Calculus · Engineering Mathematics · GATE IN

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

$$\mathop {\lim }\limits_{n \to \infty } \left( {\sqrt {{n^2} + n} - \sqrt {{n^2} + 1} } \right)\,\,$$ 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 $$x$$-yplot will be

The series $$\,\,\sum\limits_{m = 0}^\alpha {{1 \over {{4^m}}}{{\left( {x - 1} \right)}^{2m}}\,\,\,} $$ converges for

Given $$y = {x^2} + 2x + 10\,\,\,$$ the value of $$\,\,{\left. {{{dy} \over {dx}}} \right|_{x = 1}}\,\,$$ is equal to

$$\,\mathop {Lim}\limits_{x \to 0} {{\sin x} \over x}\,\,\,$$ is

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 constants then $$x{{\partial f} \over {\partial x}} + y{{\partial f} \over {\partial y}}\,\,\,$$ is

$$\mathop {Lim}\limits_{x \to {\pi \over 4}} \,\,{{Sin\,\,2\left( {x - {\pi \over 4}} \right)} \over {x - {\pi \over 4}}} = \_\_\_\_\_\_.$$

$$\mathop {Lim}\limits_{x \to 0} \,{1 \over {10}}\,\,{{1 - {e^{ - j5x}}} \over {1 - {e^{ - jx}}}} = \_\_\_\_.$$

The angle between two vectors $${X_1} = {\left[ {\matrix{ 2 & 6 & {14} \cr } } \right]^T}$$ and $${X_2} = {\left[ {\matrix{ { - 12} & 8 & {16} \cr } } \right]^T}$$ in radian is ________.

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