List of practice Questions

Let \( a, b \in \mathbb{R}, b \neq 0 \). Define a function \[ f(x) = \begin{cases} a \sin \frac{\pi}{2}(x - 1), & \text{for } x \leq 0 \\ \frac{\tan 2x - \sin 2x}{bx^3}, & \text{for } x > 0 \end{cases} \] If \( f \) is continuous at \( x = 0 \), then \( 10 - ab \) is equal to ________.
The sum of solutions of the equation \(\frac{\cos x}{1 + \sin x} = |\tan 2x|\), \(x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) - \left\{\frac{\pi}{4}, -\frac{\pi}{4}\right\}\) is :
Let \(\theta \in \left(0, \frac{\pi}{2}\right)\). If the system of linear equations
\((1 + \cos^2\theta)x + \sin^2\theta y + 4\sin3\theta z = 0\)
\(\cos^2\theta x + (1 + \sin^2\theta)y + 4\sin3\theta z = 0\)
\(\cos^2\theta x + \sin^2\theta y + (1 + 4\sin3\theta)z = 0\)
has a non-trivial solution, then the value of \(\theta\) is :
The value of \(\int_{-1/\sqrt{2}}^{1/\sqrt{2}} \left[\left(\frac{x+1}{x-1}\right)^2 + \left(\frac{x-1}{x+1}\right)^2 - 2\right]^{1/2} dx\) is :
If \( A = \begin{bmatrix} \frac{1}{\sqrt{5}} & \frac{2}{\sqrt{5}} \\ -\frac{2}{\sqrt{5}} & \frac{1}{\sqrt{5}} \end{bmatrix} \), \( B = \begin{bmatrix} 1 & 0 \\ i & 1 \end{bmatrix} \), \( i = \sqrt{-1} \), and \( Q = A^T B A \), then the inverse of the matrix \( A Q^{2021} A^T \) is equal to :

The major product formed in the following reaction is : 

The correct options for the products A and B of the following reactions are : 

Among the following compounds I-IV, which one forms a yellow precipitate on reacting sequentially with (i) NaOH (ii) dil. $HNO_3$ (iii) $AgNO_3$? 

The major product formed in the following reaction is : 

Excess of isobutane on reaction with $Br_2$ in presence of light at $125^\circ C$ gives which one of the following, as the major product ? 

Consider a badminton racket with length scales as shown in the figure. 

If the mass of the linear and circular portions of the badminton racket are same (M) and the mass of the threads are negligible, the moment of inertia of the racket about an axis perpendicular to the handle and in the plane of the ring at, \(\frac{r}{2}\) distance from the end A of the handle will be _________ \(Mr^2\). 
 

The magnitude of vectors \(\vec{OA}\), \(\vec{OB}\) and \(\vec{OC}\) in the given figure are equal. The direction of \(\vec{OA} + \vec{OB} - \vec{OC}\) with x-axis will be : 

Let the line $L$ be the projection of the line \[ \frac{x - 1}{2} = \frac{y - 3}{1} = \frac{z - 4}{2} \] in the plane $x - 2y - z = 3$. If $d$ is the distance of the point $(0, 0, 6)$ from $L$, then $d^2$ is equal to _________.
The mean and standard deviation of 20 observations were calculated as 10 and 2.5 respectively. It was found that by mistake one data value was taken as 25 instead of 35. If \(\alpha\) and \(\sqrt{\beta}\) are the mean and standard deviation respectively for correct data, then \((\alpha, \beta)\) is :
A wire of length 36 m is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is $k$ (meter), then $\left( \frac{4}{\pi} + 1 \right)k$ is equal to _________.
A plane \(P\) contains the line \(x + 2y + 3z + 1 = 0 = x - y - z - 6\), and is perpendicular to the plane \(-2x + y + z + 8 = 0\). Then which of the following points lies on \(P\) ?
If the truth value of the Boolean expression \(((p \vee q) \wedge (q \to r) \wedge (\sim r)) \to (p \wedge q)\) is false, then the truth values of the statements \(p, q, r\) respectively can be :
If the sum of an infinite GP \(a, ar, ar^2, ar^3, \dots\) is 15 and the sum of the squares of its each term is 150, then the sum of \(ar^2, ar^4, ar^6, \dots\) is :
Let \(f(x) = \cos\left(2\tan^{-1} \sin \left(\cot^{-1} \sqrt{\frac{1-x}{x}}\right)\right)\), \(0<x<1\). Then :
The value of \(\lim_{n \to \infty} \frac{1}{n} \sum_{r=0}^{2n-1} \frac{n^2}{n^2 + 4r^2}\) is :
Let \(ABC\) be a triangle with \(A(-3, 1)\) and \(\angle ACB = \theta, 0<\theta \leq \frac{\pi}{2}\). If the equation of the median through \(B\) is \(2x + y - 3 = 0\) and the equation of angle bisector of \(C\) is \(7x - 4y - 1 = 0\), then \(\tan \theta\) is equal to :
Out of all the patients in a hospital 89% are found to be suffering from heart ailment and 98% are suffering from lungs infection. If K% of them are suffering from both ailments, then K can not belong to the set :
The sum of the series \(\frac{1}{x+1} + \frac{2}{x^2+1} + \frac{2^2}{x^4+1} + \dots + \frac{2^{100}}{x^{2^{100}}+1}\) when \(x = 2\) is :
The \(OH^-\) concentration in a mixture of 5.0 mL of 0.0504 M \(NH_4Cl\) and 2 mL of 0.0210 M \(NH_3\) solution is \(x \times 10^{-6}\) M. The value of \(x\) is _________. (Nearest integer)
[Given \(K_w = 1 \times 10^{-14}\) and \(K_b = 1.8 \times 10^{-5}\)]
The number of 4f electrons in the ground state electronic configuration of \(Gd^{2+}\) is _________. [Atomic number of Gd = 64] (Integer answer)