Full Marks: 40
Pass Marks: 18

 SAQ $10 \times 3 = 30$

1) a) If $x \sqrt{1+y} + y \sqrt{1+x}=0,$ then $\frac{dy}{dx}=?$
OR
b) Let $y= \left( \mathrm{sin}^{-1}x  \right) ^2 + \left( \mathrm{cos}^{-1} x \right) ^2$ ; Show that $(1-x^2) \frac{d^2y}{dx^2} - x \frac{dy}{dx}=4$

2) Find the derivative of $\mathrm{sin}^{-1}  \left( \frac{2x}{1+x^2} \right)$ with respect to $\mathrm{tan}^{-1} \left( \frac{2x}{1-x^2} \right)$

3) Find the derivative with respect to $x$, $\mathrm{tan^{-1}} \left( \frac{ \mathrm{sin}x}{1+\mathrm{cos}x} \right)$

4) Find $\frac{dy}{dx}$
a) $x= \mathrm{cos}^{-1}(8t^4-8t^2+1), \, y=\mathrm{sin}^{-1}(3t-4t^3)$
OR b) $y=x^ { \mathrm{sin}x} + \left(  \mathrm{sin}x \right) ^{\mathrm{cos}x}$

5) a) If $y= \frac{1+\mathrm{sin} \theta + \mathrm{cos} \theta }{1+\mathrm{sin} \theta - \mathrm{cos} \theta}$, show that, $\frac{dy}{dx} + \frac{1}{1-\mathrm{cos} \theta}=0$
OR b) If $\mathrm{tan}y = \frac{\mathrm{tan}x + \mathrm{sec}x - 1}{ \mathrm{tan}x - \mathrm{sec}x + 1}$, Show that, $\frac{dy}{dx}=\frac{1}{2}$

6) Prove
a) $\int \frac{\mathrm{cos}x - \mathrm{cos}2x}{1- \mathrm{cos} x}dx = x+ 2\mathrm{sin}x+c$
OR b)$\int \mathrm{cos}x \cdot \mathrm{cos}2x \cdot \mathrm{cos}3x dx$
        $= \frac{1}{4} \left( x + \frac{\mathrm{sin}2x}{2} + \frac{\mathrm{sin}4x}{4} + \frac{\mathrm{sin}6x}{6} \right) + c$

7) Evaluate
a) $\int \frac{\mathrm{cos}x + x \mathrm{sin}x}{x(x+ \mathrm{cos}x)}dx$
OR b) $\int \frac{dx}{\mathrm{cos}x + \mathrm{sin}x }$

8) Evaluate
a) $\int \frac{e^{\mathrm{cos}^{-1} x}}{1+x^2}$
OR b) $\int \frac{dx}{(1+x^2) \sqrt{\mathrm{tan}^{-1}x + 4}}$

9) Integrate
a) $\int \frac{\mathrm{cos}x dx}{ \sqrt{6+11\mathrm{sin}x-10\mathrm{sin}^2x}}$
OR
b) $\int \frac{dx}{(1+x) \sqrt{1+x-x^2}}$

10) Integrate $\int \frac{xdx}{(1+x^2)(1+x)}$

VSQ $5 \times 2=10$
1) Differentiate (any 2)
a) $e^{{x}^9}$ b) $\mathrm{log}(\mathrm{cot}x)$ c) $\left( {\sqrt{\mathrm{log}x}} \right) ^3$

2) Integrate (Any 3)
a) $\int \frac{dx}{\sqrt{ax+b}}$,  b) $\int \frac{dx}{16-25x^2}$,  c) $\int \mathrm{log}xdx$,  d) $\int \mathrm{cos}^5xdx$

also read: Chemistry Test-(i)