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.
| 351. |
Let ar, aϕ, and az be unit vectors along r, ϕ and z directions, respectively in the cylindrical coordinate system. For the electric flux density given by D = (ar 15 + aϕ 2r - az 3rz) Coulomb/m2, the total electric flux, in Coulomb, emanating from the volume enclosed by a solid cylinder of radius 3 m and height 5 m oriented along the z-axis with its base at the origin is: |
| A. | 54 π |
| B. | 90 π |
| C. | 108 π |
| D. | 180 π |
| Answer» E. | |
| 352. |
Consider the following statements:1. f(x) = In x is an increasing function on (0, ∞).2. f(x) = ex – x (In x) is an increasing function on (1, ∞)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 | |
| 353. |
If \({\rm{f}}\left( {\rm{x}} \right) = {\rm{x}}\left( {\sqrt {\rm{x}} - \sqrt {{\rm{x}} + 1} } \right)\), then f(x) is |
| A. | continuous but not differentiable at x = 0 |
| B. | differentiable at x = 0 |
| C. | not continuous at x = 0 |
| D. | None of the above |
| Answer» C. not continuous at x = 0 | |
| 354. |
If xy = a2 and S = b2x + c2y, Where a, b and c are constants, then the minimum value of S is: |
| A. | abc |
| B. | 2 abc |
| C. | 3 abc |
| D. | a + b + c |
| Answer» C. 3 abc | |
| 355. |
Differentiate x-ln x with respect to \(\rm e^{x^{2}}\) |
| A. | \(\rm -x^{-\ln x}(\ln x)\over x^2e^{x^{2}}\) |
| B. | \(\rm -x^{-\ln x}(\ln x)\over xe^{x^{2}}\) |
| C. | \(\rm -2x^{-\ln x}(\ln x)\over x^2e^{x^{2}}\) |
| D. | \(\rm -2x^{-\ln x}(\ln x)\over xe^{x^{2}}\) |
| Answer» B. \(\rm -x^{-\ln x}(\ln x)\over xe^{x^{2}}\) | |
| 356. |
If \(y = \sqrt {x + \sqrt {x + \sqrt {x + ... \infty } } } \), then \(\frac{{dy}}{{dx}}\) is |
| A. | 1 |
| B. | \(\frac{1}{{xy}}\) |
| C. | \(\frac{1}{{2y - x}}\) |
| D. | \(\frac{1}{{2y - 1}}\) |
| Answer» E. | |
| 357. |
\(\mathop {\lim }\limits_{x \to 0} \frac{{{e^x} - \left( {1 + x + \frac{{{x^2}}}{2}} \right)}}{{{x^3}}} =\) |
| A. | 0 |
| B. | \(\frac{1}{6}\) |
| C. | \(\frac{1}{3}\) |
| D. | 1 |
| Answer» C. \(\frac{1}{3}\) | |
| 358. |
Let x be a continuous variable defined over the interval (-∞, ∞) and \(f\left( x \right) = {e^{ - x - {e^{ - x}}}}\). The integral \(g\left( x \right) = \smallint f\left( x \right)dx\) is equal to |
| A. | \({e^{{e^{ - x}}}}\) |
| B. | \({e^{ - {e^{ - x}}}}\) |
| C. | \({e^{ - {e^x}}}\) |
| D. | e-x |
| Answer» C. \({e^{ - {e^x}}}\) | |
| 359. |
\(\displaystyle\int\left(\dfrac{1}{\sin^2 x} + \dfrac{1}{\cos^2 x}\right)dx= \ ?\) |
| A. | tan x + cot x + C |
| B. | tan x - cot x + C |
| C. | (tan x + cot x)2 + C |
| D. | (tan x - cot x)2 + C |
| Answer» C. (tan x + cot x)2 + C | |
| 360. |
Let X = (3, 2, -1), Y = (2, 4, 1), Z = (4, 0, -3) and W = (10, 4, -5) be vector in R3 in real vector space. Which one of the following is correct? |
| A. | 2X + Z = W, Y + Z = W |
| B. | 2X – Y = Z, Y + 2Z = W |
| C. | X + Z = W, 2X + Y = Z |
| D. | 2Z + Y = W, X – Y = Z |
| Answer» C. X + Z = W, 2X + Y = Z | |
| 361. |
Find \(\mathop {\lim }\limits_{x \to \infty } \left( {{x^{\frac{1}{x}}}} \right)\) |
| A. | 1 |
| B. | ∞ |
| C. | 0 |
| D. | -1 |
| Answer» B. ∞ | |
| 362. |
If the curve y = a√x + bx, passes through the point (1, 2) and the area bounded by the curve, line x = 4 and x-axis is 8 sq. unit, then : |
| A. | a = 3, b = -1 |
| B. | a = 3, b = 1 |
| C. | a = -3, b = 1 |
| D. | a = -3, b = -1 |
| Answer» B. a = 3, b = 1 | |
| 363. |
If = 3xzî + 2xyĵ - yz2 k̂ then div \(\rm \vec v\) is |
| A. | 3x + 2y + 2z |
| B. | 2x + 3y + 2z |
| C. | x + 2y + z |
| D. | 3z + 2x - 2yz |
| Answer» E. | |
| 364. |
If u = eax sin bx and v = eax cos bx, then what is \({\rm{u}}\frac{{{\rm{du}}}}{{{\rm{dx}}}} + {\rm{v}}\frac{{{\rm{dv}}}}{{{\rm{dx}}}}\) equal to? |
| A. | a e2ax |
| B. | (a2 + b2)eax |
| C. | ab e2ax |
| D. | (a + b)eax |
| Answer» B. (a2 + b2)eax | |
| 365. |
For any quadratic function px2 + qx + r the value of θ in Lagrange’s theorem is always: |
| A. | ½ (whatever p, q, r, a and h may be) |
| B. | \(\frac{a+h}{2}\) (whatever p, q, r, a and h may be) |
| C. | 0 (whatever p, q, r, a and h may be) |
| D. | \(\frac{a-h}{2}\) (whatever p, q, r, a and h may be) |
| Answer» B. \(\frac{a+h}{2}\) (whatever p, q, r, a and h may be) | |
| 366. |
Let \({a_n} = \int_0^{\pi /4} {{{\tan }^n{{x}}}} dx\). Then a2 + a4, a3 + a5, a4 + a6 are in |
| A. | A. P. |
| B. | G. P. |
| C. | H. P. |
| D. | None of these |
| Answer» D. None of these | |
| 367. |
log 2 = x, log 3 = y, then log 6 is |
| A. | x - y |
| B. | xy |
| C. | x + y |
| D. | x/y |
| Answer» D. x/y | |
| 368. |
If the vectors \(\overrightarrow a = 2\widehat i - 3\widehat j - \widehat k\) and \(\overrightarrow b = \widehat i + 4\widehat j - 2\widehat k\) represent the two sides of any triangle, then the area of that triangle is: |
| A. | \(\frac{1}{2}\sqrt {232} \) square units |
| B. | \(\sqrt {234} \) square units |
| C. | \(\sqrt {250} \) square units |
| D. | \(\frac{1}{2}\sqrt {230} \) square units |
| Answer» E. | |
| 369. |
As \(\rm x\) varies from \(\rm −1\ to \ +3\), which one of the following describes the behaviour of the function \(\rm f(x) = x^3 – 3x^2 + 1\)? |
| A. | \(\rm f(x)\) increases monotonically. |
| B. | \(\rm f(x)\) increases, then decreases and increases again. |
| C. | \(\rm f(x)\) decreases, then increases and decreases again. |
| D. | \(\rm f(x)\) increases and then decreases. |
| Answer» C. \(\rm f(x)\) decreases, then increases and decreases again. | |
| 370. |
A rectangular box with square base is open at the top. The maximum volume of the box made from 1200 m2 tin is |
| A. | 2000 m3 |
| B. | 3000 m3 |
| C. | 4000 m3 |
| D. | None of the above |
| Answer» D. None of the above | |
| 371. |
In the Taylor’s series expansion of ex about x = 2, the coefficient of (x – 2)4 is |
| A. | 24/4! |
| B. | 1/4! |
| C. | e2/4! |
| D. | e4/4! |
| Answer» D. e4/4! | |
| 372. |
If f(x) is an even function, where f(x) ≠ 0, then which one of the following is correct? |
| A. | f’(x) is an even function |
| B. | f’(x) is an odd function |
| C. | f’(x) may be an even or odd function depending on the type of function |
| D. | f’(x) is a constant function |
| Answer» C. f’(x) may be an even or odd function depending on the type of function | |
| 373. |
Evaluate \(\mathop \smallint \limits_{ - \infty }^\infty {x^4}f\left( x \right)dx\), where,\(f\left( x \right) = \frac{1}{{\sqrt {2\pi } }}{e^{ - \left( {\frac{{{x^2}}}{2}} \right)}},\;x \in \left( { - \infty ,\;\infty } \right)\) |
| A. | 3 |
| B. | \(3\sqrt \pi \) |
| C. | \(\sqrt 3 \;\pi \) |
| D. | 3π |
| Answer» B. \(3\sqrt \pi \) | |
| 374. |
\(\mathop \smallint \limits_1^2 {x^2}dx =\) |
| A. | 7/3 |
| B. | 3/7 |
| C. | 2/3 |
| D. | 3/2 |
| Answer» B. 3/7 | |
| 375. |
Let \(\vec a = \hat i - \hat j + \hat k\) and \(\vec b = \hat i + \hat j - \hat k\) then unit vector perpendicular to the plane containing \(\vec a\) and \(\vec b\) is |
| A. | \(\frac{{\hat j\;+\;\hat k}}{{\sqrt 2 }}\) |
| B. | \(\sqrt 2({\hat j + \hat k} )\) |
| C. | \(\sqrt 2 ({\hat j - \hat k})\) |
| D. | \(\frac{{\hat j\;-\;\hat k}}{{\sqrt 2 }}\) |
| Answer» B. \(\sqrt 2({\hat j + \hat k} )\) | |
| 376. |
If Y = ex sin x, then which of the following differential equation holds true? |
| A. | \(\frac{{{d^2}y}}{{d{x^2}}} + \frac{{dy}}{{dx}} + y = 0\) |
| B. | \(\frac{{{d^2}y}}{{d{x^2}}}-2\frac{{dy}}{{dx}} + 2y = 0\) |
| C. | \(\frac{{{d^2}y}}{{d{x^2}}}-\frac{{dy}}{{dx}} + y = 0\) |
| D. | \(\frac{{{d^2}y}}{{d{x^2}}} + 2\frac{{dy}}{{dx}}\; - \;y = 0\) |
| Answer» C. \(\frac{{{d^2}y}}{{d{x^2}}}-\frac{{dy}}{{dx}} + y = 0\) | |
| 377. |
Consider function f(x) = (x2 - 4)2, where x is a real number. The function f(x) has |
| A. | Only one minimum |
| B. | Only two minima |
| C. | Only three maxima |
| D. | None of the above |
| Answer» C. Only three maxima | |
| 378. |
Let \(f\left( x \right)\) be a function such that\(f\left( {\frac{1}{x}} \right)\; + \;{x^3}f\left( x \right)\) = 0, what is \(\mathop \smallint \limits_{ - 1}^1 f\left( x \right)dx\) equal to? |
| A. | 2 f(1) |
| B. | 0 |
| C. | 2 f(-1) |
| D. | 4 f(1) |
| Answer» C. 2 f(-1) | |
| 379. |
If x loge (loge x) – x2 + y2 = 4 (y > 0), then dy/dx at x = e is equal to: |
| A. | \(\frac{{\left( {1 + 2e} \right)}}{{2\sqrt {4 + {e^2}} }}\) |
| B. | \(\frac{{\left( {2e - 1} \right)}}{{2\sqrt {4 + {e^2}} }}\) |
| C. | \(\frac{{\left( {1 + 2e} \right)}}{{\sqrt {4 + {e^2}} }}\) |
| D. | \(\frac{e}{{\sqrt {4 + {e^2}} }}\) |
| Answer» C. \(\frac{{\left( {1 + 2e} \right)}}{{\sqrt {4 + {e^2}} }}\) | |
| 380. |
If two vectors \(\overrightarrow a\) and \(\overrightarrow b \) be such that \(\left| {\overrightarrow a + \overrightarrow b } \right| = \left| {\overrightarrow a - \overrightarrow b } \right|\), then the angle between them is |
| A. | \(\frac{\pi }{4}\) |
| B. | \(\frac{\pi }{3}\) |
| C. | \(\frac{\pi }{2}\) |
| D. | None of these |
| Answer» D. None of these | |
| 381. |
If (sin x)y, then \(\dfrac{dy}{dx}\) is ______: |
| A. | \(\rm \dfrac{y^2\ cot\ x}{1-y\ log(sin\ x)}\) |
| B. | \(\rm \dfrac{y^2\ cot\ x}{1-y\ log\ x}\) |
| C. | \(\rm \dfrac{y^2\ cot\ x}{1+y\ log(sin\ x)}\) |
| D. | \(\rm \dfrac{y^2\ cot\ x}{1+y\ log\ x}\) |
| Answer» B. \(\rm \dfrac{y^2\ cot\ x}{1-y\ log\ x}\) | |
| 382. |
Evaluate the following integral \(\mathop{\int }_{0}^{a}\mathop{\int }_{0}^{a}\mathop{\int }_{0}^{a}\left( xy+xz+yz \right)dx~dy~dz.\) |
| A. | \(\frac{3}{4}{{a}^{3}}\) |
| B. | \(\frac{2}{3}{{a}^{5}}\) |
| C. | \(\frac{3}{4}{{a}^{5}}\) |
| D. | \(\frac{5}{3}{{a}^{3}}\) |
| Answer» D. \(\frac{5}{3}{{a}^{3}}\) | |
| 383. |
If \(\int e^x (f(x) - f'(x))dx = \phi(x),\) then the value of \(\int e^x f(x)dx\) is |
| A. | ϕ(x) + ex f(x) |
| B. | ϕ(x) - ex f(x) |
| C. | \(\frac 1 2 \left[\phi (x) + e^x f(x)\right]\) |
| D. | \(\frac 1 2 \left[\phi (x) + e^x f'(x)\right]\) |
| Answer» D. \(\frac 1 2 \left[\phi (x) + e^x f'(x)\right]\) | |
| 384. |
A function f : (0, π) → R defined by f(x) = 2 sin x + cos 2x has |
| A. | A local minimum but no local maximum |
| B. | A local maximum but no local minimum |
| C. | Both local minimum and local maximum |
| D. | Neither a local minimum nor a local maximum |
| Answer» D. Neither a local minimum nor a local maximum | |
| 385. |
Let f(x) be a real -valued function such that f'(x0) = 0 for some x0 ∈ (0, 1), and f"(x) > 0 for all x ∈ (0, 1). Then f(x) has |
| A. | exactly one local minimum in (0, 1) |
| B. | two distinct local minima in (0, 1) |
| C. | one local maximum in (0, 1) |
| D. | no local minimum in (0, 1) |
| Answer» B. two distinct local minima in (0, 1) | |
| 386. |
How much angle does the tangent at P make with y-axis? |
| A. | tan-1m2 |
| B. | cot-1 (1 + m2) |
| C. | \({\sin ^{ - 1}}\left( {\frac{1}{{\sqrt {1 + {m^4}} }}} \right)\) |
| D. | \({\sec ^{ - 1}}\sqrt {1 + {m^4}}\) |
| Answer» D. \({\sec ^{ - 1}}\sqrt {1 + {m^4}}\) | |
| 387. |
The two types of errors that are related to differentials are: |
| A. | Absolute, Relative. |
| B. | Human, Absolute. |
| C. | Controllable, Natural. |
| D. | Relative, Controllable. |
| Answer» B. Human, Absolute. | |
| 388. |
At what value of q is the concavity of w(q) = -2, if w(q) = q4 - 16? |
| A. | At q = 0. |
| B. | At q = fourth root of 14. |
| C. | Never; w(q) is always concave up. |
| D. | Never; w(q) is always concave down. |
| Answer» D. Never; w(q) is always concave down. | |
| 389. |
If f (0) = 2 and f (x) = 1 / (5-x2), then lower and upper bound of f(1) estimated by the mean value theorem are |
| A. | 1.9,2.2 |
| B. | 2.2,2.25 |
| C. | 2.25,2.5 |
| D. | None of these |
| Answer» C. 2.25,2.5 | |
| 390. |
What is the derivative of f(x) = | x | at x = 0 |
| A. | 1 |
| B. | -1 |
| C. | 0 |
| D. | Does not exist |
| Answer» E. | |
| 391. |
The function f(x) = 3x(x - 2) has a |
| A. | minimum at x = 1 |
| B. | maximum at x = 1 |
| C. | minimum at x = 2 |
| D. | maximum at x = 2 |
| Answer» B. maximum at x = 1 | |
| 392. |
The minimum value of | x2 _ 5x + 21 | is |
| A. | -5 |
| B. | 0 |
| C. | -1 |
| D. | -2 |
| Answer» C. -1 | |
| 393. |
If f(x) = | x | , then for interval [-1, 1] ,f(x) |
| A. | satisied all the conditions of Rolles Theorem |
| B. | satisfied all the conditions of Mean Value Theorem |
| C. | does not satisied the -conditions of Mean Value Theorem |
| D. | None of these |
| Answer» D. None of these | |
| 394. |
The interval in which the Lagrange's theorem is applicable for the function f(x) = 1/x is |
| A. | [-3, 3] |
| B. | [-2, 2] |
| C. | [2, 3] |
| D. | [-1, 1] |
| Answer» D. [-1, 1] | |
| 395. |
The function f(x) = x3 - 6x2 + 9x + 25 has |
| A. | a maxima at x= 1 and a minima at x = 3 |
| B. | a maxima at x = 3 and a minima at x = 1 |
| C. | no maxima, but a minima at x = 1 |
| D. | a maxima at x = 1, but no minima |
| Answer» B. a maxima at x = 3 and a minima at x = 1 | |