MCQOPTIONS
Saved Bookmarks
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.
| 301. |
Consider the function f(x) = |x| in the interval -1 ≤ x ≤ 1. At the point x =0, f(x) is |
| A. | Continuous and differentiable |
| B. | Non – continuous and differentiable |
| C. | Continuous and non – differentiable |
| D. | Neither continuous nor differentiable |
| Answer» D. Neither continuous nor differentiable | |
| 302. |
According to the Mean Value Theorem, for a continuous function f(x) in the interval [a, b], there exists a value ξ in this interval such that \(\mathop \smallint \limits_a^b f\left( x \right)dx =\) |
| A. | f (ξ) (b - a) |
| B. | f (b) (ξ - a) |
| C. | f (a) (b - ξ) |
| D. | 0 |
| Answer» B. f (b) (ξ - a) | |
| 303. |
If \(f(x)=\sin^5 x + \sin^3 x\) and \(g(x)=\cos^6 x + \sin^3 x\), then the value of \(\displaystyle\int_0^{\pi/2} [f(x) + f(-x)] [g(x) + g(-x)]dx\) is |
| A. | 0 |
| B. | > 1 |
| C. | 0 and 1 |
| D. | less than 0 |
| Answer» B. > 1 | |
| 304. |
Find the differentiation of x4 + y4 = 0 |
| A. | \( -\frac{{{x^3}}}{{{y^3}}}\) |
| B. | \( -\frac{{{x^3}}}{{{y^4}}}\) |
| C. | \( -\frac{{{x^4}}}{{{y^3}}}\) |
| D. | \( \frac{{{x^3}}}{{{y^3}}}\) |
| Answer» B. \( -\frac{{{x^3}}}{{{y^4}}}\) | |
| 305. |
A normal to the curve x2 = 4y passes through the point (1, 2). The distance of the origin from the normal is |
| A. | \(\sqrt{2}\) |
| B. | \(2\sqrt{2}\) |
| C. | \(\dfrac{1}{\sqrt{2}}\) |
| D. | \(\dfrac{3}{\sqrt{2}}\) |
| Answer» E. | |
| 306. |
If \(\rm \int_0^a \left[f(x)+f(-x)\right]dx=\int_{-a}^{\ \ a} g(x)\ dx\), then what is g(x) equal to? |
| A. | f(x) |
| B. | f(-x) + f(x) |
| C. | -f(x) |
| D. | None of the above. |
| Answer» C. -f(x) | |
| 307. |
A function f(x) is continuous in the interval [0, 2]. It is known that f(0) = f(2) =-1 and f(1) = 1. Which one of the following statements must be true? |
| A. | There exists a y in the interval (0, 1) such that f(y) = f(y + 1) |
| B. | For every y in the interval (0, 1), f(y) = f(2 – y) |
| C. | The maximum value of the function in the interval (0, 2) is 1 |
| D. | There exists a y in the interval (0, 1) such that f(y) = -f(2 – y) |
| Answer» B. For every y in the interval (0, 1), f(y) = f(2 – y) | |
| 308. |
Evaluate \(\mathop \smallint \limits_0^1 \mathop \smallint \limits_0^{\sqrt {1 + {x^2}} } \frac{{dx.dy}}{{\left( {1 + {x^2} + {y^2}} \right)}}\) |
| A. | \(\frac{\pi }{2}\left[ {\log \left( {1 + \sqrt 2 } \right)} \right]\) |
| B. | \(\frac{\pi }{4}\left[ {\log \left( {1 + \sqrt 2 } \right)} \right]\) |
| C. | \(\frac{\pi }{2}\left[ {\log \left( {1 - \sqrt 2 } \right)} \right]\) |
| D. | \(\frac{\pi }{4}\left[ {\log \left( {1 - \sqrt 2 } \right)} \right]\) |
| Answer» C. \(\frac{\pi }{2}\left[ {\log \left( {1 - \sqrt 2 } \right)} \right]\) | |
| 309. |
If f (x, y) = x3y – xy3, then what is the value of \(\left[ {\frac{1}{{\frac{{{\rm{df}}}}{{{\rm{dx}}}}}}{\rm{\;}} + \frac{1}{{\frac{{{\rm{df}}}}{{{\rm{dy}}}}}}} \right]\) x = 1, y = 2? |
| A. | 13/18 |
| B. | -9/18 |
| C. | 9/22 |
| D. | -13/22 |
| Answer» E. | |
| 310. |
\(\mathop \smallint \limits_0^{\pi /2} \log \sin x\;dx\) equals |
| A. | \( - \frac{\pi }{2}\log 2\) |
| B. | \(\frac{\pi }{2}\log 2\) |
| C. | -π log 2 |
| D. | π log 2 |
| Answer» B. \(\frac{\pi }{2}\log 2\) | |
| 311. |
Consider the function \(y = x^2 + \dfrac{250}{x}\) At x = 5, the function attains. |
| A. | Maximum |
| B. | Minimum |
| C. | 0 |
| D. | 1 |
| Answer» C. 0 | |
| 312. |
In Fourier series \(f\left( x \right) = {a_0} + \mathop \sum \limits_{n - 1}^\infty \left\{ {{a_n} + \cos \left( {nx} \right) + {b_n}\sin \left( {nx} \right)} \right\}\) |
| A. | \({a_0} = \frac{1}{{2\pi }}\mathop \smallint \limits_{ - \pi }^\pi f\left( x \right)dx\) |
| B. | \({a_0} = \mathop \smallint \limits_{ - \pi }^\pi f\left( x \right)dx\) |
| C. | \({a_0} = \frac{1}{{2\pi }}\mathop \smallint \limits_{ - \pi }^\pi f\left( x \right)\sin \left( x \right)dx\) |
| D. | None of these |
| Answer» B. \({a_0} = \mathop \smallint \limits_{ - \pi }^\pi f\left( x \right)dx\) | |
| 313. |
Given the following statements about a function f: R → R, select the right option:P: If f(x) is continuous at x = x0, then it is also differentiable at x = x0.Q: If f(x) is continuous at x = x0, then it may not be differentiable at x = x0.R: If f(x) is differentiable at x = x0, then it is also continuous at x = x0. |
| A. | P is true, Q is false, R is false |
| B. | P is false, Q is true, R is true |
| C. | P is false, Q is true, R is false |
| D. | P is true, Q is false, R is true |
| Answer» C. P is false, Q is true, R is false | |
| 314. |
If a continuous function f(x) does not have a root in the interval [a, b], then which one of the following statements is TRUE? |
| A. | \(f\left( a \right).f\left( b \right) = 0\) |
| B. | \(f\left( a \right).f\left( b \right) < 0\) |
| C. | \(f\left( a \right).f\left( b \right) > 0\) |
| D. | \(f\left( a \right)/f\left( b \right) \le 0\) |
| Answer» D. \(f\left( a \right)/f\left( b \right) \le 0\) | |
| 315. |
If \(I_1 = \displaystyle\int_e^{e^2} \dfrac{dx}{\log x}\)and \(I_2 = \displaystyle\int_1^2 \dfrac{e^x}{x} dx\) then |
| A. | I1 - I2 = 0 |
| B. | I2 = 2I1 |
| C. | I1 = 2I2 |
| D. | I1 + I2 = 0 |
| Answer» B. I2 = 2I1 | |
| 316. |
A given quantity of metal is to be cast into a half cylinder (i.e. with a rectangular base and semicircular ends). If the total surface area is to be minimum, then the ratio of the height of the half cylinder to the diameter of the semicircular ends is |
| A. | π : (π + 2) |
| B. | (π + 2) : π |
| C. | 1 : 1 |
| D. | None of the above |
| Answer» B. (π + 2) : π | |
| 317. |
Consider the following statements:1. The function f(x) = x2 + 2 cos x is increasing in the interval (0, π)2. The function \({\rm{f}}\left( {\rm{x}} \right) = {\rm{In\;}}\left( {\sqrt {1 + {{\rm{x}}^2}} - {\rm{x}}} \right)\) is decreasing in the interval (-∞, ∞)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 | |
| 318. |
For the curve xy3 - yx3 = 6, the slope of the tangent line at the point (1, -1) is: |
| A. | 1 / 2 |
| B. | -1 |
| C. | 2 |
| D. | 1 |
| Answer» C. 2 | |
| 319. |
Derivative of cos x with respect to x is |
| A. | – sin x |
| B. | sin x |
| C. | sec x |
| D. | tan x |
| Answer» B. sin x | |
| 320. |
In a vector field; Divergence of the gradient is |
| A. | curl |
| B. | unity |
| C. | zero |
| D. | Laplacian |
| Answer» E. | |
| 321. |
Let \(f\left( x \right) = \;\left\{ {\begin{array}{*{20}{c}} { - π }&{if\;}&{ - π < x \le π } \end{array}} \right.\)< span>be a periodic function of period 2π. The coefficient of sin 5x in the Fourier series expansion of f(x) in the interval [-π, π] is |
| A. | \(\frac{4}{5}\) |
| B. | \(\frac{5}{4}\) |
| C. | \(\frac{4}{3}\) |
| D. | \(\frac{3}{4}\) |
| Answer» B. \(\frac{5}{4}\) | |
| 322. |
If \(z = {e^x}\sin y,x = {\log _e}t,y = {t^2}\) then \(\frac{{dz}}{{dt}}\) is given by the expression- |
| A. | \(\frac{{{e^x}}}{t}(\sin y - 2{t^2}\cos y)\) |
| B. | \(\frac{{{e^x}}}{t}\left( {\sin y + 2{t^2}\cos y} \right)\) |
| C. | \(\frac{{{e^x}}}{t}\left( {\cos y + 2{t^2}\sin y} \right)\) |
| D. | \(\frac{{{e^x}}}{t}\left( {\cos y - 2{t^2}\sin y} \right)\) |
| Answer» C. \(\frac{{{e^x}}}{t}\left( {\cos y + 2{t^2}\sin y} \right)\) | |
| 323. |
Find dy/dx if y = ex sin x |
| A. | \({e^x}\sin x\) |
| B. | \({e^x}\cos x\) |
| C. | sin x |
| D. | \({e^x}\cos x + \sin x\;{e^x}\) |
| Answer» E. | |
| 324. |
By Lagrange’s mean value theorem which of the following statement is true:a) If a curve has a tangent at each of its points then there exists at least one-point C on this curve, the tangent at which is parallel to chord ABb) If f’(x) = 0 in the interval then f(x) has same value for every value of x in (a, b) |
| A. | (a) alone is true |
| B. | (b) alone is true |
| C. | Both (a) and (b) are true |
| D. | Neither (a) nor (b) is true |
| Answer» B. (b) alone is true | |
| 325. |
Consider the following statements :Stokes' theorem is valid irrespective of 1. Shape of closed curve C2. Type of vector A3. Type of coordinate system4. Whether the surface is closed or openWhich of the above statements are correct? |
| A. | 1, 2 and 4 |
| B. | 1, 3 and 4 |
| C. | 2, 3 and 4 |
| D. | 1, 2 and 3 |
| Answer» E. | |
| 326. |
\(\mathop {\lim }\limits_{x \to \infty } {\left( {1 + \frac{1}{x}} \right)^{2x}}\) is equal to |
| A. | e-2 |
| B. | e |
| C. | 1 |
| D. | e2 |
| Answer» E. | |
| 327. |
If \(u = {x^4} + y{x^3} + 3{x^2}.{y^2} + 3{x^2}y\), Find ∂u/∂x |
| A. | \(4{x^3} + 6x{y^2} + 6xy\) |
| B. | \(4{{\rm{x}}^3} + 6{{\rm{x}}^2}{\rm{y}} + 3{\rm{x}}{{\rm{y}}^2} + 6{\rm{xy}}\) |
| C. | \(4{{\rm{x}}^3} + 3{{\rm{x}}^2}{\rm{y}} + 6{\rm{x}}{{\rm{y}}^2} + 6{\rm{xy}}\) |
| D. | \(4{{\rm{x}}^3} + 6{\rm{x}}{{\rm{y}}^2} + 6{\rm{xy}}\) |
| Answer» D. \(4{{\rm{x}}^3} + 6{\rm{x}}{{\rm{y}}^2} + 6{\rm{xy}}\) | |
| 328. |
For the vector a̅ with initial point P(4, 0, 2) and the terminal point Q(6, -1, 2), the value of |a| will be: |
| A. | √3 |
| B. | √7 |
| C. | √5 |
| D. | √2 |
| Answer» D. √2 | |
| 329. |
For a given relation \(\sqrt {1 - {x^2}} + \sqrt {1 - {y^2}} = P\left( {x - y} \right)\), where P is a constant the value of \(\frac{{dy}}{{dx}}\) at point (0, 0) is |
| A. | -1 |
| B. | 0 |
| C. | 1 |
| D. | -2 |
| Answer» D. -2 | |
| 330. |
Let \(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}}{ - 1,}&{ - 2 \le x < 0}\\{{x^2} - 1,}&{0 \le x \le 2}\end{array}} \right.\) and g(x) = |f(x)| + f(|x|). Then in the interval (-2, 2), g is: |
| A. | Differentiable at all points |
| B. | Not continuous |
| C. | Not differentiable at two points |
| D. | Not differentiable at one point |
| Answer» E. | |
| 331. |
Match List - I with List - II and select the correct answer using the code given below the lists: List - I(Function) List - II(Maximum value)A.sin x + cos x1.\(\sqrt {10} \)B.3 sin x + 4 cos x2.\(\sqrt {2} \)C.2 sin x + cos x3. 5DSin x + 3 cos x4.\(\sqrt {5} \) |
| A. | A - 2, B - 3, C - 1, D - 4 |
| B. | A - 2, B - 3, C - 4, D - 1 |
| C. | A - 3, B - 2, C - 1, D - 4 |
| D. | A - 3, B - 2, C - 4, D - 1 |
| Answer» C. A - 3, B - 2, C - 1, D - 4 | |
| 332. |
Find the equation of tangent to the curve y ⋅ (x - 2) ⋅ (x - 3) - x + 7 = 0 at the point where it cuts the x - axis ? |
| A. | x + 20y + 7 = 0 |
| B. | x - 20y + 7 = 0 |
| C. | x + 20y - 7 = 0 |
| D. | x - 20y - 7 = 0 |
| Answer» E. | |
| 333. |
At x = 0, the function f(x) = x3 + 1 has |
| A. | A maximum value |
| B. | A minimum value |
| C. | A singularity |
| D. | A point of inflection |
| Answer» E. | |
| 334. |
Let f = yx. What is \(\frac{{{\partial ^2}f}}{{\partial x\partial y}}\) at x = 2, y = 1? |
| A. | 0 |
| B. | ln 2 |
| C. | 1 |
| D. | \(\frac{1}{{\ln 2}}\) |
| Answer» D. \(\frac{1}{{\ln 2}}\) | |
| 335. |
Equation of a line normal to f(x) = \({\left( {{\rm{x\;}} + {\rm{\;}}4} \right)^{\frac{1}{2}}} + 1\;\) at Q (0, 3) is |
| A. | y = 3 – 4x |
| B. | y = 3 + 4x |
| C. | 4y = 12 - x |
| D. | 4y = 12 + x |
| Answer» B. y = 3 + 4x | |
| 336. |
If a curve passes through the point (1, -2) and has slope of the tangent at any point(x, y)on it as \(\frac{{{x^2} - 2y}}{x}\), then the curve also passes through the point |
| A. | \(\left( {\sqrt 3 ,0} \right)\) |
| B. | (-1, 2) |
| C. | \(\left( { - \sqrt 2 ,1} \right)\) |
| D. | (3, 0) |
| Answer» B. (-1, 2) | |
| 337. |
\(\frac{d}{{dx}}cosec\;hx\) |
| A. | cosechx cothx |
| B. | – cosechx cothx |
| C. | – sechx cothx |
| D. | sechx cothx |
| Answer» C. – sechx cothx | |
| 338. |
In spherical coordinates, a vector field is given by:\(A = \frac{5}{{{r^2}}}\sin \theta {a_r} + r\cot \theta {a_\theta } + r\sin \theta \;cos\varphi \;{a_\varphi }\)Find div A. |
| A. | 1 - sin φ |
| B. | 1 + sin φ |
| C. | - 1 - sin φ |
| D. | 1 - sin θ |
| Answer» D. 1 - sin θ | |
| 339. |
If \({\rm{\vec F}} = {\rm{x\;\vec i}} + {\rm{y\;\vec j}} + {\rm{z\;\vec k}}\) and s is the closed surface of x2 + y2 + z2 = a2. Then \(\mathop \int\!\!\!\int \nolimits_{\rm{s}}^{} {\rm{\vec F}} \cdot {\rm{\hat n\;ds}}\) is |
| A. | \(\frac{4}{3}{\rm{\pi }}{{\rm{a}}^3}\) |
| B. | πa3 |
| C. | 4πa3 |
| D. | \(\frac{1}{3}{\rm{\pi }}{{\rm{a}}^3}\) |
| Answer» D. \(\frac{1}{3}{\rm{\pi }}{{\rm{a}}^3}\) | |
| 340. |
\(\int\limits_0^{\pi /2} {({{\cos }^3}x)dx = } \) |
| A. | 3/2 |
| B. | 2/3 |
| C. | 8/9 |
| D. | 8/13 |
| Answer» C. 8/9 | |
| 341. |
If \(\left| {\overrightarrow a } \right| = 5\), \(\left| {\overrightarrow a - \overrightarrow b } \right| = 8\) and \(\left| {\overrightarrow a + \overrightarrow b } \right| = 10\), then the value of \(\left| {\overrightarrow b } \right|\) is: |
| A. | 1 |
| B. | √57 |
| C. | 3 |
| D. | √42 |
| Answer» C. 3 | |
| 342. |
Let y = y(x) be the solution of the differential equation, \(\frac{{dy}}{{dx}} + y\;tan\;x = 2x + {x^2}\;tan\;x,\;\)\(x \in \left( { - \frac{\pi }{2},\frac{\pi }{2}} \right)\), such that y(0) = 1. Then: |
| A. | \(y\left( {\frac{\pi }{4}} \right) + y\left( { - \frac{\pi }{4}} \right) = \frac{{{\pi ^2}}}{2} + 2\) |
| B. | \(y'\left( {\frac{\pi }{4}} \right) + y'\left( { - \frac{\pi }{4}} \right) = - \sqrt 2 \) |
| C. | \(y\left( {\frac{\pi }{4}} \right) - y\left( { - \frac{\pi }{4}} \right) = \sqrt 2 \) |
| D. | \(y'\left( {\frac{\pi }{4}} \right) - y'\left( { - \frac{\pi }{4}} \right) = \pi - \sqrt 2 \) |
| Answer» E. | |
| 343. |
For the function \(f(x)=\left\lbrace \begin{matrix} -2, & -\pi < x < 0 \\\ 2, & 0 < x < \pi \end{matrix}\right.\) The value of \(a_n\) in the Fourier series expansion of f(x) is |
| A. | 2 |
| B. | 4 |
| C. | 0 |
| D. | -2 |
| Answer» D. -2 | |
| 344. |
Find f'(2) if \(f\left( t \right) = \frac{{{t^2} - 1}}{{{t^2} + t - 2}}\) |
| A. | 1/4 |
| B. | 1/12 |
| C. | 1/16 |
| D. | 4 |
| Answer» D. 4 | |
| 345. |
\(\mathop {\lim }\limits_{x \to 1 \\y \to2}(x^2 + 2y)\) is equal to |
| A. | 0 |
| B. | 3 |
| C. | 5 |
| D. | 6 |
| Answer» D. 6 | |
| 346. |
Find the distance from C to D if the coordinates are given as C (-3, 2, 1) and D (r = 5, θ = 20°, Φ = -70°). |
| A. | 5.99 unit |
| B. | 9.07 unit |
| C. | 7.90 unit |
| D. | 6.29 unit |
| Answer» E. | |
| 347. |
if \(y = tan^{-1}(\frac{\sqrt{1+x^2}-1}{x})\) then |
| A. | y’(0) = 1 |
| B. | y’(0) = 1/2 |
| C. | y’(0) = 0 |
| D. | does not exist |
| Answer» C. y’(0) = 0 | |
| 348. |
If f(x) and g(x) are continuous functions satisfying f(x) = f(a – x) and g(x) + g(a – x) = 2, then what is \(\mathop \smallint \nolimits_0^{\rm{a}} {\rm{f}}\left( {\rm{x}} \right){\rm{g}}\left( {\rm{x}} \right){\rm{dx}}\) equal to? |
| A. | \(\mathop \smallint \nolimits_0^{\rm{a}} {\rm{g}}\left( {\rm{x}} \right){\rm{dx}}\) |
| B. | \(\mathop \smallint \nolimits_0^{\rm{a}} {\rm{f}}\left( {\rm{x}} \right){\rm{dx}}\) |
| C. | \(2\mathop \smallint \nolimits_0^{\rm{a}} {\rm{f}}\left( {\rm{x}} \right){\rm{dx}}\) |
| D. | 0 |
| Answer» C. \(2\mathop \smallint \nolimits_0^{\rm{a}} {\rm{f}}\left( {\rm{x}} \right){\rm{dx}}\) | |
| 349. |
Find the net area between y = sin x and the x-axis between the values x = 0 and x = 2π. |
| A. | 4 |
| B. | 3 |
| C. | 2 |
| D. | 0 |
| Answer» E. | |
| 350. |
Divergence theorem is applicable for ______. |
| A. | static field only |
| B. | time varying fields only |
| C. | both static and time varying fields |
| D. | electric fields only |
| Answer» D. electric fields only | |