Quadratic Equations Exercise Quizs Question

Q 1. The roots of the equation ix² – 4x – 4i = 0 are
A. –2i
B. 2i
C. –2i, –2i
D. 2i, 2i

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Answer:-C. –2i, –2i

Q 2. Two students while solving a quadratic equation in x, one copied the constant term incorrectly and got the roots 3 and 2. The other copied the constant term and coefficient of x² correctly as –6 and 1 respectively. The correct roots are
A. 3, –2
B. –3, 2
C. 6, –1
D. -6, –1

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Answer:-C. 6, –1

Q 3. If sin A, sin B, cos A are in G.P., then roots of x² + 2x cot B + 1 = 0 are always
A. Real
B. Imaginary
C. Greater than 1
D. Equal

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Answer:-A. Real

Q 4.The  roots of the quadratic equation 2x² + 3x + 1 = 0, are
A. Irrational
B. Rational
C. Imaginary
D. None of these

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Answer:-B. Rational

Q 5. If a root of the equation x² + px + 12 = 0 is 4, while the roots of the equation x² + px + q = 0 are same, then the value of q will be
A. 4
B. 4/49
C. 49/4
D. None of these

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Answer:-C. 49/4

Q 6. Roots of the equation x² + bx – c = 0 (b, c > 0) are
A. Both positive
B. Both negative
C. Of opposite sign
D. None of these

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Answer:-C. Of opposite sign

Q 7. If the sum of the roots of the equation ax² + bx + c = 0 be equal to the sum of their squares, then
A. a(a + b) = 2bc
B. c(a + c) = 2ab
C. b(a + b) = 2ac
D. b(a + b) = ac

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Answer:-C. b(a + b) = 2ac

Q 8. If the roots of equation 5x² – 7x + k = 0 are reciprocal to each other, then value of k is
A. 5
B. 2
C. 1/5
D. 1

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Answer:-A. 5

Q 9.The roots of the equation x^2/3 + x^1/3 – 2 = 0 are
A. 1, 4
B. 1, –4
C. 1, –8
D. 1, 8

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Answer:-C. 1, –8

Q 10. The expression y = ax² + bx + c has always the same sign as c if
A. 4ac < b²
B. 4ac > b²
C. ac < b²
D. ac > b²

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Answer:-B. 4ac > b2

Q 11. If the roots of x² – bx + c = 0 are two consecutive integers, then b² – 4c is
A. 1
B. 2
C. 3
D. 4

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Answer:-A. 1

Q 12. If one of the roots of  equation x² + ax + 3 = 0 is 3 and one of the roots of the equation x² + ax + b = 0 is three times the other root, then the value of b is equal to
A. 3
B. 4
C. 2
D. 1

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Answer:-A. 3

Q 13.  If the roots of the equation ax² + bx + c = 0 are reciprocal to each other,  then
A. a – c = 0
B. b – c = 0
C. a + c = 0
D. b + c = 0

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Answer:-A. a – c = 0

Q 14. The number which exceeds its positive square root by 12 is
A. 9
B. 16
C. 25
D. None of these

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Answer:-B. 16

Q 15. What is the sum of the squares of roots of x² – 3x + 1 = 0
A. 5
B. 7
C. 9
D. 10

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Answer:-B. 7

Q 16. In the equation x³ + 3Hx + G = 0, if G and H are real and G² + H³ > 0, then the roots are
A. All real and equal
B. All real and distinct
C. One real and two imaginary
D. All real and two equal

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Answer:-C. One real and two imaginary

Q 17. Roots of ax² + b = 0 are real and distinct if
A. ab > 0
B. ab < 0
C. a, b > 0
D. a, b < 0

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Answer:-B. ab < 0

Q 18. If x² + y² = 25, xy = 12, then x =
A. {3, 4}
B. {3, –3}
C. {3, 4, –3, –4}
D. {–3, –3}

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Answer:-C. {3, 4, –3, –4}

Q 19. If the roots of equation x² + px + q = 0 differ by 1, then
A. p² = 4q
B. p² = 4q + 1
C. p² = 4q – 1
D. None of these

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Answer:-B. p2 = 4q + 1

Q 20. Two students while solving a quadratic equation in x, one copied the constant term incorrectly and got the roots 3 and 2. The other copied the constant term and coefficient of x² correctly as –6 and 1 respectively. The correct roots are
A. 3, –2
B. –3, 2
C. –6, –1
D. 6, –1

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Answer:-D. 6, –1

Q 21. If x be real, then the minimum value of x² – 8x + 17 is
A. –1
B. 0
C. 1
D. 2

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Answer:-C. 1

Q 22. Roots of the equations 2x² – 5x + 1 = 0, x² + 5x + 2 = 0 are
A. Reciprocal and of same sign
B. Reciprocal and of opposite sign
C. Equal in product
D. None of these

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Answer:-B. Reciprocal and of opposite sign

Q 23. The number of solutions of log₄ (x – 1) = log₂ (x – 3)
A. 3
B. 1
C. 2
D. 0

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Answer:-B. 1

Q 24. If the quadratic equations ax² + 2cx + b = 0 and ax² + 2bx + c = 0 have a  common root, then a + 4b + 4c =
A. –2
B. –1
C. 0
D. 1

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Answer:-C. 0

Q 25. If one root of ax² + bx + c = 0 be square of the other, then the value of b³ + ac² + a²c is
A. 3abc
B. –3abc
C. 0
D. None of these

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Answer:-A. 3abc

Q 26. If x² + px + 1 is a factor of the expression ax³ + bx + c, the
A. a² + c² = –ab
B. a² – c² = –ab
C. a² – c² = ab
D. None of these

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Answer:-C. a2 – c2 = ab

Q 27. The number of roots of the equation | x |² – 7 | x | + 12 = 0 is
A. 1
B. 2
C. 3
D. 4

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Answer:-D. 4

Q 28. The number of solutions for the equation  x² – 5 | x | + 6 = 0 is
A. 4
B. 3
C. 2
D. 1

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Answer:-A. 4

Q 29. Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation
A. x² – 18x – 16 = 0
B. x² – 18x + 16 = 0
C. x² + 18x – 16 = 0
D. x² + 18x + 16 = 0

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Answer:-B. x2 – 18x + 16 = 0

Q 30. If a + b + c = 0, then the roots of the equation 4x² + 3bx + 2c = 0 are
A. Equal
B. Imaginary
C. Real
D. None of these

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Answer:-C. Real

Q 31. If the roots of the equation x² + 2mx + m² – 2m + 6 = 0 are same, then the value of m will be
A. 3
B. 0
C. 2
D. –1

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Answer:-A. 3

Q 32. The number of values of  for which the equation x² – 3x + k = 0 has two real and distinct roots lying in the interval (0, 1), are
A. 0
B. 2
C. 3
D. Infinitely many

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Answer:-A. 0

Q 33. Product of real roots of the equation t²x² + | x | + 9 = 0
A. Is always positive
B. Is always negative
C. Does not exist
D. None of these

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Answer:-C. Does not exist

Q 34. The equation e^x – x – 1 = 0 has
A. Only one real root x = 0
B. At least two real roots
C. Exactly two real roots
D. Infinitely many real roots
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Answer:-A. Only one real root x = 0

$$\begin{gathered} e x = x + 1 ⇒ 1 + x 1 ! + x 2 2 ! + . . . . . = x + 1 \\ ⇒ x 2 2 ! + x 3 3 ! + . . . . = 0 \\ x 2 = 0 , x 3 = 0 , . . . . x n = 0 \end{gathered}$$

Q 35. If the product of roots of the equation, mx² + 6x + (2m – 1) = 0 is –1, then the value of m will be
A. 1
B. – 1
C. 1/3
D. – 1/3

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Answer:-C. 1/3

Q 36. The value of k for which 2x² – kx + x + 8 = 0 has equal and real roots are
A. –9 and –7
B. 9 and 7
C. –9 and 7
D. 9 and –7

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Answer:-D. 9 and –7

Q 37. The roots of 4x² + 6px + 1 = 0 are equal, then the value of p is
A. 4/5
B. 1/3
C. 2/3
D. 4/3

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Answer:-C. 2/3

Q 38. The number of quadratic equations which remains unchanged by squaring their roots is
A. 2
B. 4
C. 6
D. Infinite many

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Answer:-A. 2