# calculus 1 marks

GATE 2019 Set-1

1. A parabola x=y2  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

(A) π/4

(B) π/2

(C) π

(D) 2π

2. Evaluation of $$\large \int_{2}^{4}x^{3}dx$$ using a 2-equal-segment trapezoidal rule gives a value of ________

GATE 2019 Set-2

1. The directional derivative of the function f(x,y) = x2+y2 along a line directed from (0,0)  to (1,1), evaluated at the point x = 1, y = 1 is

(A) √2

(B) 2

(C) 2√2

(D) 4√2

GATE 2018 Set-1

1. According to the Mean Value Theorem, for a continuous function f(x) in the interval [a,b] there exists a value ξ in this interval such that $$\large \int_{a}^{b}f(x)dx=$$

(A) f(ξ)(b-a)

(B) f(b)(ξ-a)

(C) f(a)(b-ξ)

(D) 0

GATE 2018 Set-2

1. The Fourier cosine series for an even function $$f\left ( x \right )$$ is given by

$$\large f\left ( x \right )=a_{o}+\sum_{n=1}^{\infty }a _{n}cos\left ( nx \right )$$

The value of the coefficient $$a_{2}$$ for the function $$f\left ( x \right )=cos^{2}\left ( x \right )$$ in $$\left [ 0,\pi \right ]$$ is

(A) -0.5

(B) 0.0

(C) 0.5

(D) 1.0

2. The divergence of the vector field $$\vec {u}=e^{x}\left ( cosy\hat{i} +siny\hat{j}\right )$$ is

(A) 0

(B) $$e^{x}cos y +e^{x}sin y$$

(C) $$2e^{x}cos y$$

(D) $$2e^{x}sin y$$

GATE 2017 Set-1

1. The value of $$\large \underset{x\rightarrow 0}{lim}\;\frac{x^3-\sin x}{x}$$

(A) 0

(B) 3

(C) 1

(D) -1

GATE 2017 Set-2

1. The divergence of the vector -yi – xj is __________

GATE 2016 Set-2

1. The values of x for which the function
$$\large f(x)=\frac{x^2-3x-4}{x^2+3x-4}$$

is NOT continuous are

(A) 4 and −1

(B) 4 and 1

(C) −4 and 1

(D) −4 and −1

GATE 2016 Set-3

1. $$\large Lt_{x\rightarrow 0}\frac{\log_{e}(1+4x)}{e^{3x}-1}$$ is equal to

(A) 0

(B) 1/12

(C) 4/3

(D) 1

GATE 2015 Set-1

1. The value of $$\large lim_{x\rightarrow 0}\frac{1-\cos x^2}{2x^4}$$

(A) 0

(B) 1/2

(C) 1/4

(D) undefined

GATE 2015 Set-2

1. At x = 0, the function f(x)=|x| has

(A) a minimum

(B) a maximum

(C) a point of inflexion

(D) neither a maximum nor minimum

2. Curl of vector $$\large V(x,y,z)=2x^2\hat{i}+3z^2\hat j+y^3\hat k$$ at x=y=z=1 is

(A) −3i

(B) 3i

(C) 3i– 4j

(D) 3i– 6k