a real number if and only if
A
1≤x≤2
B
-3≤x≤3
C
1≤x≤3
D
-1≤x≤3
3. The quadratic equation \(x^2+bx+c=0\) has two roots 4a and 3a, where a is an integer. Which of the following is a possible value of \(b^2+c\ ?\)
A
3721
B
549
C
427
D
361
4. Let f(x) = ax² + bx + c, where a, b, and c are real numbers with a ≠ 0. If the graph of f(x) intersects the x-axis at two distinct points with x-coordinates p and q (p < q), then the equation ax² + bx + c + 1 = 0 has:
A
Exactly one root between p and q
B
No roots between p and q
C
Two roots, both between p and q
D
One root less than p and one root greater than q
5. Let \(x, y, z\) be real numbers such that \(4(x^2 + y^2 + z^2) = a\) and \(4(x - y - z) = 3 + a\), then find the value of \(a\).
A
1
B
1 and \(\frac{1}{3}\)
C
4
D
3
6. The roots \(\alpha\), \(\beta\) of the equation \(3x^2 + \lambda x - 1 = 0\), satisfy \( \frac{1}{\alpha^2} + \frac{1}{\beta^2} = 15 \). The value of \( (\alpha^3 + \beta^3)^2 \) is
A
16
B
9
C
1
D
4
7. If the roots of the equation \( x^2 - 2px + p^2 - 1 = 0 \) are real and distinct, what is the range of \( p \)?
A
\( p>1 \)
B
\( p<-1 \)
C
\( p>1 \text{ or } p<-1 \)
D
\( -1 < p < 1 \)
8. If the roots of the equation \[(a^2 + b^2)x^2 + 2(b^2 + c^2)x + (b^2 + c^2) = 0\] are real, which of the following must hold true?
A
\( c^2 \ge a^2 \)
B
\( c^4 \ge a^2(b^2 + c^2) \)
C
\( b^2 \ge a^2 \)
D
\( a^4 \le b^2(a^2 + c^2) \)
9. If \( \alpha \) and \( \beta \) are the roots of the quadratic equation \[x^2 - 10x + 15 = 0,\] then find the quadratic equation whose roots are \( \left( \alpha + \frac{\alpha}{\beta} \right) \) and \( \left( \beta + \frac{\beta}{\alpha} \right) \).
A
\( 15x^2 + 71x + 210 = 0 \)
B
\( 5x^2 - 22x + 56 = 0 \)
C
\( 3x^2 - 44x + 78 = 0 \)
D
Cannot be determined
10. If \( ax^2 + bx + c = 0 \) and \( 2a, b, 2c \) are in arithmetic progression, then which of the following are the roots of the equation?
A
\( \frac{a}{c} \)
B
\( \frac{a - c}{b} \)
C
\( \frac{a}{2}, \frac{c}{2} \)
D
\( \frac{c - a}{b - a} \)
11. The ratio of the roots of \( bx^2 + nx + n = 0 \) is \( p:q \), then
A
\( \frac{q \sqrt{p}}{\sqrt{q}} + \frac{p \sqrt{q}}{\sqrt{p}} = 0 \)
B
\( \frac{p}{\sqrt{q}} + \frac{q}{\sqrt{p}} = 0 \)
C
\( \frac{q}{\sqrt{p}} + \frac{p}{\sqrt{q}} = 0 \)
D
\( \frac{p}{\sqrt{q}} + \frac{q}{\sqrt{p}} eq 0 \)
12. A quadratic function \( f(x) \) attains a maximum of 3 at \( x = 1 \). The value of the function at \( x = 0 \) is 1. What is the value \( f(x) \) at \( x = 10 \)?
A
-119
B
-159
C
-110
D
-180
13. If \(x^2 - 7x + 12 = 0\), what is the value of \(x^3 - 4x^2 + 3x\)?
A
0
B
12
C
24
D
36
14. Find the roots of \( x^2 - 8x + 15 = 0\).
A
3, 5
B
2, 6
C
1, 7
D
4, 4
15. Let \( f(x) = ax^2 - b |x| \), where \(a\) and \(b\) are constants. Then at \(x = 0\), \(f(x)\) is
A
maximized whenever \( a>0, b>0 \)
B
maximized whenever \( a>0, b<0 \)
C
minimized whenever \( a>0, b>0 \)
D
minimized whenever \( a>0, b<0 \)
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