MCQOPTIONS
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MCQOPTIONS
This section includes 78 Mcqs, each offering curated multiple-choice questions to sharpen your Mathematics knowledge and support exam preparation. Choose a topic below to get started.
| 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. | |
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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. | |
| 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) | |
| 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 | |
| 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 | |
| 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 | |
| 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. \(- \frac{8}{9}\) | |
| 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 | |
| 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. | |
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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 | |
| 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. | |
| 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. \(\frac {17} 2\) | |
| 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 | |
| 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 | |
| 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 | |
| 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. \(\frac{{{2^{12}}}}{{{{({\rm{sin\theta }} - 8)}^6}}}\) | |
| 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) | |
| 76. |
If x = 3 - 2√2, then √x + (1/√x) |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | 2√2 |
| Answer» E. | |
| 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 | |
| 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 | |