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Differentiate $y=\sqrt{\log\left\{\sin\left(\frac{x^{3}}{3}-1\right)\right\}}$ with respect to x.

Let $2x+5y-1=0$ and $3x+2y-7=0$ represent the equations of two lines on which the ants are moving on the ground. Using matrix method, find a point common to the paths of the ants.

Let R be a relation defined on a set N of natural numbers such that $R=\{(x,y): xy \text{ is a square of a natural number, } x, y\in N\}$. Determine if the relation R is an equivalence relation.

Show that the function $f:R\rightarrow R$ defined by $f(x)=4x^{3}-5$, $\forall x\in R$ is one-one and onto.

10 identical blocks are marked with '0' on two of them, '1' on three of them, '2' on four of them and '3' on one of them and put in a box. If X denotes the number written on the block, then write the probability distribution of X and calculate its mean.

If $-2x^{2}-5xy+y^{3}=76$, then find $\frac{dy}{dx}$.

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.

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

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

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

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 $\frac{x+5}{1}=\frac{y+3}{4}=\frac{z-6}{-9}$.

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.

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.