Test for Coplanarity

Q. If a, b, c are respectively the $p^{th},q^{th},r^{th}$ terms of an A.P., the $\left|\begin{array}{ccc}a& p& 1\\ b& q& 1\\ c& r& 1\end{array}\right|=$

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

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

Q.

If $A$ and $B$ be points with position vectors $a$ and $b$ with respect to origin $O$.

If the point $C$ on $OA$ is such that $2AC=CO$, $CD$ is parallel to $\left|CD\right|=3\left|OB\right|$, then $\left|AD\right|$ is equal to?

1. $2b-\frac{1}{2}a$

2. $3b-\frac{1}{3}a$

3. $b-\frac{1}{2}a$

4. None of these

Q.

In the following diagrams, the universal set is represented by a triangle and set A and B are the set of points in the triangle.

Then in which of the given figures the shaded portion represents $A^{\prime }\cup B$?

1. 

2. 

3. 

4. 

Q. The vectors $\overrightarrow{a}=x\hat{i}+(x+1)\hat{j}+(x+2)\hat{k},\overrightarrow{b}=(x+3)\hat{i}+(x+4)\hat{j}+(x+5)\hat{k}$ and $\overrightarrow{c}=(x+6)\hat{i}+(x+7)\hat{j}+(x+8)\hat{k}$ are coplanar for

1. x > 0
2. None of these
3. all values of x
4. x < 0

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

Q. If $\overrightarrow{a}=2\hat{i}+3\hat{j}+\hat{k},\overrightarrow{b}=2\hat{i}+p\hat{j}+3\hat{k}$ and $\overrightarrow{c}=2\hat{i}+17\hat{j}+3\hat{k}$ are coplanar vectors, then the value of p is

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

Q. If $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ are three non – coplanar vectors, then $(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}).\left(\right(\overrightarrow{a}+\overrightarrow{b})\times (\overrightarrow{a}+\overrightarrow{c}\left)\right)$ equals

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

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

2. Zero

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

4. None of these

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

Q. If the points with position vectors $-(\hat{j}+\hat{k}),4\hat{i}+5\hat{j}+\lambda \hat{k},3\hat{i}+9\hat{j}+4\hat{k}$ and $-4\hat{i}+4\hat{j}+4\hat{k}$ are coplanar, then the value of $\lambda$ is

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

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. The points $2\hat{i}-\hat{j}-\hat{k},\hat{i}+\hat{j}+\hat{k},2\hat{i}+2\hat{j}+\hat{k},2\hat{j}+5\hat{k}$ are

1. collinear
2. coplanar but not collinear
3. noncoplanar
4. none

Q. The vectors 3a-2b-4c, -a+2c, -2a+b+3c are

1. linearly dependent
2. collinear
3. linearly independent
4. none

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 vectors 3a-2b-4c, -a+2c, -2a+b+3c are

1. linearly dependent
2. collinear
3. linearly independent
4. none

Q.

Solve $2x-y=12$ and $x+3y+1=0$ and hence find the value of m for which $y=mx+3$.

Q.

Solve:

$3-z=8$

Q. If A = {x : x = 2${}^n$, n $ϵ$ W and n < 4}, B = {x: x = 2 n, n $ϵ$ N and n $\le$ 4} and C = {0, 1, 2, 5, 6}, then verify the associative property of intersection of sets.