Calculus MCQs Question and Answer in Engineering Maths

41

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

A) 0 B) $$\frac{{\text{a}}}{2}$$ C) a D) 2a

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Answer: Option B

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42

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

A) -2 B) 2 C) 1 D) 0

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Answer: Option A

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43

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

A) 0° B) 30° C) 45° D) 90°

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Answer: Option D

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44

The expression e^{-$$l$$n x} for x > 0 is equal to

A) -x B) x C) x^-1 D) -x^-1

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Answer: Option C

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45

While minimizing the function f(x), necessary and sufficient conditions for a point x₀ to be a minima are

A) f'(x₀) > 0 and f''(x₀) = 0 B) f'(x₀) < 0 and f''(x₀) = 0 C) f'(x₀) = 0 and f''(x₀) < 0 D) f'(x₀) = 0 and f''(x₀) > 0

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Answer: Option D

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46

The function f(x) = x² = x + x ...... x times, is defined

A) at all real values of x B) only at positive integer values of x C) only at negative integer value of x D) only at rational values of x

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Answer: Option B

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47

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
Calculus mcq question image

A) (13.40, 22.32) B) (18.66, 12.32) C) (22.32, 8.26) D) (12.32, 18.66)

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Answer: Option D

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48

If z = xy $$l$$n(xy), then

A) $${\text{x}}\frac{{\partial {\text{z}}}}{{\partial {\text{x}}}} + {\text{y}}\frac{{\partial {\text{z}}}}{{\partial {\text{y}}}} = 0$$ B) $${\text{y}}\frac{{\partial {\text{z}}}}{{\partial {\text{x}}}} = {\text{x}}\frac{{\partial {\text{z}}}}{{\partial {\text{y}}}}$$ C) $${\text{x}}\frac{{\partial {\text{z}}}}{{\partial {\text{x}}}} = {\text{y}}\frac{{\partial {\text{z}}}}{{\partial {\text{y}}}}$$ D) $${\text{y}}\frac{{\partial {\text{z}}}}{{\partial {\text{x}}}} + {\text{x}}\frac{{\partial {\text{z}}}}{{\partial {\text{y}}}} = 0$$

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Answer: Option C

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49

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

A) $${{\text{a}}^2}{{\text{b}}^2} - {\left( {\overrightarrow {\text{a}} \cdot \overrightarrow {\text{b}} } \right)^2}$$ B) $${\text{ab}} - \overrightarrow {\text{a}} \cdot \overrightarrow {\text{b}} $$ C) $${{\text{a}}^2}{{\text{b}}^2} + {\left( {\overrightarrow {\text{a}} \cdot \overrightarrow {\text{b}} } \right)^2}$$ D) $${\text{ab}} + \overrightarrow {\text{a}} \cdot \overrightarrow {\text{b}} $$

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Answer: Option A

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50

The value of the integral given below is
$$\int\limits_0^\pi {{{\text{x}}^2}\cos \,{\text{x dx}}} $$

A) $$ - 2\pi $$ B) $$ - \pi $$ C) $$\pi $$ D) $$2\pi $$

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Answer & Solution

Answer: Option A

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