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.
| 201. |
Let g(x) = f(x) + f(1 – x) and f'(x) < 0 for all x ∈ (0,1). Then the interval in which g(x) is increasing is |
| A. | (1/2, 1) |
| B. | (0, 1/2) |
| C. | (0, 1/2) ∪ (1/2, 1) |
| D. | none of these |
| Answer» B. (0, 1/2) | |
| 202. |
If \(\int \frac{dx}{{{\left( {{x}^{2}}-2x+10 \right)}^{2}}}=A\left( ta{{n}^{-1}}\left( \frac{x-1}{3} \right)+\frac{f\left( x \right)}{{{x}^{2}}-2x+10} \right)+C\) where C is a constant of integration, then: |
| A. | \(A = \frac{1}{{54}}{\rm{\;and\;}}f\left( x \right) = 3\left( {x - 1} \right)\) |
| B. | \(A = \frac{1}{{81}}{\rm{\;and\;}}f\left( x \right) = 3\left( {x - 1} \right)\) |
| C. | \(A = \frac{1}{{27}}{\rm{\;and\;}}f\left( x \right) = 9\left( {x - 1} \right)\) |
| D. | \(A = \frac{1}{{54}}{\rm{\;and\;}}f\left( x \right) = 9{(x - 1)^2}\) |
| Answer» B. \(A = \frac{1}{{81}}{\rm{\;and\;}}f\left( x \right) = 3\left( {x - 1} \right)\) | |
| 203. |
If \(\begin{array}{l} \overrightarrow a = 2\widehat i - 2\widehat j - \widehat k\\ \end{array}\)and \(\begin{array}{l} \overrightarrow b = 3\widehat i + 2\widehat j + \widehat k\\ \end{array}\)then the projection of \(\begin{array}{l} \overrightarrow b \\ \end{array}\)on \(\begin{array}{l} \overrightarrow a \\ \end{array}\)is |
| A. | \(\begin{array}{l} \frac{1}{3}\\ \end{array}\) |
| B. | 1 |
| C. | \(\begin{array}{l} \frac{1}{{\sqrt {14} }}\\ \end{array}\) |
| D. | \(\begin{array}{l} \frac{{11}}{3}\\ \end{array}\) |
| Answer» B. 1 | |
| 204. |
Evaluate \(\int^1_0 x(1 - x)^n dx\) |
| A. | \(\frac {-1}{(n + 1)(n + 2)}\) |
| B. | \(\frac {1}{(n + 1)(n + 2)}\) |
| C. | (n + 1)(n + 2) |
| D. | (n - 1)(n - 2) |
| Answer» C. (n + 1)(n + 2) | |
| 205. |
If x = A cos 4t + B sin 4t, then \(\dfrac{d^2x}{dt^2}\) is equal to - |
| A. | -16 x |
| B. | 16 x |
| C. | x |
| D. | -x |
| Answer» B. 16 x | |
| 206. |
If sin y + e-x cos y = c, then \(\dfrac{dy}{dx}\)at (1, π) is: |
| A. | -x cos y |
| B. | sin y - x cos y |
| C. | e |
| D. | sin y |
| Answer» D. sin y | |
| 207. |
At x = 0, the function f(x) = |x| has |
| A. | A minimum |
| B. | A maximum |
| C. | A point of inflexion |
| D. | neither a maximum nor minimum |
| Answer» B. A maximum | |
| 208. |
Consider a cube defined byx, y, z ∈ [1, 3]If vector, \(\vec A = 2{x^2}y{\hat a_x} + 3{x^2}{y^2}{\hat a_y}\)∇.A at the center of the cube will be |
| A. | 72 |
| B. | 64 |
| C. | 60 |
| D. | 48 |
| Answer» C. 60 | |
| 209. |
If \(\begin{array}{l} H = {\tan ^{ - 1}}\frac{x}{y}\\ \end{array}\), x = u + v, y = u - v then \(\begin{array}{l} \frac{{\partial H}}{{\partial V}}\\ \end{array}\)is |
| A. | \(\frac{u}{{\mathop u\nolimits^2 + \mathop v\nolimits^2 }}\) |
| B. | \(\frac{{ - v}}{{\mathop u\nolimits^2 + \mathop v\nolimits^2 }}\) |
| C. | \(\frac{u}{{\mathop x\nolimits^2 + \mathop y\nolimits^2 }}\) |
| D. | \(\frac{{ - 2v}}{{\mathop x\nolimits^2 + \mathop y\nolimits^2 }}\) |
| Answer» B. \(\frac{{ - v}}{{\mathop u\nolimits^2 + \mathop v\nolimits^2 }}\) | |
| 210. |
If \({\rm{s}} = \sqrt {{{\rm{t}}^2} + 1} \), then \(\frac{{{{\rm{d}}^2}{\rm{s}}}}{{{\rm{d}}{{\rm{t}}^2}}}\) is equal to |
| A. | \(\frac{1}{{\rm{s}}}\) |
| B. | \(\frac{1}{{{{\rm{s}}^2}}}\) |
| C. | \(\frac{1}{{{{\rm{s}}^3}}}\) |
| D. | \(\frac{1}{{{{\rm{s}}^4}}}\) |
| Answer» D. \(\frac{1}{{{{\rm{s}}^4}}}\) | |
| 211. |
If \(f\left( x \right) = \frac{{2{x^2} - 7x + 3}}{{5{x^2} - 12x - 9}}\) , then \(\begin{array}{*{20}{c}} {limf\left( x \right)}\\ {x \to 3} \end{array}\) will be |
| A. | \(\frac{{ - 1}}{3}\) |
| B. | \(\frac{5}{{18}}\) |
| C. | 0 |
| D. | \(\frac{2}{5}\) |
| Answer» C. 0 | |
| 212. |
Maximum value of (1/x)x is |
| A. | 1 |
| B. | e - e |
| C. | e1/e |
| D. | 1/e |
| Answer» D. 1/e | |
| 213. |
Determine \({\left. {\frac{{dy}}{{dx}}} \right|_{x = 1}}\) wheny = x2 + 2x + 15 |
| A. | 8 |
| B. | 4 |
| C. | 15 |
| D. | 2 |
| Answer» C. 15 | |
| 214. |
Let \(f\left( x \right) = x{e^{ - x}}\). The maximum value of the function in the interval (\(0,\;\infty\)) is |
| A. | \({e^{ - 1}}\) |
| B. | \(e\) |
| C. | \(1\;-\;{e^{ - 1}}\) |
| D. | \(1\; + \;{e^{ - 1}}\) |
| Answer» B. \(e\) | |
| 215. |
Consider the following statements:1. The function f(x) = ln x increases in the interval (0, ∞).2. The function f(x) = tan x increases in the interval \(\rm \left(- \dfrac{\pi}{2}, \dfrac{\pi}{2}\right)\) .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» D. Neither 1 nor 2 | |
| 216. |
Let y = y(x) be the solution of the differential equation, \({\rm{}}{\left( {{{\rm{x}}^2} + 1} \right)^2}\frac{{{\rm{dy}}}}{{{\rm{dx}}}} + 2{\rm{x}}\left( {{{\rm{x}}^2} + 1} \right){\rm{y}} = 1\) is such that y(0) = 0. If \(\sqrt {\rm{a}} {\rm{\;y}}\left( 1 \right) = \frac{{\rm{\pi }}}{{32}},{\rm{\;then\;the}}\) value of ‘a’ is: |
| A. | \(\frac{1}{4}\) |
| B. | \(\frac{1}{2}\) |
| C. | 1 |
| D. | \(\frac{1}{{16}}\) |
| Answer» E. | |
| 217. |
\(\mathop \smallint \limits_0^{\pi /2} {\sin ^2}xdx =\) |
| A. | π/4 |
| B. | π/5 |
| C. | π/2 |
| D. | π/6 |
| Answer» B. π/5 | |
| 218. |
If 2î + 4ĵ - 5k̂ and î + 2ĵ + 3k̂ are two different sides of rhombus. Find the length of diagonals. |
| A. | 7, \(\sqrt{69}\) |
| B. | 6, \(\sqrt{59}\) |
| C. | 5, \(\sqrt{65}\) |
| D. | 8, \(\sqrt{45}\) |
| Answer» B. 6, \(\sqrt{59}\) | |
| 219. |
A condition that x3 + ax2 + bx + c may have no extremum is |
| A. | a2 ≥ 3b |
| B. | b2 < 3b |
| C. | a2 < 3b |
| D. | b2 ≥ 3b |
| Answer» D. b2 ≥ 3b | |
| 220. |
Let f(x) and g(x) be twice differentiable functions on [0, 2] satisfying f’’(x) = g”(x), f’(1) = 4, g’(1) = 6, f(2) = 3 and g(2) = 9. Then what is f(x) – g(x) at x = 4 equal to? |
| A. | -10 |
| B. | -6 |
| C. | -4 |
| D. | 2 |
| Answer» B. -6 | |
| 221. |
First Order Differentiation of any function at a point is |
| A. | Slope of Perpendicular to function at that point |
| B. | Slope of tangent to function at that point |
| C. | Value of function at that point |
| D. | None of these |
| Answer» C. Value of function at that point | |
| 222. |
If a continuously differentiable vector function is the gradient of a scalar function, then its curl is |
| A. | infinite |
| B. | indeterminate |
| C. | unity |
| D. | zero |
| Answer» E. | |
| 223. |
Given the vector field \(A = 5\;{x^2}\left( {\sin \left( {\frac{{\pi x}}{2}} \right)} \right){a_x},\) find div A at x = 1. |
| A. | 5 |
| B. | 20 |
| C. | 10 |
| D. | 15 |
| Answer» D. 15 | |
| 224. |
If \(\displaystyle\int \dfrac{(x - 1)^2}{(x^2 + 1)^2} dx = tan^{-1} x + g(x) + k,\)the g(x) = |
| A. | None of these |
| B. | \(\dfrac{1}{2(x^2+1)}\) |
| C. | \(\dfrac{1}{(x^2+1)}\) |
| D. | \(tan^{-1} \dfrac{x}{2}\) |
| Answer» D. \(tan^{-1} \dfrac{x}{2}\) | |
| 225. |
If \(y = {\tan ^{ - 1}}\left( {\frac{{5 - 2\tan \sqrt x }}{{2 + 5\tan \sqrt x }}} \right)\) then what is \(\frac{{dy}}{{dx}}\) equal to? |
| A. | \( - \frac{1}{{2\sqrt x }}\) |
| B. | 1 |
| C. | -1 |
| D. | \(\frac{1}{{2\sqrt x }}\) |
| Answer» B. 1 | |
| 226. |
Find the point at which the tangent to the curve y = \(\rm \sqrt{4x-3}-1\) has its slope \(\dfrac{2}{3}\). |
| A. | (3, 3) |
| B. | (3, 2) |
| C. | (2, 3) |
| D. | (2, 2) |
| Answer» C. (2, 3) | |
| 227. |
If ∫ cos x cos 2x cos 5x dx = A1 sin 2x + A2 sin 4x + A3 sin 6x + A4 sin 8x + c, then the values of A1, A2, A3, A4 are |
| A. | \(A_1 = \frac 1 2,\;A_2 = \frac 1 4,\; A_3 = \frac 1 6,\; A_4 = \frac 1 8\) |
| B. | \(A_1 = \frac 1 8,\;A_2 = \frac 1 {16},\; A_3 = \frac 1 {24},\; A_4 = \frac 1 {32}\) |
| C. | \(A_1 = \frac 1 6,\;A_2 = \frac 1 {12},\; A_3 = \frac 1 {18},\; A_4 = \frac 1 {24}\) |
| D. | \(A_1 = \frac 1 4,\;A_2 = \frac 1 8,\; A_3 = \frac 1 {12},\; A_4 = \frac 1 {16}\) |
| Answer» C. \(A_1 = \frac 1 6,\;A_2 = \frac 1 {12},\; A_3 = \frac 1 {18},\; A_4 = \frac 1 {24}\) | |
| 228. |
If the radius of the circle changes at the rate of \(\rm-\frac{2}{\pi}\ m/sec\), at what rate does the circle's area change when the radius is 10 m? |
| A. | 40 m2/sec |
| B. | 30 m2/sec |
| C. | -30 m2/sec |
| D. | -40 m2/sec |
| Answer» E. | |
| 229. |
Find the area bounded by the line y = 3 - x, the parabola y = x2 - 9 and x ≥ -4, y ≥ 0. |
| A. | \(\dfrac{7}{2}\) |
| B. | \(\dfrac{11}{2}\) |
| C. | \(\dfrac{9}{2}\) |
| D. | None of these |
| Answer» E. | |
| 230. |
Let y = 3x2 + 2. If x changes from 10 to 10.1, then what is the total change in y? |
| A. | 4.71 |
| B. | 5.23 |
| C. | 6.03 |
| D. | 8.01 |
| Answer» D. 8.01 | |
| 231. |
A vector P is given by \({\rm{\vec P}} = {{\rm{x}}^3}{\rm{y}}{{\rm{\vec a}}_{\rm{x}}} - {{\rm{x}}^2}{{\rm{y}}^2}{{\rm{\vec a}}_{\rm{y}}} - {{\rm{x}}^2}{\rm{yz}}{{\rm{\vec a}}_{\rm{z}}}\) . Which one of the following statements is TRUE? |
| A. | P is solenoidal, but not irrotational |
| B. | P is irrotational, but not solenoidal |
| C. | P is neither solenoidal nor irrotational |
| D. | P is both solenoidal and irrotational |
| Answer» B. P is irrotational, but not solenoidal | |
| 232. |
Find the equations of the normals to the curve 3x2 - y2 = 8, parallel to the line x + 3y = 4 ? |
| A. | x - 3y - 8 = 0 and x + 3y + 8 = 0 |
| B. | x + 3y - 8 = 0 and x + 3y + 8 = 0 |
| C. | x + 3y - 8 = 0 and - x + 3y + 8 = 0 |
| D. | None of these |
| Answer» C. x + 3y - 8 = 0 and - x + 3y + 8 = 0 | |
| 233. |
Integral of sec2 x with respect to sec x is |
| A. | tan x + C |
| B. | sec x + C |
| C. | \(\rm\frac{{ta{n^3}x}}{{3 }} + c\) |
| D. | \(\rm\frac{{sec{^3}x}}{{3 }} + c\) |
| Answer» E. | |
| 234. |
A cylindrical jar without a lid has to be constructed using a given surface area of a metal sheet. If the capacity of the jar is to be maximum, then the diameter of the jar must be k times the height of the jar. The value of k is |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | 4 |
| Answer» C. 3 | |
| 235. |
In the Taylor series expansion of ex + sin x about the point x = π, the coefficient of (x – π)2 is |
| A. | eπ |
| B. | 0.5 eπ |
| C. | eπ+1 |
| D. | eπ-1 |
| Answer» C. eπ+1 | |
| 236. |
\(\lim x \to \;\infty \;{x^{\frac{1}{x}}}\) is |
| A. | ∞ |
| B. | 0 |
| C. | 1 |
| D. | Not defined |
| Answer» D. Not defined | |
| 237. |
Minimum value of 27cos x 81sin x |
| A. | -5 |
| B. | 1/5 |
| C. | 1/243 |
| D. | 1/27 |
| Answer» D. 1/27 | |
| 238. |
If the interval of differencing is 1, then the value of Δ sin 4 x will be |
| A. | 2(sin 2) (cos 2 (2x + 1)) |
| B. | 2(sin 2) (sin 2 (x + 1)) |
| C. | 2(cos 4 ) (sin 2 (x + 2)) |
| D. | 2(sin 2) (cos 2 (2x - 1)) |
| Answer» B. 2(sin 2) (sin 2 (x + 1)) | |
| 239. |
Consider the functionsI. e-xII. x2 – sin xIII. \(\sqrt {{x^3} + 1} \)Which of the above functions is/are increasing everywhere in [0, 1]? |
| A. | III only |
| B. | II only |
| C. | II and III only |
| D. | I and III only |
| Answer» B. II only | |
| 240. |
In which of the following cases the divergence of the electric field is zero |
| A. | When electric field flows uniform through open tubes |
| B. | When electric field is released from the tube |
| C. | When electric field is within the closed tube |
| D. | None of above |
| Answer» D. None of above | |
| 241. |
If eθϕ = c + 4θϕ, where c is an arbitrary constant and ϕ is a function of θ, then what is ϕ dθ equal to? |
| A. | θ dϕ |
| B. | - θdϕ |
| C. | 4θ dϕ |
| D. | -4θ dϕ |
| Answer» C. 4θ dϕ | |
| 242. |
\(\displaystyle\lim_{x \rightarrow 0} \dfrac{\sqrt{1+x}-\sqrt{1-x}}{x}\) is given by |
| A. | 0 |
| B. | –1 |
| C. | 1 |
| D. | \(\frac{1}{2}\) |
| Answer» D. \(\frac{1}{2}\) | |
| 243. |
If \(\rm \displaystyle\int\dfrac{xe^x}{\sqrt{1+e^x}}dx=f(x)\sqrt{1+e^x}- \rm 2 \log \frac{\sqrt{1+e^x}-1}{\sqrt{1+e^x}+1}+C\), then f(x) is |
| A. | 2x - 1 |
| B. | 2x - 4 |
| C. | x + 4 |
| D. | x - 4 |
| Answer» C. x + 4 | |
| 244. |
If \(\mathop \smallint \nolimits_0^{\frac{\pi }{2}} \left( {\frac{{cotx}}{{cotx + cosecx}}} \right)dx = m\left( {\pi + n} \right)\), then m⋅n is equal to: |
| A. | \(-\frac{1}{2}\) |
| B. | 1 |
| C. | 1/2 |
| D. | -1 |
| Answer» E. | |
| 245. |
Consider a rotating disk cam and a translating roller follower with zero offset. Which one of the following pitch curves, parameterized by t, lying in the interval 0 to 2π, is associated with the maximum translation of the follower during one full rotation of the cam rotating about the center at (x, y) = (0, 0)? |
| A. | x(t) = cos t, y(t) = sin t |
| B. | x(t) = cos t, y(t) = 2 sin t |
| C. | x(t) = ½ + cos t, y(t) = 2 sin t |
| D. | x(t) = ½ + cost t, y(t) = sin t |
| Answer» D. x(t) = ½ + cost t, y(t) = sin t | |
| 246. |
If \(\smallint {x^5}{e^{ - {x^2}}}dx = g\left( x \right){e^{ - {x^2}}} + c\), where c is a constant of integration, then g(-1) is equal to: |
| A. | -1 |
| B. | 1 |
| C. | \( - \frac{5}{2}\) |
| D. | \( - \frac{1}{2}\) |
| Answer» D. \( - \frac{1}{2}\) | |
| 247. |
If x = et cost and y = et sint, then what is \(\dfrac{dx}{dy}\) at t = 0 equal to? |
| A. | 0 |
| B. | 1 |
| C. | 2e |
| D. | -1 |
| Answer» C. 2e | |
| 248. |
Let the slope of the curve y = cos-1 (sin x) be tan θ, then the value of θ in the interval (0, π) is |
| A. | π/6 |
| B. | 3π/4 |
| C. | π/4 |
| D. | π/2 |
| Answer» C. π/4 | |
| 249. |
Let n ≥ 2 be a natural number and \(0 < \theta < \frac{\pi }{2}.{\rm{\;Then\;}}\smallint \left( {\frac{{{{\left( {{\rm{si}}{{\rm{n}}^n}\theta - {\rm{sin}}\theta } \right)}^{\frac{1}{n}}}{\rm{cos}}\theta }}{{{\rm{si}}{{\rm{n}}^{n + 1}}\theta }}} \right)d\theta\) is equal to:(Where C is a constant of integration) |
| A. | \(\frac{{\rm{n}}}{{{{\rm{n}}^2} - 1}}{\left( {1 - \frac{1}{{{\rm{si}}{{\rm{n}}^{{\rm{n}} - 1}}\theta }}} \right)^{\frac{{{\rm{n}} + 1}}{{\rm{n}}}}} + {\rm{C}}\) |
| B. | \(\frac{{\rm{n}}}{{{{\rm{n}}^2} + 1}}{\left( {1 - \frac{1}{{{\rm{si}}{{\rm{n}}^{{\rm{n}} - 1}}{\rm{\theta }}}}} \right)^{\frac{{{\rm{n}} + 1}}{{\rm{n}}}}} + {\rm{C}}\) |
| C. | \(\frac{{\rm{n}}}{{{{\rm{n}}^2} - 1}}{\left( {1 + \frac{1}{{{\rm{si}}{{\rm{n}}^{{\rm{n}} - 1}}{\rm{\theta }}}}} \right)^{\frac{{{\rm{n}} + 1}}{{\rm{n}}}}} + {\rm{C}}\) |
| D. | \(\frac{{\rm{n}}}{{{{\rm{n}}^2} - 1}}{\left( {1 - \frac{1}{{{\rm{si}}{{\rm{n}}^{{\rm{n}} + 1}}\theta }}} \right)^{\frac{{{\rm{n}} + 1}}{{\rm{n}}}}} + {\rm{C}}\) |
| Answer» B. \(\frac{{\rm{n}}}{{{{\rm{n}}^2} + 1}}{\left( {1 - \frac{1}{{{\rm{si}}{{\rm{n}}^{{\rm{n}} - 1}}{\rm{\theta }}}}} \right)^{\frac{{{\rm{n}} + 1}}{{\rm{n}}}}} + {\rm{C}}\) | |
| 250. |
Evaluate the following integral:\(\smallint \cos {\rm{x}}{{\rm{e}}^{\sin {\rm{x}}}}{\rm{dx}}\) |
| A. | \({{\rm{e}}^{\sin {\rm{x}}}} + {\rm{C}}\) |
| B. | \({{\rm{e}}^{\cos {\rm{x}}}} + {\rm{C}}\) |
| C. | \(\frac{{\cos {\rm{x}}}}{{{{\rm{e}}^{\sin {\rm{x}}}}}} + {\rm{C}}\) |
| D. | \(\frac{{\sin {\rm{x}}}}{{{{\rm{e}}^{\cos {\rm{x}}}}}} + {\rm{C}}\) |
| Answer» B. \({{\rm{e}}^{\cos {\rm{x}}}} + {\rm{C}}\) | |