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Differentiate $\left(\frac{5^{x}}{x^{5}}\right)$ with respect to x.

If $A=\begin{bmatrix}2&3\\ -1&2\end{bmatrix}$, then show that $A^{2}-4A+7I=0$.

Let $f:A\rightarrow B$ be defined by $f(x)=\frac{x-2}{x-3}$ where $A=R-\{3\}$ and $B=R-\{1\}$. Discuss the bijectivity of the function.

Let the polished side of the mirror be along the line $\frac{x}{1}=\frac{1-y}{-2}=\frac{2z-4}{6}$. A point $P(1,6,3)$, some distance away from the mirror, has its image formed behind the mirror. Find the coordinates of the image point and the distance between the point P and its image.

Solve the differential equation $(1+x^{2})\frac{dy}{dx}+2xy-4x^{2}=0$ subject to initial condition $y(0)=0$.

Solve the differential equation: $x^{2}y~dx-(x^{3}+y^{3})dy=0$.

Draw a rough sketch for the curve $y=2+|x+1|$. Using integration, find the area of the region bounded by the curve $y=2+|x+1|$, $x=-4$, $x=3$ and $y=0$.

Evaluate: $\int_{0}^{\pi/2}\frac{x}{\sin x+\cos x}dx$.

Find: $\int\frac{x^{2}+1}{(x-1)^{2}(x+3)}dx$.

A person is Head of two independent selection committees I and II. If the probability of making a wrong selection in committee I is 0.03 and that in committee II is 0.01, then find the probability that the person makes the correct decision of selection: (i) in both committees (ii) in only one committee.

Let the position vectors of the points A, B and C be $3\hat{i}-\hat{j}-2\hat{k}$, $\hat{i}+2\hat{j}-\hat{k}$ and $\hat{i}+5\hat{j}+3\hat{k}$ respectively. Find the vector and cartesian equations of the line passing through A and parallel to line BC.

Find the distance of the point $P(2,4,-1)$ from the line $\frac{x+5}{1}=\frac{y+3}{4}=\frac{z-6}{-9}$.

Consider the Linear Programming Problem, where the objective function $Z=(x+4y)$ needs to be minimized subject to constraints $2x+y\ge1000$, $x+2y\ge800$, $x,y\ge0$. Draw a neat graph of the feasible region and find the minimum value of Z.

Show that the function $f:N\rightarrow N$, where N is a set of natural numbers, given by $f(n) = n-1$, if n is even, $n+1$, if n is odd, is a bijection.

A student wants to pair up natural numbers in such a way that they satisfy the equation $2x+y=41$, $x, y\in N$. Find the domain and range of the relation. Check if the relation thus formed is reflexive, symmetric and transitive. Hence, state whether it is an equivalence relation or not.

Differentiate $y=\cos^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right)$ with respect to x, when $x\in(0,1)$.

Differentiate $y=\sin^{-1}(3x-4x^{3})$ w.r.t. x, if $x\in[-\frac{1}{2},\frac{1}{2}]$.

Let $A=\begin{bmatrix}1\\ 4\\ -2\end{bmatrix}$ and $C=\begin{bmatrix}3&4&2\\ 12&16&8\\ -6&-8&-4\end{bmatrix}$ be two matrices. Then, find the matrix B if $AB=C$.

Determine if the lines $\vec{r}=(\hat{i}+\hat{j}-\hat{k})+\lambda(3\hat{i}-\hat{j})$ and $\vec{r}=(4\hat{i}-\hat{k})+\mu(2\hat{i}+3\hat{k})$ intersect with each other.

Vector $\vec{r}$ is inclined at equal angles to the three axes x, y and z. If magnitude of $\vec{r}$ is $5\sqrt{3}$ units, then find $\vec{r}$.