A rectangle of perimeter 36 cm is revolved around one of its sides to sweep out a cylinder of maximum volume.
Find the dimensions of the rectangle.
Find the co-ordinates of the point on the line \(\vec{r} = -\hat{j} + 3\hat{k} + \lambda(2\hat{i} - 2\hat{j} + \hat{k})\) such that the sum of co-ordinates is 3.
Find the domain of \(g(x) = \cos^{-1}(x^{2} - 1)\). Hence, find the value of \(x\) for which \(g(x) = \dfrac{\pi}{3}\).
Also, write the range of \(\cos^{-1}x\) other than its principal branch.
Find the sub intervals in which \(f(x) = \cot^{-1}(\sin x + \cos x),\ x \in (0, \pi)\) is increasing and decreasing.
The general solution of the differential equation \(\frac{dy}{dx} = \frac{\sqrt{y}}{\sqrt{x}}\) is
Direction cosines of the line given by equations : \(\frac{2x-1}{4} = \frac{1-y}{3} = \frac{-z}{6}\) are
If \(e^{-x} + e^{-y} = 2\), then \(\frac{dy}{dx}\) is
If \(x = a\sin^{3}t,\ y = b\cos^{3}t\), then find \(\frac{dy}{dx}\) at \(t = \frac{\pi}{4}\).
If \(\displaystyle\int_{0}^{2a}\frac{1}{1+4x^{2}}\,dx = \frac{\pi}{6}\), then the value of a is
A relation R on set A = {1, 2, 3} defined as R = {(1, 1), (2, 2), (1, 2)} is
If \( A = \begin{bmatrix} \cos x & -\sin x \\ \sin x & \cos x \end{bmatrix} \) and \( A + A^{\prime} = I \), then the value of \( x \in \left[ 0, \dfrac{\pi}{2} \right] \) is
If \( x = 3\sin t - \sin 3t,\ y = 3\cos t - \cos 3t \), find \( \dfrac{dy}{dx} \) and prove that \( \dfrac{d^2y}{dx^2} = \dfrac{-\operatorname{cosec}^3 2t \cdot \operatorname{cosec} t}{3} \).
Prove that the line through points A(0, -1, -1) and B(4, 5, 1) intersects the line through points C(3, 9, 4) and D(-4, 4, 4). Hence, write the equation of line passing through the point of intersection of lines AB and CD as well as origin.
Evaluate : \(\sin\left(\tan^{-1}(-\sqrt{3}) - \sec^{-1} 2\right)\)
Find the general solution of the differential equation :
\(y^2\,dx + (x^2 - xy + y^2)\,dy = 0\)
Evaluate : \( \displaystyle\int_{0}^{1} x \sin^{-1}x \, dx \)
Find the particular solution of the differential equation \(\dfrac{dy}{dx} = y\tan x\), given that \(y = 2\) if \(x = 0\).
A vector \(\vec{a}\) of magnitude 14 has direction ratios \(\langle 2, 3, -6 \rangle\). Find the projection of the vector \(\vec{a}\) on \(\hat{i}\).
Simplify : \(\cot^{-1}\sqrt{\dfrac{1 + \cos 2x}{1 - \cos 2x}}\), \(x \in \left(0, \dfrac{\pi}{2}\right)\).
If \(\tan^{-1} x = 3y\), then
The absolute maximum value of \(f(x) = x^2 + 1\) in \([-5, 2]\) is
The length of perpendicular drawn from the point \((1, 2, 3)\) on line \(\dfrac{x}{0} = \dfrac{y}{1} = \dfrac{z}{0}\) is
If A and B are symmetric matrices of same order, then which of the following matrices is a skew-symmetric matrix ?
Assertion (A) : Lines given by \(x = py + q\), \(z = ry + s\) and \(x = p^{\prime}y + q^{\prime}\), \(z = r^{\prime}y + s^{\prime}\) are perpendicular to each other when \(pp^{\prime} + rr^{\prime} = 1\).
Reason (R) : Two lines \(\vec{r} = \vec{a}_1 + \lambda\vec{b}_1\) and \(\vec{r} = \vec{a}_2 + \mu\vec{b}_2\) are perpendicular to each other if \(\vec{b}_1 \cdot \vec{b}_2 = 0\).
Which of the following cannot be an order of a column-matrix ?
