GATE IN
Engineering Mathematics
Calculus
Previous Years Questions

## Marks 1

$$\mathop {\lim }\limits_{n \to \infty } \left( {\sqrt {{n^2} + n} - \sqrt {{n^2} + 1} } \right)\,\,$$ is ________.
A straight line of the form $$y=mx+c$$ passes through the origin and the point $$(x, y)=(2,6).$$ The value of $$m$$ is
The double integral $$\,\,\int_0^a {\int_0^y {f\left( {x,y} \right)\,dx\,dy\,\,\,} }$$ is equivalent to
Given $$x\left( t \right) = 3\,\sin \,\left( {1000\pi t} \right)\,\,$$ and $$\,\,y\left( t \right) = 5\cos \,\left( {1000\pi t{\pi \over t}} \right)... The series$$\,\,\sum\limits_{m = 0}^\alpha {{1 \over {{4^m}}}{{\left( {x - 1} \right)}^{2m}}\,\,\,} $$converges for$$\,\mathop {Lim}\limits_{x \to 0} {{\sin x} \over x}\,\,\,$$is Given$$y = {x^2} + 2x + 10\,\,\,$$the value of$$\,\,{\left. {{{dy} \over {dx}}} \right|_{x = 1}}\,\,$$is equal to For real$$x,$$the maximum value of$${{{e^{Sin\,x}}} \over {{e^{Cos\,x}}}}\,\,$$is Consider the function$$\,\,f\left( x \right) = {\left| x \right|^3},\,\,\,$$where$$x$$is real. Then the function$$f(x)$$at$$x=0$$is The value of$$\,\int\limits_0^\infty {\int\limits_0^\infty {{e^{ - {x^2}}}{e^{ - {y^2}}}} dx\,dy\,\,\,\,} $$is The value of the integral$$\int\limits_{ - 1}^1 {{1 \over {{x^2}}}dx} \,\,\,$$is$$f = {a_0}\,{x^n} + {a_1}\,{x^{n - 1}}\,y + - - + \,{a_{n - 1}}\,x\,{y^{n - 1}} + {a_n}\,{y^n}$$where$$\,\,{a_i}\,\,$$($$i=0$$to$$n$$) are c...$$\mathop {Lim}\limits_{x \to {\pi \over 4}} \,\,{{Sin\,\,2\left( {x - {\pi \over 4}} \right)} \over {x - {\pi \over 4}}} = \_\_\_\_\_\_.\mathop {Lim}\limits_{x \to 0} \,{1 \over {10}}\,\,{{1 - {e^{ - j5x}}} \over {1 - {e^{ - jx}}}} = \_\_\_\_.$$## Marks 2 The angle between two vectors$${X_1} = {\left[ {\matrix{ 2 & 6 & {14} \cr } } \right]^T}$$and$${X_2} = {\left[ {\matrix{ { - 12} ...
Let $$\,\,f:\left[ { - 1, - } \right] \to R,\,\,$$ where $$\,f\left( x \right) = 2{x^3} - {x^4} - 10.$$ The minimum value of $$f(x)$$ is _______.
The maximum value of $$f\left( x \right) = {x^3} - 9{x^2} + 24x + 5$$ in the interval $$\left[ {1,6} \right]$$ is
The expression $${e^{ - ln\,x}}$$ for $$x > 0$$ is equal to
Consider the function $$\,\,y = {x^2} - 6x + 9.\,\,\,$$ The maximum value of $$y$$ obtained when $$x$$ varies over the interval $$2$$ to $$5$$ is
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