Calculus — Part 1 MCQs with Explanations

Limit does not exit is the correct answer to “What is the value of \[\mathop {\mathop {\lim }\limits_{{\text{x}} \to 0} }\limits_{{\text{y}} \to 0} \frac{{{\text{xy}}}}{{{{\text{x}}^2} + {{\text{y}}^2}}}?\]”. This question is standard in competitive examination syllabi for Calculus.

2 + 3x - x2 - \[\frac{{{{\text{x}}^3}}}{2}\] + ... is the correct answer to “The Taylor series expansion of 3 sinx + 2 cosx is . . . . . . . .”. This question is standard in competitive examination syllabi for Calculus.

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

\[ - 1 - \frac{{{{\left( {{\text{x}} - \pi } \right)}^2}}}{{3!}} + ...\] is the correct answer to “The Taylor series expansion of \[\frac{{\sin {\text{x}}}}{{{\text{x}} - \pi }}\] at \[{\text{x}} = \pi \] is given by”. This question is standard in competitive examination syllabi for Calculus.

tan\[\left( {\frac{\theta }{2}} \right)\] is the correct answer to “If x = a (θ + sin θ) and y = a (1 - cos θ), then \[\frac{{{\text{dy}}}}{{{\text{dx}}}}\] will be equal to”. This question is standard in competitive examination syllabi for Calculus.

\[12 + \pi \] is the correct answer to “Consider the line integral \[\int\limits_{\text{C}} {\left( {{\text{xdy}} - {\text{ydx}}} \right)} \] the integral being taken in a counterclockwise direction over the closed curve C that forms the boundary of the region R shown in the figure below. The region R is the area enclosed by the union of a 2 × 3 rectangle and a semi-circle of radius 1. The line integral evaluates to”. This question is standard in competitive examination syllabi for Calculus.

\[\frac{1}{6}\] is the correct answer to “Consider the shaded triangular region P shown in the figure. What is If $$\iint\limits_{\text{P}} {{\text{xydxdy}}\,?}$$”. This question is standard in competitive examination syllabi for Calculus.

\[{\text{Div Curl }}\overrightarrow {\text{V}} = 0\] is the correct answer to “Let \[\phi \] be an arbitrary smooth real valued scalar function and V be an arbitrary smooth vector valued function in a three-dimensional space. Which one of the following is an identity”. This question is standard in competitive examination syllabi for Calculus.

4√2 is the correct answer to “The magnitude of the directional derivative of the function f(x, y) = x 2 + 3y 2 in a direction normal to the circle x 2 + y 2 = 2, at the point (1, 1), is”. This question is standard in competitive examination syllabi for Calculus.

$$\frac{1}{2}$$ is the correct answer to “If $${\text{S}} = \int\limits_1^\infty {{{\text{x}}^{ - 3}}{\text{dx,}}} $$ then S has the value”. This question is standard in competitive examination syllabi for Calculus.

x = -2 only is the correct answer to “The function f(x) = 2x 3 - 3x 2 - 36x + 2 has its maxima at”. This question is standard in competitive examination syllabi for Calculus.

x = 1 is the correct answer to “The minimum value of the function $${\text{f}}\left( {\text{x}} \right) = \left( {\frac{{{{\text{x}}^3}}}{3}} \right) - {\text{x}}$$ occurs at”. This question is standard in competitive examination syllabi for Calculus.

only a maxima at x = 1 is the correct answer to “The function f(x) = 2x - x 2 + 3 has”. This question is standard in competitive examination syllabi for Calculus.

is continuous $$\forall {\text{x}} \in {\text{R}}$$ and differentiable $$\forall {\text{x}} \in {\text{R}}$$ except at x = $$\frac{2}{3}$$ is the correct answer to “The function y = |2 - 3x|”. This question is standard in competitive examination syllabi for Calculus.

$$\int\limits_0^{\text{a}} {\int\limits_{\text{x}}^{\text{a}} {{\text{f}}\left( {{\text{x, y}}} \right){\text{dy dx}}} } $$ is the correct answer to “The double integral $$\int\limits_0^{\text{a}} {\int\limits_0^{\text{y}} {{\text{f}}\left( {{\text{x, y}}} \right){\text{dx dy}}} } $$ is equivalent to”. This question is standard in competitive examination syllabi for Calculus.

-4 and 1 is the correct answer to “The values of x for which the function $${\text{f}}\left( {\text{x}} \right) = \frac{{{{\text{x}}^2} - 3{\text{x}} - 4}}{{{{\text{x}}^2} + 3{\text{x}} - 4}}$$ is NOT continuous are”. This question is standard in competitive examination syllabi for Calculus.

$$4\int_0^{\text{R}} {\pi {\text{rH}}{{\left( {1 - \frac{{\text{r}}}{{\text{R}}}} \right)}^2}} {\text{dr}}$$ is the correct answer to “The expression $${\text{V}} = \int_0^{\text{H}} {\pi {{\text{R}}^2}{{\left( {1 - \frac{{\text{h}}}{{\text{H}}}} \right)}^2}} {\text{dh}}$$ for the volume of a cone is equal to”. This question is standard in competitive examination syllabi for Calculus.

continuous and non-differentiable is the correct answer to “Consider the function f(x) = |x| in the interval -1 < x ≤ 1. At the point x = 0, f(x) is”. This question is standard in competitive examination syllabi for Calculus.

$$2\overrightarrow {\text{i}} + 12\overrightarrow {\text{j}} - 4\overrightarrow {\text{k}} $$ is the correct answer to “For a scalar function f(x, y, z) = x 2 + 3y 2 + 2z 2 , the gradient at the point P(1, 2, -1) is”. This question is standard in competitive examination syllabi for Calculus.

1 is the correct answer to “The value of the quantity P, where $${\text{P}} = \int\limits_0^1 {{\text{x}}{{\text{e}}^{\text{x}}}{\text{dx,}}} $$ is equal to”. This question is standard in competitive examination syllabi for Calculus.

-√2 is the correct answer to “What is the value of $$\mathop {\lim }\limits_{{\text{x}} \to \frac{\pi }{4}} \frac{{\cos {\text{x}} - \sin {\text{x}}}}{{{\text{x}} - \frac{\pi }{4}}}$$”. This question is standard in competitive examination syllabi for Calculus.

1 is the correct answer to “The value of $$\mathop {\lim }\limits_{{\text{x}} \to \infty } {\left( {1 + {{\text{x}}^2}} \right)^{{{\text{e}}^{ - {\text{x}}}}}}$$ is”. This question is standard in competitive examination syllabi for Calculus.

60° is the correct answer to “For a right angled triangle, if the sum of the lengths of the hypotenuse and a side is kept constant, in order to have maximum area of the triangle, the angle between the hypotenuse and the side is”. This question is standard in competitive examination syllabi for Calculus.

$$\frac{1}{{12}}$$ is the correct answer to “The value of $$\mathop {\lim }\limits_{{\text{x}} \to 8} \frac{{{{\text{x}}^{\frac{1}{3}}} - 2}}{{\left( {{\text{x}} - 8} \right)}}$$”. This question is standard in competitive examination syllabi for Calculus.

(4i + 3j + 5k) is the correct answer to “The vector that is NOT perpendicular to the vectors (i + j + k) and (i + 2j + 3k) is . . . . . . . .”. This question is standard in competitive examination syllabi for Calculus.

i is the correct answer to “Given $${\text{i}} = \sqrt { - 1} ,$$ what will be the evaluation of the definite integral $$\int_0^{\frac{\pi }{2}} {\frac{{\cos {\text{x}} + {\text{i}}\sin {\text{x}}}}{{\cos {\text{x}} - {\text{i}}\sin {\text{x}}}}} {\text{dx}}\,?$$”. This question is standard in competitive examination syllabi for Calculus.

Continuous but not differentiable is the correct answer to “At x = 0, the function $${\text{f}}\left( {\text{x}} \right) = \left| {\sin \frac{{2\pi {\text{x}}}}{{\text{L}}}} \right|$$ , (-$$\infty $$ < x < $$\infty $$, L > 0) is”. This question is standard in competitive examination syllabi for Calculus.

$$\pi $$ is the correct answer to “The value of $$\int_0^\infty {\frac{1}{{1 + {{\text{x}}^2}}}} {\text{dx}} + \int_0^\infty {\frac{{\sin {\text{x}}}}{{\text{x}}}} {\text{dx}}$$ is”. This question is standard in competitive examination syllabi for Calculus.

0.5 and -0.5 respectively is the correct answer to “The values of the integrals $$\int\limits_0^1 {\left( {\int\limits_0^1 {\frac{{{\text{x}} - {\text{y}}}}{{{{\left( {{\text{x}} + {\text{y}}} \right)}^3}}}{\text{dy}}} } \right){\text{dx}}} $$ and $$\int\limits_0^1 {\left( {\int\limits_0^1 {\frac{{{\text{x}} - {\text{y}}}}{{{{\left( {{\text{x}} + {\text{y}}} \right)}^3}}}{\text{dx}}} } \right){\text{dy}}} $$ are”. This question is standard in competitive examination syllabi for Calculus.

\[{\rm{2\hat i}} - {\rm{\hat j}}\] is the correct answer to “$$\overrightarrow {\text{a}} ,\,\overrightarrow {\text{b}} ,\,\overrightarrow {\text{c}} $$ are three orthogonal vectors, Given that \[\overrightarrow {\rm{a}} = {\rm{\hat i}} + 2{\rm{\hat j}} + 5{\rm{\hat k}}\] and \[\overrightarrow {\rm{b}} = {\rm{\hat i}} + 2{\rm{\hat j}} - {\rm{\hat k}}\] , the vector $$\,\overrightarrow {\text{c}} $$ is parallel to”. This question is standard in competitive examination syllabi for Calculus.

-1 is the correct answer to “Given $$\overrightarrow {\text{F}} = \left( {{{\text{x}}^2} - 2{\text{y}}} \right)\overrightarrow {\text{i}} - 4{\text{yz}}\overrightarrow {\text{j}} + 4{\text{x}}{{\text{z}}^2}\overrightarrow {\text{k}} ,$$ the value of the line integral $$\int\limits_{\text{c}} {\overrightarrow {\text{F}} \cdot d\overrightarrow l } $$ along the straight line c from (0, 0, 0) to (1,1,1) is”. This question is standard in competitive examination syllabi for Calculus.

a minimum is the correct answer to “At x = 0, the function f(x) = |x| has”. This question is standard in competitive examination syllabi for Calculus.

$$\frac{\pi }{2}$$ is the correct answer to “A parabola x = y 2 with 0 ≤ x ≤ 1 is shown in the figure. The volume of the solid of rotation obtained by rotating the shaded area by 360° around the x-axis is”. This question is standard in competitive examination syllabi for Calculus.

x2y2 - x1y1 is the correct answer to “The line integral $$\int\limits_{{{\text{P}}_1}}^{{{\text{P}}_2}} {\left( {{\text{ydx}} + {\text{xdy}}} \right)} $$ for P 1 (x 1 , y 1 ) to P 2 (x 2 , y 2 ) along the semicircle P 1 , P 2 shown in the figure is”. This question is standard in competitive examination syllabi for Calculus.

45° is the correct answer to “The angle between two unit-magnitude coplanar vectors P(0.866, 0.500, 0) and Q(0.259, 0.966, 0) will be”. This question is standard in competitive examination syllabi for Calculus.

$$\pi $$ is the correct answer to “The following surface integral is to be evaluated over a sphere for the given steady velocity vector field F = xi + yj + zk defined with respect to a Cartesian coordinate system having i, j and k as unit base vectors. $$\iint\limits_{\text{S}} {\frac{1}{4}\left( {{\text{F}} \cdot {\text{n}}} \right){\text{dA}}}$$ where S is the sphere, x 2 + y 2 + z 2 = 1 and n is the outward unit normal vector to the sphere. The value of the surface integral is”. This question is standard in competitive examination syllabi for Calculus.

the limit must exist at the point and the value of limit should be same as the value of the function at that point is the correct answer to “If a function is continuous at a point,”. This question is standard in competitive examination syllabi for Calculus.

7i - 3j + 8k is the correct answer to “The vector that is normal to the surface 2xz 2 - 3xy - 4x = 7 at the point (1, -1, 2) is”. This question is standard in competitive examination syllabi for Calculus.

1 real and 4 complex roots is the correct answer to “The polynomial p(x) = x 5 + x + 2 has”. This question is standard in competitive examination syllabi for Calculus.

25 is the correct answer to “A function y = 5x 2 + 10x is defined over an open interval x = (1, 2). At least at one point in this interval, $$\frac{{{\text{dy}}}}{{{\text{dx}}}}$$ is exactly”. This question is standard in competitive examination syllabi for Calculus.