# GATE Questions & Answers of Calculus Electronics and Communication Engg

#### Calculus 43 Question(s)

Consider the following statement about the linear dependence of the real valued function $y_1=1,y_2=x$ and $y_3=x^2$, over the field of real of real numbers.

I.           $y_1,\;y_2$ and $y_3$ are linearly independent on $-1\le x\le 0$

II.          $y_1,\;y_2$ and $y_3$ are linearly dependent on $0\leq x\leq1$

III.         $y_1,\;y_2$ and $y_3$ are linearly independent on $0\le x\le 1$

IV.         $y_1,\;y_2$ and $y_3$ and  are linearly dependent on $-1\le x\le 0$

Which one among the following is correct?

Let $f\left(x\right)={e}^{x+{x}^{2}}$ for real $x$. From among the following, choose the Taylor series approximation of $f\left(x\right)$ around $x=0$, which includes all power of $x$ less than or equal to 3.

A three dimension region R of finite volume is described by where x, y, z are real. The volume of R (up to two decimal places ) is_____________.

Let where $x, y, z$ are real, and let C be the straight line segment from point A:(0,2,1) to point B:(4,1,-1). The value of l is_________.

The smaller angle (in degrees) between the planes x + y + z = 1 and 2x - y + 2z = 0 is___________.

The values of the integrals

$\int\limits_0^1\left(\int\limits_0^1\frac{x-y}{(x+y)^3}\operatorname dy\right)\operatorname dx$

$\int\limits_0^1\left(\int\limits_0^1\frac{x-y}{(x+y)^3}\operatorname dx\right)\operatorname dy$

are

The minimum value of the function $f\left(x\right)=\frac{1}{3}x\left({x}^{2}-3\right)$ in the interval occurs at x=_________.

Given the following statements about a function $\style{font-family:'Times New Roman'}{f:\mathbb{R}\rightarrow\mathbb{R}}$, select the right option:

P: If $f(x)$is continuous at $x={x}_{0}$, then it is also differentiable at $x={x}_{0}$.
Q: If $f(x)$ is continuous at $x={x}_{0}$, then it may not be differentiable at $x={x}_{0}$.
R: If $f(x)$is differentiable at $x={x}_{0}$, then it is also continuous at $x={x}_{0}$.

Consider the plot of $f\left(x\right)$ versus $x$ as shown below.

Suppose $F\left(x\right)={\int }_{-5}^{x}f\left(y\right)dy.$ Which one of the following is a graph of $F\left(x\right)?$

The integral $\frac1{2\mathrm\pi}\iint_D\left(x+y+10\right)dx\;dy$ where D denotes the disc: ${x}^{2}+{y}^{2}\le 4$, evaluates to_____

The region specified by $\left\{\left(\rho ,\phi ,z\right):3\le \rho \le 5,\frac{\mathrm{\pi }}{8}\le \phi \le \frac{\mathrm{\pi }}{4},3\le \mathrm{z}\le 4.5\right\}$ in cylindrical coordinates has volume of _______

As $x$ varies from -1 to +3,  which one of the following describes the behaviour of the function $f\left(x\right)=x^3-3x^2+1?$

How many distinct values of $x$ satisfy the equation $\sin\left(x\right)=x/2,$ where $x$ is in radians?

Consider the time-varying vector $\mathbf I=\widehat{\mathrm x}\;15\;\cos\left(\mathrm{ωt}\right)+\widehat{\mathrm y}\;5\;\sin\left(\mathrm{ωt}\right)$ in Cartesian coordinates, where $\mathrm{\omega }>0$ is a constant. When the vector magnitude is at its minimum value, the angle $\mathrm{\theta }$ that $\mathrm{I}$ makes with the x axis (in degrees, such that $0\le \mathrm{\theta }\le 180$) is ________

Suppose C is the closed curve defined as the circle ${x}^{2}+{y}^{2}=1$ with C oriented anti-clockwise. The value of $\oint\left(xy^2\;dx\;+x^2y\;dy\right)$ over the curve C equals ________

The integral $\int\limits_0^1\frac{dx}{\sqrt{\left(1-x\right)}}$ is equal to __________

If the vectors e1 = (1, 0, 2), e2 = (0, 1, 0) and e3 = (−2, 0, 1) form an orthogonal basis of the threedimensional real space$\mathbb{R}^3$, then the vector u = (4, 3,−3) $\in\mathbb{R}^3$ can be expressed as

A triangle in the xy-plane is bounded by the straight lines 2x = 3y, y = 0 and x = 3. The volume above the triangle and under the plane x + y +z = 6 is __________

A function (x) = 1 – x2 + x3 is defined in the closed interval [–1,1]. The value of x, in the open interval (–1,1) for which the mean value theorem is satisfied, is

Which one of the following graphs describes the function ?

The maximum area (in sqare units) of a rectangle whose vertices lie on the ellipse x2 + 4y2 = 1 is ______.

The value of the integral is ________ .

The contour on the x-y plane, where the partial derivative of x2+y2 with respect to y is equal to the partial derivative of 6y+4x with respect to x, is

The volume under the surface z(x, y) = x + y and above the triangle in the x-y plane defined by is______

For, the maximum value of the function $f\left(t\right)={e}^{-t}-2{e}^{-2t}$ occurs at

The value of

is

The maximum value of the function occurs at x =______.

If then

The maximum value of $f\left(x\right)=2{x}^{3}-9{x}^{2}+12x-3$ in the interval $0\le x\le 3$ is _______.

The series $\sum _{n=0}^{\infty }\frac{1}{n!}$ converges 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

The maximum value of $\theta$ until which the approximation $\mathrm{sin}\theta \approx \theta$ holds to within 10% error is

The maximum value of f(x) = x3 - 9x2 + 24x + 5 in the interval [1, 6] is

If ey=${x}^{\frac{1}{x}}$ then y has a

If then ${\oint_c}$$\stackrel{\to }{A}.d\stackrel{\to }{l}$ over the path shown in the figure is

For real values of x, the minimum value of the function f (x) = exp (x) + exp (-x) is

The value of the integral of the function g(x, y) = 4x3 + 10y4 along the straight line segment from the point (0, 0) to the point (1, 2) in the x-y plane is

Consider points P and Q in the x-y plane, with P = (1, 0) and Q = (0, 1). The line integral $2\int\limits_P^Q\left(xdx+ydy\right)$ along the semicircle with the line segment PQ as its diameter

The following plot shows a function y which varies linearly with x. The value of the integral $I=\int\limits_1^2y\;dx$ is

$\underset{\theta \to 0}{\mathrm{lim}}\frac{\mathrm{sin}\left(\theta /2\right)}{\theta }$ is

Which one of the following functions is strictly bounded?