Calculus
Practice QuestionsMarks 1
1
Consider the set $S$ of points $(x, y) \in R^2$ which minimize the real valued function
$$ f(x, y)=(x+y-1)^2+(x+y)^2 $$
Which of the following statements is true about the set $S$ ?
GATE EE 2025
2
Let $$x$$ and $$y$$ be integers satisfying the following equations
$$$2{x^2} + {y^2} = 34$$$
$$$x + 2y = 11$$$
The value of $$(x+y)$$ is _________.
The value of $$(x+y)$$ is _________.
GATE EE 2017 Set 2
3
Consider a function $$f\left( {x,y,z} \right)$$ given by $$f\left( {x,y,z} \right) = \left( {{x^2} + {y^2} - 2{z^2}} \right)\left( {{y^2} + {z^2}} \right).$$ The partial derivative of this function with respect to $$x$$ at the point $$x=2, y=1$$ and $$z=3$$ is _______.
GATE EE 2017 Set 2
4
Let $$\,{y^2} - 2y + 1 = x$$ and $$\,\sqrt x + y = 5.\,\,$$ The value of $$\,x + \sqrt y \,\,$$ equals ________. (Given the answer up to three decimal places)
GATE EE 2017 Set 2
5
Let $${\rm I} = c\int {\int {_Rx{y^2}dxdy,\,\,} } $$ where $$R$$ is the region shown in the figure and $$c = 6 \times {10^{ - 4}}.\,\,$$ The value of $${\rm I}$$ equals ___________.
GATE EE 2017 Set 1
6
The maximum value attained by the function $$f(x)=x(x-1) (x-2)$$ in the interval $$\left[ {1,2} \right]$$ is _________.
GATE EE 2016 Set 1
7
Minimum of the real valued function $$f\left( x \right) = {\left( {x - 1} \right)^{2/3}}$$ occurs at $$x$$ equal to
GATE EE 2014 Set 2
8
A particle, starting from origin at $$t=0$$ $$s,$$ is traveling along $$x$$-axis with velocity $$v = {\pi \over 2}\cos \left( {{\pi \over 2}t} \right)m/s$$
At $$t=3$$ $$s,$$ the difference between the distance covered by the particle and the magnitude of displacement from the origin is _________.
At $$t=3$$ $$s,$$ the difference between the distance covered by the particle and the magnitude of displacement from the origin is _________.
GATE EE 2014 Set 3
9
Let $$f\left( x \right) = x{e^{ - x}}.$$ The maximum value of the function in the interval $$\left( {0,\infty } \right)$$ is
GATE EE 2014 Set 1
10
A function $$y = 5{x^2} + 10x\,\,$$ is defined over an open interval $$x=(1,2).$$ At least at one point in this interval, $${{dy} \over {dx}}$$ is exactly
GATE EE 2013
11
Roots of the algebraic equation $${x^3} + {x^2} + x + 1 = 0$$ are
GATE EE 2011
12
The function $$f\left( x \right) = 2x - {x^2} + 3\,\,$$ has
GATE EE 2011
13
At $$t=0,$$ the function $$f\left( t \right) = {{\sin t} \over t}\,\,$$ has
GATE EE 2010
14
Consider the function $$f\left( x \right) = {\left( {{x^2} - 4} \right)^2}$$ where $$x$$ is a real number. Then the function has
GATE EE 2007
15
For the function $$f\left( x \right) = {x^2}{e^{ - x}},$$ the maximum occurs when $$x$$ is equal to
GATE EE 2005
16
If $$S = \int\limits_1^\infty {{x^{ - 3}}dx} $$ then $$S$$ has the value
GATE EE 2005
17
The area enclosed between the parabola $$y = {x^2}$$ and the straight line $$y=x$$ is _______.
GATE EE 2004
18
$$\mathop {Lim}\limits_{\theta \to 0} \,{{\sin \,m\,\theta } \over \theta },$$ where $$m$$ is an integer, is one of the following :
GATE EE 1997
19
$$\mathop {Lim}\limits_{x \to \infty } \,x\sin {\raise0.5ex\hbox{$\scriptstyle 1$}
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbox{$\scriptstyle x$}} = \_\_\_\_\_.$$
GATE EE 1995
20
If $$f(0)=2$$ and $$f'\left( x \right) = {1 \over {5 - {x^2}}},$$ then the lower and upper bounds of $$f(1)$$ estimated by the mean value theorem are ______.
GATE EE 1995
21
The volume generated by revolving the area bounded by the parabola $${y^2} = 8x$$ and the line $$x=2$$ about $$y$$-axis is
GATE EE 1994
22
The integration of $$\int {{\mathop{\rm logx}\nolimits} \,dx} $$ has the value
GATE EE 1994
Marks 2
1
Let $f(t)$ be a real-valued function whose second derivative is positive for $- \infty < t < \infty$. Which of the following statements is/are always true?
GATE EE 2024
2
Consider the function $f(t) = (\text{max}(0,t))^2$ for $- \infty < t < \infty$, where $\text{max}(a,b)$ denotes the maximum of $a$ and $b$. Which of the following statements is/are true?
GATE EE 2024
3
Consider the following equation in a 2-D real-space.
$$|{x_1}{|^p} + |{x_2}{|^p} = 1$$ for $$p > 0$$
Which of the following statement(s) is/are true.
GATE EE 2023
4
Let $$f(x) = \int\limits_0^x {{e^t}(t - 1)(t - 2)dt} $$. Then f(x) decreases in the interval.
GATE EE 2022
5
Let R be a region in the first quadrant of the xy plane enclosed by a closed curve C considered in counter-clockwise direction. Which of the following expressions does not represent the area of the region R?
GATE EE 2022
6
Let $$g\left( x \right) = \left\{ {\matrix{
{ - x} & {x \le 1} \cr
{x + 1} & {x \ge 1} \cr
} } \right.$$ and
$$f\left( x \right) = \left\{ {\matrix{ {1 - x,} & {x \le 0} \cr {{x^{2,}}} & {x > 0} \cr } } \right..$$
Consider the composition of $$f$$ and $$g,$$ i.e., $$\left( {f \circ g} \right)\left( x \right) = f\left( {g\left( x \right)} \right).$$ The number of discontinuities in $$\left( {f \circ g} \right)\left( x \right)$$ present in the interval $$\left( { - \infty ,0} \right)$$ is
$$f\left( x \right) = \left\{ {\matrix{ {1 - x,} & {x \le 0} \cr {{x^{2,}}} & {x > 0} \cr } } \right..$$
Consider the composition of $$f$$ and $$g,$$ i.e., $$\left( {f \circ g} \right)\left( x \right) = f\left( {g\left( x \right)} \right).$$ The number of discontinuities in $$\left( {f \circ g} \right)\left( x \right)$$ present in the interval $$\left( { - \infty ,0} \right)$$ is
GATE EE 2017 Set 2
7
A function $$f(x)$$ is defined as
$$f\left( x \right) = \left\{ {\matrix{ {{e^x},x < 1} \cr {\ln x + a{x^2} + bx,x \ge 1} \cr } \,\,,\,\,} \right.$$ where $$x \in R.$$
$$f\left( x \right) = \left\{ {\matrix{ {{e^x},x < 1} \cr {\ln x + a{x^2} + bx,x \ge 1} \cr } \,\,,\,\,} \right.$$ where $$x \in R.$$
Which one of the following statements is TRUE?
GATE EE 2017 Set 1
8
The value of the integral $$\,\,2\int_{ - \infty }^\infty {\left( {{{\sin \,2\pi t} \over {\pi t}}} \right)} dt\,\,$$ is equal to
GATE EE 2016 Set 2
9
Let $$\,\,S = \sum\limits_{n = 0}^\infty {n{\alpha ^n}} \,\,$$ where $$\,\,\left| \alpha \right| < 1.\,\,$$ The value of $$\alpha $$ in the range $$\,\,0 < \alpha < 1,\,\,$$ such that $$\,\,S = 2\alpha \,\,$$ is ___________.
GATE EE 2016 Set 1
10
The volume enclosed by the surface $$f\left( {x,y} \right) = {e^x}$$ over the triangle bounded by the lines $$x=y;$$ $$x=0;$$ $$y=1$$ in the $$xy$$ plane is ________.
GATE EE 2015 Set 2
11
To evaluate the double integral $$\int\limits_0^8 {\left( {\int\limits_{y/2}^{\left( {y/2} \right) + 1} {\left( {{{2x - y} \over 2}} \right)dx} } \right)dy,\,\,} $$ we make the substitution $$u = \left( {{{2x - y} \over 2}} \right)$$ and $$v = {y \over 2}.$$ The integral will reduce to
GATE EE 2014 Set 2
12
The minimum value of the function $$f\left( x \right) = {x^3} - 3{x^2} - 24x + 100$$ in the interval $$\left[ { - 3,3} \right]$$ is
GATE EE 2014 Set 2
13
The maximum value of $$f\left( x \right) = {x^3} - 9{x^2} + 24x + 5$$ in the interval $$\left[ {1,6} \right]$$ is
GATE EE 2012
14
The value of the quantity, where $$P = \int\limits_0^1 {x{e^x}\,dx\,\,\,} $$ is
GATE EE 2010
15
If $$(x, y)$$ is continuous function defined over $$\left( {x,y} \right) \in \left[ {0,1} \right] \times \left[ {0,1} \right].\,\,\,$$ Given two constants, $$\,x > {y^2}$$ and $$\,y > {x^2},$$ the volume under $$f(x, y)$$ is
GATE EE 2009
16
The integral $$\,\,{1 \over {2\pi }}\int\limits_0^{2\Pi } {Sin\left( {t - \tau } \right)\cos \tau \,d\tau \,\,\,} $$ equals
GATE EE 2007
17
The expression $$V = \int\limits_0^H {\pi {R^2}{{\left( {1 - {h \over H}} \right)}^2}dh\,\,\,} $$ for the volume of a cone is equal to _________.
GATE EE 2006