GATE Questions & Answers of Calculus Electronics and Communication Engg

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 -1x0

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 0x1

IV.         $ y_1,\;y_2 $ and $y_3$ and  are linearly dependent on -1x0

Which one among the following is correct?

Let f(x)=ex+x2 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 x2+y2z3; 0z1 where x, y, z are real. The volume of R (up to two decimal places ) is_____________.

Let l=c(2z dx+2y dy+2x dz) 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(x)=13x(x2-3) in the interval -100  x  100 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=x0, then it is also differentiable at x=x0.
Q: If $f(x)$ is continuous at x=x0, then it may not be differentiable at x=x0.
R: If $f(x)$is differentiable at x=x0, then it is also continuous at x=x0.

Consider the plot of fx versus x as shown below.

Suppose Fx=-5xfydy. 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: x2+y24, evaluates to_____

The region specified by ρ,φ,z:3ρ5,π8φπ4,3z4.5 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 ω>0 is a constant. When the vector magnitude is at its minimum value, the angle θ that I makes with the x axis (in degrees, such that 0θ180) is ________

Suppose C is the closed curve defined as the circle x2+y2=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 fx = e-xx2+ x + 1?

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

The value of the integral -12cos2πtsin4πt4πtd 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 0yx and 0x12 is______

For  0t<, the maximum value of the function ft=e-t-2e-2t occurs at

The value of

                l i m x ? 8 1 + 1 x x

is

The maximum value of the function fx=ln1+x-xwhere x>-1 occurs at x =______.

If z=xy lnxythen

The maximum value of fx=2x3-9x2+12x-3 in the interval 0x3 is _______.

The series n=01n! 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 θ until which the approximation sinθθ holds to within 10% error is

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

If ey=x1x then y has a

If A=xy ax^+x2ay^then ${\oint_c}$A.dl 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

limθ0sinθ/2θ is

Which one of the following functions is strictly bounded?

It is given that X1, X2···XM are M non-zero, orthogonal vectors. The dimension of the vector space spanned by the 2M vectors X1, X2···XM, -X1, -X2···-XM is

Consider the function f(t) = x2 - x - 2. The maximum value of f(x) in the closed interval [-4, 4] is: