NDA MATHEMATICS QUIZ-5

If $\mathrm{y}=\sin ^{-1} \sqrt{\frac{1+x^{2}+1}{2 \sqrt{1+x^{2}}}}$ then $\frac{d y}{d x}=\ldots \ldots$

$\int \frac{x \cdot d x}{x^4+x^2+1}$ is equal to

If $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}=1$ then $\frac{a^{2}}{b+c}+\frac{b^{2}}{c+a}+\frac{c^{2}}{a+b}$ equals

Projection of the vector $3 \hat{i}+\hat{j}+\hat{k}$ on $-2 \hat{i}+\hat{j}-2 \hat{k}$ is equal to

If position vector of a point $\mathrm{A}$ is $\vec{r}=$ $a \vec{i}+b \vec{j}+c \vec{k}$ where $a, b, c$ all are natural numbers it $\bar{X}=\vec{i}+\vec{j}+\vec{k}$ and $\vec{r} \cdot \vec{X}=12$ than number of possible values for $\vec{A}$ will be.

A funciton $f(x)$ be defined in such a way that

$f^{\prime}(x)=\frac{x ^3}{3}+c x+2$ where

$f(0)=0$ and $f(6)=210$ then the value of $\mathrm{c}$ is.

$\int_{\log \frac{3}{4}}^{\log _{2}} \sin \frac{a^{x}-1}{a^{x}+1} \cdot d x$ is.

If $y={(\tan^{-1} x)^{2}}$ then $\left(y^{2}+1\right)^{2} y_{2}$ $+2 x\left(x^{2}+1\right) y$$_{1}$ is:

What is the angle between the lines whose direction cosines are proportional to $(2,3,4)$ and $(1,-2,1)$ respectively?

The average salary of male employees in a firm was Rs. 560 and that of females was Rs.360. The mean salary of all the employees was Rs. 520 than percentages of males and females will be respectively.