Q. If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are the position vectors of the vertices of the triangle $ABC$, then write the formula of the area of $△ABC$.

Q. Related to the numerical method, the formula by the trapezoidal rule is _________.

Q. Write the value of $\int \frac{dx}{ax+1}$.

Q. If $\overrightarrow{a}=2\hat{i}-3\hat{j}+\hat{k}$ and $\overrightarrow{b}=3\hat{i}+2\hat{j}$, then find $\overrightarrow{a}\times \overrightarrow{b}$

Q. Prove that vectors $2\hat{i}-3\hat{j}+5\hat{k}$ and $-2\hat{i}+2\hat{j}+2\hat{k}$ are mutually perpendicular.

Q. Find the differential coefficient of $\sin x$ by first principle.

Q. The unit vector in the direction of $\hat{i}+\hat{j}+\hat{k}$ is be:

1. $\frac{1}{\sqrt{3}}(\hat{i}+\hat{j}+\hat{k})$
2. $\sqrt{3}(\hat{i}+\hat{j}+\hat{k})$
3. $\frac{1}{\sqrt{2}}(\hat{i}+\hat{j}+\hat{k})$
4. $\sqrt{2}(\hat{i}+\hat{j}+\hat{k})$

Q. If ${\tan }^{-1}x+{\tan }^{1}y+{\tan }^{1}z=\frac{\pi }{2}$, then prove that, $xy+yz+zx=1$.

Q. Fraction form of $\frac{1}{(x+3)(x+4)}$ is.

1. $\frac{1}{(x+3)}+\frac{1}{(x+4)}$
2. $\frac{1}{(x+3)}-\frac{1}{(x+4)}$
3. $\frac{1}{(x+4)}-\frac{1}{(x+3)}$
4. $\frac{1}{2}[\frac{1}{(x+3)}+\frac{1}{(x+4)}]$

Q. Evaluate : $\int \frac{dx}{1+\cos 2x}$.

Q. Define the positive co-relation.

Q. Write the equation of a straight line which passes through the point $(2,1,3)$ and has direction rations $(1,3,2)$.

Q. ${\sin }^{-1}x+{\cos }^{-1}x=$ ________

Q. Find the vector equation of sphere whose centre is $(2,-3,4)$ and radius is $5$.

Q. Sphere $3x^2+3y^2+3z^2-6x-12y+6z+2=0$ has centre ________.

Q. Prove that:
$\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CA}=0$

Q. The arithmetic mean of regression coefficients is always _______ the correlation.

Q. Differential coefficient of $\log \left(\sin x\right)$ with respect to $x$ is:

1. $\text{cosec}x$
2. $\tan x$
3. $\sec x$
4. $\cot x$

Q. Prove that : ${\sin }^{-1}x+{\sin }^{-1}y={\sin }^{-1}[x\sqrt{1-y^2}+y\sqrt{1-x^2}]$

Q. Match the correct pair.

Q. Evaluate ${\int }_0^{\pi /4}\sin 2xdx$.

Q.

Q. Resolve the following fraction into partial fractions : $\frac{16}{(x+2)(x^2-4)}$.

Q. Resolve the following fraction into partial fractions : $\frac{2x+1}{(x-1)(x^2+1)}$.

Q. Evaluate $\int \frac{\sec x}{(\sec x-\tan x)}dx$.

Q. Find the distance of point $(2,-1,3)$ from the plane $\overrightarrow{r}.(3\hat{i}+2\hat{j}-6\hat{k})+15=0$

Q. The perpendicular distance of the plane $3x-6y+5z=12$ from origin is be:

1. $\frac{-\sqrt{70}}{12}$
2. $\frac{-12}{\sqrt{70}}$
3. $\frac{\sqrt{70}}{12}$
4. $\frac{12}{\sqrt{70}}$

Q. If $y=\log \left(\log \sin x\right)$, then evaluate $\frac{dy}{dx}$.

Q. If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are coplanar then $[\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}]$ will be ______.

Q. If $\overrightarrow{OP}=\hat{i}+4\hat{j}-3\hat{k}$ and $\overrightarrow{OQ}=2\hat{i}-2\hat{j}-\hat{k}$ then find the modulus of $\overrightarrow{PQ}$.