Calculus — Part 2 MCQs with Explanations

is positive when x and yare linearly independent is the correct answer to “Let x and y be two vectors in a 3 dimensional space and <x, y> denote their dot product. Then the determinant det \[\left[ {\begin{array}{*{20}{c}} { < {\text{x}},{\text{x}} > }&{ < {\text{x}},{\text{y}} > } \\ { < {\text{y}},{\text{x}} > }&{ < {\text{y}},{\text{y}} > } \end{array}} \right].\]”. This question is standard in competitive examination syllabi for Calculus.

1 is the correct answer to “$$\mathop {{\text{Lim}}}\limits_{{\text{x}} \to \infty } \left( {\frac{{{\text{x}} + \sin {\text{x}}}}{{\text{x}}}} \right)$$ equal to”. This question is standard in competitive examination syllabi for Calculus.

If $$\nabla $$ · E = 0, E is called irrotational is the correct answer to “For a vector E, which one of the following statements is NOT TRUE”. This question is standard in competitive examination syllabi for Calculus.

\[{\text{f}}\left( {\text{x}} \right) = \left\{ {\begin{array}{*{20}{c}} {2,}&{{\text{if}}}&{{\text{x}} = 3} \\ {{\text{x}} - 1,}&{{\text{if}}}&{{\text{x}} > 3} \\ {\frac{{{\text{x}} + 3}}{3},}&{{\text{if}}}&{{\text{x}} < 3} \end{array}} \right.\] is the correct answer to “Which one of the following functions is continuous at x = 3”. This question is standard in competitive examination syllabi for Calculus.

\[16{\rm{\hat i}} - 9{\rm{\hat j}} - 12{\rm{\hat k}}\] is the correct answer to “If A(0, 4, 3), B(0, 0, 0) and C(3, 0, 4) are three points defined in x, y, zeo-ordinate system, then which of the following vector is perpendicular to both vectors \[\overrightarrow {{\text{AB}}} \] and \[\overrightarrow {{\rm{BC}}} .\]”. This question is standard in competitive examination syllabi for Calculus.

f(x) is NOT differentiable at x = 1 for any values of a and b is the correct answer to “A function f(x) is defined as \[{\text{f}}\left( {\text{x}} \right) = \left\{ {\begin{array}{*{20}{c}} {{{\text{e}}^x}}&{{\text{x}} < 1} \\ {\ln {\text{x}} + {\text{a}}{{\text{x}}^2} + {\text{bx}},}&{{\text{x}} \geqslant 1} \end{array}} \right.\] where x\[ \in \] R which one of the following statements is TRUE”. This question is standard in competitive examination syllabi for Calculus.

90 is the correct answer to “The inner (dot) product of two non zero vectors \[\overrightarrow {\text{P}} \] and \[\overrightarrow {\text{Q}} \] is zero. The angle (degrees) between the two vectors is”. This question is standard in competitive examination syllabi for Calculus.

1 + x + \[\frac{3}{2}\]x2 + \[\frac{7}{6}\]x3 is the correct answer to “Let f(x) = e x + x 2 for real x. From among the following, choose the Taylor series approximation of f(x) around x = 0, which includes all powers of x less than or equal to 3,”. This question is standard in competitive examination syllabi for Calculus.

(3 - x)e-2 is the correct answer to “For the function e -x , the linear approximation around x = 2 is”. This question is standard in competitive examination syllabi for Calculus.

sin(x) is the correct answer to “The infinite series \[{\text{f}}\left( {\text{x}} \right) = {\text{x}} - \frac{{{{\text{x}}^3}}}{{3!}} + \frac{{{{\text{x}}^5}}}{{5!}} - \frac{{{{\text{x}}^7}}}{{7!}}\,...\,\infty \] converges to”. This question is standard in competitive examination syllabi for Calculus.

0.0, 3.0, 2.0 is the correct answer to “If the vector function \[\overrightarrow {\rm{F}} = {{\rm{\hat a}}_{\rm{x}}}\left( {3{\rm{y}} - {{\rm{k}}_1}{\rm{z}}} \right) + {{\rm{\hat a}}_{\rm{y}}}\left( {{{\rm{k}}_2}{\rm{x}} - 2{\rm{z}}} \right) - {{\rm{\hat a}}_{\rm{z}}}\left( {{{\rm{k}}_3}{\rm{y}} + {\rm{z}}} \right)\] is irrotational, then the values of the constants k 1 , k 2 and k 3 respectively, are”. This question is standard in competitive examination syllabi for Calculus.

f(x) ≤ 2 |x + 1| is the correct answer to “Consider a differentiable function f(x) on the set of real numbers such that f(-1) = 0 and |f'(x)| ≤ 2. Given these conditions, which one of the following inequalities is necessarily true for all x\[ \in \] [-2, 2]”. This question is standard in competitive examination syllabi for Calculus.

concave upward for all values of x is the correct answer to “The curve y = x 4 is”. This question is standard in competitive examination syllabi for Calculus.

e-x2 is the correct answer to “Which one of the following functions is strictly bounded”. This question is standard in competitive examination syllabi for Calculus.

\[2\pi \] is the correct answer to “A parametric curve defined by \[{\rm{x}} = \cos \left( {\frac{{\pi {\rm{u}}}}{2}} \right),\,{\rm{y}} = \sin \left( {\frac{{\pi {\rm{u}}}}{2}} \right)\] in the range 0 ≤ u ≤ 1 is rotated about the x-axis by 360°. Area of the surface generated is”. This question is standard in competitive examination syllabi for Calculus.

0 is the correct answer to “Let \[{\rm{g}}\left( {\rm{x}} \right) = \left\{ {\begin{array}{*{20}{c}} { - {\rm{x,}}}&{{\rm{x}} \le 1}\\ {{\rm{x}} + 1,}&{{\rm{x}} \ge 1} \end{array}} \right.\] and \[{\rm{f}}\left( {\rm{x}} \right) = \left\{ {\begin{array}{*{20}{c}} {1 - {\rm{x,}}}&{{\rm{x}} \le 0}\\ {{{\rm{x}}^2},}&{{\rm{x}} > 0} \end{array}} \right..\] 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 (\[ - \infty ,\] 0) is:”. This question is standard in competitive examination syllabi for Calculus.

1 is the correct answer to “What is \[\mathop {\lim }\limits_{\theta \to 0} \frac{{\sin \theta }}{\theta }\] equal to”. This question is standard in competitive examination syllabi for Calculus.

cannot have more than two extrema and more than three zero crossings is the correct answer to “A cubic polynomial with real coefficients”. This question is standard in competitive examination syllabi for Calculus.

33 is the correct answer to “The value of the integral of the function g(x, y) = 4x 3 + 10y 4 along the straight line segment from the point (0, 0) to the point (1, 2) in the x - y plane is”. This question is standard in competitive examination syllabi for Calculus.

\[\frac{1}{4}\] is the correct answer to “The value of \[\mathop {\lim }\limits_{{\rm{x}} \to 0} \frac{{1 - \cos \left( {{{\rm{x}}^2}} \right)}}{{2{x^4}}}\] is”. This question is standard in competitive examination syllabi for Calculus.

is 0 is the correct answer to “Consider points P and Q in the x-y plane, with P = (1, 0) and Q = (0, 1). The line integral \[2\int\limits_{\rm{P}}^{\rm{Q}} {\left( {{\rm{xdx}} + {\rm{ydy}}} \right)} \] along the semicircle with the line segment PQ as its diameter”. This question is standard in competitive examination syllabi for Calculus.

0 is the correct answer to “A surface S(x, y) = 2x + 5y - 3 is integrated once over a path consisting of the points that satisfy (x + 1) 2 + (y - 1) 2 = √2. The integral evaluates to”. This question is standard in competitive examination syllabi for Calculus.

\[\frac{{\partial {\rm{w}}}}{{\partial {\rm{x}}}}\frac{{{\rm{dx}}}}{{{\rm{dt}}}} + \frac{{\partial {\rm{w}}}}{{\partial {\rm{y}}}}\frac{{{\rm{dy}}}}{{{\rm{dt}}}}\] is the correct answer to “Let w = f(x, y), where x and y are functions of t. Then, according to the chain rule, \[\frac{{{\rm{dw}}}}{{{\rm{dt}}}}\] is equal”. This question is standard in competitive examination syllabi for Calculus.

ad - bc is the correct answer to “For the parallelogram OPQR shown in the sketch, \[\overline {{\rm{OP}}} = {\rm{a\hat t}} + {\rm{b\hat j}}\] and \[\overline {{\rm{OR}}} = {\rm{c\hat t}} + {\rm{d\hat j}}{\rm{.}}\] The area of the parallelogram is”. This question is standard in competitive examination syllabi for Calculus.

1 is the correct answer to “The distance between the origin and the point nearest to it on the surface z 2 = 1 + xy is”. This question is standard in competitive examination syllabi for Calculus.

13.5 is the correct answer to “The value of \[\int\limits_0^3 {\int\limits_0^{\rm{x}} {\left( {6 - {\rm{x}} - {\rm{y}}} \right)} {\rm{dx\,dy}}} \] is”. This question is standard in competitive examination syllabi for Calculus.

\[\frac{1}{{{{\rm{a}}^2} - {{\rm{b}}^2}}}\left[ {a\left( {\ln 2 - 25} \right) + \frac{{47{\rm{b}}}}{2}} \right]\] is the correct answer to “If for non-zero x, \[{\rm{af}}\left( {\rm{x}} \right) + {\rm{bf}}\left( {\frac{1}{{\rm{x}}}} \right) = \frac{1}{{\rm{x}}} - 25\] where a ≠ b then \[\int\limits_1^2 {{\rm{f}}\left( {\rm{x}} \right){\rm{dx}}} \] is”. This question is standard in competitive examination syllabi for Calculus.

0.5 is the correct answer to “The improper integral \[\int\limits_0^\infty {{{\rm{e}}^{ - 2{\rm{t}}}}} {\rm{dt}}\] converges to”. This question is standard in competitive examination syllabi for Calculus.

monotonically increases is the correct answer to “As x increased from \[ - \infty \] to \[\infty \] , the function \[{\rm{f}}\left( {\rm{x}} \right) = \frac{{{{\rm{e}}^{\rm{x}}}}}{{1 + {{\rm{e}}^{\rm{x}}}}}\]”. This question is standard in competitive examination syllabi for Calculus.

is 1 is the correct answer to “The value of \[\mathop {\lim }\limits_{{\text{x}} \to 1} \frac{{{{\text{x}}^7} - 2{{\text{x}}^5} + 1}}{{{{\text{x}}^3} - 3{{\text{x}}^2} + 2}}\]”. This question is standard in competitive examination syllabi for Calculus.

0 is the correct answer to “\[\mathop {{\text{Lim}}}\limits_{{\text{x}} \to 0} \frac{{{\text{x}} - \sin {\text{x}}}}{{1 - \cos {\text{x}}}}\]”. This question is standard in competitive examination syllabi for Calculus.

1 is the correct answer to “Let f = y x . What is \[\frac{{{\partial ^2}{\text{f}}}}{{\partial {\text{x}}\partial {\text{y}}}}\] at x = 2, y = 1”. This question is standard in competitive examination syllabi for Calculus.

4 is the correct answer to “Given y = x 2 + 2x + 10, the value of \[{\left. {\frac{{{\text{dy}}}}{{{\text{dx}}}}} \right|_{{\text{x}} = 1}}\] is equal to”. This question is standard in competitive examination syllabi for Calculus.

\[\left( {{\text{x, y, z}}} \right)\] is the correct answer to “A sphere of unit radius is centered at the origin. The unit normal at a point (x, y, z) on the surface of the sphere is the vector”. This question is standard in competitive examination syllabi for Calculus.

xdy + ydx is the correct answer to “The total derivative of function xy is”. This question is standard in competitive examination syllabi for Calculus.

0.5 is the correct answer to “\[\mathop {{\text{Lim}}}\limits_{{\text{x}} \to 0} \left( {\frac{{{{\text{e}}^{2{\text{x}}}} - 1}}{{\sin \left( {4{\text{x}}} \right)}}} \right)\] is equal to”. This question is standard in competitive examination syllabi for Calculus.

\[ - \frac{1}{7}\] is the correct answer to “The value of the function \[{\text{f}}\left( {\text{x}} \right) = \mathop {\lim }\limits_{{\text{x}} \to 0} \frac{{{{\text{x}}^3} + {{\text{x}}^2}}}{{2{{\text{x}}^3} - 7{{\text{x}}^2}}}\] is”. This question is standard in competitive examination syllabi for Calculus.

twice differentiable but not thrice is the correct answer to “Consider the function f(x) = |x| 3 , where x is real. Then the function f(x) at x = 0 is”. This question is standard in competitive examination syllabi for Calculus.

\[\frac{4}{3}\] is the correct answer to “The integral \[\int\limits_0^\pi {{{\sin }^3}\theta \,{\text{d}}\theta } \] is given by”. This question is standard in competitive examination syllabi for Calculus.

\[\frac{\pi }{3}\] is the correct answer to “The volume of an object expressed in spherical co-ordinates is given by \[{\text{V}} = \int_0^{2\pi } {\int_0^{\frac{\pi }{3}} {\int_0^1 {{{\text{r}}^2}\sin \phi \,{\text{dr d}}\phi \,{\text{d}}\theta .} } } \] The value of the integral is”. This question is standard in competitive examination syllabi for Calculus.