ii) if  , prove that $AA^T=I$. Hence find $A^{-1}$

i) Evaluate $\lim _{x\rightarrow 0}\frac{log(\sin x+ \cos x)}{x}$

ii) Find the value of $\int e^x \left( \frac{x+2}{x+4}\right)^2dx$

iii) find the derivative of $e^{sin^{-1}x}$ w.r.t $e^{-\cos ^ {-1}x}$

iv) form differential equation of $y=ax+bx^3$

v) on applying lagrange's mean value theorem of $f(x)=x^2-4x+5$ at [1,2] we get C $\in$ (1,2). then find the value of C 

vi)if $f(x)=x$ for x$\leq0$

i) $|\vec{\alpha}|=4$ , $|\vec{\beta}|=3$  and $|\vec{\alpha}\times\vec{\beta}| =6$, then find the angle between $|\vec{\alpha}|$  and $|\vec{\beta}|$ 

ii) if a straight line makes angles $\alpha, \beta$ and $\gamma$ respectively with the coordinate axis, then prove that $\sin ^2 \alpha+ \sin^2 \beta + \sin^2 \gamma=2$

i) the mean and variance of a binomial distribution are 6 and 4 respectively. find the value of the parameters of that distribution.

ii)Akhil and Vijay appear for an interview for two vacancies. the probability of Akhil's selection is $\frac{1}{4}$ and Vijay's selection is $\frac{2}{3}$. Find the probability that only one of them will be selected.

i) Prove that: $\tan \left( \frac{\pi}{4}+\frac{1}{2}\cos ^{-1} \frac{a}{b} \right)+\tan \left( \frac{\pi}{4}-\frac{1}{2}\cos ^{-1} \frac{a}{b} \right)=\frac{2b}{a}$

ii) Let $A= \{-1,1,-2,2 \}, \ B= \{3,4,5,6 \}$ and $f:A\rightarrow B$ be the mapping defined by 

i) Show that $[\vec{a}+\vec{b}$    $\vec{b}+\vec{c}$   $\vec{c}+\vec{a}]$ $=2[\vec{a}\vec{b}\vec{c}]$

ii) if the points (2-x,2,2),(2,2-y,2) and (2,2,2-z) are coplanar then prove that $\frac{2}{x}+\frac{2}{y}+\frac{2}{z}=1$

iii) if $\vec{a}+\vec{b}+\vec{c}=\vec{0}$ and $|\vec{a}|=3, |\vec{b}|=5, |\vec{c}|=7$, find the angle between $\vec{a}$ and $\vec{b}$. 

i) $\int^{\frac{\pi}{2}}_0 log( \sin x) dx$ OR $\int^{1}_0 \frac{log (1+x)}{1+x^2}dx$

ii) $\int^\pi _0 \frac{xdx}{a^2 \cos ^2 x + b^2 \sin ^2 x}=\frac{\pi ^2}{2ab}$

i) in a box there are 5 watches of which 2 are known to be defective. two watches are taken at random. Let X denote the number of defective watches selected. obtain the probability distribution of X. also calculate the mean of X.

ii) a man speaks the truth 3 out of 4 times. he throws an unbiased die and reports that it is a six. Find the probability that it is actually six.

i) A small firm manufactures A and B. The total number of items it can manufacture in a day is at the most 24. Item A takes an hour to make while item B takes only half an hour. the maximum time available per day is 16 hours. if the profit on one unit pt item A be Rs. 300 and that on one unit of item B be Rs 160. Formulate the problem.

ii)Solve graphically the LPP given below: