Find the equation of a line passing through the point \((1, -1, 2)\) and parallel to x-axis. Also, find the shortest distance between this line and the line \(\vec{r} = 2\hat{i} + \hat{j} - \hat{k} + \mu(3\hat{i} - 5\hat{j} + 2\hat{k})\).
Find the equations of the lines AC and BD, if coordinates of points A, B, C and D respectively, are \((3, -4, 2)\), \((-1, -2, 0)\), \((4, 1, -1)\) and \((4, 0, 5)\). Show that the lines AC and BD are perpendicular to each other.
Find the values of \(a\) and \(b\), if \(y = \frac{ax - b}{(x - 1)(x - 4)}\) has a critical point \((3, 1)\).
A box contains cards numbered 1 to 22. Two cards are drawn simultaneously without replacement and their numbers are noted. What is the probability that both numbers are even ?
Students are trying to fix thumb pins on a graph sheet on the class notice board. A relation \(R\) is defined on the set \(A\) of thumb pins as
\[ R = \{(P, Q) : P, Q \in A \text{ and distance between } P \text{ and } Q \text{ is less than } 3 \text{ cm}\} \]
Determine if \(R\) is an equivalence relation.
If \(2y^3 + y = 2x\), then show \(\dfrac{d^2y}{dx^2} = \dfrac{-48y}{(6y^2 + 1)^3}\).
Determine \(a\) and \(b\) if \(f(x)\) is a continuous function, where
\[ f(x) = \begin{cases} 2, & x \le 3 \\ ax + b, & 3 < x < 5 \\ 5, & x \ge 5 \end{cases} \]
For an LPP with constraints
\[ \begin{aligned} 2x + y &\le 70 \\ x + y &\le 40 \\ x, y &\ge 0, \end{aligned} \]
shade the feasible region on the graph and write the coordinates of its corner points.
Assertion (A) : Points \(A(6, -7, 0)\), \(B(16, -19, -4)\) and \(C(1, -1, 2)\) are collinear.
Reason (R) : Three points \(A\), \(B\) and \(C\) are collinear, when point \(C\) is the mid-point of line segment joining points \(A\) and \(B\).
If \(A\) and \(B\) are independent events such that \(P(A) = 0{\cdot}45\) and \(P(A\cup B) = 0{\cdot}5\), then \(P(B)\) is equal to :
Assertion (A) : Let \(R\) be a relation on set \(A\) of all books in a school library given by \(R = \{(x, y) : x, y \in A,\ x \text{ and } y \text{ have the same number of pages}\}\) and \(S\) is a relation on set \(A\) from the same library such that \(S = \{(p, q) : p \text{ and } q \text{ are Mathematics books}\}\). Also, \(R\cap S \neq \phi\). Then, \(R\cap S\) is an equivalence relation.
Reason (R) : The intersection of two equivalence relations on a set is an equivalence relation on the set.
If two events \(A\) and \(B\) are independent, then which of the following statements is not correct ?
If \(\displaystyle\int_{0}^{4}\frac{1}{x+\sqrt{x}}\,dx = k\log 3\), then the value of \(k\) is :
Solve the matrix equation for \(x\) and \(y\).
\[ \begin{bmatrix} x^2 \\ y^2 \end{bmatrix} - 3\begin{bmatrix} x \\ 3y \end{bmatrix} = \begin{bmatrix} -2 \\ -20 \end{bmatrix} \]
\( \displaystyle\int \left[ \dfrac{1}{\log x} - \dfrac{1}{(\log x)^{2}} \right] dx \) is equal to :
If \( \displaystyle\int_{0}^{1} (9x^{2} - 4x - 2k)\,dx = 0 \), then the value of \( k \) is :
If \(y = \log \sqrt{\tan x}\), then the value of \(\frac{dy}{dx}\) at \(x = \frac{\pi}{4}\) is :
If vectors \(\vec{a} = 3\hat{i} - 2\hat{j} + 4\hat{k}\) and \(\vec{b} = \lambda\hat{i} + \hat{j}\) are perpendicular to each other, then the value of \(\lambda\) is :
The values of \( x \) for which the rate of increase of \( f(x) = x^{3} - 5x^{2} + 6x - 7 \) is thrice the rate of increase of \( x \), are :
Differential of \( \cos^{-1}(e^{x}) \) with respect to \( x \) is :
For \( f(x) = -|x+1| + 5 \), which of the following statements is true ?
\( \left[ \tan^{-1}\sqrt{3} - \sec^{-1}(-2) \right] \) is equal to :
If \( A \) and \( B \) are square matrices of order 3 such that \( |A| = 4 \) and \( |B| = 7 \), then \( |2AB| \) is :
If \( A = \begin{bmatrix} 1 & -2 & 3 \\ 3 & 2 & -1 \end{bmatrix} \), \( B = \begin{bmatrix} 2 & 3 \\ -1 & 2 \\ 4 & -5 \end{bmatrix} \) and \( C = [C_{ij}] = AB \), then \( C_{12} \) is :
If \(A\) is a square matrix, then which of the following is not true ?