List of practice Questions

Let y = y(x), x > 1, be the solution of the differential equation
\((x-1)\frac{dy}{dx} + 2xy = \frac{1}{x-1}\)with \(y(2) = \frac{1+e^4}{2e^4}\). If \(y(3) = \frac{e^α + 1}{βe^α}\) , 
then the value of α + β is equal to ____.

The number of solutions of the equation sin x = cos² x in the interval (0, 10) is _____.

Let f and g be twice differentiable even functions on (–2, 2) such that
\(ƒ(\frac{1}{4})=0, ƒ(\frac{1}{2})=0, ƒ(1) =1\) and \(g(\frac{3}{4}) = 0 , g(1)=2\)
.Then, the minimum number of solutions of f(x)g′′(x) + f′(x)g′(x) = 0 in (–2, 2) is equal to_____.

Let the coefficients of x^–1 and x^–3 in the expansion of
\((2x^{\frac{1}{5}} - \frac{1}{x^{\frac{1}{5}}} )^{15} , x > 0\)be m and n respectively. If r is a positive integer such that
\(mn² = ^{15}C_r.2^r\)
then the value of r is equal to ______.

Let
\(M = \begin{bmatrix}   0 & -\alpha \\   \alpha & 0 \\ \end{bmatrix}\)
where α is a non-zero real number an
\(N =   \sum\limits_{k=1}^{49} M^{2k}. \) If \((I - M^2)N = -2I\)
then the positive integral value of α is ____ .

Let f(x) and g(x) be two real polynomials of degree 2 and 1 respectively. If f(g(x)) = 8x² – 2x and g(f(x)) = 4x² + 6x + 1, then the value of f(2) + g(2) is ____________ .

For real number a, b (a > b > 0), let
\(\text{{Area}} \left\{ (x, y) : x^2 + y^2 \leq a^2 \text{{ and }} \frac{x^2}{a^2} + \frac{y^2}{b^2} \geq 1 \right\} = 30\pi\)
and 
\(\text{{Area}} \left\{ (x, y) : x^2 + y^2 \geq b^2 \text{{ and }} \frac{x^2}{a^2} + \frac{y^2}{b^2} \leq 1 \right\} = 18\pi\)
Then the value of (a – b)² is equal to _____.

Let
\(A = \{z \in \mathbb{C} : |\frac{z+1}{z-1}| < 1\}\)
and
\(B = \{z \in \mathbb{C} : \text{arg}(\frac{z-1}{z+1}) = \frac{2\pi}{3}\}\)
Then \(A∩B\) is :

The current density in a cylindrical wire of radius \(r = 4.0\) mm is \(1.0 \times 10^6\) \(A/m^2\). The current through the outer portion of the wire between radial distances \(\frac{r}{2}\) and \(r\) is \(xπ\) \(A\); where \(x\) is _________.