Quadratic Equations · NDA previous year questions

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2025 9 questions

Q212025 · 14 Sept 2025
For the following two (02) items: Let $α$ and $β$ be the roots of the quadratic equation $x^{2} + (lo g_{0.5} (α^{2})) x + (lo g_{0.5} (α^{2}))^{4} = 0$ where $α^{2} \neq = 1$ and $lo g_{0.5} (α^{2}) > 0$ Further, $β^{2} = α (lo g_{α^{2}} (0.5))$ What is $β$ equal to?
1. A. $lo g_{α^{2}} (0.5)$
2. B. $lo g_{0.5} (α^{2})$
3. C. $2 (lo g_{α^{2}} (0.5))$
4. D. $2 lo g_{0.5} (α^{2})$
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Answer: B
Q222025 · 14 Sept 2025
For the following two (02) items: Let $α$ and $β$ be the roots of the quadratic equation $x^{2} + (lo g_{0.5} (α^{2})) x + (lo g_{0.5} (α^{2}))^{4} = 0$ where $α^{2} \neq = 1$ and $lo g_{0.5} (α^{2}) > 0$ Further, $β^{2} = α (lo g_{α^{2}} (0.5))$ What is the relation between $α$ and $β$ ?
1. A. $α = 2 β$
2. B. $2 α = β$
3. C. $α = - 2 β$
4. D. $2 α = - β$
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Answer: A
Q232025 · 14 Sept 2025
For the following two (02) items: Let $p = \sum_{j = 1}^{n} lo g_{10} 2^{j} and q = \sum_{j = 1}^{n} lo g_{10} 5^{j}$ If p + q = 66 then which one of the following is correct?
1. A. $n < 7$
2. B. $7 < n < 9$
3. C. $9 < n < 12$
4. D. $n > 12$
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Answer: C
Q242025 · 14 Sept 2025
For the following two (02) items: Let $p = \sum_{j = 1}^{n} lo g_{10} 2^{j} and q = \sum_{j = 1}^{n} lo g_{10} 5^{j}$ If p + q = 15 then what is equal to? q - p
1. A. $lo g_{10} 2 \cdot 5$
2. B. $5 lo g_{10} 2 \cdot 5$
3. C. $10 lo g_{10} 2 \cdot 5$
4. D. $15 lo g_{10} 2 \cdot 5$
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Answer: D
Q312025 · 14 Sept 2025
Passage
For the following two (02) items: Consider the equation abx² + bcx + ca = cax² + abx + bc
If the roots of the equation are equal, then which one of the following is correct?
1. A. $a c = b^{2}$
2. B.a + c = 2b
3. C. $\frac{1}{a} + \frac{1}{c} = \frac{1}{2 b}$
4. D. $\frac{1}{a} + \frac{1}{c} = \frac{2}{b}$
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Answer: D
Q372025 · 14 Sept 2025
For the following two (02) items: Let $f (x) = a x^{2} + b x + c$ be a quadratic polynomial such that f(1) = f(4) = 2 Further, 2 is a root of f(x) = 0 What is the other root of f(x) = 0
1. A.1
2. B.2
3. C.3
4. D.Cannot be determined
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Answer: C
Q382025 · 14 Sept 2025
For the following two (02) items: Let $f (x) = a x^{2} + b x + c$ be a quadratic polynomial such that f(1) = f(4) = 2 Further, 2 is a root of f(x) = 0 What is (a + b + c) equal to?
1. A.0
2. B.1
3. C.2
4. D.Cannot be determined
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Answer: C
Q32025 · 13 Apr 2025
If one root of the equation $x^{2} - k x + k = 0$ exceeds the other by $23$ then which one of the following is a value of k?
1. A.3
2. B.6
3. C.9
4. D.12
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Answer: B
Q42025 · 13 Apr 2025
If $x + \frac{5}{y} = 4$ , and $y + \frac{5}{x} = - 4$ , then what is (x + y) equal to?
1. A.0
2. B.1
3. C.4
4. D.5
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Answer: A