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Calculus · Engineering Mathematics · GATE CE

Marks 1
35
Marks 2
14
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Marks 1

1

The function $f(x) = x^3 - 27x + 4$, $1 \leq x \leq 6$ has

GATE CE 2024 Set 2
2

The smallest positive root of the equation $$x^5 - 5 x^4 - 10 x^3 + 50 x^2 + 9 x - 45 = 0$$ lies in the range

GATE CE 2024 Set 1
3

For the integral $\rm I=\displaystyle\int^1_{-1}\frac{1}{x^2}dx$

which of the following statements is TRUE? 

GATE CE 2023 Set 1
4

The following function is defined over the interval [-L, L]:

f(x) = px4 + qx5.

If it is expressed as a Fourier series, 

$\rm f(x)=a_0 +\displaystyle\sum^\infty_{n=1} \left\{a_n \sin\left( \frac{\pi x}{L} \right) +b_n\cos\left( \frac{\pi x}{L} \right) \right\} $,

which options amongst the following are true?

GATE CE 2023 Set 1
5

Consider the polynomial f(x) = x3 $$-$$ 6x2 + 11x $$-$$ 6 on the domain S, given by 1 $$\le$$ x $$\le$$ 3. The first and second derivatives are f'(x) and f''(x).

Consider the following statements :

I. The given polynomial is zero at the boundary points x = 1 and x = 3.

II. There exists one local maxima of f(x) within the domain S.

III. The second derivative f''(x) > 0 throughout the domains S.

IV. There exists one local minima f(x) within the domain S.

GATE CE 2022 Set 2
6

$$\int {\left( {x - {{{x^2}} \over 2} + {{{x^3}} \over 3} - {{{x^4}} \over 4} + ....} \right)dx} $$ is equal to :

GATE CE 2022 Set 2
7

Let max {a, b} denote the maximum of two real numbers a and b. Which of the following statements is/are TRUE about the function f(x) = max{3 $$-$$ x, x $$-$$ 1}?

GATE CE 2022 Set 1
8

A set of observations of independent variable (x) and the corresponding dependent variable (y) is given below.

x 5 2 4 3
y 16 10 13 12

Based on the data, the coefficient a of the linear regression model

y = a + bx

is estimated as 6.1. The coefficient b is _________. (round off to one decimal place)

GATE CE 2022 Set 1
9
Let $$\,\,W = f\left( {x,y} \right),\,\,$$ where $$x$$ and $$y$$ are functions of $$t.$$ Then, according to the chain rule, $${{dw} \over {dt}}$$ is equal to
GATE CE 2017 Set 2
10
$$\mathop {Lim}\limits_{x \to 0} \left( {{{\tan x} \over {{x^2} - x}}} \right)$$ is equal to _________.
GATE CE 2017 Set 1
11
Let $$x$$ be a continuous variable defined over the interval $$\left( { - \infty ,\infty } \right)$$, and $$f\left( x \right) = {e^{ - x - {e^{ - x}}}}.$$
The integral $$g\left( x \right) = \int {f\left( x \right)dx\,\,} $$ is equal to
GATE CE 2017 Set 1
12
$$\mathop {\lim }\limits_{x \to \infty } {\left( {1 + {1 \over x}} \right)^{2x}}\,\,$$ is equal to
GATE CE 2015 Set 2
13
Given $$i = \sqrt { - 1} ,$$ the value of the definite integral, $$\,{\rm I} = \int\limits_0^{\pi /2} {{{\cos x + \sin x} \over {\cos x - i\,\sin x}}dx\,\,} $$ is :
GATE CE 2015 Set 2
14
$$\,\,\mathop {Lim}\limits_{x \to \infty } \left( {{{x + \sin x} \over x}} \right)\,\,$$ equal to
GATE CE 2014 Set 1
15
With reference to the conventional Cartesian $$(x,y)$$ coordinate system, the vertices of a triangle have the following coordinates: $$\,\left( {{x_1},{y_1}} \right) = \left( {1,0} \right);\,\,\,\left( {{x_2},{y_2}} \right) = \left( {2,2} \right);\,\,\,$$ and $$\,\left( {{x_3},{y_3}} \right) = \left( {4,3} \right).$$ The area of the triangle is equal to
GATE CE 2014 Set 1
16
The solution $$\int\limits_0^{\pi /4} {{{\cos }^4}3\theta {{\sin }^3}\,6\theta d\theta \,\,} $$ is :
GATE CE 2013
17
The infinite series $$1 + x + {{{x^2}} \over {2!}} + {{{x^3}} \over {3!}} + {{{x^4}} \over {4!}} + ........$$ corresponds to
GATE CE 2012
18
What should be the value of $$\lambda $$ such that the function defined below is continuous at $$x = {\pi \over 2}$$?
$$f\left( x \right) = \left\{ {\matrix{ {{{\lambda \,\cos x} \over {{\pi \over 2} - x}},} & {if\,\,x \ne {\pi \over 2}} \cr {1\,\,\,\,\,\,\,\,\,\,,} & {if\,\,x = {\pi \over 2}} \cr } } \right.$$
GATE CE 2011
19
The $$\mathop {Lim}\limits_{x \to 0} {{\sin \left( {{2 \over 3}x} \right)} \over x}\,\,\,$$ is
GATE CE 2010
20
The value of the function, $$f\left( x \right) = \mathop {Lim}\limits_{x \to 0} {{{x^3} + {x^2}} \over {2{x^3} - 7{x^2}}}\,\,\,$$ is
GATE CE 2004
21
The following function has local minima at which value of $$x,$$ $$f\left( x \right) = x\sqrt {5 - {x^2}} $$
GATE CE 2002
22
The value of the following definite integral in $$\int\limits_{{\raise0.5ex\hbox{$\scriptstyle { - \pi }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}^{{\raise0.5ex\hbox{$\scriptstyle \pi $} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}} {{{Sin2x} \over {1 + \cos x}}dx = \_\_\_\_\_\_\_\_.} $$
GATE CE 2002
23
Limit of the following series as $$x$$ approaches $${\pi \over 2}$$ is
$$f\left( x \right) = x - {{{x^3}} \over {3!}} + {{{x^5}} \over {5!}} - {{{x^7}} \over {7!}} + - - - - - $$
GATE CE 2001
24
Consider the following integral $$\mathop {Lim}\limits_{x \to 0} \int\limits_1^a {{x^{ - 4}}} dx$$ ________.
GATE CE 2000
25
The function $$f\left( x \right) = {e^x}$$ is _________.
GATE CE 1999
26
Number of inflection points for the curve $$\,\,\,y = x + 2{x^4}\,\,\,\,$$ is_______.
GATE CE 1999
27
Limit of the function, $$\mathop {Lim}\limits_{n \to \infty } {n \over {\sqrt {{n^2} + n} }}$$ is _______.
GATE CE 1999
28
The continuous function $$f(x, y)$$ is said to have saddle point at $$(a, b)$$ if
GATE CE 1998
29
A discontinuous real function can be expressed as
GATE CE 1998
30
The Taylor's series expansion of sin $$x$$ is ______.
GATE CE 1998
31
If $$\varphi \left( x \right) = \int\limits_0^{{x^2}} {\sqrt t \,dt\,} $$ then $${{d\varphi } \over {dx}} = \_\_\_\_\_\_\_.$$
GATE CE 1997
32
If $$y = \left| x \right|$$ for $$x < 0$$ and $$y=x$$ for $$x \ge 0$$ then
GATE CE 1997
33
The function $$f\left( x \right) = \left| {x + 1} \right|$$ on the interval $$\left[ { - 2,0} \right]$$ is __________.
GATE CE 1995
34
The function $$f\left( x \right) = {x^3} - 6{x^2} + 9x + 25$$ has
GATE CE 1995
35
The value of $$\varepsilon $$ in the mean value theoram of $$f\left( b \right) - f\left( a \right) = \left( {b - a} \right)\,\,f'\left( \varepsilon \right)$$ for $$f\left( x \right) = A{x^2} + Bx + C$$ in $$(a, b)$$ is
GATE CE 1994

Marks 2

1

The expression for computing the effective interest rate $(i_{eff})$ using continuous compounding for a nominal interest rate of 5% is

$i_{eff} = \lim\limits_{m \to \infty} \left(1 + \frac{0.05}{m}\right)^m - 1$

The effective interest rate (in percentage) is ___________ (rounded off to 2 decimal places).

GATE CE 2024 Set 2
2
For the function f(x) = ex |sin x|; x ∈ ℝ, which of the following statements is/are TRUE?
GATE CE 2023 Set 1
3
Consider the following definite integral $$${\rm I} = \int\limits_0^1 {{{{{\left( {{{\sin }^{ - 1}}x} \right)}^2}} \over {\sqrt {1 - {x^2}} }}dx} $$$
The value of the integral is
GATE CE 2017 Set 2
4
The tangent to the curve represented by $$y=x$$ $$ln$$ $$x$$ is required to have $${45^ \circ }$$ inclination with the $$x-$$axis. The coordinates of the tangent point would be
GATE CE 2017 Set 2
5
The expression $$\mathop {Lim}\limits_{a \to 0} \,{{{x^a} - 1} \over a}\,\,$$ is equal to
GATE CE 2014 Set 2
6
What is the value of the definite integral? $$\,\,\int\limits_0^a {{{\sqrt x } \over {\sqrt x + \sqrt {a - x} }}dx\,\,} $$?
GATE CE 2011
7
A parabolic cable is held between two supports at the same level. The horizontal span between the supports is $$L.$$ The sag at the mid-span is $$h.$$ The equation of the parabola is $$y = 4h{{{x^2}} \over {{L^2}}},\,\,$$ where $$x$$ is the horizontal coordinate and $$y$$ is the vertical coordinate with the origin at the centre of the cable. The expanssion for the total length of the cable is
GATE CE 2010
8
Given a function $$f\left( {x,y} \right) = 4{x^2} + 6{y^2} - 8x - 4y + 8,$$ the optimal values of $$f(x,y)$$ is
GATE CE 2010
9
The function $$f\left( x \right) = 2{x^3} - 3{x^2} - 36x + 2\,\,\,$$ has its maxima at
GATE CE 2004
10
Limit of the following sequence as $$n \to \infty $$ $$\,\,\,$$ is $$\,\,\,$$ $${x_n} = {n^{{1 \over n}}}$$
GATE CE 2002
11
The value of the following improper integral is $$\,\int\limits_0^1 {x\,\log \,x\,dx} = \_\_\_\_\_.$$
GATE CE 2002
12
Limit of the function
$$f\left( x \right) = {{1 - {a^4}} \over {{x^4}}}\,\,as\,\,x \to \infty $$ is given by
GATE CE 2000
13
The Taylor series expansion of sin $$x$$ about $$x = {\pi \over 6}$$ is given by
GATE CE 2000
14
If $$f\left( {x,y,z} \right) = $$
$${\left( {{x^2} + {y^2} + {z^2}} \right)^{{\raise0.5ex\hbox{$\scriptstyle { - 1}$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}},$$ $${{{\partial ^2}f} \over {\partial {x^2}}} + {{{\partial ^2}f} \over {\partial {y^2}}} + {{{\partial ^2}f} \over {\partial {z^2}}}$$ is equal to _______.
GATE CE 2000
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