Applications Scalar Triple Product

Q.

The set of all points where the function$f\left(x\right)=\sqrt{1-e^{-x^2}}$ is differentiable is:

1. $[0,\infty )$

2. $(–\infty ,\infty )$

3. $(–\infty ,\infty )-\left\{0\right\}$

4. $(–1,\infty )$

Q. If $\overrightarrow{a}=-\hat{i}+\hat{j}+\hat{k}$ and $\overrightarrow{b}=2\hat{i}+0\hat{j}+\hat{k}$ then the vectors $\overrightarrow{c}$ satisfying the conditions, (i) that it is coplanar with $\overrightarrow{a}$ and $\overrightarrow{b}$ (ii) that it is perpendicular to $\overrightarrow{b}$, and (iii) that $\overrightarrow{a}.\overrightarrow{c}=7$, is

1. $-\frac{3}{2}\hat{i}+\frac{5}{2}\hat{j}+3\hat{k}$
2. $-3\hat{i}+5\hat{j}+6\hat{k}$
3. $-6\hat{i}+0\hat{j}+5\hat{k}$
4. $-\hat{i}+2\hat{j}+2\hat{k}$

Q. If $\overrightarrow{a}{}^{\prime }=\hat{i}+\hat{j},\overrightarrow{b}{}^{\prime }=\hat{i}+\hat{j}+2\hat{k}$ and $\overrightarrow{c}{}^{\prime }=2\hat{i}+\hat{j}-\hat{k}$. Then altitude of the parallelopiped formed by the vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ having base formed by $\overrightarrow{b}$ and $\overrightarrow{c}$ is $(\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ and ${\overrightarrow{a}}^{\prime },{\overrightarrow{b}}^{\prime },{\overrightarrow{c}}^{\prime }$ are reciprocal system of vectors)

1. $\frac{3\sqrt{2}}{2}$
2. $\frac{1}{\sqrt{6}}$
3. $1$
4. $\frac{1}{\sqrt{2}}$

Q. If a, b, c are linearly independent, then $\frac{[2a+b,2b+c,2c+a]}{[a,b,c]}=$

1. 8
2. 7
3. 9
4. None

Q. Find the surface area of the rectangular prism : $=1m^2$

1. $180m^2$
2. $202m^2$
3. $313m^2$
4. $188m^2$

Q. $[b\times cc\times aa\times b]=$

1. [a, b, c]
2. 0
3. 2 [a, b, c]
4. $[a,b,c{]}^2$

Q. If a, b, c are linearly independent, then $\frac{[2a+b,2b+c,2c+a]}{[a,b,c]}=$

1. 9
2. 7
3. 8
4. None

Q. State whether the following statement is True or False
$x+3\sqrt{y}$ and $\sqrt{x}-3y$ are conjugate pairs

1. True
2. False

Q. The volume of the tetrahedron with vertices at (1, 2, 3), (4, 3, 2), (5, 2, 7), (6, 4, 8) is

1. $\frac{22}{3}$
2. $\frac{11}{3}$
3. $\frac{1}{3}$
4. $\frac{16}{3}$

Q. The volume of the parallellopiped whose coterminal edges are $2\hat{i}-3\hat{j}+4\hat{k},\hat{i}+2\hat{j}-2\hat{k},3\hat{i}-\hat{j}+\hat{k}$ is

1. 5
2. 6
3. 7
4. 8