GATE CSE
Discrete Mathematics
Calculus
Previous Years Questions

## Marks 1

The value of the following limit is _____________. $$\mathop {\lim }\limits_{x \to {0^ + }} {{\sqrt x } \over {1 - {e^{2\sqrt x }}}}$$
Consider the functions I. $${e^{ - x}}$$ II. $${x^2} - \sin x$$ III. $$\sqrt {{x^3} + 1}$$ Which of the above functions is/are increasing everywhere ...
Compute $$\mathop {\lim }\limits_{x \to 3} {{{x^4} - 81} \over {2{x^2} - 5x - 3}}$$
If $$f\left( x \right)\,\,\, = \,\,\,R\,\sin \left( {{{\pi x} \over 2}} \right) + S.f'\left( {{1 \over 2}} \right) = \sqrt 2$$ and $$\int_0^1 {f\left... Consider a quadratic equation$${x^2} - 13x + 36 = 0$$with coefficients in a base$$b.$$The solutions of this equation in the same base$$b$$are... Let$$f(x)$$be a polynomial and$$g\left( x \right) = f'\left( x \right)$$be its derivative. If the degree of$$\left( {f\left( x \right) + f\left( ...
$$\mathop {\lim }\limits_{x \to 4} {{\sin \left( {x - 4} \right)} \over {x - 4}} = \_\_\_\_\_\_\_.$$
A function $$f(x)$$ is linear and has a value of $$29$$ at $$x=-2$$ and $$39$$ at $$x=3.$$ Find its value at $$x=5.$$
The value of $$\mathop {\lim }\limits_{x \to \alpha } {\left( {1 + {x^2}} \right)^{{e^{ - x}}}}\,\,$$ is
Choose the most appropriate equation for the function drawn as a thick line, in the plot below. ...
$$\,\,\mathop {\lim }\limits_{x \to \infty } \,{x^{1/x}}\,\,$$ is
If $$g(x)=1-x$$ & $$h\left( x \right) = {x \over {x - 1}}\,\,$$ then $$\,\,{{g\left( {h\left( x \right)} \right)} \over {h\left( {g\left( x \righ... If$$\int_0^{2\pi } {\left| {x\sin x} \right|dx = k\pi ,} $$then the values of$$k$$is equal to _________ . Let the function$$f\left( \theta \right) = \left| {\matrix{ {\sin \,\theta } & {\cos \,\theta } & {\tan \,\theta } \cr {\sin \left(...
Function $$f$$ is known at the following points: The value of $$\int\limits_0^3 {f\left( x \right)dx}$$ computed using the trapezoidal rule is ...
Which one of the following functions is continuous at $$x=3?$$
Consider the function $$f\left( x \right) = \sin \left( x \right)$$ in the interval $$x \in \left[ {\pi /4,\,\,7\pi /4} \right].$$ The number and loca...
What is the value of $$\mathop {\lim }\limits_{n \to \infty } {\left[ {1 - {1 \over n}} \right]^{2n}}?$$
$$\mathop {\lim }\limits_{x \to \infty } {{x - \sin x} \over {x + \cos \,x}}\,\,Equals$$
The value of $$\int\limits_0^3 {\int\limits_0^x {\left( {6 - x - y} \right)dxdy\,\,\,} }$$ is _____.
Consider the following two statements about the function $$f\left( x \right) = \left| x \right|:$$$$$P.\,\,f\left( x \right)$$ is continuous for al... $$\mathop {Lim}\limits_{x \to 0} \,{{Si{n^2}x} \over x} = \_\_\_\_.$$ The value of the integral is $${\rm I} = \int\limits_0^{{\raise0.5ex\hbox{\scriptstyle \pi } \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{\scriptstyl... Consider the function$$y = \left| x \right|$$in the interval$$\left[ { - 1,1} \right]$$. In this interval, the function is The formula used to compute an approximation for the second derivative of a function$$f$$at a point$${x_0}$$is If at every point of a certain curve, the slope of the tangent equals$${{ - 2x} \over y}$$the curve is$$\mathop {Lim}\limits_{x \to \infty } {{{x^3} - \cos x} \over {{x^2} + {{\left( {\sin x} \right)}^2}}} = \_\_\_\_\_\_.$$The value of the double integral$$\int\limits_0^1 {\int\limits_x^{{1 \over x}} {{x \over {1 + {y^2}}}\,\,dx\,\,dy = \_\_\_\_\_.} } $$## Marks 2 Consider the following expression$$\mathop {\lim }\limits_{x \to -3} \frac{{\sqrt {2x + 22} - 4}}{{x + 3}}$$The value of the above expression (rou... The value of$$\int_0^{\pi /4} {x\cos \left( {{x^2}} \right)dx} $$correct to three decimal places (assuming that$$\pi = 3.14$$) is ________. The value of$$\mathop {\lim }\limits_{x \to 1} {{{x^7} - 2{x^5} + 1} \over {{x^3} - 3{x^2} + 2}}.$$If for non-zero$$x,af\left( x \right) + bf\left( {{1 \over x}} \right) = {1 \over x} - 25$$where$$a \ne b$$then$$\int\limits_1^2 {f\left( ... Let $$\,\,f\left( x \right) = {x^{ - \left( {1/3} \right)}}\,\,$$ and $${\rm A}$$ denote the area of the region bounded by $$f(x)$$ and the $$X-$$axis... $$\,\int\limits_{1/\pi }^{2/\pi } {{{\cos \left( {1/x} \right)} \over {{x^2}}}dx = }$$ __________. The value of the integral given below is $$\int_0^\pi {{x^2}\,\cos \,x\,dx}$$$
Suppose you want to move from 0 to 100 on the number line. In each step, you either move right by a unit distance or you take a shortcut. A shortcut i...
A function $$f(x)$$ is continuous in the interval $$\left[ {0,2} \right]$$. It is known that $$f(0)$$ $$=$$ $$f(2)$$ $$= -1$$ and $$f(1)$$ $$= ... The function$$f(x) =xsinx$$satisfies the following equation:$$f$$"$$\left( x \right) + f\left( x \right) + t\,\cos \,x\,\, = \,\,0$$. Th... Given$$i = \sqrt { - 1} ,$$what will be the evaluation of the definite integral$$\int\limits_0^{\pi /2} {{{\cos x +i \sin x} \over {\cos x - i\,\si...
$$\int\limits_0^{\pi /4} {\left( {1 - \tan x} \right)/\left( {1 + \tan x} \right)dx}$$ $$\,\,\,\,\,\,$$ evaluates to
If $$\,\,\,\,f\,\,\,\,\left( x \right)$$ is defined as follows, what is the minimum value of $$f\,\left( x \right)$$ for $$x \in \left( {0,2} \right)... A point on a curve is said to be an extremum if it is a local minimum or a local maximum. The number of distinct extrema for the curve$$3{x^4} - 16{x...
What is the value of $$\int\limits_0^{2\pi } {{{\left( {x - \pi } \right)}^3}\left( {\sin x} \right)dx}$$
The function $$f\left( {x,y} \right) = 2{x^2} + 2xy - {y^3}$$ has
Find the points of local maxima and minima if any of the following function defined in $$0 \le x \le 6,$$ $$\,\,\,\,f\left( x \right) = {x^3} - 6{x^2... What is the maximum value of the function$$f\left( x \right) = 2{x^2} - 2x + 6$$in the interval$$\left[ {0,2} \right]$$? ## Marks 5 (a) Find the points of local maxima and minima, if any, of the following function defined$$0 \le x \le 6.\,\,\,{x^3} - 6{x^2} + 9x + 15 (b) Integra...
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