JEE Mathematics Quadratic Equations MCQs

Question. The roots of the quadratic equation 2x² – 7x + 4 = 0 are -
(a) Irrational and different
(b) Rational and equal
(c) Imaginary and different
(d) Rational and different

Answer: A

Question. The roots of the quadratic equation
x² – 2 (a + b) x + 2 (a² + b²) = 0 are -
(a) Imaginary and different
(b) Irrational and different
(c) Rational and equal
(d) Rational and different

Answer: A

Question. The roots of the equation x² – 2 √2 x + 1 = 0 are –
(a) Real and different
(b) Real and equal
(c) Imaginary and different
(d) Rational and different

Answer: A

Question. The roots of the equation x² –3x – 4 = 0 are–
(a) Opposite and greater root in magnitude is positive
(b) Opposite and greater root in magnitude is negative
(c) Reciprocal to each other
(d) None of these

Answer: A

Question. The roots of the equation 2x² – 3x + 2 = 0 are -
(a) Reciprocal to each other
(b) Both roots are zero
(c) None of these
(d) Negative of each other

Answer: A

Question. If equation x²-bx/ax-c = k-1/k+1 has equal and opposite roots then the value of k is -
(a) a-b/a+b
(b) a-b/a-b
(c) a+b/a-b
(d) None of these

Answer: A

Question. If the roots of the equation x² + 2x + P = 0 are real then the value of P is -
(a) P≤1
(b) P≤2
(c) P≤3
(d) P≤4

Answer: A

Question. If the product of the roots of the quadratic equation mx² – 2x + (2m – 1) = 0 is 3 then the value of m is -
(a) –1
(b) 1
(c) 2
(d) 3

Answer: A

Question. If the equation (k – 2)x² – (k – 4) x – 2 = 0 has difference of roots as 3 then the value of k is-
(a) 3,3/2
(b) 3/2, 1
(c) 1,3
(d) 2, 3/2

Answer: A

Question. The roots of the equation x² – 2x – 8 = 0 are -
(a) 4, – 2
(b) – 4,– 2
(c) – 4, 2
(d) 4, 2

Answer: A

Question. The roots of the equation x² – 4x + 1 = 0 are –
(a) 2 ± √3
(b) -2 ± 3
(c) 2, 4
(d) None of these

Answer: A

Question. The roots of the quadratic equation 7x² – 9x + 2 = 0 are -
(a) Rational and different
(b) Irrational and different
(c) Rational and equal
(d) Imaginary and different

Answer: A

Question. The equation whose roots are 3 and 4 will be-
(a) x² – 7x + 12 = 0
(b) x² + 7x – 12=0
(c) x² – x + 12 = 0
(d) x² + 7x + 12 = 0

Answer: A

Question. The quadratic equation whose one root is 2 – i √3 is -
(a) x² – 4x + 7 = 0
(b) x² – 4x – 7=0
(c) x² + 4x – 7 = 0
(d) None of these

Answer: A

Question. If r and s are positive, then roots of the equation x² – rx – s = 0 are -
(1) real                  (2) imaginary
(3) opposite signs (4) both negative
(a) 1 and 3 are correct
(b) 1 and 2 are correct
(c) 1, 2 and 3 are correct
(d) 2 and 4 are correct

Answer: A

Question. If a < b < c < d, then roots of equation (x – a)(x – c) + 2 (x – b) (x – d) = 0 are
(1) real (2) unequal
(3) imaginary (4) equal
(a) 1 and 2 are correct
(b) 1 and 3 are correct
(c) 1, 2 and 3 are correct
(d) 2 and 4 are correct

Answer: A

Question. If p and q are roots of the equation x2 – 2x + A = 0 and r and s be roots of the equation x2 – 18 x + B = 0 if p < q < r < s be in A.P., then choose the correct options –
(a) 2 and 4 are correct
(b) 1, 2 and 3 are correct
(c) 1 and 2 are correct
(d) 1 and 3 are correct

Answer: A

Question. Let a, b, c be real such that ax² + bx + c = 0 and x² + x + 1= 0 have a common root
Statement–1 : a = b = c
Statement–2 : Two quadratic equations with real coefficients can not have only one imaginary root common.
(a) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.
(c) Statement -1 is False, Statement-2 is True.
(d) Statement -1 is True, Statement-2 is False.

Answer: A

Question. Statement-1 : If one roots is √5 - √2 then the equation of lowest degree with rational coefficient is x⁴–14x²+ 9 = 0
Statement-2 : For a polynomial equation with rational coefficient irrational roots occurs in pairs.
(a) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.
(c) Statement -1 is False, Statement-2 is True.
(d) Statement -1 is True, Statement-2 is False.

Answer: A

Question. If f (x) is a quadratic expression which is positive for all real values of x and g(x) = f (x) + f '(x) + f "(x) , then for any real value of x-
(a) g (x) > 0
(b) g (x) < 0
(c) g (x) = 0
(d) None of these

Answer: C

Question. For real values of x, 2x² + 5x – 3 > 0, if-
(a) x > 1
(b) x < – 2
(c) x > 0
(d) None of these

Answer: A

Question. For what value of m the expression y² + 4xy + 4x + my – 2 can be resolved into two rational factors-
(a) –1
(b) –2
(c) 1
(d) 2

Answer: A

Question. The quadratic equation whose one root is 1/2 + √5 will be-
(a) x² + 4x – 1 = 0
(b) x² + 4x + 1= 0
(c) x² – 4x – 1 = 0
(d) None of these

Answer: A

Question. If the expression x²–11x + a and x² – 14x + 2a must have a common factor and a ≠ 0, then, the common factor is –
(a) (x – 8)
(b) (x – 3)
(c) (x – 6)
(d) None of these

Answer: A

Question. If x is real then the value of the expression x²+14x+9/ x²+2x+3 between –
(a) –5 and 4
(b) – 4 and 5
(c) –3 and 3
(d) – 4 and 4

Answer: A

Question. The real values of a for which the quadratic equation 2x² – (a³ + 8a – 1) x + a²– 4a = 0 possesses roots of opposite signs are given by-
(a) 0 < a < 4
(b) a > 7
(c) a > 5
(d) a > 0

Answer: A

Question. The value of the expression x² + 2bx + c will be positive, if-
(a) b² < c
(b) b²–4c < 0
(c) b²–4c > 0
(d) c² < b

Answer: A

Question. If roots of the equation x² + ax + 25 = 0 are in the ratio of 2 : 3 then the value of a is -
(a) ±25/√6
(b) ±5/√6
(c) ±6/√6
(d) None of these

Answer: A

Question. If the roots of the equations x² + 3x + 2 = 0 and x² –x +λ=0 are in the same ratio then the value of λ is given by-
(a) 2/9
(b) 7/2
(c) 2/7
(d) 9/2

Answer: A

Question. The sum of all real roots of the equation |x – 2|2 + | x – 2 | – 2 = 0, is -
(a) 4
(b) 0
(c) 8
(d) None of these

Answer: A

Question. If p, q, r are in H.P. and p and r be different having same sign, then the root of the equation px² + 2qx + r = 0 will be
(a) Imaginary
(b) Equal
(c) Real
(d) None of these

Answer: A

Question. If x = 2 + √3 then the value of x³ – 7x² + 13 x – 12 is –
(a) –9
(b) 9
(c) 6
(d) 3

Answer: A

Question. If every pair from among the equations x² + px + qr = 0, x² + qx + rp= 0 and x² + rx + pq = 0 has a common root, then the sum of the three common roots is-
(a) p + q + r
(b) pqr
(c) 2 (p + q+ r)
(d) – (p + q + r)

Answer: A

Question. If the quadratic equations ax² + 2cx + b = 0 and ax² + 2bx + c = 0 (b≠c) have a common root, then a + 4b + 4c is equal to-
(a) 0
(b) –2
(c) –1
(d) 1

Answer: A

Question. The value of m for which one of the roots of x² – 3x + 2m = 0 is double of one of the roots of x² – x + m = 0 is-
(a) 0, 2
(b) 2, –2
(c) 0, –2
(d) None of these

Answer: A