Calculus MCQ

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• If $$\overrightarrow {\text{r}} $$ is the position vector of any point on a closed surface S that encloses volume V then $$\iint\limits_{\text{S}} {\overrightarrow {\text{r}} \cdot {\text{d}}\overrightarrow {\text{S}} }$$  is equal to
  ক

  $$\frac{1}{2}$$V

  খ

  V

  গ

  2V

  ঘ

  3V
• The value of the integral given below is<br>$$\int\limits_0^\pi {{{\text{x}}^2}\cos \,{\text{x dx}}} $$
  ক

  $$ - 2\pi $$

  খ

  $$ - \pi $$

  গ

  $$\pi $$

  ঘ

  $$2\pi $$
• If $$\overrightarrow {\text{a}} $$ and $$\overrightarrow {\text{b}} $$ are two arbitrary vectors with magnitudes a and b, respectively, $${\left| {\overrightarrow {\text{a}} \times \overrightarrow {\text{b}} } \right|^2}$$  will be equal to
  ক

  $${{\text{a}}^2}{{\text{b}}^2} - {\left( {\overrightarrow {\text{a}} \cdot \overrightarrow {\text{b}} } \right)^2}$$

  খ

  $${\text{ab}} - \overrightarrow {\text{a}} \cdot \overrightarrow {\text{b}} $$

  গ

  $${{\text{a}}^2}{{\text{b}}^2} + {\left( {\overrightarrow {\text{a}} \cdot \overrightarrow {\text{b}} } \right)^2}$$

  ঘ

  $${\text{ab}} + \overrightarrow {\text{a}} \cdot \overrightarrow {\text{b}} $$
• If z = xy $$l$$n(xy), then
  ক

  $${\text{x}}\frac{{\partial {\text{z}}}}{{\partial {\text{x}}}} + {\text{y}}\frac{{\partial {\text{z}}}}{{\partial {\text{y}}}} = 0$$

  খ

  $${\text{y}}\frac{{\partial {\text{z}}}}{{\partial {\text{x}}}} = {\text{x}}\frac{{\partial {\text{z}}}}{{\partial {\text{y}}}}$$

  গ

  $${\text{x}}\frac{{\partial {\text{z}}}}{{\partial {\text{x}}}} = {\text{y}}\frac{{\partial {\text{z}}}}{{\partial {\text{y}}}}$$

  ঘ

  $${\text{y}}\frac{{\partial {\text{z}}}}{{\partial {\text{x}}}} + {\text{x}}\frac{{\partial {\text{z}}}}{{\partial {\text{y}}}} = 0$$
• The position vector $$\overrightarrow {{\text{OP}}} $$ of P(20, 10) is rotated anti-clockwise in X-Y plane by an angle θ = 30° such that the point P occupies position Q, as shown in the figure. The coordinates (x, y) of Q are<br><img src="/images/question-image/engineering-maths/calculus/1689232201-the-position-vector-overrightarrow-textop-of-p20-10-is-rotated-anti-clockwise.png" title="Calculus mcq question image" alt="Calculus mcq question image">
  ক

  (13.40, 22.32)

  খ

  (18.66, 12.32)

  গ

  (22.32, 8.26)

  ঘ

  (12.32, 18.66)
• The function f(x) = x² = x + x ...... x times, is defined
  ক

  at all real values of x

  খ

  only at positive integer values of x

  গ

  only at negative integer value of x

  ঘ

  only at rational values of x
• While minimizing the function f(x), necessary and sufficient conditions for a point x₀ to be a minima are
  ক

  f'(x₀) &gt; 0 and f''(x₀) = 0

  খ

  f'(x₀) 0) = 0

  গ

  f'(x₀) = 0 and f''(x₀)

  ঘ

  f'(x₀) = 0 and f''(x₀) &gt; 0
• The expression e^{-$$l$$n x} for x &gt; 0 is equal to
  ক

  -x

  খ

  x

  গ

  x^-1

  ঘ

  -x^-1
• The angle of intersection of the curves x² = 4y and y² = 4x at point (0, 0) is
  ক

  0°

  খ

  30°

  গ

  45°

  ঘ

  90°
• The direction of vector A is radially outward from the origin, with |A| = krⁿ where r² = x² + y² + z² and k is a constant. The value of n for which $$\nabla \cdot {\text{A}} = 0$$   is
  ক

  -2

  খ

  2

  গ

  1

  ঘ

  0
• What is the value of the definite integral, $$\int\limits_0^{\text{a}} {\frac{{\sqrt {\text{x}} }}{{\sqrt {\text{x}} + \sqrt {{\text{a}} - {\text{x}}} }}} {\text{dx}}\,{\text{?}}$$
  ক

  0

  খ

  $$\frac{{\text{a}}}{2}$$

  গ

  a

  ঘ

  2a
• Equation of the line normal to function $${\text{f}}\left( {\text{x}} \right) = {\left( {{\text{x}} - 8} \right)^{\frac{2}{3}}} + 1$$    at P(0, 5) is
  ক

  y = 3x - 5

  খ

  y = 3x + 5

  গ

  3y = x + 15

  ঘ

  3y = x - 15
• $$\mathop {\lim }\limits_{{\text{x}} \to 0} \frac{{{{\sin }^2}{\text{x}}}}{{\text{x}}}$$   is equal to
  ক

  0

  খ

  $$\infty $$

  গ

  1

  ঘ

  -1
• If $${{\text{e}}^{\text{y}}} = {{\text{x}}^{\frac{1}{{\text{x}}}}},$$  then y has a
  ক

  maximum at x = e

  খ

  minimum at x = e

  গ

  maximum at x = e^-1

  ঘ

  minimum at x = e^-1
• The value of $$\mathop {\lim }\limits_{\left( {{\text{x,}}\,{\text{y}}} \right) \to \left( {0,\,0} \right)} \frac{{{{\text{x}}^2} - {\text{xy}}}}{{\sqrt {\text{x}} - \sqrt {\text{y}} }}$$    is
  ক

  0

  খ

  $$\frac{1}{2}$$

  গ

  1

  ঘ

  100
• The curl of the gradient of the scalar field defined by V = 2x²y + 3y²z + 4z²x is
  ক

  4xy a_x + 6yz a_y + 8zx a_z

  খ

  4a_x + 6a_y + 8a_z

  গ

  (4xy + 4z²)a_x + (2x² + 6yz)a_y + (3y² + 8zx)a_z

  ঘ

  0
• If f(x) is an even function and a is a positive real number, then $$\int_{ - {\text{a}}}^{\text{a}} {{\text{f}}\left( {\text{x}} \right){\text{dx}}} $$   equals
  ক

  0

  খ

  a

  গ

  2a

  ঘ

  $$2\int_0^{\text{a}} {{\text{f}}\left( {\text{x}} \right){\text{dx}}} $$
• The curve y = f(x) is such that the tangent to the curve at every point (x, y) has a Y-axis intercept c, given by c = -y. Then f(x) is proportional to
  ক

  x^-1

  খ

  x²

  গ

  x³

  ঘ

  x⁴
• Let $${\text{f}}\left( {{\text{x, y}}} \right) = \frac{{{\text{a}}{{\text{x}}^2} + {\text{b}}{{\text{y}}^2}}}{{{\text{xy}}}},$$     where a and b are constants. If $$\frac{{\partial {\text{f}}}}{{\partial {\text{x}}}} = \frac{{\partial {\text{f}}}}{{\partial {\text{y}}}}$$   at x = 1 and y = 2, then the relation between a and b is
  ক

  $${\text{a}} = \frac{{\text{b}}}{4}$$

  খ

  $${\text{a}} = \frac{{\text{b}}}{2}$$

  গ

  a = 2b

  ঘ

  a = 4b
• Let $$\nabla \cdot \left( {{\text{f}}\overrightarrow {\text{v}} } \right) = {{\text{x}}^2}{\text{y}} + {{\text{y}}^2}{\text{z}} + {{\text{z}}^2}{\text{x}},$$      where f and v are scalar and vector fields respectively. If $$\overrightarrow {\text{v}} = {\text{y}}\overrightarrow {\text{i}} + {\text{z}}\overrightarrow {\text{j}} + {\text{x}}\overrightarrow {\text{k}} ,$$     then $$\overrightarrow {\text{v}} \cdot \nabla {\text{f}}$$   is
  ক

  x²y + y²z + z²x

  খ

  2xy + 2yz + 2zx

  গ

  x + y + z

  ঘ

  0
• $$\mathop {\lim }\limits_{{\text{x}} \to \infty } {{\text{x}}^{\frac{1}{{\text{x}}}}}$$  is
  ক

  $$\infty $$

  খ

  0

  গ

  1

  ঘ

  Not defined
• The area between the parabolas y² = 4ax and x² = 4ay is
  ক

  $$\frac{2}{3}{{\text{a}}^2}$$

  খ

  $$\frac{{14}}{3}{{\text{a}}^2}$$

  গ

  $$\frac{{16}}{3}{{\text{a}}^2}$$

  ঘ

  $$\frac{{17}}{3}{{\text{a}}^2}$$
• Consider a vector field $$\overrightarrow {\text{A}} \left( {\overrightarrow {\text{r}} } \right).$$  The closed loop line integral $$\oint {\overrightarrow {\text{A}} \cdot \overrightarrow {{\text{d}}l} } $$  can be expressed as
  ক

  $$\mathop{{\int\!\!\!\!\!\int}\mkern-21mu \bigcirc} {\left( {\nabla \times \overrightarrow {\text{A}} } \right) \cdot \overrightarrow {{\text{ds}}} } $$    over the closed surface bounded by the loop

  খ

  $$\mathop{{\int\!\!\!\!\!\int\!\!\!\!\!\int}\mkern-31.2mu \bigodot} {\left( {\nabla \cdot \overrightarrow {\text{A}} } \right){\text{dv}}} $$    over the closed volume bounded by the top

  গ

  $$\iiint {\left( {\nabla \cdot \overrightarrow {\text{A}} } \right){\text{dv}}}$$    over the open volume bounded by the loop

  ঘ

  $$\iint {\left( {\nabla \times \overrightarrow {\text{A}} } \right) \cdot \overrightarrow {{\text{ds}}} }$$    over the open surface bounded by the loop
• The value of the line integral $$\int\limits_{\text{c}} {\left( {2{\text{x}}{{\text{y}}^2}{\text{dx}} + 2{{\text{x}}^2}{\text{ydy}} + {\text{dz}}} \right)} $$     along a path joining the origin (0, 0, 0) and the point (1, 1, 1) is
  ক

  0

  খ

  2

  গ

  4

  ঘ

  6
• The value of the integral $$\int_0^\pi {{\text{x}}{{\cos }^2}{\text{xdx}}} $$   is
  ক

  $$\frac{{{\pi ^2}}}{8}$$

  খ

  $$\frac{{{\pi ^2}}}{4}$$

  গ

  $$\frac{{{\pi ^2}}}{2}$$

  ঘ

  p²
• In the Taylor series expansion of e^x about x = 2, the coefficient of (x - 2)⁴ is
  ক

  $$\frac{1}{{4!}}$$

  খ

  $$\frac{{{2^4}}}{{4!}}$$

  গ

  $$\frac{{{{\text{e}}^2}}}{{4!}}$$

  ঘ

  $$\frac{{{{\text{e}}^4}}}{{4!}}$$
• The value of the definite integral $$\int_1^{\text{e}} {\sqrt {\text{x}} } \ln \left( {\text{x}} \right){\text{dx}}$$    is
  ক

  $$\frac{4}{9}\sqrt {{{\text{e}}^3}} + \frac{2}{9}$$

  খ

  $$\frac{2}{9}\sqrt {{{\text{e}}^3}} - \frac{4}{9}$$

  গ

  $$\frac{2}{9}\sqrt {{{\text{e}}^3}} + \frac{4}{9}$$

  ঘ

  $$\frac{4}{9}\sqrt {{{\text{e}}^3}} - \frac{2}{9}$$
• For two non-zero vectors $$\overrightarrow {\text{A}} $$ and $$\overrightarrow {\text{B}} $$, if $$\overrightarrow {\text{A}} $$ + $$\overrightarrow {\text{B}} $$ is perpendicular to $$\overrightarrow {\text{A}} $$ - $$\overrightarrow {\text{B}} $$ then,
  ক

  Magnitude of $$\overrightarrow {\text{A}} $$ is twice magnitude of $$\overrightarrow {\text{B}} $$

  খ

  Magnitude of $$\overrightarrow {\text{A}} $$ is half the magnitude of $$\overrightarrow {\text{B}} $$

  গ

  $$\overrightarrow {\text{A}} $$ and $$\overrightarrow {\text{B}} $$ cannot be orthogonal

  ঘ

  the magnitudes of $$\overrightarrow {\text{A}} $$ and $$\overrightarrow {\text{B}} $$ are equal
• If $${\text{f}}\left( {\text{x}} \right) = \frac{{2{{\text{x}}^2} - 7{\text{x}} + 3}}{{5{{\text{x}}^2} - 12{\text{x}} - 9}},$$     then $$\mathop {\lim }\limits_{{\text{x}} \to 3} {\text{f}}\left( {\text{x}} \right)$$  will be
  ক

  $$ - \frac{1}{3}$$

  খ

  $$\frac{5}{{18}}$$

  গ

  0

  ঘ

  $$\frac{2}{5}$$
• The value of the integral $$\int\limits_{ - \infty }^\infty {\frac{{{\text{dx}}}}{{1 + {x^2}}}} $$  is
  ক

  $$ - \pi $$

  খ

  $$ - \frac{\pi }{2}$$

  গ

  $$\frac{\pi }{2}$$

  ঘ

  $$\pi $$