Scalar Triple Product

Q. If $\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]=-4,$ then the Volume (in cubic units) of the parallelopiped with coterminous edges $\overrightarrow{a}+2\overrightarrow{b},2\overrightarrow{b}+\overrightarrow{c},3\overrightarrow{c}+\overrightarrow{a}$ is

1. $64$
2. $32$
3. $8$
4. $16$

Q. Let $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ be three vectors mutually perpendicular to each other and have same magnitude. If a vector $\overrightarrow{r}$ satisfies
$\overrightarrow{a}\times \{(\overrightarrow{r}-\overrightarrow{b})\times \overrightarrow{a}\}+\overrightarrow{b}\times \{(\overrightarrow{r}-\overrightarrow{c})\times \overrightarrow{b}\}+\overrightarrow{c}\times \{(\overrightarrow{r}-\overrightarrow{a})\times \overrightarrow{c}\}=\overrightarrow{0}$
then $\overrightarrow{r}$ is equal to

1. $\frac{1}{3}(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c})$
2. $\frac{1}{2}(\overrightarrow{a}+\overrightarrow{b}+2\overrightarrow{c})$
3. $\frac{1}{2}(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c})$
4. $\frac{1}{3}(2\overrightarrow{a}+\overrightarrow{b}-\overrightarrow{c})$

Q.

What is $a$ cross $b$ cross $c$ ?

Q. Let $\hat{a},\hat{b}$ be unit vectors. If $\overrightarrow{c}$ be a vector such that the angle between $\hat{a}$ and $\overrightarrow{c}$ is $\frac{\pi }{12},$ and $\hat{b}=\overrightarrow{c}+2(\overrightarrow{c}\times \hat{a})$, then ${\left|6\overrightarrow{c}\right|}^2$ is equal to

1. $6(3+\sqrt{3})$
2. $6(3-\sqrt{3})$
3. $3+\sqrt{3}$
4. $6(\sqrt{3}+1)$

Q. Two fair dice are thrown. The numbers on them are taken as $\lambda$ and $\mu ,$ and a system of linear equations
$x+y+z=5$
$x+2y+3z=\mu$
$x+3y+\lambda z=1$
is constructed. If $p$ is the probability that the system has a unique solution and $q$ is the probability that the system has no solution, then

1. $p=\frac{1}{6}\text{and}q=\frac{1}{36}$
2. $p=\frac{5}{6}\text{and}q=\frac{5}{36}$
3. $p=\frac{1}{6}\text{and}q=\frac{5}{36}$
4. $p=\frac{5}{6}\text{and}q=\frac{1}{36}$

Q.

If the vectors $c,a=x\hat{i}+y\hat{j}+z\hat{k}$ and $b=\hat{j}$ are such that $a,c,$ and $b$ form a right handed system. Then $c$ is?

1. $z\hat{i}-x\hat{k}$

2. $0$

3. $y\hat{j}$

4. $-z\hat{i}+x\hat{k}$

Q. If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are three non coplanar, non zero vectors and $\overrightarrow{r}$ is any vector in space, then $(\overrightarrow{a}\times \overrightarrow{b})\times (\overrightarrow{r}\times \overrightarrow{c})+(\overrightarrow{b}\times \overrightarrow{c})\times (\overrightarrow{r}\times \overrightarrow{a})+(\overrightarrow{c}\times \overrightarrow{a})\times (\overrightarrow{r}\times \overrightarrow{b})$ is equal to $\lambda \left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]$. Then the value of $\lambda$ is

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

Q. If $\overrightarrow{p}=\frac{\overrightarrow{b}\times \overrightarrow{c}}{\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]},\overrightarrow{q}=\frac{\overrightarrow{c}\times \overrightarrow{a}}{\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]}$ and $\overrightarrow{r}=\frac{\overrightarrow{a}\times \overrightarrow{b}}{\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]}$, where $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ are three non-coplanar vectors, then the value of the expression $(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c})\cdot (\overrightarrow{p}+\overrightarrow{q}+\overrightarrow{r})$ is

1. $3$
2. $1$
3. $0$
4. $2$

Q. If $\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}$ are three non-coplanar unit vectors then $\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]$ is equal to

1. $\pm 2$
2. $\pm 1$
3. $2$
4. $\pm 3$

Q.

If $u,v$ and $w$ are the three non-coplanar vectors, then $\left(u+v–w\right)\bullet \left(u–v\right)\times \left(v–w\right)$ is equal to

1. $u\bullet w\times u$

2. $u\bullet v\times w$

3. $0$

4. $3u\bullet v\times w$

Q. Let vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ make an angle $\theta =\frac{2\pi }{3}$ between them. If $\left|\overrightarrow{a}\right|=1$ and $\left|\overrightarrow{b}\right|=2,$ then ${|(\overrightarrow{a}+3\overrightarrow{b})\times (3\overrightarrow{a}-\overrightarrow{b})|}^2$ is

1. $300$
2. $75$
3. $400$
4. $350$

Q. Value of scalar triple product is zero if any 2 vectors are identical.

1. False
2. True

Q. The cube roots of unity when represented on Argand diagram form the vertices of an equilateral triangle.

1. TRUE
2. FALSE

Q. Two perpendicular unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are such that $\left[\overrightarrow{r}\overrightarrow{a}\overrightarrow{b}\right]=\frac{5}{4},$ $\overrightarrow{r}\cdot (3\overrightarrow{a}+2\overrightarrow{b})=0$ and $\underset{-2\overrightarrow{r}.\overrightarrow{a}}{\overset{\frac{-4}{3}\overrightarrow{r}.\overrightarrow{b}}{\int }}\frac{x+1}{x^2+1}dx=\frac{\pi }{2}$. Then which of the following is(are) correct ?

1. $|\overrightarrow{r}{|}^2=\frac{19}{8}$
2. $|\overrightarrow{r}{|}^2=\frac{7}{4}$
3. $\overrightarrow{r}=\frac{\overrightarrow{a}}{2}-\frac{3\overrightarrow{b}}{4}+\frac{5}{4}(\overrightarrow{a}\times \overrightarrow{b})$
4. $\overrightarrow{r}=-\frac{\overrightarrow{a}}{2}+\frac{3\overrightarrow{b}}{2}+\frac{5}{4}(\overrightarrow{a}\times \overrightarrow{b})$

Q. Which of the following represent(s) a vector

1. $\left[\begin{array}{ccc}1& 2& -1\end{array}\right]$
2. $\left[\begin{array}{c}2\\ -3\end{array}\right]$
3. $\left[\begin{array}{ccc}0& 1& 0\\ 0& 2& 0\\ 0& 0& 3\end{array}\right]$
4. $\left[\begin{array}{cc}2& 3\\ -1& 0\end{array}\right]$

Q. Let $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ be three non-coplanar vectors and $\overrightarrow{d}=\sin x(\overrightarrow{a}\times \overrightarrow{b})+\cos y(\overrightarrow{b}\times \overrightarrow{c})+2(\overrightarrow{c}\times \overrightarrow{a})$ be a non-zero vector, which is perpendicular to $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}.$ If the minimum value of $x^2+y^2$ is equal to $\frac{\lambda {\pi }^2}{4},$ then the value of $\lambda$ is

Q.

The volume of tetrahedron whose vertices are

A = (3, 2, 1) , ~B = (1, 2, 4), ~ C = (4, 0, 3), ~ D = (1, 1, 7)~will be $\underset{–}{}$cubic units

1. 

2. 5

3. 

4. None of the above

Q. $ABCD$ is a regular tetrahedron. $M(0,0,0)$ is the midpoint of the altitude $DN$ of the tetrahedron. Coordinates of the points $A$ and $B$ are $(0,0,-2)$ and $(b,b,0)$ respectively (with $b>0$). Also $x$ - coordinate of the point $C$ is given to be positive. Then which of the following is/are correct?

1. Image of point $D$ in the plane generated by $ABC$ is $(2\sqrt{2},0,-2)$
2. Image of point $D$ in the plane generated by $ABC$ is $(2,0,-2\sqrt{2})$
3. Volume of tetrahedron $ACBM$ is $\frac{4\sqrt{2}}{3}$
4. Volume of tetrahedron $ACBM$ is $\frac{4}{3}$

Q. If $\overrightarrow{A},\overrightarrow{B},\overrightarrow{C}$ are non-zero vectors. Then the scalar $\overrightarrow{A}\cdot (\overrightarrow{B}+\overrightarrow{C})\times (\overrightarrow{A}+\overrightarrow{B}+\overrightarrow{C})$ equals to:

1. $0$
2. $\left[\overrightarrow{A}\overrightarrow{B}\overrightarrow{C}\right]+\left[\overrightarrow{B}\overrightarrow{C}\overrightarrow{A}\right]$
3. $\left[\overrightarrow{A}\overrightarrow{B}\overrightarrow{C}\right]$
4. None of these

Q.

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are unit vectors, then the greatest value of $\sqrt{3}\left|\begin{array}{c}\overrightarrow{a}+\overrightarrow{b}\end{array}\right|+\left|\begin{array}{c}\overrightarrow{a}-\overrightarrow{b}\end{array}\right|$ is

Q.

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are non-coplanar vectors and $\overrightarrow{d}=\lambda \overrightarrow{a}+\mu \overrightarrow{b}+\nu \overrightarrow{c},$ then $\lambda$ equal to

1. $\frac{\left[\overrightarrow{d}\overrightarrow{b}\overrightarrow{c}\right]}{\left[\overrightarrow{b}\overrightarrow{a}\overrightarrow{c}\right]}$

2. $\frac{\left[\overrightarrow{b}\overrightarrow{c}\overrightarrow{d}\right]}{\left[\overrightarrow{b}\overrightarrow{c}\overrightarrow{a}\right]}$

3. $\frac{\left[\overrightarrow{b}\overrightarrow{d}\overrightarrow{c}\right]}{\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]}$

4. $\frac{\left[\overrightarrow{c}\overrightarrow{b}\overrightarrow{d}\right]}{\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]}$

Q. For non-zero vectors $\overrightarrow{p},\overrightarrow{q}$ and $\overrightarrow{r},$ let $(\overrightarrow{p}\times \overrightarrow{q})\times \overrightarrow{r}+(\overrightarrow{q}\cdot \overrightarrow{r})\overrightarrow{q}=(x^2+y^2)\overrightarrow{q}+(14-4x-6y)\overrightarrow{p}$ and $(\overrightarrow{r}\cdot \overrightarrow{r})\overrightarrow{p}=\overrightarrow{r}$ where $\overrightarrow{p}$ and $\overrightarrow{q}$ are non-collinear, and $x$ and $y$ are scalars. Then the value of $(x+y)$ is

Q. If $\frac{1}{{\alpha }_k+i}$ are 8 vertices of a regular octogon where ${\alpha }_kϵR,1,2,3,....8$ (where $i=\left(\sqrt{-1}\right)$ then area of the regular octagon is:-

1. $1$
2. $\sqrt{2}$
3. None of these
4. $\frac{1}{\sqrt{2}}$

Q. If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are three non coplanar vectors and a vector $\overrightarrow{\alpha }$ is such that $\overrightarrow{\alpha }=p(\overrightarrow{b}\times \overrightarrow{c})+q(\overrightarrow{c}\times \overrightarrow{a})+r(\overrightarrow{a}\times \overrightarrow{b})\text{and}\overrightarrow{\alpha }\cdot (\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c})=1$ $,\text{then}\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]$ is equal to

1. $p+q+r$
2. $\frac{1}{p+q+r}$
3. $2(p+q+r)$
4. $\frac{2}{p+q+r}$

Q. The value of $[\overrightarrow{a}-\overrightarrow{b}\overrightarrow{b}-\overrightarrow{c}\overrightarrow{c}-\overrightarrow{a}]$ where $|\overrightarrow{a}|=1,|\overrightarrow{b}|=5,|\overrightarrow{c}|=3$, is

1. 0
2. 6
3. None of these
4. 1

Q. Let $\overrightarrow{u}$ and $\overrightarrow{v}$ be the unit vectors such that $\overrightarrow{u}\times \overrightarrow{v}+\overrightarrow{u}=\overrightarrow{w}$ and $\overrightarrow{w}\times \overrightarrow{u}=\overrightarrow{v}$. Then the value of $\left[\overrightarrow{u}\overrightarrow{v}\overrightarrow{w}\right]$ is equal to

1. $-1$
2. $0$
3. $1$
4. $2$

Q. If $\overrightarrow{a}$ and $\overrightarrow{b}$ are perpendicular vectors, $|\overrightarrow{a}+\overrightarrow{b}|=13$ and $|\overrightarrow{a}|=5$, find the value of $|\overrightarrow{b}|$.

Q. A function is selected randomly from the set of functions defined from set $A$ to set $B$, where $n\left(A\right)=4$ and $n\left(B\right)=7$. Then the probability that the selected function is not injection, is

1. $\frac{223}{343}$
2. $\frac{120}{343}$
3. $\frac{210}{343}$
4. $\frac{123}{343}$

Q. $\text{If}\overrightarrow{a},\overrightarrow{b}\text{and}\overrightarrow{c}$ are non-coplanar vectors and $\overrightarrow{a}\times \overrightarrow{c}$ is perpendicular to $\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c}),$ then the value of $[\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c}\left)\right]\times \overrightarrow{c}$ is equal to

1. $\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]\overrightarrow{c}$
2. $\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]\overrightarrow{b}$
3. $\overrightarrow{0}$
4. $\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]\overrightarrow{a}$

Q. For any three vectors, $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$, the value of $(\overrightarrow{a}-\overrightarrow{b})\cdot (\overrightarrow{b}-\overrightarrow{c})\times (\overrightarrow{c}-\overrightarrow{a})$ is:

1. $\overrightarrow{b}\cdot (\overrightarrow{c}\times \overrightarrow{a})$
2. $2\overrightarrow{b}\cdot (\overrightarrow{a}\times \overrightarrow{c})$
3. $0$
4. $2\overrightarrow{a}\cdot (\overrightarrow{b}\times \overrightarrow{c})$