Test for Coplanarity

Q. Four points given by position vectors $2\overrightarrow{a}+3\overrightarrow{b}-\overrightarrow{c},\overrightarrow{a}-2\overrightarrow{b}+3\overrightarrow{c},3\overrightarrow{a}+4\overrightarrow{b}-2\overrightarrow{c}and\overrightarrow{a}-6\overrightarrow{b}+6\overrightarrow{c}$ are coplanar, where $\overrightarrow{a},\overrightarrow{b}and\overrightarrow{c}$ are non-coplanar vectors.

1. True
2. False

Q.

$A\left(\overrightarrow{a}\right),B\left(\overrightarrow{b}\right),C\left(\overrightarrow{c}\right)$ are the vertices of a triangle ABC and $R\left(\overrightarrow{r}\right)$ is any point in the plane of triangle ABC, then $\overrightarrow{r}.(\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a})$ is always equal to

1. Zero

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

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

4. None of these

Q. Given: $\overrightarrow{a},\overrightarrow{b}and\overrightarrow{c}$ are coplanar.
Vectors $\overrightarrow{a}-2\overrightarrow{b}+3c,-\overrightarrow{2a}+\overrightarrow{3b}-\overrightarrow{4c}and-\overrightarrow{b}+\overrightarrow{2c}$ are non-coplanar vectors.

1. True
2. False

Q.

The function $f\left(x\right)$ is defined by $f\left(x\right)=(x+2)e^{-x}$is :

1. decreasing for all $x$

2. decreasing in $(-\infty ,-1)$ and increasing in $(-1,\infty )$

3. increasing for all $x$

4. decreasing in $(-1,\infty )$ and increasing in $(-\infty ,-1)$

Q.

The coefficient of correlation (${\mathrm{r}}_{\mathrm{xy}}$) for the following will be approximately

$x$	$2$	$4$	$5$	$6$	$3$	$6$	$8$	$10$
$y$	$5$	$6$	$6$	$8$	$4$	$8$	$12$	$15$

1. $0.90$

2. $0.96$

3. $-0.90$

4. $-0.96$

Q. Let $\hat{a}=\hat{i}+\hat{j}+\hat{k},\hat{b}=\hat{i}-\hat{j}+2\hat{k}$ and $\hat{c}=x\hat{i}+(x-2)\hat{j}-\hat{k}$. If the vector $\hat{c}$ lies in the plane of $\hat{a}$ and $\hat{b}$, then x equals

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

Q. If the vectors $\overrightarrow{a}+\lambda \overrightarrow{b}+3\overrightarrow{c},-2\overrightarrow{a}+3\overrightarrow{b}-4\overrightarrow{c}$ and $\overrightarrow{a}-3\overrightarrow{b}+5\overrightarrow{c}$ are coplanar, then the value of $\lambda$ is

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

Q.

$y=5x-1;-15x-3y=3$

How many solution does this linear system have?

1. one solution: $\left(0,-1\right)$

2. one solution: $\left(1,4\right)$

3. no solution

4. infinite number of solutions

Q. Let a, b, c be distinct non - negative numbers. If the vectors $a\hat{i}+a\hat{j}+c\hat{k},\hat{i}+\hat{k}$ and $c\hat{i}+c\hat{j}+b\hat{k}$, are coplanar then c is

1. GM of a and b
2. AM of a and b
3. HM of a and b
4. None of the above