Volume of Parallelopiped

Q. Consider the cube in the first octant with sides $OP,OQ$ and $OR$ of length $1,$ along the x-axis, y-axis and z-axis, respectively, where $O(0,0,0)$ is the origin. Let $S(\frac{1}{2},\frac{1}{2},\frac{1}{2})$ be the centre of the cube and $T$ be the vertex of the cube opposite to the origin $O$ such that $S$ lies on the diagonal $OT.$If $\overrightarrow{p}=\overrightarrow{SP},\overrightarrow{q}=\overrightarrow{SQ},\overrightarrow{r}=\overrightarrow{SR}$ and $\overrightarrow{t}=\overrightarrow{ST},$ then the value of $|(\overrightarrow{p}\times \overrightarrow{q})\times (\overrightarrow{r}\times \overrightarrow{t})|$ is

Q.

State and prove parallelogram law of vector addition and write its special cases.

Q. If the volume of a parallelepiped, whose coterminous edges are given by the vectors $\overrightarrow{a}=\hat{i}+\hat{j}+n\hat{k},\overrightarrow{b}=2\hat{i}+4\hat{j}-n\hat{k}$ and $\overrightarrow{c}=\hat{i}+n\hat{j}+3\hat{k}(n\ge 0),$ is $158$ cu.units, then

1. $n=9$
2. $\overrightarrow{b}.\overrightarrow{c}=10$
3. $\overrightarrow{a}.\overrightarrow{c}=17$
4. $n=7$

Q. Let three points $A(2,3,4),B(3,4,2)$ and $C(4,2,3)$ in space are given. A point $D$ in space is such that it is at a distance of $\sqrt{6}$ units from the three given points. The volume of tetrahedron $ABCD$ is -

1. $1$
2. $\sqrt{3}$
3. $\sqrt{13}$
4. $2$

Q. ${(4\frac{11}{15}+\frac{15}{71})}^2-{(4\frac{11}{15}-\frac{15}{71})}^2$ is equal to

1. $1$
2. $2$
3. $3$
4. $4$

Q. Let $f:R\to R$ defined by $f\left(x\right)=\frac{x^2}{1+x^2}$
f is

1. injective
2. bijective
3. neither injective nor surjective
4. surjective

Q. Find all points of discontinuity of $f$, where $f$ is defined by
$f\left(x\right)=$ $\{\begin{array}{c}{x}^3-3,ifx\le 2\\ x^2+1,ifx>2\end{array}$

Q. If some three consecutive in the binomial expansion of ${(x+1)}^n$ is powers of $x$ are in the ratio $2:15:70$, then the average of these three coeffcient is:

1. $625$
2. $227$
3. $964$
4. $232$

Q. (A) : $\int e^x(\log x+x^{-2})dx=e^x(\log x-\frac{1}{x})+c$
(R): $\int e^x\left[f\right(x)+f^{\prime }(x\left)\right]dx=e^{x}f\left(x\right)+c$

1. Both A and R are true and R is the correct explanation of A
2. Both A and R are true but R is not correct explanation of A
3. A is true but R is false
4. A is false but R is true.

Q. If either $\overrightarrow{a}=\overrightarrow{0}$ or $\overrightarrow{b}=\overrightarrow{0}$, then $\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{0}$. Is the converse true? Justify your answer with an example .

Q. Let P point on the circle $x^2+y^2=9$, Q a point on the line $7x+y+3=0$, and the perpendicular bisector of PQ be the line $x-y+1=0$. Then the coordinate of P are

1. $(\frac{72}{25},-\frac{21}{25})$
2. (0, -3)
3. (0, 3)
4. $(-\frac{72}{25},\frac{21}{25})$

Q. Integrate the rational function $\frac{x^3+x+1}{x^2-1}$

Q. $\int \left(\begin{array}{c}\frac{1}{x-3}-\frac{1}{x^2-3x}\end{array}\right)dx=$ ____ $+C,x>3$

1. $\frac{1}{3}\log \left(x\right(x-3\left)\right)$
2. $\frac{1}{3}\log \left(\sqrt{x}\right(x-3\left)\right)$
3. $\frac{2}{3}\log \left(x\right(x-3\left)\right)$
4. $\frac{2}{3}\log \left(\sqrt{x}\right(x-3\left)\right)$

Q. If the image of a point $P(2,3)$ in the line mirror $y=x$ is the point $Q$ and the image $Q$ in the line mirror $y=0$ is the point $R(x,y)$, then the coordinates of $R$ are

1. $(3,2)$
2. $(-2,3)$
3. $(-3,-2)$
4. $(3,-2)$

Q. Two rods of lengths $a$ and $b$ slide along coordinate axes such that their ends are concyclic. Locus of the center of the circle is

1. $4(x^2+y^2)=a^2+b^2$
2. $4(x^2+y^2)=a^2-b^2$
3. $4(x^2-y^2)=a^2-b^2$
4. $xy=ab$

Q. If the position vectors of $P$ and $Q$ are $\overline{i}+2\overline{j}-7\overline{k}$ and $5\overline{i}-3\overline{j}+4\overline{k}$ respectively then the cosine of the angle between $\overline{PQ}$ and $z$-axis is

1. $\frac{11}{\sqrt{162}}$
2. $\frac{5}{\sqrt{162}}$
3. $\frac{4}{\sqrt{162}}$
4. $\frac{-5}{\sqrt{162}}$

Q. The abscissas of points $P$ and $Q$ on the curve $y=e^x+e^{-x}$ such that tangents at $P$ and $Q$ make ${60}^{\circ }$ with the $x$-axis are

1. $\ln \left(\frac{\sqrt{3}+\sqrt{7}}{7}\right)$ and $\ln \left(\frac{\sqrt{3}+\sqrt{5}}{2}\right)$
2. $\ln \left(\frac{\sqrt{3}+\sqrt{7}}{2}\right)$
3. $\ln \left(\frac{\sqrt{7}-\sqrt{3}}{7}\right)$
4. $\pm \ln \left(\frac{\sqrt{3}+\sqrt{7}}{2}\right)$

Q. If the volume of a parallelopiped formed by the vectors $\hat{i}+\lambda \hat{j}+\hat{k},\hat{j}+\lambda \hat{k}$ and $\lambda \hat{i}+\hat{k}$ is minimum, then $\lambda$ is equal to :

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