If α and β are the roots of the quadratic equation, \({{\rm{x}}^2} + {\rm{x\;sin\;\theta }} - 2{\rm{\;sin\;\theta }} = 0,{\rm{\theta }} \in \left( {0,\frac{{\rm{\pi }}}{2}} \right)\) then \(\frac{{{{\rm{\alpha }}^{12}} + {{\rm{\beta }}^{12}}}}{{\left( {{{\rm{\alpha }}^{ - 12}} + {{\rm{\beta }}^{ - 12}}} \right)\cdot{{({\rm{\alpha }} - {\rm{\beta }})}^{24}}}}\) is equal to:

\(\frac{{{2^{12}}}}{{{{({\rm{sin\theta }} - 8)}^6}}}\)

\(\frac{{{2^{12}}}}{{{{({\rm{sin\theta }} - 4)}^{12}}}}\)

\(\frac{{{2^{12}}}}{{{{({\rm{sin\theta }} + 8)}^{12}}}}\)

\(\frac{{{2^{12}}}}{{{{({\rm{sin\theta }} - 8)}^6}}}\)

\(\frac{{{2^6}}}{{{{({\rm{sin\theta }} + 8)}^{12}}}}\)

78 + Mcqs in Quadratic Equations in Mathematics Page 2 McqOptions

51.	Let p, q ∈ R. If \(2 - \sqrt 3\) is a root of the quadratic equation,x2 + px + q = 0, then:
A.	p2 – 4q + 12 = 0
B.	q2 – 4p – 16 = 0
C.	q2 + 4p + 14 = 0
D.	p2 – 4q – 12 = 0
Answer» E.

Discussion

52.	If x2 + 1/x2 = 7/4, then what is the value of x + 1/x?
A.	2
B.	√15/2
C.	√15
D.	√3
Answer» C. √15

Discussion

53.	Let p and q be non-zero integers. Consider the polynomial A(x) = x2 + px + q. It is given that (x - m) and (x - km) are simple factors of A(x), where m is a non-zero integer and k is positive integer, k ≥ 2. Which one of the following is correct?
A.	(k + 1)2p2 = kq
B.	(k + 1)2q = kp2
C.	k2q = (k + 1) p2
D.	k2p2 = (k + 1)2q
Answer» C. k2q = (k + 1) p2

Discussion

54.	If the sum of the roots of the equation ax2 + bx + c = 0 is equal to the sum of their squares then
A.	a2 + b2 = c2
B.	a2 + b2 = a + b
C.	ab + b2 = 2ac
D.	ab - b2 = 2ac
Answer» D. ab - b2 = 2ac

Discussion

55.	If 4x + 3a = 0, then what is the value of \(\frac{{{x^2}\; + \;ax\; + \;{a^2}}}{{{x^3} - {a^3}}} - \;\frac{{{x^2} - \;ax\; + \;{a^2}}}{{{x^3}\; + \;{a^3}}}\;\)?
A.	-4/7a
B.	7/a
C.	-32/7a
D.	24/7a
Answer» D. 24/7a

Discussion

56.	If the roots of the quadratic equation x2 + 2x + k = 0 are real, then
A.	k < 0
B.	k ≤ 0
C.	k < 1
D.	k ≤ 1
Answer» E.

Discussion

57.	If the coefficients of x2 and x3 are both zero, in the expansion of the expression (1 + ax + bx2)(1 – 3x)15 in powers of x, then the ordered pair (a, b) is equal to:
A.	(28, 861)
B.	(-54, 315)
C.	(28, 315)
D.	(-21, 714)
Answer» D. (-21, 714)

Discussion

58.	In solving a problem that reduces to a quadratic equation, one student makes a mistake in the constant term and obtains 8 and 2 for roots. Another student makes a mistake only in the coefficient of first-degree term and finds -9 and -1 for roots.The correct equation is
A.	x2 – 10x + 9 = 0
B.	x2 + 10x + 9 = 0
C.	x2 – 10x + 16 = 0
D.	x2 – 8x – 9 = 0
Answer» B. x2 + 10x + 9 = 0

Discussion

59.	If p and q are the roots of the equation x2 – 30x + 221 = 0, what is the value of p3 + q3?
A.	7010
B.	7110
C.	7210
D.	7240
Answer» C. 7210

Discussion

60.	If the roots of the equation x2 + px + q = 0 are tan 19° and tan 26°, then which one of the following is correct?
A.	q – p = 1
B.	p – q = 1
C.	p + q = 2
D.	p + q = 3
Answer» B. p – q = 1

Discussion

61.	If one root of the equation \(\left( {{\rm{l}} - {\rm{m}}} \right){{\rm{x}}^2} + {\rm{lx}} + 1 = 0\) is double the other and l is real, then what is the greatest value of m?
A.	\(- \frac{9}{8}\)
B.	\(\frac{9}{8}\)
C.	\(- \frac{8}{9}\)
D.	\(\frac{8}{9}\)
Answer» C. \(- \frac{8}{9}\)

Discussion

62.	If equation 2x2 + 3x + 5λ = 0 and x2 + 2x + 3λ = 0 have a common root then λ is equal to:
A.	2
B.	1
C.	0, -1
D.	2, -1
Answer» D. 2, -1

Discussion

63.	If x + 4 is a factor of 3x2 + kx + 8 then what is the value of k?
A.	4
B.	-4
C.	-14
D.	14
Answer» E.

Discussion

64.	A real root equation x3 – 5x – 7 = 0 by the method of false position correct to three decimal places is
A.	2.7472
B.	2.084
C.	2.077
D.	None of these
Answer» B. 2.084

Discussion

65.	If α and β are the roots of the quadratic equation 2x2 + 6x + k = 0, where k < 0, then what is the maximum value of (α/β + β/α)?
A.	2
B.	-2
C.	9
D.	-9
Answer» C. 9

Discussion

66.	How many real roots does the equation x2 + 3\|x\| + 2 = 0 have?
A.	Zero
B.	One
C.	Two
D.	Four
Answer» B. One

Discussion

67.	If the point (2, α, β) lies on the plane which passes through the points (3, 4, 2) and (7, 0, 6) and is perpendicular to the plane 2x - 5y = 15, then 2α - 3β is equal to:
A.	12
B.	7
C.	5
D.	17
Answer» C. 5

Discussion

68.	Consider the following:1. β < - α2. β < \|α\|Which of the above is/are correct?
A.	1 only
B.	2 only
C.	Both 1 and 2
D.	Neither 1 nor 2
Answer» D. Neither 1 nor 2

Discussion

69.	Factorisation of x3 - y3 is
A.	(x + y) (x2 - xy + y2)
B.	(x - y) (x2 - xy + y2)
C.	(x - y) (x2 - xy - y2)
D.	(x - y) (x2 + xy + y2)
Answer» E.

Discussion

70.	Let f(x) = x2 - bx + c, b is an odd positive integer. If f(x) = 0 has to prime numbers as roots and b + c = 35, then the global minimum value of f(x) is
A.	\(- \frac {183} 4\)
B.	\(\frac {173} {16}\)
C.	\(- \frac {81} 4\)
D.	\(\frac {17} 2\)
Answer» D. \(\frac {17} 2\)

Discussion

71.	Consider the following1. α + β + αβ > 02. α2β + β2α > 0Which of the above is/are correct?
A.	1 only
B.	2 only
C.	Both 1 and 2
D.	Neither 1 nor 2
Answer» C. Both 1 and 2

Discussion

72.	In ΔPQR, \(\angle R = \frac{\pi }{2}\). If \(\tan \left( {\frac{P}{2}} \right)\) and \(\tan \left( {\frac{Q}{2}} \right)\) are the roots of the equation ax2 + bx + c = 0, then which one of the following is correct?
A.	a = b + c
B.	b = c + a
C.	c = a + b
D.	b = c
Answer» D. b = c

Discussion

73.	If α and β are the roots of the equation x2 + px + q = 0, then what is α2 + β2 equal to?
A.	p2 – 2q
B.	q2 – 2p
C.	p2 + 2q
D.	q2 – q
Answer» B. q2 – 2p

Discussion

74.	If α and β are the roots of the quadratic equation, \({{\rm{x}}^2} + {\rm{x\;sin\;\theta }} - 2{\rm{\;sin\;\theta }} = 0,{\rm{\theta }} \in \left( {0,\frac{{\rm{\pi }}}{2}} \right)\) then \(\frac{{{{\rm{\alpha }}^{12}} + {{\rm{\beta }}^{12}}}}{{\left( {{{\rm{\alpha }}^{ - 12}} + {{\rm{\beta }}^{ - 12}}} \right)\cdot{{({\rm{\alpha }} - {\rm{\beta }})}^{24}}}}\) is equal to:
A.	\(\frac{{{2^{12}}}}{{{{({\rm{sin\theta }} - 4)}^{12}}}}\)
B.	\(\frac{{{2^{12}}}}{{{{({\rm{sin\theta }} + 8)}^{12}}}}\)
C.	\(\frac{{{2^{12}}}}{{{{({\rm{sin\theta }} - 8)}^6}}}\)
D.	\(\frac{{{2^6}}}{{{{({\rm{sin\theta }} + 8)}^{12}}}}\)
Answer» C. \(\frac{{{2^{12}}}}{{{{({\rm{sin\theta }} - 8)}^6}}}\)

Discussion

75.	If the difference between the roots of the equation x2 + kx + 1 = 0 is strictly less than √5, where \|k\| ≥ 2, then k can be any element of the interval
A.	(-3, -2] ∪ [2, 3)
B.	(-3, 3)
C.	[-3, -2] ∪ [2, 3]
D.	None of the above
Answer» B. (-3, 3)

Discussion

76.	If x = 3 - 2√2, then √x + (1/√x)
A.	0
B.	1
C.	2
D.	2√2
Answer» E.

Discussion

77.	Consider the following expressions: 1) \(x + {x^2} - \frac{1}{x}\)2) \(\sqrt {a{x^2} + bx + x - c + \frac{d}{x} - \frac{e}{{{x^2}}}} \)3) 3x2 - 5x + ab4) \(\frac{2}{{{x^2} - ax + {b^3}}}\)5) \(\frac{1}{x} - \frac{2}{{x + 5}}\)Which of the above are rational expressions?
A.	1, 4 and 5 only
B.	1, 3, 4 and 5 only
C.	2, 4 and 5 only
D.	1 and 2 only
Answer» B. 1, 3, 4 and 5 only

Discussion

78.	Consider the following statements in respect of the given equation:\({\left( {{x^2} + 2} \right)^2} + 8{x^2} = 6x\left( {{x^2} + 2} \right)\)1. All the roots of the equation are complex.2. The sum of all the roots of the equation is 6.Which of the above statements is/are correct?
A.	1 only
B.	2 only
C.	Both 1 and 2
D.	Neither 1 nor 2
Answer» C. Both 1 and 2

Discussion

Explore topic-wise MCQs in Mathematics.

Let p, q ∈ R. If \(2 - \sqrt 3\) is a root of the quadratic equation,x2 + px + q = 0, then:

If x2 + 1/x2 = 7/4, then what is the value of x + 1/x?

Let p and q be non-zero integers. Consider the polynomial A(x) = x2 + px + q. It is given that (x - m) and (x - km) are simple factors of A(x), where m is a non-zero integer and k is positive integer, k ≥ 2. Which one of the following is correct?

If the sum of the roots of the equation ax2 + bx + c = 0 is equal to the sum of their squares then

If 4x + 3a = 0, then what is the value of \(\frac{{{x^2}\; + \;ax\; + \;{a^2}}}{{{x^3} - {a^3}}} - \;\frac{{{x^2} - \;ax\; + \;{a^2}}}{{{x^3}\; + \;{a^3}}}\;\)?

If the roots of the quadratic equation x2 + 2x + k = 0 are real, then

If the coefficients of x2 and x3 are both zero, in the expansion of the expression (1 + ax + bx2)(1 – 3x)15 in powers of x, then the ordered pair (a, b) is equal to:

In solving a problem that reduces to a quadratic equation, one student makes a mistake in the constant term and obtains 8 and 2 for roots. Another student makes a mistake only in the coefficient of first-degree term and finds -9 and -1 for roots.The correct equation is

If p and q are the roots of the equation x2 – 30x + 221 = 0, what is the value of p3 + q3?

If the roots of the equation x2 + px + q = 0 are tan 19° and tan 26°, then which one of the following is correct?

If one root of the equation \(\left( {{\rm{l}} - {\rm{m}}} \right){{\rm{x}}^2} + {\rm{lx}} + 1 = 0\) is double the other and l is real, then what is the greatest value of m?

If equation 2x2 + 3x + 5λ = 0 and x2 + 2x + 3λ = 0 have a common root then λ is equal to:

If x + 4 is a factor of 3x2 + kx + 8 then what is the value of k?

A real root equation x3 – 5x – 7 = 0 by the method of false position correct to three decimal places is

If α and β are the roots of the quadratic equation 2x2 + 6x + k = 0, where k < 0, then what is the maximum value of (α/β + β/α)?

How many real roots does the equation x2 + 3|x| + 2 = 0 have?

If the point (2, α, β) lies on the plane which passes through the points (3, 4, 2) and (7, 0, 6) and is perpendicular to the plane 2x - 5y = 15, then 2α - 3β is equal to:

Consider the following:1. β < - α2. β < |α|Which of the above is/are correct?

Factorisation of x3 - y3 is

Let f(x) = x2 - bx + c, b is an odd positive integer. If f(x) = 0 has to prime numbers as roots and b + c = 35, then the global minimum value of f(x) is

Consider the following1. α + β + αβ > 02. α2β + β2α > 0Which of the above is/are correct?

In ΔPQR, \(\angle R = \frac{\pi }{2}\). If \(\tan \left( {\frac{P}{2}} \right)\) and \(\tan \left( {\frac{Q}{2}} \right)\) are the roots of the equation ax2 + bx + c = 0, then which one of the following is correct?

If α and β are the roots of the equation x2 + px + q = 0, then what is α2 + β2 equal to?

If the difference between the roots of the equation x2 + kx + 1 = 0 is strictly less than √5, where |k| ≥ 2, then k can be any element of the interval

If x = 3 - 2√2, then √x + (1/√x)

Consider the following expressions: 1) \(x + {x^2} - \frac{1}{x}\)2) \(\sqrt {a{x^2} + bx + x - c + \frac{d}{x} - \frac{e}{{{x^2}}}} \)3) 3x2 - 5x + ab4) \(\frac{2}{{{x^2} - ax + {b^3}}}\)5) \(\frac{1}{x} - \frac{2}{{x + 5}}\)Which of the above are rational expressions?

Consider the following statements in respect of the given equation:\({\left( {{x^2} + 2} \right)^2} + 8{x^2} = 6x\left( {{x^2} + 2} \right)\)1. All the roots of the equation are complex.2. The sum of all the roots of the equation is 6.Which of the above statements is/are correct?