List of practice Questions

The value of \( \int_{0}^{20\pi} \left( \sin^4 x + \cos^4 x \right)\, dx \) is equal to
A regular polygon with \( n \) sides is given. \( P_n \) denotes number of triangles formed by joining any three points of given regular polygon. If \( P_{n+1} - P_n = 66 \), then the sum of all prime divisors of \( n \) is
Let \( A = \{2,3,4,5,6\} \) be a set. Consider \( R \) be a relation on \( A \times A \) such that \( (x,y) \, R \, (a,b) \) implies \( x \) divides \( a \) and \( y \leq b \), then total number of elements in \( R \) is:
Find number of points of discontinuity of the function \( f(x) = [x^2 - x + 2] \) in \( x \in [2,4] \) (where \( [\ ] \) denotes greatest integer function).
If the lines \( x + (k - 1)y + 3 = 0 \) \& \( 2x + k^2y - 4 = 0 \) are perpendicular and their point of intersection is the centre of a circle which passes through origin. If chord \( x - y + 2 = 0 \) intersects this circle at \( A \) \& \( B \) then \( (AB)^2 = ? \)
Let \( x(y) \) be the solution of the given differential equation \( 2y^2 \dfrac{dx}{dy} - 2xy + x^2 = 0 \). If \( x(e) = e \), then \( \dfrac{3x(e^2)}{e^2} \) equals.
If the area bounded by two curves \( \dfrac{x^2}{9} - \dfrac{y^2}{16} = 1 \) and \( 8x - 3y = 24 \) is \( A - 6\log_e 3 \), then \( A \) is equal to
Let \( P = \{\theta \in [0, 4\pi] : \tan^2\theta \neq 1\} \)
\( S = \{ a \in \mathbb{Z} : (\cos^2\theta - \sin^2\theta)\sec 2\theta = a^2,\ \theta \in P \} \)
then \( n(S) \) equals
If \( {}^{30}C_{30-r} + 3 \cdot {}^{30}C_{31-r} + 3 \cdot {}^{30}C_{32-r} + {}^{30}C_{33-r} = {}^{n}C_{r} \), then value of \( n \) is
If matrices \( A = \begin{bmatrix} 2 & -2 \\ 4 & -2 \end{bmatrix} \) and \( B = \begin{bmatrix} 1 & 3 \\ 3 & 9 \end{bmatrix} \) are such that \( PA = B \) and \( AQ = B \), then \( \mathrm{tr}\bigl(2(P+Q)\bigr) \) is -
Ram is tossing a coin. If head comes then 10 points will be given and if tail comes then 5 points will be given. If the probability of getting exactly 30 point is \( \dfrac{m}{n} \), then \( (m+n) \) equals (Where \( m \) \& \( n \) are co-prime numbers).
Match List I with List II.
Predict expression for \( \alpha \) in terms of \( K_{eq} \) and concentration
C: \( A_2B_3 \, (aq) \rightleftharpoons 2A^{3+} (aq) + 3B^{2-} (aq) \)
For a transparent prism, if the angle of minimum deviation is equal to its refracting angle, the refractive index \(n\) of the prism satisfies:
The internal energy of a monoatomic gas is 3nRT. One mole of helium... heated slowly by supplying 126 J heat... piston will move ___ cm.
Why are the means of transportation and communication called the lifelines of National Economy?

Consider the following reaction sequence.

For the thermal decomposition of reactant AB(g), the following plot is constructed. 

The half life of the reaction is 'x' min.
x =_______} min. (Nearest integer)} 
 

What is the equivalent gate for the following digital circuit?

If a random variable \( x \) has the probability distribution 

then \( P(3<x \leq 6) \) is equal to

Two distinct numbers $a$ and $b$ are selected at random from $1, 2, 3, \ldots, 50$. The probability that their product $ab$ is divisible by $3$ is
Let \( A, B, C \) be three \( 2\times2 \) matrices with real entries such that \[ B=(I+A)^{-1} \quad \text{and} \quad A+C=I. \] If \[ BC=\begin{bmatrix}1 & -5 \\-1 & 2\end{bmatrix} \quad \text{and} \quad B\begin{bmatrix}x_1\\x_2\end{bmatrix} =\begin{bmatrix}12\\-6\end{bmatrix}, \] then \( x_1+x_2 \) is:
In an equilateral prism the path of a ray is shown in the figure. Determine the refractive index of the prism.
From the following, the least stable structure is:
(Assume standard resonance structures)

Consider two arrangements of wires. Find the ratio of magnetic field at the centre of the semi–circular part.