MCQOPTIONS
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MCQOPTIONS
This section includes 395 Mcqs, each offering curated multiple-choice questions to sharpen your UPSEE knowledge and support exam preparation. Choose a topic below to get started.
| 51. |
If ∫esec x (secx tanx f(x) + (secx tanx + sec2 x)) dx = esec x f(x) + C, then a possible choice of f(x) is: |
| A. | \({\rm{secx}} + {\rm{tanx}} + \frac{1}{2}\) |
| B. | \({\rm{secx}} - {\rm{tanx}} - \frac{1}{2}\) |
| C. | \({\rm{secx}} + {\rm{tanx}} - \frac{1}{2}\) |
| D. | \({\rm{xsecx}} + {\rm{tanx}} + \frac{1}{2}\) |
| Answer» B. \({\rm{secx}} - {\rm{tanx}} - \frac{1}{2}\) | |
| 52. |
Find the area bounded between the curve y = x and \(\rm y=x^{2}\). |
| A. | 1/2 |
| B. | 1/4 |
| C. | 1 |
| D. | 1/6 |
| Answer» E. | |
| 53. |
If u = x log xy, where x3 + y3 + 3xy = 1, then \(\frac{{du}}{{dx}}\) is |
| A. | \(1 - \log xy + \frac{{x\left( {{x^2} + y} \right)}}{{y\;\left( {{y^2} + x} \right)}}\) |
| B. | \(1 - \log xy - \frac{{x\left( {{x^2} + y} \right)}}{{y\;\left( {{y^2} + x} \right)}}\) |
| C. | \(1 + \log xy - \frac{{x\left( {{x^2} + y} \right)}}{{y\;\left( {{y^2} + x} \right)}}\) |
| D. | \(1 + \log xy + \frac{{x\left( {{x^2} + y} \right)}}{{y\;\left( {{y^2} + x} \right)}}\) |
| Answer» D. \(1 + \log xy + \frac{{x\left( {{x^2} + y} \right)}}{{y\;\left( {{y^2} + x} \right)}}\) | |
| 54. |
Let l be the length and b be the breadth of a rectangle such that l + b = k. What is the maximum area of rectangle? |
| A. | 2k2 |
| B. | k2 |
| C. | \(\rm \dfrac{k^2}{2}\) |
| D. | \(\rm \dfrac{k^2}{4}\) |
| Answer» E. | |
| 55. |
If \(\displaystyle\int_0^1 \dfrac{e^t}{1+t}dt=a\), then \(\displaystyle\int_0^1 \dfrac{e^t}{(1+t)^2}dt=\) |
| A. | \(a-1+\dfrac{e}{2}\) |
| B. | \(a+1+\dfrac{e}{2}\) |
| C. | \(a-1-\dfrac{e}{2}\) |
| D. | \(a+1-\dfrac{e}{2}\) |
| Answer» E. | |
| 56. |
A particle moves along a curve whose parametric equations are x = e-t, y = 2 cos t, z = sin t. Its velocity is: |
| A. | \( - {e^{ - t}}\hat i - 2\sin t\,\hat j + \cos t\,\hat k\) |
| B. | \( - {e^{ - t}}\hat i + 2\cos t\,\hat j + \sin t\,\hat k\) |
| C. | \( {e^t}\hat i - 2\sin t\,\hat j + \cos t\,\hat k\) |
| D. | \( {e^t}\hat i + - 2\sin t\,\hat j - \cos t\,\hat k\) |
| Answer» B. \( - {e^{ - t}}\hat i + 2\cos t\,\hat j + \sin t\,\hat k\) | |
| 57. |
A function f (x) is defined as \(g\left( t \right) = \left\{ {\begin{array}{*{20}{c}} {{e^x},}&{x < 1}\\ {\ln x + a{x^2}+bx,}&{x \ge 1} \end{array}} \right.\), where x ϵ R. Which one of the following statements is TRUE? |
| A. | f(x) is NOT differentiable at x = 1 for any values of a and b |
| B. | f(x) is differentiable at x = 1 for the unique value of a and b. |
| C. | f(x) is differentiable at x = 1 for all values of a and b such that a + b = e |
| D. | f(x) is differentiable at x = 1 for all values of a and b. |
| Answer» C. f(x) is differentiable at x = 1 for all values of a and b such that a + b = e | |
| 58. |
If \({\rm{y}} = {{\rm{e}}^{{{\rm{x}}^2}}}\sin 2{\rm{x}}\), then what is \(\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\) at x = π equal to? |
| A. | \(\left( {1 + {\rm{\pi }}} \right){{\rm{e}}^{{{\rm{\pi }}^2}}}\) |
| B. | \(2{\rm{\pi \;}}{{\rm{e}}^{{{\rm{\pi }}^2}}}\) |
| C. | \(2{{\rm{e}}^{{{\rm{\pi }}^2}}}\) |
| D. | \({{\rm{e}}^{{{\rm{\pi }}^2}}}\) |
| Answer» D. \({{\rm{e}}^{{{\rm{\pi }}^2}}}\) | |
| 59. |
A real-valued function y of real variable x is such that \(y = 5 |x|\). At x = 0, the function is |
| A. | discontinuous but differentiable |
| B. | both continuous and differentiable |
| C. | discontinuous and not differentiable |
| D. | continuous but not differentiable |
| Answer» E. | |
| 60. |
\(\mathop {\lim }\limits_{x \to \infty } \left( {\frac{{x + \sin x}}{x}} \right)\) equals to |
| A. | -∞ |
| B. | 0 |
| C. | 1 |
| D. | ∞ |
| Answer» D. ∞ | |
| 61. |
For a small value of h, the Taylor series expansion for f (x + h) is |
| A. | \(f\left( x \right) - hf'\left( x \right) + \frac{{{h^2}}}{2}f''\left( x \right) - \frac{{{h^3}}}{3}f'''\left( x \right) + \ldots \infty \) |
| B. | \(f\left( x \right) - hf'\left( x \right) + \frac{{{h^2}}}{2}f''\left( x \right) - \frac{{{h^3}}}{{3!}}f'''\left( x \right) + \ldots \infty \) |
| C. | \(f\left( x \right) + hf'\left( x \right) + \frac{{{h^2}}}{2}f''\left( x \right) + \frac{{{h^3}}}{3}f'''\left( x \right) + \ldots \infty \) |
| D. | \(f\left( x \right) + hf'\left( x \right) + \frac{{{h^2}}}{{2!}}f''\left( x \right) + \frac{{{h^3}}}{{3!}}f'''\left( x \right) + \ldots \infty \) |
| Answer» E. | |
| 62. |
If \(cosx\frac{{dy}}{{dx}} - ysinx = 6x,\left( {0 < x < \frac{\pi }{2}} \right)\;{\rm{and}}\;y\left( {\frac{\pi }{3}} \right) = 0,\;{\rm{then}}\;y\left( {\frac{\pi }{6}} \right)\) is equal to: |
| A. | \(\frac{{{\pi ^2}}}{{2\sqrt 3 }}\) |
| B. | \(- \frac{{{\pi ^2}}}{2}\) |
| C. | \(- \frac{{{\pi ^2}}}{{2\sqrt 3 }}\) |
| D. | \(- \frac{{{\pi ^2}}}{{4\sqrt 3 }}\) |
| Answer» D. \(- \frac{{{\pi ^2}}}{{4\sqrt 3 }}\) | |
| 63. |
\(\frac{d}{{dx}}\left( {\frac{{1 + {x^2} + {x^4}}}{{1 + x + {x^2}}}} \right) = Ax + B.\)What is the value of A? |
| A. | -1 |
| B. | 1 |
| C. | 2 |
| D. | 4 |
| Answer» D. 4 | |
| 64. |
\(\frac{d}{{dx}}\left( {\frac{{1 + {x^2} + {x^4}}}{{1 + x + {x^2}}}} \right) = Ax + B.\)What is the value of B? |
| A. | -1 |
| B. | 1 |
| C. | 2 |
| D. | 4 |
| Answer» B. 1 | |
| 65. |
If f(x) = x + |x2 - 8| then the derivative of f(x) at x = 3 is |
| A. | 6 |
| B. | 7 |
| C. | 8 |
| D. | -8 |
| Answer» C. 8 | |
| 66. |
Equation of the line normal to function \(f\left( x\right) = {\left( {4 - x} \right)^{\frac{1}{2}}} + 1\) at Q(0,3) is |
| A. | y = 3 – 4x |
| B. | y = 3 + 4x |
| C. | 4y = 12 - x |
| D. | 4y = 12 + x |
| Answer» C. 4y = 12 - x | |
| 67. |
If f(x, y) = xy, then the differential df is equal to |
| A. | xdx + ydy |
| B. | ydx + xdy |
| C. | dx + dy |
| D. | dx - dy |
| Answer» C. dx + dy | |
| 68. |
Evaluation of \(\int{x\ \sec^2x\ dx}\) is: |
| A. | x tan x + log sin x + c |
| B. | x tan x + log cos x + c |
| C. | x tan x - log sin x + c |
| D. | x tan x - log cos x + c |
| Answer» C. x tan x - log sin x + c | |
| 69. |
If f(1) = 1, f'(1) = 3, then the value of derivative of f(f(f(x))) + (f(x))2 at x = 1 is: |
| A. | 3 |
| B. | 12 |
| C. | 15 |
| D. | 9 |
| Answer» B. 12 | |
| 70. |
If f(x) = xm sin(1/x), x ≠ 0, f(0) = 0 then the minimum value of m for which f is derivable at x = 0 and also \(\frac{{df}}{{dx}}\) is continuous at x = 0 is |
| A. | m = 1 |
| B. | m = 4 |
| C. | m = 2 |
| D. | m = 3 |
| Answer» E. | |
| 71. |
If \(x=e^{y+e^{y+e^{y+...}}}\) then \(\frac {dy}{dx}\) is |
| A. | (1 - x) |
| B. | (1 - x) / x |
| C. | 1 / x |
| D. | x / (1 - x) |
| Answer» C. 1 / x | |
| 72. |
Let \(f\left( x \right) = \;{e^{x + {x^2}}}\) for real x. From among the following, choose the Taylor series approximation of f(x) around x = 0, which includes all powers of x less than or equal to 3. |
| A. | 1 + x + x2 + x3 |
| B. | 1 + x + \(\frac{3}{2}\) x2 + x3 |
| C. | 1 + x + \(\frac{3}{2}\) x2 + \(\frac{7}{6}\) x3 |
| D. | 1 + x + 3x2 + 7x3 |
| Answer» D. 1 + x + 3x2 + 7x3 | |
| 73. |
If \(\tan x = \dfrac{-3}{4}\) and \(\dfrac{3\pi}{2}< x < 2\pi\), then the value of sin 2x is |
| A. | 7/25 |
| B. | -7/25 |
| C. | 24/25 |
| D. | -24/25 |
| Answer» E. | |
| 74. |
Differentiate (a cos3t) w.r.t. to (a sin3 t) |
| A. | cot t |
| B. | –cot t |
| C. | tan t |
| D. | –tan t |
| Answer» C. tan t | |
| 75. |
A parabola x = y2 with 0 ≤ x ≤ 1 is shown in the figure. The volume of the solid of rotation obtained by rotating the shaded area by 360° around the x-axis is |
| A. | \(\frac{\pi }{4}\) |
| B. | \(\frac{\pi }{2}\) |
| C. | π |
| D. | 2π |
| Answer» C. π | |
| 76. |
Let \(f\left( x \right) = \mathop \smallint \nolimits_0^x g\left( t \right)dt\), where g is a non-zero even function. If f(x + 5) = g(x), then \(\mathop \smallint \nolimits_0^x f\left( t \right)dt\) equals: |
| A. | \(\mathop \smallint \nolimits_{x + 5}^5 g\left( t \right)dt\) |
| B. | \(\mathop \smallint \nolimits_5^{x + 5} g\left( t \right)dt\) |
| C. | \(2\mathop \smallint \nolimits_5^{x + 5} g\left( t \right)dt\) |
| D. | \(5\mathop \smallint \nolimits_{x + 5}^5 g\left( t \right)dt\) |
| Answer» B. \(\mathop \smallint \nolimits_5^{x + 5} g\left( t \right)dt\) | |
| 77. |
For the two functions f (x, y) = x3 – 3xy2 and g(x,y) = 3x2y – y3. Which one of the following options is correct? |
| A. | \(\frac {\partial f}{\partial x} = \frac {\partial g}{\partial x}\) |
| B. | \(\frac {\partial f}{\partial x} = - \frac {\partial g}{\partial y}\) |
| C. | \(\frac {\partial f}{\partial y} = - \frac {\partial g}{\partial x}\) |
| D. | \(\frac {\partial f}{\partial y} = \frac {\partial g}{\partial x}\) |
| Answer» D. \(\frac {\partial f}{\partial y} = \frac {\partial g}{\partial x}\) | |
| 78. |
If a vector gets multiplied by a positive number, then its direction |
| A. | remain same |
| B. | reversed |
| C. | gets half |
| D. | gets double |
| Answer» B. reversed | |
| 79. |
\(\mathop \smallint \limits_0^{\frac{\pi }{2}} {e^{\sin x}}\cos x\;dx\) is equal to |
| A. | e + 1 |
| B. | e - 1 |
| C. | e + 2 |
| D. | e |
| Answer» C. e + 2 | |
| 80. |
\(\int {\frac{{{e^x}\left( {1 + \sin x} \right)}}{{1 + \cos x}}} dx\) is equal to |
| A. | \(\log \tan x + c\) |
| B. | \({e^x}\tan \frac{x}{2} + c\) |
| C. | \(\sin \log x + c\) |
| D. | \({e^x} \cot x + c\) |
| Answer» C. \(\sin \log x + c\) | |
| 81. |
Given integral \(\mathop \smallint \limits_0^\infty \frac{{sinx}}{x}dx\), Then the integral |
| A. | Is not convergent |
| B. | Converges |
| C. | Converges absolutely |
| D. | Converges but not absolutely |
| Answer» D. Converges but not absolutely | |
| 82. |
If p(x) = (4e)2x, then what is ∫ p(x) dx equal to? |
| A. | \(\rm \dfrac{p(x)}{1+2 \ln 2}+c\) |
| B. | \(\rm \dfrac{p(x)}{2(1+2 \ln 2)}+c\) |
| C. | \(\rm \dfrac{2p(x)}{1+ \ln 4}+c\) |
| D. | \(\rm \dfrac{p(x)}{1+\ln 2}+c\) |
| Answer» C. \(\rm \dfrac{2p(x)}{1+ \ln 4}+c\) | |
| 83. |
If \(\rm x^m y^n =a^{m+n}\), then what is \(\rm \dfrac{dy}{dx}\) equal to? |
| A. | \(\rm \dfrac{my}{nx}\) |
| B. | \(-\rm \dfrac{my}{nx}\) |
| C. | \(\rm \dfrac{mx}{ny}\) |
| D. | \(-\rm \dfrac{ny}{mx}\) |
| Answer» C. \(\rm \dfrac{mx}{ny}\) | |
| 84. |
Consider the following statements in respect of the function f(x) = sin x:1. f(x) increases in the interval (0, π).2. f(x) decreases in the interval \(\left(\dfrac{5\pi}{2},3\pi\right).\)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 | |
| 85. |
A water tank has the shape of an inverted right circular cone, whose semi-vertical angle is, \({\rm{ta}}{{\rm{n}}^{ - 1}}\left( {\frac{1}{2}} \right)\). Water is poured into it, at a constant rate of 5 cubic meter per minute. Then, the rate (in m/min.), at which the level of water is rising at the instant when the depth of water in the tank is 10 m; is: |
| A. | \(\frac{1}{{15\pi }}\) |
| B. | \(\frac{1}{{10\pi }}\) |
| C. | \(\frac{2}{\pi }\) |
| D. | \(\frac{1}{{5{\rm{\pi }}}}\) |
| Answer» E. | |
| 86. |
Consider the sequence xn = 0. 5xn−1 + 1, n = 1, 2, ... ... with x0 = 0. Then \(\mathop {\lim }\limits_{n \to \infty } {x_n}\) |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | ∞ |
| Answer» D. ∞ | |
| 87. |
In which one of the following intervals is the function f(x) = x2 – 5x + 6 decreasing? |
| A. | (-∞, 2.5) |
| B. | (3, ∞) |
| C. | (-∞,∞) |
| D. | (2, 3) |
| Answer» B. (3, ∞) | |
| 88. |
If \(f\left( x \right) = 3{x^2},\) then F(x) = |
| A. | 6x |
| B. | x3 |
| C. | x3 + C |
| D. | 6x + C |
| Answer» D. 6x + C | |
| 89. |
Let f: (-1, 1) → R be a function defined by \(f\left( x \right) = {\rm{max\;}}\left\{ { - \left| x \right|, - \sqrt {1 - {x^2}} } \right\}\). If K be the set of all points at which f is not differentiable, then K has exactly: |
| A. | Five elements |
| B. | One elements |
| C. | Three elements |
| D. | Two elements |
| Answer» D. Two elements | |
| 90. |
Find the gradient of the curve y = 3x2 - 7x + 2 at the point (1, -2): |
| A. | 1 |
| B. | -2 |
| C. | 2 |
| D. | -1 |
| Answer» E. | |
| 91. |
Let \(I = \mathop \oint \limits_C \left( {{x^2}y\;dy - {y^2}x\;dx} \right)\), where C is the boundary of square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1. Then I equals |
| A. | \( {1 \over 4}\) |
| B. | 4 |
| C. | 1 |
| D. | 2 |
| Answer» D. 2 | |
| 92. |
If the derivative of \(\dfrac{e^{3x} + e^x}{e^{4x} - e^{2x} + 1}\) if f(ex - e-x) + C, then f(x) is equal to: |
| A. | tan-1 x |
| B. | tan x |
| C. | cos-1 x |
| D. | sin x |
| Answer» B. tan x | |
| 93. |
For x2 ≠ nπ + 1, n ∈ N (the set of natural numbers), the integral \(\smallint {\rm{x}}\sqrt {\frac{{2{\rm{sin}}\left( {{{\rm{x}}^2} - 1} \right) - {\rm{sin}}2\left( {{{\rm{x}}^2} - 1} \right)}}{{2{\rm{sin}}\left( {{{\rm{x}}^2} - 1} \right) + {\rm{sin}}2\left( {{{\rm{x}}^2} - 1} \right)}}} {\rm{dx}}\) is equal to:(where c is a constant of integration) |
| A. | \({\rm{lo}}{{\rm{g}}_{\rm{e}}}\left| {\frac{1}{2}{\rm{se}}{{\rm{c}}^2}\left( {{{\rm{x}}^2} - 1} \right)} \right| + {\rm{c}}\) |
| B. | \(\frac{1}{2}{\rm{lo}}{{\rm{g}}_{\rm{e}}}\left| {{\rm{sec}}\left( {{{\rm{x}}^2} - 1} \right)} \right| + {\rm{c}}\) |
| C. | \(\frac{1}{2}{\rm{lo}}{{\rm{g}}_{\rm{e}}}\left| {{\rm{se}}{{\rm{c}}^2}\left( {\frac{{{{\rm{x}}^2} - 1}}{2}} \right)} \right| + {\rm{c}}\) |
| D. | \({\rm{lo}}{{\rm{g}}_{\rm{e}}}\left| {{\rm{sec}}\left( {\frac{{{{\rm{x}}^2} - 1}}{2}} \right)} \right| + {\rm{c}}\) |
| Answer» E. | |
| 94. |
If \(u = log\left( {\frac{{{x^2} + {y^2}}}{{x + y}}} \right)\), what is the value of \(x\frac{{\partial u}}{{\partial x}} + y\frac{{\partial u}}{{\partial y}}?\) |
| A. | 0 |
| B. | 1 |
| C. | u |
| D. | eu |
| Answer» C. u | |
| 95. |
\(\mathop \smallint \limits_0^{\frac{\pi }{2}} |\sin x - \cos x|dx\) is equal to |
| A. | 0 |
| B. | \(2\left( {\sqrt 2 - 1} \right)\) |
| C. | \(2\sqrt 2 \) |
| D. | \(2\left( {\sqrt 2 + 1} \right)\) |
| Answer» C. \(2\sqrt 2 \) | |
| 96. |
Maximum slope of the curve y = -x3 + 3x2 + 9x - 27 is |
| A. | 0 |
| B. | 12 |
| C. | 16 |
| D. | 32 |
| Answer» C. 16 | |
| 97. |
If \(\mathop \smallint \limits_0^{{\rm{\pi }}/2} \frac{{{\rm{dx}}}}{{3\cos {\rm{x}} + 5}} = {\rm{k}}{\cot ^{ - 1}}2\), then what is the value of k? |
| A. | 1/4 |
| B. | 1/2 |
| C. | 1 |
| D. | 2 |
| Answer» C. 1 | |
| 98. |
\(\mathop \smallint \limits_{ - 1}^1 {\rm{x}}\left| {\rm{x}} \right|{\rm{dx}}\) is equal to |
| A. | 0 |
| B. | \(\frac{2}{3}\) |
| C. | 2 |
| D. | -2 |
| Answer» B. \(\frac{2}{3}\) | |
| 99. |
Consider the following statements:1. The function f(x) is differentiable at x = 02. The function f(x) is differentiable at \({\rm{x}} = \frac{{\rm{\pi }}}{2}\).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» E. | |
| 100. |
Consider the following statements:1. The function f(x) is continuous at x = 02. The function f(x) is continuous at \({\rm{x}} = \frac{{\rm{\pi }}}{2}\)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» D. Neither 1 nor 2 | |