MCQOPTIONS
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MCQOPTIONS
This section includes 657 Mcqs, each offering curated multiple-choice questions to sharpen your Testing Subject knowledge and support exam preparation. Choose a topic below to get started.
| 1. |
Let \(P = \left[ {\begin{array}{*{20}{c}} 3&1\\ 1&3 \end{array}} \right]\) consider the set \(S\) of all vectors \(\left( {\begin{array}{*{20}{c}} x\\ y \end{array}} \right)\) such that \({a^2} + {b^2} = 1\) where \(\left( {\begin{array}{*{20}{c}} a\\ b \end{array}} \right) = P\left( {\begin{array}{*{20}{c}} x\\ y \end{array}} \right)\). This \(S\) is |
| A. | A circle of radius \(\surd 10\) |
| B. | A circle of radius \(\frac{1}{{\sqrt {10} }}\) |
| C. | An ellipse with major axis along \(\left( {\begin{array}{*{20}{c}} 1\\ 1 \end{array}} \right)\) |
| D. | An ellipse with minor axis along \(\left( {\begin{array}{*{20}{c}} 1\\ 1 \end{array}} \right)\) |
| Answer» E. | |
| 2. |
Find the absolute maximum and minimum values off(x, y) = 2 + 2x + 2y – x2 – y2on triangular plate in the first quadrant, bounded by the lines x = 0, y = 0 and y = 9 – x. |
| A. | -4 |
| B. | -2 |
| C. | 4 |
| D. | 2 |
| Answer» D. 2 | |
| 3. |
If \(\rm z = \tan^{-1} \frac y x\) then the value of \(\rm \frac {\partial^2z}{\partial x^2} + \frac {\partial^2 z}{\partial y^2}\) is equal to- |
| A. | \(\rm \frac {-y}{x^2 + y^2}\) |
| B. | \(\rm \frac {x}{x^2 + y^2}\) |
| C. | \(\rm \frac {2xy}{x^2 + y^2}\) |
| D. | 0 |
| Answer» E. | |
| 4. |
If y = |sin x||x| then what is the value of \(\frac{{dy}}{{dx}}{\rm{\;}}at{\rm{\;}}x = {\rm{\;}} - \frac{\pi }{6}?\) |
| A. | \(\frac{{{2^{ - \frac{\pi }{6}}}\left( {6\ln 2 - \sqrt {3\pi } } \right)}}{6}\) |
| B. | \(\frac{{{2^{\frac{\pi }{6}}}\left( {6{\rm{\;}}ln2 + \sqrt 3 \pi } \right)}}{6}\) |
| C. | \(\frac{{{2^{ - \frac{\pi }{6}}}\left( {6{\rm{\;}}ln2 + \sqrt 3 \pi } \right)}}{6}\) |
| D. | \(\frac{{{2^{\frac{\pi }{6}}}\left( {6{\rm{\;}}ln2 - \sqrt 3 \pi } \right)}}{6}\) |
| Answer» B. \(\frac{{{2^{\frac{\pi }{6}}}\left( {6{\rm{\;}}ln2 + \sqrt 3 \pi } \right)}}{6}\) | |
| 5. |
For 0 ≤ |
| A. | t = loge 4 |
| B. | t = loge 2 |
| C. | t = 0 |
| D. | t = loge 8 |
| Answer» B. t = loge 2 | |
| 6. |
A particle starts at the origin and moves along the x-axis in such a way that its velocity at the point (x, 0) is given by the formula \(\dfrac{dx}{dt}=\cos^2 \pi x\). Then the particle never reaches the point on - |
| A. | \(x=\dfrac{1}{4}\) |
| B. | \(x=\dfrac{3}{4}\) |
| C. | \(x=\dfrac{1}{2}\) |
| D. | \(x=1\) |
| Answer» D. \(x=1\) | |
| 7. |
If \({\rm{y}} = {\cos ^{ - 1}}\left( {\frac{{2{\rm{x}}}}{{1 + {{\rm{x}}^2}}}} \right)\), then \(\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\) is equal to |
| A. | \(- \frac{2}{{1 + {{\rm{x}}^2}}}\) for all |x| < 1 |
| B. | \(- \frac{2}{{1 + {{\rm{x}}^2}}}\) for all |x| > 1 |
| C. | \(\frac{2}{{1 + {{\rm{x}}^2}}}\) for all |x| < 1 |
| D. | None of the above |
| Answer» B. \(- \frac{2}{{1 + {{\rm{x}}^2}}}\) for all |x| > 1 | |
| 8. |
If the ratio of surface area to volume of a cuboid is √3, then the sum of reciprocals of the three sides (length, breadth and height) of the cuboid will be: |
| A. | \(\sin \frac \pi 2\) |
| B. | sin π |
| C. | \(\sin \frac \pi 4\) |
| D. | \(\sin \frac \pi 3\) |
| Answer» E. | |
| 9. |
Let \(\overrightarrow a = \widehat i + \widehat j - \widehat k\) and \(\overrightarrow b = 2\widehat i + 2\widehat j + \widehat k\) be the two sides of the triangle. Then the area of the triangle is |
| A. | 7 |
| B. | \(\sqrt {13} \) |
| C. | \(\frac{{\sqrt {13} }}{2}\) |
| D. | \(\frac{{3\sqrt 2 }}{2}\) |
| Answer» E. | |
| 10. |
\(\displaystyle\int_{a+c}^{b+c}f(x)dx = \ ?\) |
| A. | \(\displaystyle\int_a^b f(x-c)dx\) |
| B. | \(\displaystyle\int_a^b f(x+c)dx\) |
| C. | \(\displaystyle\int_a^b f(x)dx\) |
| D. | \(\displaystyle\int_{a-c}^{b-c} f(x)dx\) |
| Answer» C. \(\displaystyle\int_a^b f(x)dx\) | |
| 11. |
At the point x = 0, the function f(x) = x3 has |
| A. | Local maximum |
| B. | Local minimum |
| C. | Both local maximum and minimum |
| D. | Neither local maximum nor local minimum |
| Answer» E. | |
| 12. |
Given that f(x) = x1/x, x > 0 has the maximum value at x = e, then |
| A. | eπ > πe |
| B. | eπ < πe |
| C. | eπ = πe |
| D. | eπ ≤ πe |
| Answer» B. eπ < πe | |
| 13. |
Consider the following functions:1. \({\rm{f}}\left( {\rm{x}} \right) = \left\{ {\begin{array}{*{20}{c}} {\frac{1}{{\rm{x}}}{\rm{\;\;if\;\;x}} \ne 0}\\ {0{\rm{\;\;if\;\;x}} = 0} \end{array}} \right.\)2. \({\rm{f}}\left( {\rm{x}} \right) = \left\{ {\begin{array}{*{20}{c}} {2{\rm{x}} + 5{\rm{\;\;if\;\;x}} > 0}\\ {{{\rm{x}}^2} + 2{\rm{x}} + 5{\rm{\;\;if\;\;x}} \le 0} \end{array}} \right.\)Which of the above functions is/are derivable at x = 0? |
| A. | 1 only |
| B. | 2 only |
| C. | Both 1 and 2 |
| D. | Neither 1 nor 2 |
| Answer» C. Both 1 and 2 | |
| 14. |
For a complex number z, \(\begin{array}{*{20}{c}} {lim}\\ {z \to i} \end{array}\frac{{{z^2} + 1}}{{{z^3} + 2z - i\left( {{z^2} + 2} \right)}}\) is |
| A. | -2i |
| B. | -i |
| C. | i |
| D. | 2i |
| Answer» E. | |
| 15. |
If f(x) is an even function and a is a positive real number, then \(\mathop \smallint \limits_{ - a}^a f\left( x \right)dx\) equals |
| A. | 0 |
| B. | a |
| C. | 2a |
| D. | \(2\mathop \smallint \limits_0^a f\left( x \right)dx\) |
| Answer» E. | |
| 16. |
If \(\smallint {x^5}{e^{ - 4{x^3}}}dx = \frac{1}{{48}}{e^{ - 4{x^3}}}f\left( x \right) + C\) where C is a constant of integration, then f(x) is equal to: |
| A. | -2x3 - 1 |
| B. | -4x3 - 1 |
| C. | -2x3 + 1 |
| D. | 4x3 + 1 |
| Answer» C. -2x3 + 1 | |
| 17. |
Evaluate\(\mathop{\int }_{0}^{1}\mathop{\int }_{{{y}^{2}}}^{1}\mathop{\int }_{0}^{1-x}x~dz~dx~dy\) |
| A. | \(\frac{{2}}{{35}}\) |
| B. | \(\frac{{4}}{{35}}\) |
| C. | \(\frac{{4}}{{17}}\) |
| D. | \(\frac{{2}}{{17}}\) |
| Answer» C. \(\frac{{4}}{{17}}\) | |
| 18. |
Let \(I = \mathop \smallint \nolimits_{x = 0}^1 \mathop \smallint \nolimits_{y = 0}^{{x^2}} x{y^2}dydx.\) Then, I may also be expressed as |
| A. | \(\mathop \smallint \nolimits_{y = 0}^1 \mathop \smallint \nolimits_{x = 0}^{\sqrt y } x{y^2}dxdy\) |
| B. | \(\mathop \smallint \nolimits_{y = 0}^1 \mathop \smallint \nolimits_{x = \sqrt y }^1 y{x^2}dxdy\) |
| C. | \(\mathop \smallint \nolimits_{y = 0}^1 \mathop \smallint \nolimits_{x = \sqrt y }^1 x{y^2}dxdy\) |
| D. | \(\mathop \smallint \nolimits_{y = 0}^1 \mathop \smallint \nolimits_{x = 0}^{\sqrt y } y{x^2}dxdy\) |
| Answer» D. \(\mathop \smallint \nolimits_{y = 0}^1 \mathop \smallint \nolimits_{x = 0}^{\sqrt y } y{x^2}dxdy\) | |
| 19. |
Equation of the line normal to the function F(x) = (x - 8)2/3 + 1, at P(0, 5) is |
| A. | y = 3x – 5 |
| B. | 3y = x – 15 |
| C. | 3y = x + 15 |
| D. | y = 3x + 5 |
| Answer» E. | |
| 20. |
If y(x) is the solution of the differential equation \(\frac{{dy}}{{dx}} + \left( {\frac{{2x + 1}}{x}} \right)y = {e^{ - 2x}}\), x > 0, where \(y\left( 1 \right) = \frac{1}{2}{e^{ - 2}}\) is: |
| A. | y(loge 2) = loge 4 |
| B. | \(y\left( {{\rm{lo}}{{\rm{g}}_e}2} \right) = \frac{{{\rm{lo}}{{\rm{g}}_e}2}}{4}\) |
| C. | y(x) is decreasing in \(\left( {\frac{1}{2},1} \right)\) |
| D. | y(x) is decreasing in (0,1) |
| Answer» D. y(x) is decreasing in (0,1) | |
| 21. |
Choose the correct option for the following sentences.a. A function f(x, y) is said to have a maximum value at x = a, y = b if f(a, b) > f(a + h, b + k) and f(a, b) > f(a - h, b + k)b. A function f(x, y) is said to have a maximum value at x = a, y = b if f(a, b) < f(a + h, b + k) and f(a, b) > f(a - h, b + k) |
| A. | Both a and b are wrong |
| B. | Both a and b are true |
| C. | a is true, b is wrong |
| D. | b is true, a is wrong |
| Answer» D. b is true, a is wrong | |
| 22. |
Let S(α) = {(x, y) : y2 ≤ x, 0 ≤ x ≤ α} and A(α) is area of the region S(α). If for aλ, 0 < λ < 4, A(λ) : A(4) = 2 : 5, then λ equals: |
| A. | \(2{\left( {\frac{4}{{25}}} \right)^{\frac{1}{3}}}\) |
| B. | \(2{\left( {\frac{2}{5}} \right)^{\frac{1}{3}}}\) |
| C. | \(4{\left( {\frac{2}{5}} \right)^{\frac{1}{3}}}\) |
| D. | \(4{\left( {\frac{4}{{25}}} \right)^{\frac{1}{3}}}\) |
| Answer» E. | |
| 23. |
\(\int \sec x\; dx\) is |
| A. | \(\log \left( {\sec x + \tan x} \right) + c\) |
| B. | \(\tan x + c\) |
| C. | \(\sec x + \tan x + c\) |
| D. | \(\sec x\tan x + c\) |
| Answer» B. \(\tan x + c\) | |
| 24. |
For real numbers, x and y with y = 3x2 + 3x + 1, the maximum and minimum value of y for x ∈ [-2, 0] are respectively, ______ |
| A. | 7 and 1/4 |
| B. | 7 and 1 |
| C. | -2 and -1/2 |
| D. | 1 and 1/4 |
| Answer» B. 7 and 1 | |
| 25. |
Evaluate: \(\int \rm \frac{{dx}}{{(x-2)\;\left( {x - 1} \right)}}\) |
| A. | \(\rm log \left|{(x-2)\over (x + 1)}\right| + c\) |
| B. | \(\rm log \left|{(x-2)\over (x - 1)}\right|+c\) |
| C. | \(\rm log \left|{(x-1)\over (x - 2)}\right|+c\) |
| D. | \(\rm log \left|{(x+2)\over (x - 1)}\right|+c\) |
| Answer» C. \(\rm log \left|{(x-1)\over (x - 2)}\right|+c\) | |
| 26. |
\(\mathop {\lim }\limits_{x \to 0} \left( {\frac{{1 - cosx}}{{{x^2}}}} \right)\) |
| A. | 1/4 |
| B. | 1/2 |
| C. | 1 |
| D. | 2 |
| Answer» C. 1 | |
| 27. |
If \(\mathop a\limits^ \to = 6\hat i + 4\hat j + 4\hat k\) and \(\mathop b\limits^ \to = 2\hat i + 4\hat j + 3\hat k\), then the value of \(\mathop b\limits^ \to \times \mathop a\limits^ \to\) is: |
| A. | \(4\hat i + 10\hat j - 16\hat k\) |
| B. | \(4\hat i + 10\hat j + 16\hat k\) |
| C. | \(4\hat i + 10\hat j - 12\hat k\) |
| D. | \(4\hat i + 10\hat j + 3\hat k\) |
| Answer» B. \(4\hat i + 10\hat j + 16\hat k\) | |
| 28. |
A political party orders an arch for the entrance to the ground in which the annual convention is being held. The profile of the arch follows the equation y = 2x - 0.1x2 where y is the height of the arch in meters. The maximum possible height of the arch is |
| A. | 8 meters |
| B. | 10 meters |
| C. | 12 meters |
| D. | 14 meters |
| Answer» C. 12 meters | |
| 29. |
How many distinct values of \(\rm x\) satisfy the equation \(\rm sin(x) = x/2\), where \(\rm x\) is in radians? |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | 4 or more |
| Answer» D. 4 or more | |
| 30. |
Consider the following statements:Statement I: x > sin x for all x > 0Statement II: f(x) = x - sin x is an increasing function for all x > 0Which one of the following is correct is respect of the above statements? |
| A. | Both Statement I and Statement II are true and Statement II is the correct explanation of Statement I |
| B. | Both Statement I and Statement II are true and Statement II is not the correct explanation of Statement I |
| C. | Statement I is true but Statement II is false |
| D. | Statement I is false but Statement II is true |
| Answer» B. Both Statement I and Statement II are true and Statement II is not the correct explanation of Statement I | |
| 31. |
\(\mathop {\lim }\limits_{x \to 0} \frac{{{x^2} + x - \sin x}}{{{x^2}}}\) |
| A. | 0 |
| B. | ∞ |
| C. | 1 |
| D. | None of these |
| Answer» D. None of these | |
| 32. |
If \(2{\rm{y}} = {\left( {{\rm{co}}{{\rm{t}}^{ - 1}}\left( {\frac{{\sqrt 3 {\rm{cos\;x}} + {\rm{sin\;x}}}}{{{\rm{cos\;x}} - \sqrt 3 {\rm{sin\;x}}}}} \right)} \right)^2},\;x \in \left( {0,\frac{\pi }{2}} \right)\;{\rm{then}}\;\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\) is equal to |
| A. | \({\rm{\;}}\frac{{\rm{\pi }}}{6} - x\) |
| B. | \(x - \frac{{\rm{\pi }}}{6}\) |
| C. | \({\rm{\;}}\frac{{\rm{\pi }}}{3} - x\) |
| D. | \({\rm{\;}}2x - \frac{{\rm{\pi }}}{3}\) |
| Answer» C. \({\rm{\;}}\frac{{\rm{\pi }}}{3} - x\) | |
| 33. |
Let f be a real-valued function of a real variable defined as f(x) = x2 for x ≥ 0, and f(x) = -x2 for x < 0.Which one of the following statements is true? |
| A. | f(x) is discontinuous at x = 0 |
| B. | f(x) is continuous but not differentiable at x = 0. |
| C. | f(x) is differentiable but its first derivative is not continuous x = 0. |
| D. | f(x) is differentiable but its first derivative is not differentiable at x = 0. |
| Answer» E. | |
| 34. |
If \({\rm{f}}\left( {\rm{x}} \right) = \frac{{2 - {\rm{xcos\;x}}}}{{2 + {\rm{xcos\;x}}}}\;\)and g(x) = loge x, (x > 0) then the value of the integral \(\mathop \smallint \limits_{ - \pi /4}^{{\rm{\pi }}/4} {\rm{g}}\left( {{\rm{f}}\left( {\rm{x}} \right)} \right){\rm{dx}}\) is: |
| A. | log_e 3 |
| B. | log_e e |
| C. | log_e 2 |
| D. | log_e 1 |
| Answer» E. | |
| 35. |
If the function \(\sqrt {{X^2} - 4} \) in [2, 4] satisfies the Lagrange’s mean value theorem, then there exists some c ∈ [2, 4]. The value of c is |
| A. | 12 |
| B. | 6 |
| C. | √2 |
| D. | √6 |
| Answer» E. | |
| 36. |
A vector field is defined as\(\vec f\left( {x,y,z} \right) = \frac{x}{{{{\left[ {{x^2} + {y^2} + {z^2}} \right]}^{\frac{3}{2}}}}}\hat i + \frac{y}{{{{\left[ {{x^2} + {y^2} + {z^2}} \right]}^{\frac{3}{2}}}}}\hat j + \frac{z}{{{{\left[ {{x^2} + {y^2} + {z^2}} \right]}^{\frac{3}{2}}}}}\hat k\)where î, ĵ, k̂ are unit vectors along the axes of a right-handed rectangular/Cartesian coordinate system. The surface integral \(\smallint \smallint \vec f.d\vec S\) (Where \(d\vec S\) is an elemental surface area vector) evaluated over the inner and outer surfaces of a spherical shell formed by two concentric spheres with origin as the centre, and internal and external radii of 1 and 2, respectively, is |
| A. | 0 |
| B. | 2π |
| C. | 4π |
| D. | 8π |
| Answer» B. 2π | |
| 37. |
Consider the shaded triangular region P shown in the figure. What is \(\smallint \mathop \smallint \limits_P^\; xydxdy\) ? |
| A. | 1/6 |
| B. | 2/9 |
| C. | 7/16 |
| D. | 1 |
| Answer» B. 2/9 | |
| 38. |
In an inorganic reaction, the compound concentration (y) is mathematically represented as \(\frac{{dx}}{{dt}} = - k{x^2}\), where k is a positive constant. If t = 0, x = c, then variation of x as a function of t will be represented by: |
| A. | \(\frac{1}{x} = \frac{1}{c} + kt\) |
| B. | x = c - kt |
| C. | x = 1 – ce-kt |
| D. | x = ce-kt |
| Answer» B. x = c - kt | |
| 39. |
If y = y(x) is the solution of the differential equation \(\frac{dy}{dx}=\left( tanx-y \right)se{{c}^{2}}x,~x\in \left( -\frac{\pi }{2},\frac{\pi }{2} \right)\) such that y(0) = 0, then \(y\left( -\frac{\pi }{4} \right)\) is equal to: |
| A. | e - 2 |
| B. | \(\frac{1}{2}-e\) |
| C. | \(2+\frac{1}{e}\) |
| D. | \(\frac{1}{e}-2\) |
| Answer» B. \(\frac{1}{2}-e\) | |
| 40. |
∫ eax dx is equal to: |
| A. | \(\frac{1}{a}{e^{ - ax}} - c\) |
| B. | e-ax + c |
| C. | \(\frac{1}{a}{e^{ax}} + c\) |
| D. | eax + c |
| Answer» D. eax + c | |
| 41. |
If \(\rm y = \sqrt {\sin x + \sqrt {\sin x + \sqrt {\sin x + ... \infty}}}\) then \(\rm \frac {dy}{dx}\) is equal to: |
| A. | \(\rm \frac {\cos x}{2y - 1}\) |
| B. | \(\rm \frac {-\cos x}{2y - 1}\) |
| C. | \(\rm \frac {\sin x}{1 - 2y}\) |
| D. | \(\rm \frac {-\sin x}{1 - 2y}\) |
| Answer» B. \(\rm \frac {-\cos x}{2y - 1}\) | |
| 42. |
If \(\rm I_n = \displaystyle\int_0^{\tfrac{\pi}{4}} \tan^n \theta \ d\theta \), then I8 + I6 equals: |
| A. | \(\dfrac14\) |
| B. | \(\dfrac15\) |
| C. | \(\dfrac16\) |
| D. | \(\dfrac17\) |
| Answer» E. | |
| 43. |
If [x] represents the greatest integer not exceeding x, then \(\rm \int_0^9 [x]\ dx\) is |
| A. | 32 |
| B. | 36 |
| C. | 40 |
| D. | 28 |
| Answer» C. 40 | |
| 44. |
Let \(g\left( x \right) = \left\{ {\begin{array}{*{20}{c}} { - x,}&{x \le 1}\\ {x + 1,}&{x \ge 1} \end{array}} \right.\) and \(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {1 - x,}&{x \le 0}\\ {{x^2},}&{x > 0} \end{array}} \right.\).Consider the composition of f and g, i.e. (fog)(x) = f(g(x)). The number of discontinuities in (fog)(x) present in the interval (-∞, 0) is: |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | 4 |
| Answer» B. 1 | |
| 45. |
If A = 5t2 I + t J – t3 K, B = sin t I – cos t J, find\(\frac{d}{{dt}}\left( {A.B} \right)\) |
| A. | 5t2 cos t + 9t sin t + cos t |
| B. | 5t2 cos t + 11t sin t – cos t |
| C. | 5t2 cos t + 11t sin t + cos t |
| D. | 5t2 cos t + 9t sin t – cost |
| Answer» C. 5t2 cos t + 11t sin t + cos t | |
| 46. |
If f is twice differential such that f"(x) = -f(x) and f'(x) = g(x), h(x) = [f(x)]2 + [g(x)]2, then the value of h(10) if h(5) = 11 is equal to |
| A. | 5 |
| B. | 8 |
| C. | 11 |
| D. | 1 |
| Answer» D. 1 | |
| 47. |
Minimum of the real valued function \(f\left( x \right) = 10{\left( {x - 1} \right)^{\frac23}}\) occurs at x equal to |
| A. | - ∞ |
| B. | 0 |
| C. | 1 |
| D. | ∞ |
| Answer» D. ∞ | |
| 48. |
If m, n are integers and m + n is odd then the value of \(\mathop{\int }_{0}^{\pi }\sin mx.\cos nx.dx\) is |
| A. | 0 |
| B. | π/2 |
| C. | π |
| D. | 1 |
| Answer» B. π/2 | |
| 49. |
If \(\rm \bar r = \left( {x\bar i + y\bar j + z\bar k} \right)\) then div\(\rm \;\bar r\) = |
| A. | 0 |
| B. | 3 |
| C. | (x + y + z) |
| D. | (x2 +y2 + z2) |
| Answer» C. (x + y + z) | |
| 50. |
If φ is a scalar point function, the value of curl Grad φ is |
| A. | 0 |
| B. | 1 |
| C. | 2 |
| D. | 3 |
| Answer» B. 1 | |