List of practice Questions

Suppose \( \theta_1 \) and \( \theta_2 \) are such that \( (\theta_1 - \theta_2) \) lies in the 3rd or 4th quadrant. If \[ \sin\theta_1 + \sin\theta_2 = \frac{21}{65} \quad \text{and} \quad \cos\theta_1 + \cos\theta_2 = \frac{27}{65} \] then \[ \cos\left(\frac{\theta_1 - \theta_2}{2}\right) = \] 

A test containing 3 objective type of questions is conducted in a class. Each question has 4 options and only one option is the correct answer. No two students of the class have answered identically and no student has written all correct answers. If every student has attempted all the questions, then the maximum possible number of students who have written the test is:
In the expansion of \( \frac{2x+1}{(1+x)(1-2x)} \), the sum of the coefficients of the first 5 odd powers of \( x \) is:

'कल' का अनेकार्थक शब्द है- 
(a) बीता हुआ कल (दिन) 
(b) आने वाला दिन 
(c) चैन 
(d) मशीन

The number of ways a committee of 8 members can be formed from a group of 10 men and 8 women such that the committee contains at most 5 men and at least 5 women is:

The range of the real valued function \( f(x) =\) \(\sin^{-1} \left( \frac{1 + x^2}{2x} \right)\) \(+ \cos^{-1} \left( \frac{2x}{1 + x^2} \right)\) is:

There were two women participating with some men in a chess tournament. Each participant played two games with the other. The number of games that the men played among themselves is 66 more than the number of games the men played with the women. Then the total number of participants in the tournament is:
If the line \( 5x - 2y - 6 = 0 \) is a tangent to the hyperbola \( 5x^2 - ky^2 = 12 \), then the equation of the normal to this hyperbola at \( (\sqrt{6}, p) \) is:
If a random variable \( X \) has the following probability distribution, then its variance is nearly: 
random variable
If the probability distribution of a random variable \( X \) is given as follows, then find \( k \):
probability distribution of a random variable X
In \( \triangle ABC \), if \( (a+c)^2 = b^2 + 3ca \), then \( \frac{a+c}{2R} \) is:
The probability that A speaks truth is 75\% and the probability that B speaks truth is 80\%. The probability that they contradict each other when asked to speak on a fact is:
If \[ y = \tan^{-1} \left( \frac{2 - 3\sin x}{3 - 2\sin x} \right), \] then find \( \frac{dy}{dx} \).
Match List-I with List-II:
Table
The sum of number of angular nodes and radial nodes for 4d orbital is:
The set of all real values of \( c \) for which the equation \[ zz' + (4 - 3i)z + (4+3i)z + c = 0 \] represents a circle is:
The equations of the perpendicular bisectors of the sides AB and AC of \( \triangle ABC \) are \( x - y + 5 = 0 \) and \( x + 2y = 0 \) respectively. If the coordinates of \( A \) are \( (1, -2) \), then the equation of the line BC is:
\(\int\frac{2x^{2}\cos(x^{2})-\sin(x^{2})}{x^{2}}dx=\)
Under which of the following systems is all work broken down into its smallest repetitive elements and divided amongst the workers?
The oxidation number of Co in complex [Co(H$_2$NCH$_2$CHNH$_2$)$_3$]$_2$(SO$_4$)$_3$ is:
A battery of emf \( 8V \) and internal resistance \( 0.5 \Omega \) is being charged by a \( 120V \) DC supply using a series resistor of \( 15.5 \Omega \). The terminal voltage of the \( 8V \) battery during charging is:
A ray of light passes through a triangular prism. Show graphically how the angle of deviation varies with the angle of incidence. Hence define the angle of minimum deviation
If \( x_1, x_2, x_3, \dots, x_n \) are \( n \) observations such that \( \sum (x_i + 2)^2 = 28n \) and \( \sum (x_i - 2)^2 = 12n \), then the variance is:
Evaluate the sum: \[ \frac{1}{3 \cdot 7} + \frac{1}{7 \cdot 11} + \frac{1}{11 \cdot 15} + \dots \text{ up to 50 terms=} \]
If \( P(x, y) \) represents the complex number \( z = x + iy \) in the Argand plane and \[ \arg \left( \frac{z - 3i}{z + 4} \right) = \frac{\pi}{2}, \] then the equation of the locus of \( P \) is: