CAT sample papers

Quadratic Equations Exercise

Quadratic Equations questions

Quadratic Equations Test:

Ques: The roots of the equation ix2 – 4x – 4i = 0 are
(a) –2i
(b) 2i
(c) –2i, –2i
(d) 2i, 2i
Ans. (c)

Related: probability question examples

Ques: 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
Ans. (a)

Ques: The  roots of the quadratic equation 2x2 + 3x + 1 = 0, are
(a) Irrational
(b) Rational
(c) Imaginary
(d) None of these
Ans. (b)

Ques: 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
Ans. (c)

Related: Mensurations formulas questions

Ques: 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
Ans. (c)

Ques: 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
Ans. (c)

Ques: 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
Ans. (a)

Related: HCF and LCM word problems

Ques: The roots of the equation x2/3 + x1/3 – 2 = 0 are
(a) 1, 4
(b) 1, –4
(c) 1, –8
(d) 1, 8
Ans. (c)

Ques: 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
Ans. (b)

Ques: If the roots of x2 – bx + c = 0 are two consecutive integers, then b2 – 4c is
(a) 1
(b) 2
(c) 3
(d) 4
Ans. (a)

Related: Coding decoding

Ques: 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
Ans. (a)

Ques: 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
Ans. (a)

Ques: The number which exceeds its positive square root by 12 is
(a) 9
(b) 16
(c) 25
(d) None of these
Ans. (b)

Related: Ratio and proportions problems

Ques: What is the sum of the squares of roots of x2 – 3x + 1 = 0
(a) 5
(b) 7
(c) 9
(d) 10
Ans. (b)

Ques: Roots of ax2 + b = 0 are real and distinct if
(a) ab > 0
(b) ab < 0
(c) a, b > 0
(d) a, b < 0
Ans. (b)

Ques: If x2 + y2 = 25, xy = 12, then x =
(a) {3, 4}
(b) {3, –3}
(c) {3, 4, –3, –4}
(d) {–3, –3}
Ans. (c)

Related: Average problems for practice

Ques: 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
Ans. (b)

Ques: If x be real, then the minimum value of x2 – 8x + 17 is
(a) –1
(b) 0
(c) 1
(d) 2
Ans. (c)

Ques: 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
Ans. (b)

Related: arithmetic operation questions and answers

Ques: The number of solutions of log4 (x – 1) = log2 (x – 3)
(a) 3
(b) 1
(c) 2
(d) 0
Ans. (b)

Ques: 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
Ans. (a)

Ques: 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
Ans. (c)

Related: Percentage questions

Ques: The number of roots of the equation | x |2 – 7 | x | + 12 = 0 is
(a) 1
(b) 2
(c) 3
(d) 4
Ans. (d)

Ques: The number of solutions for the equation  x2 – 5 | x | + 6 = 0 is
(a) 4
(b) 3
(c) 2
(d) 1
Ans. (a)

Ques: 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
Ans. (b)

Related: Quadrilaterals questions

Ques: 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
Ans. (c)

Ques: 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
Ans. (a)

Ques: 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
Ans. (c)

Related: Polynomials questions

Ques: The equation ex – 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
Ans. (a)

Ques: 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
Ans. (c)

Ques: 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
Ans. (d)

Ques: 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
Ans. (c)

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Vishal Arora

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