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.
| 151. |
If \(y = {\left( {\cos x} \right)^{{{(\cos x)}^{{{(\cos x)}^.}{{^.}^.}^\infty }}}}\) then \(\frac{{dy}}{{dx}}\) is equal to |
| A. | \(- \frac{{{y^2}\tan x}}{{1 - y\;In\;(\cos x)}}\) |
| B. | \(\frac{{{y^2}\tan x}}{{1 + y\;In\;(\cos x)}}\) |
| C. | \(\frac{{{y^2}\tan x}}{{1 - y\;In\;(\sin x)}}\) |
| D. | \(\frac{{{y^2}\sin x}}{{1 + y\;In\;(\sin x)}}\) |
| Answer» B. \(\frac{{{y^2}\tan x}}{{1 + y\;In\;(\cos x)}}\) | |
| 152. |
\(\mathop a\limits^ \to = 2\hat i + \hat j + 3\hat k\) and \(\mathop b\limits^ \to = 3\hat i - 2\hat j + \hat k\) are two vectors. The angle between them is: |
| A. | 30° |
| B. | 45° |
| C. | 90° |
| D. | 60° |
| Answer» E. | |
| 153. |
If f1 and f2 are two linear functions on vector space V defined as f1(a, b) = a + 2b, f2(a, b) = 3a - b, then (3f1 - 4f2) (a, b) is equal to: |
| A. | a + b |
| B. | 2a + 6b |
| C. | 9a + 10b |
| D. | -9a + 10b |
| Answer» E. | |
| 154. |
At what value of x does the function attain maximum value? |
| A. | e |
| B. | \(\sqrt {\rm{e}} \) |
| C. | \(\frac{1}{{\sqrt {\rm{e}} }}\) |
| D. | \(\frac{1}{\rm{e}}\) |
| Answer» D. \(\frac{1}{\rm{e}}\) | |
| 155. |
Angle made between vector \(\vec c = 2\vec i - 3\vec j + 4\vec k\) and the Z-axis |
| A. | \(\frac{4}{{\sqrt {29} }}\) |
| B. | \({\sin ^{ - 1}}\frac{4}{{\sqrt {29} }}\) |
| C. | \({\cos ^{ - 1}}\frac{4}{{\sqrt {29} }}\) |
| D. | \({\tan ^{ - 1}}\frac{4}{{\sqrt {29} }}\) |
| Answer» D. \({\tan ^{ - 1}}\frac{4}{{\sqrt {29} }}\) | |
| 156. |
Let f(x) = x2 - 2x + 2 be a continuous function defined on x ∈ [1, 3]. The point x at which the tangent of f(x) becomes parallel to the straight line joining f(1) and f(3) is |
| A. | 3 |
| B. | 0 |
| C. | 2 |
| D. | 1 |
| Answer» D. 1 | |
| 157. |
Given, \(i = \sqrt { - 1} \), the value of the definite integral, \(I = \mathop \smallint \limits_0^{\pi /2} \frac{{\cos x + i\sin x}}{{\cos x - i\sin x}}dx\) is: |
| A. | 1 |
| B. | -1 |
| C. | i |
| D. | -i |
| Answer» D. -i | |
| 158. |
If, f (x, y) = x2 + y3; x = t2 + t3; y = 1 + t3Value of df/dt at t = 1 is: |
| A. | 0 |
| B. | 2 |
| C. | 19 |
| D. | 56 |
| Answer» E. | |
| 159. |
If x = a cos t, y = b sin t, then \(\rm \dfrac{d^2y}{dx^2}\) is: |
| A. | \(-\rm \dfrac{b^4}{a^2y^3}\) |
| B. | \(-\rm \dfrac{b^4}{a^2x^3}\) |
| C. | \(\rm \dfrac{b}{ay^4}\) |
| D. | \(\rm \dfrac{a^4}{bx^3}\) |
| Answer» B. \(-\rm \dfrac{b^4}{a^2x^3}\) | |
| 160. |
Let f be a function defined on the interval [0, 1] as \(f(x)=\left\lbrace \begin{matrix} 0, \ \text{if} \ x \ \text{is rational} \\\ 1, \ \text{if} \ x \ \text{is irrational}\end{matrix}\right.\) Then f is - |
| A. | Not R - integrable on [0, 1] |
| B. | R - integrable on [0, 1] |
| C. | May or may not be R - integrable on [0, 1] |
| D. | None of these |
| Answer» B. R - integrable on [0, 1] | |
| 161. |
If \(\mathop \smallint \nolimits_0^x f\left( t \right)dt = {x^2} + \mathop \smallint \nolimits_x^1 {t^2}f\left( t \right)dt\), then f'(1/2) is: |
| A. | \(\frac{{24}}{{25}}\) |
| B. | \(\frac{{18}}{{25}}\) |
| C. | \(\frac{4}{5}\) |
| D. | \(\frac{6}{{25}}\) |
| Answer» B. \(\frac{{18}}{{25}}\) | |
| 162. |
If f(x) = X5 – 20X 3 + 240X then f(x) is |
| A. | Monotonically decreasing everywhere |
| B. | Monotonically decreasing on (0, ∞) |
| C. | Monotonically increasing only in (-∞, 0) |
| D. | Monotonically increasing everywhere |
| Answer» E. | |
| 163. |
\(\int\limits_{ - 2}^2 {\left| {1 - {x^2}} \right|} dx\) is: |
| A. | 0 |
| B. | 2 |
| C. | -2 |
| D. | 4 |
| Answer» E. | |
| 164. |
If θ denotes the acute angle between the curves, y = 10 – x2 and y = 2 + x2 at a point of their intersection, then |tan θ| is equal to: |
| A. | \(\frac{4}{9}\) |
| B. | \(\frac{8}{15}\) |
| C. | \(\frac{7}{17}\) |
| D. | \(\frac{8}{17}\) |
| Answer» C. \(\frac{7}{17}\) | |
| 165. |
If \(\mathop {\lim }\limits_{x \to \infty } \left( {ax + \left( {\frac{{7 - \sqrt 3 {x^2}}}{{3 - x}}} \right)} \right) = b,\) a finite number, then the values of a and b are: |
| A. | a = -√3, b = 3√3 |
| B. | a = 3, b = √3 |
| C. | a = 2√3, b = -3 |
| D. | a = -2√3, b = 2√3 |
| Answer» B. a = 3, b = √3 | |
| 166. |
In the Taylor series expansion of ex about x = 2, the coefficient of (x - 2)4 is |
| A. | \(1/4!\) |
| B. | \({2^4}/4!\) |
| C. | \({e^2}/4!\) |
| D. | \({e^4}/4!\) |
| Answer» D. \({e^4}/4!\) | |
| 167. |
A rectangular sheet of metal of length 6 m and width 2 m is given. Four equal squares are removed from the corners. The sides of this sheet are now turned up to form an open rectangular box. Find the height of the box (in m) such that the volume of the box is maximum. |
| A. | 45 cms |
| B. | 40 cms |
| C. | 35 cms |
| D. | 20 cms |
| Answer» B. 40 cms | |
| 168. |
Evaluate \(\displaystyle\lim_{x \rightarrow \infty}\left(\dfrac{x+1}{x+2}\right)^{2x+1}\) |
| A. | 0 |
| B. | e |
| C. | e-1 |
| D. | e-2 |
| Answer» E. | |
| 169. |
limx→∞ x1/x is |
| A. | ∞ |
| B. | 0 |
| C. | 1 |
| D. | Not defined |
| Answer» D. Not defined | |
| 170. |
If \(\int^x_{\log 2} \frac 1 {\sqrt {e^x - 1}} dx = \frac \pi 6,\) then x = |
| A. | log 2 |
| B. | 2 log 2 |
| C. | 3 log 2 |
| D. | 4 log 2 |
| Answer» C. 3 log 2 | |
| 171. |
For the function \(f\left( t \right) = {e^{ - \frac{t}{\tau }}}\), the Taylor series approximation for t ≪ τ is |
| A. | 1 + t/τ |
| B. | 1 – 2/2τ2 |
| C. | 1 – t/τ |
| D. | 1 + t |
| Answer» D. 1 + t | |
| 172. |
Consider the line integral \(I = \mathop \smallint \limits_C \left( {{x^2} + i{y^2}} \right)dz\), where z = x + iy. The line C is shown in figure below.The value of I is |
| A. | \(\frac{1}{2}i\) |
| B. | \(\frac{2}{3}i\) |
| C. | \(\frac{3}{4}i\) |
| D. | \(\frac{4}{5}i\) |
| Answer» C. \(\frac{3}{4}i\) | |
| 173. |
A̅ × B̅ is a vector |
| A. | Parallel to A̅, but perpendicular to B |
| B. | Parallel to B̅ , but perpendicular to A |
| C. | Perpendicular to both A̅ and B̅ |
| D. | Parallel to both A̅ and B̅ |
| Answer» D. Parallel to both A̅ and B̅ | |
| 174. |
Let the function \(f\left( \theta \right) = \left| {\begin{array}{*{20}{c}}{\sin \theta }&{\cos \theta }&{\tan \theta }\\{\sin \left( {\frac{\pi }{6}} \right)}&{\cos \left( {\frac{\pi }{6}} \right)}&{\tan \left( {\frac{\pi }{6}} \right)}\\{\sin \left( {\frac{\pi }{3}} \right)}&{\cos \left( {\frac{\pi }{3}} \right)}&{\tan \left( {\frac{\pi }{3}} \right)}\end{array}} \right|\)Where \(\theta \in \left[ {\frac{\pi }{6},\frac{\pi }{3}} \right]\) and f’(θ) denote the derivative of f with respect to θ. Which of the following statements is/are TRUE?(I) There exists \(\theta \in \left( {\frac{\pi }{6},\frac{\pi }{3}} \right)\) such that f’(θ) = 0(II) There exists \(\theta \in \left( {\frac{\pi }{6},\frac{\pi }{3}} \right)\) such that f’(θ) ≠ 0 |
| A. | I only |
| B. | II only |
| C. | Both I and II |
| D. | Neither I nor II |
| Answer» D. Neither I nor II | |
| 175. |
Let x, y be positive real numbers and m, n positive integers. The maximum value of the expression \(\frac{{{x^{\rm{m}}}{y^{\rm{n}}}}}{{\left( {1 + {x^{2{\rm{m}}}}} \right)\left( {1 + {y^{2{\rm{n}}}}} \right)}}\) is: |
| A. | 1 |
| B. | \(\frac{1}{2}\) |
| C. | \(\frac{1}{4}\) |
| D. | \(\frac{{m + n}}{{6\:mn}}\) |
| Answer» D. \(\frac{{m + n}}{{6\:mn}}\) | |
| 176. |
If y = (sin-1x)2 then: |
| A. | (1 - x2)yn + 2 - (2n + 1)xyn + 1 - n2yn = 0 |
| B. | (1 - x2)yn + 2 - (2n + 1)xyn + 1 + n2yn = 0 |
| C. | (1 - x2)yn + 2 + (2n + 1)xyn + 1 - n2yn = 0 |
| D. | (1 - x2)yn + 2 - (2n - 1)xyn + 1 - n2yn = 0 |
| Answer» B. (1 - x2)yn + 2 - (2n + 1)xyn + 1 + n2yn = 0 | |
| 177. |
If ey + xy = e, the ordered pair \(\left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}},\frac{{{{\rm{d}}^2}{\rm{y}}}}{{{\rm{d}}{{\rm{x}}^2}}}} \right)\) at x = 0 is equal to: |
| A. | \(\left( {\frac{1}{{\rm{e}}}, - \frac{1}{{{{\rm{e}}^2}}}} \right)\) |
| B. | \(\left( { - \frac{1}{{\rm{e}}},\frac{1}{{{{\rm{e}}^2}}}} \right)\) |
| C. | \(\left( {\frac{1}{{\rm{e}}},\frac{1}{{{{\rm{e}}^2}}}} \right)\) |
| D. | \(\left( { - \frac{1}{{\rm{e}}}, - \frac{1}{{{{\rm{e}}^2}}}} \right)\) |
| Answer» C. \(\left( {\frac{1}{{\rm{e}}},\frac{1}{{{{\rm{e}}^2}}}} \right)\) | |
| 178. |
consider the following statements:1. The function attains local minima at x = -2 and x = 3.2. The function increases in the interval (-2, 0).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 | |
| 179. |
If y = xn-1 ln x, then the nth order derivative of y with respect to x at \(x=\dfrac{1}{2}\) is: |
| A. | 3 ⋅ [n!] |
| B. | 2 ⋅ [(n + 1)!] |
| C. | 3(n - 1) ⋅ [n!] |
| D. | 2 ⋅ [(n - 1)!] |
| Answer» E. | |
| 180. |
Consider the function \(f\left( x \right) = \frac{{\left| x \right|}}{x}:\)a) \(\mathop {\lim }\limits_{x \to {0^ + }} f\left( x \right) = 1\)b) \(\mathop {\lim }\limits_{x \to {0^{ -}}} f\left( x \right) = - 1\)c) \(\mathop {\lim }\limits_{x \to 0} f\left( x \right)\) does not exists |
| A. | All (a), (b) and (c) are true |
| B. | Both (a) and (b) are false |
| C. | (c) alone true |
| D. | (a) and (c) are true |
| Answer» B. Both (a) and (b) are false | |
| 181. |
Consider points A, B, C and D on a circle of radius 2 units as in the above figure.The items in List II are the values of \({\bar a_\phi }\) at different points on the circle. Match List I with List II and select the correct answer using the code given below the lists:List IList IIA1 \({\bar a_X}\)B2. \({\bar a_Y}\)C3. \(-{\bar a_X}\)D4. \(\left( {{{\bar a}_X} + {{\bar a}_Y}} \right)/\sqrt 2 \) 5. \(-\left( {{{\bar a}_X} + {{\bar a}_Y}} \right)/\sqrt 2 \) 6. \(\left( {{{\bar a}_X} - {{\bar a}_Y}} \right)/\sqrt 2 \)Code: |
| A. | A - 3, B - 4, C - 5, D - 2 |
| B. | A - 1, B - 6, C - 5, D - 2 |
| C. | A - 1, B - 6, C - 2, D - 4 |
| D. | A - 3, B - 5, C - 4, D - 2 |
| Answer» E. | |
| 182. |
If \(\mathop \smallint \nolimits_{\rm{a}}^{\rm{b}} {{\rm{x}}^3}{\rm{dx}} = 0{\rm{\;}}\) and \(\mathop \smallint \nolimits_{\rm{a}}^{\rm{b}} {{\rm{x}}^2}{\rm{dx}} = \frac{2}{3},\) then what are the values of a and b respectively? |
| A. | -1, 1 |
| B. | 1, 1 |
| C. | 0, 0 |
| D. | 2, -2 |
| Answer» B. 1, 1 | |
| 183. |
If \(\frac{\text{d}y}{\text{d}x}+\frac{3}{\text{co}{{\text{s}}^{2}}x}y=\frac{1}{\text{co}{{\text{s}}^{2}}x},~x~\epsilon \left( \frac{-\pi }{3},\frac{\pi }{3} \right),\text{ }\!\!~\!\!\text{ and }\!\!~\!\!\text{ }y\left( \frac{\pi }{4} \right)=\frac{4}{3},\text{ }\!\!~\!\!\text{ then }\!\!~\!\!\text{ }y\left( -\frac{\pi }{4} \right)\) equals: |
| A. | \(\frac{1}{3}+{{\text{e}}^{6}}\) |
| B. | 1/3 |
| C. | \(-\frac{4}{3}\) |
| D. | \(\frac{1}{3}+{{\text{e}}^{3}}\) |
| Answer» B. 1/3 | |
| 184. |
If \(\smallint \frac{{x + 1}}{{\sqrt {2x - 1} }}{\rm{d}}x = f\left( x \right)\sqrt {2x - 1} + {\rm{C}}\), where C is a constant of integration, then f(x) is equal to: |
| A. | \(\frac{1}{3}\left( {x + 1} \right)\) |
| B. | \(\frac{2}{3}\left( {x + 2} \right)\) |
| C. | \(\frac{2}{3}\left( {x - 4} \right)\) |
| D. | \(\frac{1}{3}\left( {x + 4} \right)\) |
| Answer» E. | |
| 185. |
Differential coefficient of log10 x with respect to logx 10 is |
| A. | \(-\dfrac{(\log x)^2}{(\log 10)^2}\) |
| B. | \(\dfrac{(\log_{10}x)^2}{(\log 10)^2}\) |
| C. | \(\dfrac{(\log_x 10)^2}{(\log 10)^2}\) |
| D. | \(-\dfrac{(\log 10)^2}{(\log x)^2}\) |
| Answer» B. \(\dfrac{(\log_{10}x)^2}{(\log 10)^2}\) | |
| 186. |
In computation, the function f is defined by\(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {a{x^2} + 2x - 1}&{x \le 1}\\ {b - c{x^2}}&{x > 1} \end{array}} \right.\)where a, b, c are constants. It is given that f(x) is differentiable at x = 1 and f'(0) = f'(2). Then the value of b is |
| A. | 0 |
| B. | 14 |
| C. | 16 |
| D. | 17 |
| Answer» B. 14 | |
| 187. |
If f'(x) = g(x) and g'(x) = f(x2), then f"(x2) is equal to |
| A. | g(x2) |
| B. | f(x4) |
| C. | f(x3) |
| D. | g(x4) |
| Answer» C. f(x3) | |
| 188. |
Let f and g be continuous function on [0, a] such that f(x) = f(a – x) and g(x) + g (a – x) = 4, then \(\mathop \smallint \limits_0^a f\left( x \right){\rm{\;}}g\left( x \right){\rm{\;}}dx\) is equal to |
| A. | \(4\mathop \smallint \nolimits_0^{\;a} f\left( x \right)dx\) |
| B. | \(\mathop \smallint \limits_0^a f\left( x \right){\rm{dx}}\) |
| C. | \(2\mathop \smallint \nolimits_0^{\rm{\;a}} {\rm{f}}\left( {\rm{x}} \right){\rm{dx}}\) |
| D. | \(- 3\mathop \smallint \nolimits_0^{\rm{\;a}} {\rm{f}}\left( {\rm{x}} \right){\rm{dx}}\) |
| Answer» D. \(- 3\mathop \smallint \nolimits_0^{\rm{\;a}} {\rm{f}}\left( {\rm{x}} \right){\rm{dx}}\) | |
| 189. |
\(\mathop {\lim }\limits_{\theta \to \frac{\pi }{2}} \frac{{\log \left( {\theta - \frac{\pi }{2}} \right)}}{{\tan \theta }}\) |
| A. | 1 |
| B. | π/2 |
| C. | π/4 |
| D. | 0 |
| Answer» E. | |
| 190. |
Given x01234y481576then the value of \(\rm\dfrac{dy}{dx}\) at x = 0 will be equal to - |
| A. | -13.5 |
| B. | -27.5 |
| C. | 0 |
| D. | 57.6667 |
| Answer» B. -27.5 | |
| 191. |
A flower-bed in the form of a sector has been fenced by a wire of 40m length. If the flower-bed has the greatest possible area, then what is the radius of the sector? |
| A. | 25 m |
| B. | 20 m |
| C. | 10 m |
| D. | 5 m |
| Answer» D. 5 m | |
| 192. |
If \(\mathop \smallint \nolimits_0^{\pi /3} \frac{{{\rm{tan}}\theta }}{{\sqrt {2k{\rm{sec}}\theta } }}d\theta = 1 - \frac{1}{{\sqrt 2 }},(k > 0)\), then the value of k is: |
| A. | 4 |
| B. | 1/2 |
| C. | 1 |
| D. | 2 |
| Answer» E. | |
| 193. |
Find the limit, \(\mathop {\lim }\limits_{x \to 4} \frac{x^2-16}{\sqrt{x^2+9} \;-5}\) |
| A. | 0 |
| B. | 10 |
| C. | 5 |
| D. | 20 |
| Answer» C. 5 | |
| 194. |
Divergence of the curl of a twice differentiable continuous vector function is |
| A. | Unity |
| B. | Infinity |
| C. | Zero |
| D. | A unit vector |
| Answer» D. A unit vector | |
| 195. |
Evaluation of \(\displaystyle\int{\dfrac{1-\tan x}{1+\tan x}}\ dx\) is: |
| A. | log (sin x - cos x) + c |
| B. | log (sin x - cot x) + c |
| C. | log (cos x + sin x) + c |
| D. | log (cos x - cot x) + c |
| Answer» D. log (cos x - cot x) + c | |
| 196. |
If the function \(u = \ln \left( {\frac{{{x^3} + {x^2}y - {y^3}}}{{x - y}}} \right)\) then \(x\frac{{\delta u}}{{\delta x}} + y\frac{{\delta u}}{{\delta y}}\) is |
| A. | 2eu |
| B. | e2u |
| C. | 2 |
| D. | 1/2 |
| Answer» D. 1/2 | |
| 197. |
Consider the differential equation, \({{\rm{y}}^2}{\rm{dx}} + \left( {{\rm{x}} - \frac{1}{{\rm{y}}}} \right){\rm{dy}} = 0\). If value of y is 1 when x = 1, then the value of x for which y = 2, is: |
| A. | \(\frac{5}{2} + \frac{1}{{\sqrt {\rm{e}} }}\) |
| B. | \(\frac{3}{2} - \frac{1}{{\sqrt {\rm{e}} }}\) |
| C. | \(\frac{1}{2} + \frac{1}{{\sqrt {\rm{e}} }}\) |
| D. | \(\frac{3}{2} - \sqrt {\rm{e}}\) |
| Answer» C. \(\frac{1}{2} + \frac{1}{{\sqrt {\rm{e}} }}\) | |
| 198. |
f (x, y, z) = x2 + xyz + zFind: fx at (1, 1, 1) |
| A. | 0 |
| B. | 3 |
| C. | 1 |
| D. | -1 |
| Answer» C. 1 | |
| 199. |
\(\mathop {\lim }\limits_{x \to \infty } \sqrt {{x^2} + x - 1} - x\;is\) |
| A. | 0 |
| B. | ∞ |
| C. | 1/2 |
| D. | -∞ |
| Answer» D. -∞ | |
| 200. |
Green's theorem is used to- |
| A. | transform the line integral in xy - plane to a surface integral on the same xy - plane. |
| B. | transform double integrals into triple integral in a region v. |
| C. | transform surface integral into line integral. |
| D. | None of these |
| Answer» B. transform double integrals into triple integral in a region v. | |