List of practice Questions

Let \(x_1, x_2, x_3, \ldots, x_n\) be ‘\(n\)’ observations such that \( \sum_{i=1}^{n-1} x_i = 48 \) and \( \sum_{i=1}^{n-1} x_i^2 = 496 \). If mean and variance of the distribution are 8 and 16 respectively then value of \(n\) is:

Let \( f(x) \) be a polynomial of degree \( 5 \) having extreme values at \( x = 1 \) and \( x = -1 \). If \[ \lim_{x \to 0} \frac{f(x)}{x^3} = -5, \] then the value of \( f(B) - f(-2) \) is

Let \( \overrightarrow{PS} = \hat{i} + \hat{j} \) and \( \overrightarrow{PQ} = -\hat{j} + \hat{k} \). If \( \overrightarrow{PS} \) must be rotated by an angle \( \alpha \) such that \( \overrightarrow{PS} \) is perpendicular to \( \overrightarrow{PQ} \), then \( \left( \sin^2 \frac{5\alpha}{2} - \sin^2 \frac{\alpha}{2} \right) \) equals

If \( z_1 \) lies on curve \( |z| = r \) and \( z_2 \) lies on curve \( |z - 3 - 4i| = 5 \), if minimum of \( |z_1 - z_2| = 2 \), then the maximum of \( |z_1 - z_2| \) is

Parabola \( y = x^2 + px + q \) is passing through \( (1,-1) \) and vertex of parabola is at minimum distance from x-axis then \( p^2 + q^2 \) is

Let `C' be a circle with radius `6' units centred at origin. Let \( A(3,0) \) be a point. If \( B \) is a variable point in xy-plane such that circle drawn taking \( AB \) as diameter touches the circle \( C \), then eccentricity of the locus of point `B' is

Find sum up to 8 terms of the series
\[ \frac{1^3}{1}+\frac{1^3+2^3}{1+3}+\frac{1^3+2^3+3^3}{1+3+5}+\cdots \]

If the system of equations \( x + 5y + 6z = 4 \), \( 2x + 2y + 4z = 1 \) and \( x + y + az = b \) has infinite numbers of solutions then point \( (a,b) \) lies on-

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?