List of practice Questions

If $2(\vec a \times \vec c)+3(\vec b \times \vec c)=0$, where $\vec a=2\hat i-5\hat j+5\hat k$, $\vec b=\hat i-\hat j+3\hat k$ and $(\vec a-\vec b)\cdot\vec c=-97$, find $|\vec c \times \vec k|^2$.

\[ \left(\frac{1}{^{15}C_0}+\frac{1}{^{15}C_1}\right) \left(\frac{1}{^{15}C_1}+\frac{1}{^{15}C_2}\right) \cdots \left(\frac{1}{^{15}C_{12}}+\frac{1}{^{15}C_{13}}\right) = \frac{\alpha^{13}}{^{14}C_0\cdot {}^{14}C_1\cdot {}^{14}C_2\cdots {}^{14}C_{12}} \] If so, then find the value of \(30\alpha\).

If \( A = \{ 1, 2, 3, 4, 5, 6 \}, B = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \} \), then the number of strictly increasing functions from \( A \to B \) such that \( f(i) \neq i \) for \( i = 1, 2, 3, 4, 5, 6 \) is

Number of 4 letter words with or without meaning formed from the letters of the word PQRSTTUVV is:

Let \( S \) be the number of 4-digit numbers \( abcd \), where \[ a>b>c>d \] and let \( P \) be the number of 5-digit numbers \( abcde \), where the product of digits is 20. Find \( S + P \):

Let \( y^2 = 16x \), from point \( (16, 16) \) a focal chord is passing. Point \( (\alpha, \beta) \) divides the focal chord in the ratio 2:3, then the minimum value of \( \alpha + \beta \) is:

Ellipse \( E: \frac{x^2}{36} + \frac{y^2}{25} = 1 \), A hyperbola confocal with ellipse \( E \) and eccentricity of hyperbola is equal to 5. The length of latus rectum of hyperbola is, if principle axis of hyperbola is x-axis?

Consider an ellipse \[ E_1:\ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \ (a>b) \quad \text{and} \quad E_2:\ \frac{x^2}{A^2}+\frac{y^2}{B^2}=1 \ (B>A), \] where $e=\dfrac{4}{5}$ for both the curves and $\ell_1$ is the length of latus rectum of $E_1$ and $\ell_2$ is the length of latus rectum of $E_2$. Let the distance between the foci of the first curve be $8$. Find the distance between the foci of the second curve. (Given $2\ell_1^2=9\ell_2$).

Let the curve $z(1+i) + z(1-i) = 4$, $z \in \mathbb{C}$, divide the region $|z-3| \le 1$ into two parts of areas $\alpha$ and $\beta$. Then $|\alpha - \beta|$ equals:

If \( z = \dfrac{\sqrt{3}}{2} + \dfrac{i}{2} \), then the value of \[ \left(z^{201} - i\right)^8 \] is:

If $x^2 + x + 1 = 0$, find the value of
$\sum_{k=1}^{15} \left(x^k + \frac{1}{x^k}\right)^4$

If domain of \(f(x) = \sin^{-1}\left(\frac{5-x}{2x+3}\right) + \frac{1}{\log_{e}(10-x)}\) is \((-\infty, \alpha] \cup (\beta, \gamma) - \{\delta\}\) then value of \(6(\alpha + \beta + \gamma + \delta)\) is equal to :

If the domain of the function \[ f(x) = \frac{1}{\ln(10-x)} + \sin^{-1} \left( \frac{x+2}{2x+3} \right) \] is \( (-\infty, -1) \cup (-1, b) \cup (b, c) \cup (c, \infty) \), then \( (b + c + 3a) \) is equal to:

The number of solution(s) of the equation \[ x |x + 4| + 3 |x + 2| = 0 \] is/are equal to:

Consider a set \( S = \{ a, b, c, d \} \). Then the number of reflexive as well as symmetric relations from \( S \to S \) are

If the domain of the function \[ \cos^{-1} \left( \frac{2x - 5}{11x - 7} \right) + \sin^{-1} \left( 2x^2 - 3x + 1 \right) \] is \[ [0, a] \cup [12/13, b] \] then \( \frac{1}{ab} \) is equal to}

The sum of roots of the equation \[ |x - 1|^2 - 5 |x - 1| + 6 = 0 \] is

\( y = \log_5 \log_3 \log_7 (9x - x^2 - 13) \), If its domain is \( (m, n) \) and \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] is a hyperbola having eccentricity \( \frac{n}{3} \) and length of the latus rectum is \( \frac{8m}{3} \), find \( b^2 - a^2 \):

If a line \(ax + y = 1\) does not intersect the hyperbola \(x^2 - 9y^2 = 9\) then a possible value of \(\alpha\) is :

If probability distribution is given by \[ P(x) = \begin{array}{c|c|c|c|c|c|c|c} x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline P(x) & k & 2k^2 & 6k^2 & 2k^2 + k & 4k & k & k \\ \end{array} \] Then, the value of \( P(3<x \leq 6) \) is:

If two numbers \( a \) and \( b \) are selected from \( S = \{1, 2, 3, \dots, 100\} \), then the probability that \( |a - b| \geq 10 \) is: