1. f(x)=x2 f(x)=x^{2} f(x)=x2.
Determine: ∫5e2x1+e4xdx \int \frac{5 e^{2 x}}{1+e^{4 x}} d x ∫1+e4x5e2xdx.
Determine : ∫[1+{f(x)}21+f(x)+1f(x)+f(x)+1] \int\left[\frac{1+\{f(x)\}^{2}}{1+f(x)}+\frac{1}{f(x)+\sqrt{f(x)}+1}\right] ∫[1+f(x)1+{f(x)}2+f(x)+f(x)+11] dx.
Find the value of {[f(x)ef(x)+{f(x)−1}2f(x)] \left\{\left[\sqrt{f(x)} e^{f(x)}+\frac{\{f(x)-1\}^{2}}{f(x)}\right]\right. {[f(x)ef(x)+f(x){f(x)−1}2] dx.
2.
Find the distance between the parallel lines 4x−3y+ 4 x-3 y+ 4x−3y+ 2=0 2=0 2=0 and 8x−6y−9=0 8 x-6 y-9=0 8x−6y−9=0.
If P \mathrm{P} P be the middle point of AB \mathrm{AB} AB, find the equation of the straight line CD.
Find the bisectors of the obtuse angle between the lines AE A E AE and CD C D CD.
3. f(x)=tanx \mathrm{f}(\mathrm{x})=\tan \mathrm{x} f(x)=tanx.
If y=sin{2tan−1(1−x1+x)} y=\sin \left\{2 \tan ^{-1} \sqrt{\left(\frac{1-x}{1+x}\right)}\right\} y=sin{2tan−1(1+x1−x)}, find dydx \frac{d y}{d x} dxdy.
Find the derivative of f(3x) f(3 x) f(3x) using the first principle with respect to x x x.
If y=f(x)+f′(x) y=f(x)+\sqrt{f^{\prime}(x)} y=f(x)+f′(x), prove that, (1−sinx)y2−y=0. (1-\sin x) y_{2}-y=0. (1−sinx)y2−y=0.
4. f(x)=tan−1x \mathrm{f}(\mathrm{x})=\tan ^{-1} \mathrm{x} f(x)=tan−1x and 2 g(x)=2sinx+sin2x 2 \mathrm{~g}(\mathrm{x})=2 \sin \mathrm{x}+\sin 2 \mathrm{x} 2 g(x)=2sinx+sin2x.
If y=cosec−11+x22x y=\operatorname{cosec}^{-1} \frac{1+x^{2}}{2 x} y=cosec−12x1+x2, find dydx \frac{d y}{d x} dxdy.
If y=tan{mf(x)} y=\tan \{m f(x)\} y=tan{mf(x)}. Prove that, (1+x2)y2+2xy1= \left(1+x^{2}\right) y_{2}+2 x y_{1}= (1+x2)y2+2xy1= 2 myy 1 _{1} 1.
Find the extreme value of g(x) g(x) g(x) in the interval 0<x<π0<x<\pi0<x<π.
5. P=[1+x2−y22xy2y2xy1−x2+y2−2x−2y2x1−x2−y2] P=\left[\begin{array}{ccc}1+x^{2}-y^{2} & 2 x y & 2 y \\ 2 x y & 1-x^{2}+y^{2} & -2 x \\ -2 y & 2 x & 1-x^{2}-y^{2}\end{array}\right] P=1+x2−y22xy−2y2xy1−x2+y22x2y−2x1−x2−y2 and f(x) f(x) f(x) =x3−3x+2I =\mathrm{x}^{3}-3 \mathrm{x}+2 \mathrm{I} =x3−3x+2I.
If A=[123456],B=[02120−1] A=\left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6\end{array}\right], B=\left[\begin{array}{rr}0 & 2 \\ 1 & 2 \\ 0 & -1\end{array}\right] A=[142536],B=01022−1 find BA B A BA.
If det(P)=0 \operatorname{det}(P)=0 det(P)=0, prove that x2+y2=−1 x^{2}+y^{2}=-1 x2+y2=−1.
If x=1,y=2 x=1, y=2 x=1,y=2, find f(p) f(p) f(p), where I I I is a identity matrix.
