If \( A = \begin{bmatrix} 2 & 0 & 1 \\ 2 & 1 & 3 \\ 1 & -1 & 0 \end{bmatrix} \), then compute \( A^{2} - 5A + 4I \).
Using integration, find the area of the region bounded by \( y = 5x + 4 \), \( y = 0 \), \( x = -1 \) and \( x = 1 \).
Solve the following Linear Programming Problem graphically :
Maximize \( Z = 20x + 10y \)
subject to constraints
\( x + 2y \leq 28 \)
\( 3x + y \leq 24 \)
\( x \geq 2 \)
\( x, y \geq 0 \)
If in a parallelogram PQRS, \( \overrightarrow{PQ} = 2\hat{i} + 3\hat{j} + \hat{k} \) and \( \overrightarrow{QR} = \hat{i} - 2\hat{j} + \hat{k} \), then find the unit vectors parallel to the diagonals \( \overrightarrow{PR} \) and \( \overrightarrow{SQ} \).
The probability that a particular item is available in three shops A, B and C is \( \dfrac{4}{5}, \dfrac{3}{4} \) and \( \dfrac{2}{3} \) respectively. If a person visits all the three shops to buy the item, then what is the probability that it will be available in at least one of the shops ?
Let \( A = [a_{ij}] \) be a \( 2 \times 2 \) matrix whose elements are given by \( a_{ij} = \frac{(i + 2j)^2}{3} \). Then \( A^{\prime} \) is given by :
A cylindrical tank is being filled with sand at a rate of \( 314 \text{ m}^3/\text{h} \). If the radius of the tank is 10 m, then the height of sand in the tank increases at the rate of :
The sum of the order and the degree of the differential equation \( y = x\dfrac{dy}{dx} + 3\left[2 - \left(\dfrac{dy}{dx}\right)^{2}\right] \) is :
The following graph represents :
The sum of the order and the degree of the differential equation
\[ 4\left(\dfrac{d^2y}{dx^2}\right)^2 + 3\left[1 + \left(\dfrac{dy}{dx}\right)^2 - y\right] = 0 \]
is :
Solve the differential equation \(\sin x \cos y\,dx + \cos x \sin y\,dy = 0\), given that \(y = \frac{\pi}{4}\) when \(x = 0\).
Show that a function \( f: R_{+} \to A \subset N \), defined as \( f(x) = 4x^2 + 12x + 15 \) is one-one. Find set \( A \) so that \( f \) is onto where \( R_{+} = [0, \infty) \). Also, find if there exists \( a \in R_{+} \) such that \( f(a) = 7 \). Justify.
Maximize \( Z = 8x + 8y \)
subject to the constraints
\[ \begin{aligned} x - 20 &\leq 0 \\ 2x + 3y &\leq 120 \\ 2x + y &\leq 60 \\ x \geq 0,\ y &\geq 0 \end{aligned} \]
Find the general solution of the differential equation \((x^2 - y^2)\,dx + 2xy\,dy = 0\).
Evaluate :
\[ \int_0^{\frac{\pi}{2}} \frac{\sin^2 x}{1 + \sin 2x}\,dx \]
\(\displaystyle\int \frac{\cos x}{\sqrt{\sin^2 x + 1}}\,dx\) is equal to :
Find the sub-interval of \((0, \infty)\) in which \(f(x) = x \log x\) is increasing.
If \(2\cos^{-1}x=y\), then \(\dfrac{dy}{dx}\) is :
Direction ratios of lines \(l_1\) and \(l_2\) respectively are \(\langle 1, 0, 0\rangle\) and \(\langle 0, -1, 0\rangle\). The direction ratios of the line perpendicular to both \(l_1\) and \(l_2\) are :
If \(A(\operatorname{adj}A)=\begin{bmatrix}3 & 0 & 0\\0 & 3 & 0\\0 & 0 & 3\end{bmatrix}\), then the value of \(|2A|\) is :
The domain of \(\cos^{-1}(4x+1)\) is :
If a matrix X is such that \(\begin{bmatrix}2 & 1\end{bmatrix}X=\begin{bmatrix}3 & 4 & 5\end{bmatrix}\), then the order of matrix X is :
If \(e^{x+y} = 3x\), then \(\frac{dy}{dx}\) is
Solve the following linear programming problem graphically :
Maximize \(Z = 10500x + 9000y\)
Subject to constraints
\(x + y \le 50\)
\(2x + y \le 80\)
\(x, y \ge 0\)
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.
