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.
| 101. |
If \(\mathop {\lim }\limits_{n \to \infty } {a_n} = 0\), then the series \(\sum {{a_n}}\) is: |
| A. | Oscillatory |
| B. | Divergent |
| C. | Convergent |
| D. | None of these |
| Answer» D. None of these | |
| 102. |
If the area of the trapezium is the maximum possible, then what is α equal to? |
| A. | \(\frac{{\rm{\pi }}}{6}\) |
| B. | \(\frac{{\rm{\pi }}}{4}\) |
| C. | \(\frac{{\rm{\pi }}}{3}\) |
| D. | \(\frac{{2\rm{\pi }}}{5}\) |
| Answer» D. \(\frac{{2\rm{\pi }}}{5}\) | |
| 103. |
Consider the function\({\rm{f}}\left( {\rm{x}} \right) = \left\{ {\begin{array}{*{20}{c}} {{{\rm{x}}^2}{\rm{ln}}\left| {\rm{x}} \right|}&{{\rm{x}} \ne 0}\\ 0&{{\rm{x}} = 0} \end{array}} \right.\)What is f’(0) equal to? |
| A. | 0 |
| B. | 1 |
| C. | -1 |
| D. | It does not exist |
| Answer» B. 1 | |
| 104. |
Let ϕ be an arbitrary smooth real valued scalar function and V be an arbitrary smooth vector valued function in a three-dimensional space. Which one of the following is an identity? |
| A. | \({\rm{Curl}}\left( {\phi {\rm{\vec V}}} \right) = \nabla \left( {\phi~ {\rm{Div\vec V}}} \right)\) |
| B. | \({\rm{Div\vec V}} = 0\) |
| C. | \({\rm{Div~Curl\vec V}} = 0\) |
| D. | \({\rm{Div}}\left( {\phi {\rm{\vec V}}} \right) = \phi ~{\rm{Div\vec V}}\) |
| Answer» D. \({\rm{Div}}\left( {\phi {\rm{\vec V}}} \right) = \phi ~{\rm{Div\vec V}}\) | |
| 105. |
Let f : R → R be defined by \(\rm f(x)=\left\{\begin{matrix} \rm x+2 &\rm if\ x |
| A. | 0.5 |
| B. | 2.5 |
| C. | 4.5 |
| D. | 6.5 |
| Answer» D. 6.5 | |
| 106. |
A series expansion for the function sin θ is |
| A. | \(1 - \frac{{{\theta ^2}}}{{2!}} + \frac{{{\theta^4}}}{{4!}} - \ldots\) |
| B. | \(\theta - \frac{{{\theta ^3}}}{{3!}} + \frac{{{\theta^5}}}{{5!}} - \ldots\) |
| C. | \(1 + \theta + \frac{{{\theta ^2}}}{{2!}} + \frac{{{\theta^3}}}{{3!}} + \ldots\) |
| D. | \(\theta + \frac{{{\theta ^3}}}{{3!}} + \frac{{{\theta^5}}}{{5!}} + \ldots\) |
| Answer» C. \(1 + \theta + \frac{{{\theta ^2}}}{{2!}} + \frac{{{\theta^3}}}{{3!}} + \ldots\) | |
| 107. |
Let ∇⋅(fv) = x2y + y2z + z2x, where f and v are scalar and vector fields respectively. If v = yi + zj + xk, then v⋅∇f is |
| A. | x2y + y2z + z2x |
| B. | 2xy + 2yz + 2zx |
| C. | x + y + z |
| D. | 0 |
| Answer» B. 2xy + 2yz + 2zx | |
| 108. |
\(\mathop \smallint \limits_0^{\frac{\pi }{2}} \mathop \smallint \limits_0^{\frac{\pi }{2}} \sin \left( {x + y} \right)dxdy\) is |
| A. | 0 |
| B. | π |
| C. | \(\frac{\pi }{2}\) |
| D. | 2 |
| Answer» E. | |
| 109. |
If \(\rm f(x) = \log_{x^2 } x\), then f'(x) at x = e, is |
| A. | 0 |
| B. | 1 |
| C. | 1/e |
| D. | 1/(2e) |
| Answer» B. 1 | |
| 110. |
If \(\vec{a}, \vec{b}, \vec{c}\) are three vectors such that \(\vec{a}+ \vec{b}+\vec{c}=0\) and \(|\vec{a}|=2, |\vec{b}|=3, |\vec{c}|=5\) then the value of \(\vec{a} \ \bullet \ \vec{b} + \vec{b} \ \bullet \ \vec{c} + \vec{c} \ \bullet \ \vec{a}\) is: |
| A. | 0 |
| B. | 1 |
| C. | -19 |
| D. | 38 |
| Answer» D. 38 | |
| 111. |
If x + y = 20 and P = xy, then what is the maximum value of P? |
| A. | 100 |
| B. | 96 |
| C. | 84 |
| D. | 50 |
| Answer» B. 96 | |
| 112. |
Find the angle between two vectors \(\vec a\) and \(\vec b\) with magnitudes 1 and 2 respectively and \(\vec a.\vec b = 1\) |
| A. | 30 degrees |
| B. | 60 degrees |
| C. | 45 degrees |
| D. | zero degrees |
| Answer» C. 45 degrees | |
| 113. |
If \(y = {e^{{e^x}}}\) then \(\frac{{dy}}{{dx}}\) is equal to? |
| A. | \({e^{{e^x}}}\) |
| B. | \({e^{({e^x}+ x)}}\) |
| C. | \({e^x} + x\) |
| D. | None of these |
| Answer» C. \({e^x} + x\) | |
| 114. |
Let f: [0, 1] → R be such that f(xy) = f(x)∙f(y) for all x, y ∈ [0, 1], and f(0) ≠ 0. If y = y(x) satisfies the differential equation\(\text{, }\!\!~\!\!\text{ }\frac{dy}{dx}=f\left( x \right)\text{ }\!\!~\!\!\)with y(0) = 1, then \(y\left( \frac{1}{4} \right)+y\left( \frac{3}{4} \right)\) is equal to: |
| A. | 3 |
| B. | 4 |
| C. | 2 |
| D. | 5 |
| Answer» B. 4 | |
| 115. |
In the open interval (0, 1), the polynomial p(x) = x4 - 4x3 + 2 has |
| A. | one real root |
| B. | three real roots |
| C. | two real roots |
| D. | no real roots |
| Answer» B. three real roots | |
| 116. |
If \(\left[ {\overrightarrow a \overrightarrow b \overrightarrow c } \right] = 1\), then the value of \(\frac{{\overrightarrow a \cdot \overrightarrow b \times \overrightarrow c }}{{\overrightarrow c \times \overrightarrow a \cdot \overrightarrow b }} + \frac{{\overrightarrow b \cdot \overrightarrow c \times \overrightarrow a }}{{\overrightarrow a \times \overrightarrow b \cdot \overrightarrow c }} + \frac{{\overrightarrow c \cdot \overrightarrow a \times \overrightarrow b }}{{\overrightarrow b \times \overrightarrow c \cdot \overrightarrow a }}\) is: |
| A. | 1 |
| B. | -1 |
| C. | 2 |
| D. | 3 |
| Answer» E. | |
| 117. |
In the given partial differential equation if B ≠ 0, the equation is known as:\(\frac{{{\partial ^2}u}}{{\partial {t^2}}} - {B^2}\left( {\frac{{{\partial ^2}u}}{{\partial {x^2}}} + \frac{{{\partial ^2}u}}{{\partial {y^2}}}} \right) = 0\) |
| A. | Poisson’s equation |
| B. | Laplace equation |
| C. | Wave equation |
| D. | Thermal equation |
| Answer» D. Thermal equation | |
| 118. |
Find the value of \(I=\int_1^2\frac{\log x}{x^2}\) |
| A. | \(\frac{1}{4}\log \frac{e}{2}\) |
| B. | \(\frac{1}{2}\log \frac{e}{2}\) |
| C. | log e |
| D. | \(\frac{1}{2}\log \frac{e}{4}\) |
| Answer» C. log e | |
| 119. |
For the spherical surface x2 + y2 + z2 – 1, the unit outward normal vector at the point \(\left( {\frac{1}{{\sqrt 2 }},\frac{1}{{\sqrt 2 }},0} \right)\) is given by |
| A. | \(\frac{1}{{\sqrt 2 }}\hat i + \frac{1}{{\sqrt 2 }}\hat j\;\) |
| B. | \(\frac{1}{{\sqrt 2 }}\hat i - \frac{1}{{\sqrt 2 }}\hat j\) |
| C. | \(\hat k\) |
| D. | \(\frac{1}{{\sqrt 3 }}\hat i + \frac{1}{{\sqrt 3 }}\hat j + \frac{1}{{\sqrt 3 }}\hat k\) |
| Answer» B. \(\frac{1}{{\sqrt 2 }}\hat i - \frac{1}{{\sqrt 2 }}\hat j\) | |
| 120. |
If value of ∫ \(\frac{2x}{(x^2+1)(x^2+3)}dx\) will be _______ where c is an arbitary constant. |
| A. | \(\frac{1}{2}log\frac{x^2 +1}{x^2+3}+c\) |
| B. | \(\frac{1}{2}log\frac{x^2 -1}{x^2+1}+c\) |
| C. | \(2log\frac{x +1}{x-3}+c\) |
| D. | \(2log\frac{2x +1}{2x-1}+c\) |
| Answer» B. \(\frac{1}{2}log\frac{x^2 -1}{x^2+1}+c\) | |
| 121. |
Equation of a plane passing through the point \(3\widehat i - \widehat j + \widehat k\) and perpendicular to the vector \(4\widehat i + 2\widehat j - \widehat k\) is |
| A. | 2x + 4y + z = 9 |
| B. | 2x + 4y - z + 9 = 0 |
| C. | 4x + 2y + z = 9 |
| D. | 4x + 2y - z = 9 |
| Answer» E. | |
| 122. |
Area enclosed between the curve y2(2a - x) = x3 and the line x = 2a above x -axis is |
| A. | πa2 |
| B. | \(\frac{{3\pi {a^2}}}{2}\) |
| C. | 2πa2 |
| D. | 3πa2 |
| Answer» C. 2πa2 | |
| 123. |
Consider the line integral \(\mathop \smallint \limits_c \left( {xdy - ydx} \right)\) the integral being taken in a counter clockwise direction over the closed curve C that forms the boundary of the region R shown in the figure below. The region R is the area enclosed by the union of a 2 × 3 rectangle and a semi-circle of radius 1. The line integral evaluates to |
| A. | 6 + π/2 |
| B. | 8 + π |
| C. | 12 + π |
| D. | 16 + 2π |
| Answer» D. 16 + 2π | |
| 124. |
If the area of the trapezium is maximum, what is the length of fourth side? |
| A. | 8 cm |
| B. | 9 cm |
| C. | 10 cm |
| D. | 12 cm |
| Answer» E. | |
| 125. |
If y = (1 + x)(1 + x2) ..... (1 + x100) then \(\dfrac{dy}{dx}\) at x = 0 is |
| A. | 100 |
| B. | 0 |
| C. | 1 |
| D. | 5050 |
| Answer» D. 5050 | |
| 126. |
If x = 3 tan t and y = 3 sec t, then the value of \(\frac{{{d}^{2}}y}{d{{x}^{2}}}\text{ }\!\!~\!\!\text{ at }\!\!~\!\!\text{ }t=\frac{\pi }{4},\) is: |
| A. | \(\frac{1}{3\sqrt{2}}\) |
| B. | \(\frac{1}{6\sqrt{2}}\) |
| C. | \(\frac{3}{2\sqrt{2}}\) |
| D. | 186 |
| Answer» C. \(\frac{3}{2\sqrt{2}}\) | |
| 127. |
∫ (In x)-1 dx - ∫ (In x)-2 dx is equal to |
| A. | x (In x)-1 + c |
| B. | x (In x)-2 + c |
| C. | x (In x) + c |
| D. | x (In x)2 + c |
| Answer» B. x (In x)-2 + c | |
| 128. |
Curl of vector V(x,y,z) = 2x2i + 3z2j + y3k at x = y = z = 1 is |
| A. | – 3i |
| B. | 3i |
| C. | 3i – 4j |
| D. | 3i – 6k |
| Answer» B. 3i | |
| 129. |
For a right-angled triangle, if the sum of the lengths of the hypotenuse and a side is kept constant, in order to have a maximum area of the triangle, the angle between the hypotenuse and the side is |
| A. | 12° |
| B. | 36° |
| C. | 60° |
| D. | 45° |
| Answer» D. 45° | |
| 130. |
A primitive of |x|, when x < 0, is |
| A. | \(\frac{{{x^2}}}{2} + c\) |
| B. | \( - \frac{{{x^2}}}{2} + c\) |
| C. | x + c |
| D. | -x + c |
| Answer» C. x + c | |
| 131. |
\(\mathop {\lim }\limits_{x \to - 5} \frac{{\sqrt {\left( {2x + 35} \right)} - 5}}{{x + 5}}\) |
| A. | \(\frac{1}{5}\) |
| B. | \(\frac{1}{6}\) |
| C. | \(\frac{1}{{\sqrt 5 }}\) |
| D. | \(\frac{{\sqrt 5 }}{2}\) |
| Answer» B. \(\frac{1}{6}\) | |
| 132. |
A value of α such that \(\mathop \smallint \nolimits_\alpha ^{\alpha + 1} \frac{{{\rm{d}}x}}{{\left( {x + \alpha } \right)\left( {x + \alpha + 1} \right)}} = {\rm{lo}}{{\rm{g}}_{\rm{e}}}\left( {\frac{9}{8}} \right)\) is: |
| A. | -2 |
| B. | \(\frac{1}{2}\) |
| C. | \(- \frac{1}{2}\) |
| D. | 2 |
| Answer» B. \(\frac{1}{2}\) | |
| 133. |
Find the area of triangle whose two sides are represented by the vectors 3i + 4j and 5i + 7j + k is |
| A. | \(\sqrt {26}\over 2\) |
| B. | \(\sqrt{26}\) |
| C. | 13 |
| D. | \(\sqrt{13}\over 2\) |
| Answer» B. \(\sqrt{26}\) | |
| 134. |
If x = -1 and x = 2 are extreme points of f(x) = a log |x| + βx2 + x then |
| A. | α = -6, β = -½ |
| B. | α = 2, β = -½ |
| C. | α = 2, β = ½ |
| D. | α = -6, β = ½ |
| Answer» C. α = 2, β = ½ | |
| 135. |
Divergence of the vector field \({x^2}z\hat i + xy\hat j - y{z^2}\hat k\) at (1, -1, 1) is |
| A. | 0 |
| B. | 3 |
| C. | 5 |
| D. | 6 |
| Answer» D. 6 | |
| 136. |
A function is defined in (0, ∞) by \({\rm{f}}\left( {\rm{x}} \right) = \left\{ {\begin{array}{*{20}{c}} {1 - {{\rm{x}}^2}{\rm{\;for\;}}0 < {\rm{x}} \le 1}\\ {{\rm{In\;x\;for\;}}1 < {\rm{x}} \le 2}\\ {{\rm{In\;}}2 - 1 + 0.5{\rm{x\;for\;}}2 < {\rm{x}} < \infty } \end{array}} \right.\)Which one of the following is correct in respect of the derivative of the function, i.e. f’(x)? |
| A. | f’(x) = 2x for 0 < x ≤ 1 |
| B. | f’(x) = -2x for 0 < x ≤ 1 |
| C. | f’(x) = -2x for 0 < x < 1 |
| D. | f’(x) = 0 for 0 < x < ∞ |
| Answer» C. f’(x) = -2x for 0 < x < 1 | |
| 137. |
A path AB in the form of one quarter of a circle of unit radius is shown in the figure. Integration of (x+y)2 on path AB traversed in a counter-clockwise sense is |
| A. | \(\frac{x}{2} - 1\) |
| B. | \(\frac{\pi }{2} + 1\) |
| C. | \(\frac{\pi }{2}\) |
| D. | 1 |
| Answer» C. \(\frac{\pi }{2}\) | |
| 138. |
At point (1, 0, 3) on the surface 2x2 + 3y2 + z2 – 11 = 0, the directional derivative in the direction \(\vec a = \hat i + 2\hat j + \hat k\) is |
| A. | 10 |
| B. | \(\frac{{5}}{{3 }}\) |
| C. | \(\frac{{-5}}{{3 }}\) |
| D. | \(\frac{{10}}{{\sqrt 6 }}\) |
| Answer» E. | |
| 139. |
If \(v = {\left( {{x^2} + {y^2} + {z^2}} \right)^{ - \frac{1}{2}}},~then~~\frac{{{\partial ^2}v}}{{\partial {x^2}}} + \frac{{{\partial ^2}v}}{{\partial {y^2}}} + \frac{{{\partial ^2}v}}{{\partial {z^2}}}\) is |
| A. | -1/2 |
| B. | -1 |
| C. | 0 |
| D. | 1 |
| Answer» D. 1 | |
| 140. |
Let f(x + y) = f(x)f(y) and f(x) = 1 + xg(x)φ(x), where \(\mathop {\lim }\limits_{{\rm{x}} \to 0} {\rm{g}}\left( {\rm{x}} \right) = {\rm{a}}\) and \(\mathop {\lim }\limits_{{\rm{x}} \to 0} \phi \left( {\rm{x}} \right) = {\rm{b}}\) then what is f’(x) equal to? |
| A. | 1 + abf(x) |
| B. | 1 + ab |
| C. | ab |
| D. | abf(x) |
| Answer» E. | |
| 141. |
If \(\rm \displaystyle I_n = \int_0^a(a^2 - x^2)^n \ dx\), where n is a positive integer, then the relation between In and In-1 is: |
| A. | \(\rm I_n = \left(\dfrac{2na^2}{2n+1}\right)I_{n-1}\) |
| B. | \(\rm I_n = \left(\dfrac{2n^2a^2}{2n+1}\right)I_{n-1}\) |
| C. | \(\rm I_n = \left(\dfrac{2na^2}{2n-1}\right)I_{n-1}\) |
| D. | \(\rm I_n = \left(\dfrac{2n^2a^2}{2n-1}\right)I_{n-1}\) |
| Answer» D. \(\rm I_n = \left(\dfrac{2n^2a^2}{2n-1}\right)I_{n-1}\) | |
| 142. |
Let \(I = \mathop \smallint \nolimits_a^b \left( {{x^4} - 2{x^2}} \right)dx\). If ‘I’ is minimum then the ordered pair (a, b) is: |
| A. | (0, √2) |
| B. | (-√2, 0) |
| C. | (√2, -√2) |
| D. | (-√2, -√2) |
| Answer» D. (-√2, -√2) | |
| 143. |
Find the divergence of a vector \(\vec A = {x^2}{\hat a_x} + 6{y^2}{\hat a_y} + {z^3}{\hat a_z}\) at point P (2, 4, 1). |
| A. | 10 |
| B. | 24 |
| C. | 16 |
| D. | 55 |
| Answer» E. | |
| 144. |
Consider a function \(\bar f = \frac{1}{{{r^2}}}\hat r\), where r is the distance from the origin and \(\hat r\) is the unit vector in the radial direction. The divergence of the function over a sphere of radius R, which includes the origin, is |
| A. | 0 |
| B. | 2π |
| C. | 4π |
| D. | Rπ |
| Answer» B. 2π | |
| 145. |
Find the derivation of f(x) = 1/x2 |
| A. | -2/x3 |
| B. | 2/x3 |
| C. | -1/2x |
| D. | 1/2x |
| Answer» B. 2/x3 | |
| 146. |
If the vectors \(2\widehat i + \widehat j + \widehat k\) and \(\widehat i - 4\widehat j + λ \widehat k\) are mutually perpendicular, then the value of λ is: |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | 4 |
| Answer» C. 3 | |
| 147. |
A parametric curve defined by \(x = \cos \left( {\frac{{\pi u}}{2}} \right),y = \sin \left( {\frac{{\pi u}}{2}} \right)\)in the range of 0 ≤ u ≤ 1 is rotated about the X – axis by 360 degrees. Area of the surface generated is |
| A. | \(\frac{\pi }{2}\) |
| B. | π |
| C. | 2π |
| D. | 4π |
| Answer» D. 4π | |
| 148. |
Compute \(\begin{array}{*{20}{c}}{{\rm{lim}}}\\{x \to 3}\end{array}\frac{{{x^4} - 81}}{{2{x^2} - 5x - 3}}\) |
| A. | 1 |
| B. | 53/12 |
| C. | 108/7 |
| D. | Limit does not exist |
| Answer» D. Limit does not exist | |
| 149. |
Let f : R → R be a continuously differentiable function such that f(2) = 6 and \({\rm{f'}}\left( 2 \right) = \frac{1}{{48}}\). If \(\mathop \smallint \nolimits_6^{{\rm{f}}\left( {\rm{x}} \right)} \left( {4{{\rm{t}}^3}} \right){\rm{dt}} = \left( {{\rm{x}} - 2} \right){\rm{g}}\left( {\rm{x}} \right),{\rm{\;then\;}}\mathop {{\rm{lim}}}\limits_{{\rm{x}} \to 2} {\rm{\;g}}\left( {\rm{x}} \right)\) is equal to: |
| A. | 18 |
| B. | 24 |
| C. | 12 |
| D. | 36 |
| Answer» B. 24 | |
| 150. |
Consider two functions: \(x\; = \;\psi ln\phi\) and \(y\; = \;\phi ln\psi \). Which one of the following is the correct expression for ∂ψ/∂x? |
| A. | \(\frac{{ln\psi }}{{ln\phi ln\psi - 1}}\) |
| B. | \(\frac{{ln\phi }}{{ln\phi \psi - 1}}\) |
| C. | \(\frac{{xln\psi }}{{ln\phi \psi - 1}}\) |
| D. | \(\frac{{xln\phi }}{{ln\phi ln\psi - 1}}\) |
| Answer» B. \(\frac{{ln\phi }}{{ln\phi \psi - 1}}\) | |