List of practice Questions

Domain \( A \) is bounded by the curve \( x^2 = 4y \), the ordinate \( x = 2 \), and the \( x \) axis.
The value of \( \iint_A y \, dx \, dy \) is

If \[ f(x,y,z)=e^{\,1-y\cos x}+yze^{\frac{-1}{1+x^2}}, \] then \[ \frac{\partial f}{\partial y} \] at \((0,1,e)\) is

Let \[ \vec{F}=x^2yz\,\hat{i}+y^2z\,\hat{j}+zx^3\,\hat{k} \] be a vector point function. Let \(S\) be the surface of the sphere \[ x^2+y^2+z^2=a^2. \] Then \[ \iint_{S}(\nabla\times\vec{F})\cdot\vec{N}\,dS \] is equal to (\(\vec{N}\) is any unit outward normal vector to \(S\)).

Let \(f(x) = x^3 - 3x^2 + 2\) be a function defined on \((-1, 3]\).

Which of the following statements is/are correct?

The value of \(\displaystyle\sum_{i=0}^{\infty}\sum_{j=1}^{\infty} 2^{-i}\,3^{-j}\) is ________. (Answer in integer)

Periodic function \(f(x)\) is given below.
\[ f(x)=\begin{cases}-1, & -\pi < x < 0 \\ 1, & 0 < x < \pi\end{cases}; \quad f(x+2\pi)=f(x) \]
The CORRECT option representing the Fourier series expansion of \(f(x)\) is:

If the sum of the three numbers which are in A.P. is \(27\) and the product of the first and last is \(77\), then the numbers are

Evaluate \[ \int \frac{dx}{10\cos^2x+\cos2x+5} \]

If \([x]\) denotes the greatest integer function, then \[ \int_{0}^{5}[x-2]\,dx= \]

Evaluate \[ \int_{0}^{2}x^{3}(4-x^{2})^{\frac52}\,dx \]

The area of the region bounded by the curves \[ y=2^x,\qquad y^2=4x \] and the lines \[ x=\frac12,\qquad x=1 \] is

The differential equation corresponding to the family of curves \[ y=\log_e(ax+3), \] where \(a\) is an arbitrary constant, is

The general solution of the differential equation \[ x^{3}\,dx-xy^{2}\,dy+y^{3}\,dx=0 \] is

If \[ f(x)=xe^{x^{2}-2x-3} \] is a real valued function, then \(f\) is

The local minimum value of the function \[ f(x)=2x-3\tan^{-1}x \] is

If \[ f(x)=x^{3}-10x^{2}+31x-30 \] is a real valued function, then the number of values of \(c\), as stated in Rolle's theorem, that lie in the interval \((2,5)\) is

Evaluate \[ \int \frac{\tan 2x}{\cos^4 x}\,dx \]

If \[ \int\frac{e^x+3}{\sqrt{e^x+4}}\,dx = 2\sqrt{e^x+4} +\frac32 \log\left| \frac{f(x)-2}{f(x)+2} \right| +c \] and \[ f(0)=\sqrt5, \] then \[ f(\log_e5)= \]

If \[ \int_{x}^{2x}\left(1+(\log 2)\log x\right)^{x}\,dx=f(x)+c \] and \[ f(1)=0, \] then \(f(e)=\)

If \[ f(x)=\left(\log x\right)^{\sin x},\qquad x>e, \] then \(f'(\pi)=\)

If \[ y=\left(x+\sqrt{x^{2}+1}\right)^{5}, \] then \(25y=\)

If \[ y=\sin^{-1}\!\left(\frac{1-\cos2x}{1+\sin^{4}x}\right), \] then \[ \frac{dy}{dx}= \]

Approximate volume of a cone whose semi-vertical angle is \[ \tan^{-1}(3) \] and base radius is \(15.001\) is

If the parabola \[ y^{2}=9x \] cuts the ellipse \[ \frac{x^{2}}{9}+\frac{y^{2}}{b}=1 \] orthogonally, then the length of the latus rectum of the given parabola is

If \[ y=\tan^{-1}\!\left(\frac{\sqrt{1+9x^{2}}-1}{3x}\right), \] then \[ \left(\frac{dy}{dx}\right)_{x=\frac1{\sqrt3}} = \]