Engineering Maths · all questions

Calculus
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Q1

What is the value of

y \to 0 x \to 0 lim \frac{xy}{x ^{2} + y ^{2}} ?

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Q2

The Taylor series expansion of 3 sinx + 2 cosx is . . . . . . . .

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Q3

x \to \infty lim x^{2} + x - 1 - x is

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Q4

The Taylor series expansion of

\frac{sin x}{x - π}

at

x = π

is given by

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Q5

If x = a (θ + sin θ) and y = a (1 - cos θ), then

\frac{dy}{dx}

will be equal to

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Q6

Consider the line integral

C \int (xdy - ydx)

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
Calculus mcq question image

Calculus mcq question image

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Q7

Consider the shaded triangular region P shown in the figure. What is If

P \iint xydxdy ?

Calculus mcq question image

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Q8

Let

ϕ

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?

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Q9

The magnitude of the directional derivative of the function f(x, y) = x² + 3y² in a direction normal to the circle x² + y² = 2, at the point (1, 1), is

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Q10

If

S = 1 \int \infty x^{- 3} dx,

then S has the value

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Q11

The function f(x) = 2x³ - 3x² - 36x + 2 has its maxima at

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Q12

The minimum value of the function

f (x) = (\frac{x ^{3}}{3}) - x

occurs at

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Q13

The function f(x) = 2x - x² + 3 has

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Q14

The function y = |2 - 3x|

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Q15

Which of the following is correct?

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Q16

The double integral

0 \int a 0 \int y f (x, y) dx dy

is equivalent to

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Q17

The values of x for which the function

f (x) = \frac{x ^{2} - 3 x - 4}{x ^{2} + 3 x - 4}

is NOT continuous are

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Q18

The expression

V = \int_{0}^{H} π R^{2} (1 - \frac{h}{H})^{2} dh

for the volume of a cone is equal to

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Q19

Consider the function f(x) = |x| in the interval -1 < x ≤ 1. At the point x = 0, f(x) is

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Q20

For a scalar function f(x, y, z) = x² + 3y² + 2z², the gradient at the point P(1, 2, -1) is

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