Find : $\large\textstyle\int$$(x+3){\sqrt{3-4x-x^{2}}}\ d x\ .$

Find : $\large\textstyle\int$$\large{\frac{x}{(x^{2}+1)(x-1)}}\,d x.$

Find the equation of tangent to the curve y = cos (x + y), -2π ≤ x ≤ 0, that is parallel to the line x + 2y = 0.

Find the equation of tangents to the curve y = x³ + 2x – 4 which are perpendicular to the line x + 14y – 3 = 0.

Find the equation(s) of the tangent(s) to the curve y = (x³ – 1) (x – 2) points where the curve intersects the x-axis.

Find the particular solution of the differential equation ${\frac{d y}{d x}}={\frac{x\left(2\log x+1\right)}{\sin y+y\cos y}}$ given that $y={\frac{\pi}{2}}$ where x = 1.

Let f : N → N be defined as $f(n)={\left\{\begin{array}{l l}{\displaystyle{\frac{n+1}{2}},{\mathrm{when~}}n{\mathrm{~is~odd}}}\\ {\displaystyle{\frac{n}{2}},{\mathrm{when~}}n{\mathrm{iseven}}}\end{array}\right.}$ for all n ∈ N. State whether the function f is bijective. Justify your answer.

Show that f : N → N, given by f(x) = |``x+1, if x is odd x- 1, if x is even is both one-one and onto.``|

Using properties of determinants, prove that $\begin{vmatrix}x+y & x & x \\ 5x+4y & 4x & 2x \\ 10x+8y & 8x & 3x\end{vmatrix}$ = x³

Using properties of determinants, prove that : $\begin{vmatrix}1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c\end{vmatrix}$ = $a b c+b c+c a+a b.$

Using properties of determinants, prove that : $\begin{vmatrix}b+c & c+a & a+b \\ q+r & r+p & p+q \\ y+z & z+x & x+y\end{vmatrix}$ = 2 $\begin{vmatrix}a & b & c \\ p & q & r \\ x & y & z\end{vmatrix}$ .

Evaluate : $\large\textstyle\int$$\large{\frac{x}{(x-1)^{2}(x+2)}}d x\ .$

Find : $\large\textstyle\int(3x+1)\large{\sqrt{4-3x-2x^{2}}}d x\,.$

Find the equations of the normal to the curve x² = 4y which passes through the point (1, 2).

Find the particular solution of the differential equation satisfying the given condition ${\frac{d y}{d x}}=y\tan x,$ given that y = 1, when x = 0.