Test for Coplanarity

Q. The sum of the distinct real values of $\mu ,$ for which the vectors, $\mu \hat{i}+\hat{j}+\hat{k},\hat{i}+\mu \hat{j}+\hat{k},\hat{i}+\hat{j}+\mu \hat{k}$ are co-planar, is:

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

Q. If the vector $\overrightarrow{p}=(a+1)\hat{i}+a\hat{j}+a\hat{k},\overrightarrow{q}=a\hat{i}+(a+1)\hat{j}+a\hat{k}$ and $\overrightarrow{r}=a\hat{i}+a\hat{j}+(a+1)\hat{k},(a\in R)$ are coplanar and $3(\overrightarrow{p}.\overrightarrow{q}{)}^2-\lambda |\overrightarrow{r}\times \overrightarrow{q}{|}^2=0,$ then value of` $\lambda$ is

Q. If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are three non coplanar vectors and $\overrightarrow{p}=\frac{\overrightarrow{b}\times \overrightarrow{c}}{\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]},\overrightarrow{q}=\frac{\overrightarrow{c}\times \overrightarrow{a}}{\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]},\overrightarrow{r}=\frac{\overrightarrow{a}\times \overrightarrow{b}}{\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]},$ then $(2\overrightarrow{a}+3\overrightarrow{b}+4\overrightarrow{c})\cdot \overrightarrow{p}+(2\overrightarrow{b}+3\overrightarrow{c}+4\overrightarrow{a})\cdot \overrightarrow{q}+(2\overrightarrow{c}+3\overrightarrow{a}+4\overrightarrow{b})\cdot \overrightarrow{r}=$

Q. Let $a,b$ and $c$ be distinct positive 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 co-planar, then $c$ is equal to

1. $\frac{2}{\frac{1}{a}+\frac{1}{b}}$
2. $\sqrt{ab}$
3. $\frac{a+b}{2}$
4. $\frac{1}{a}+\frac{1}{b}$

Q.

The value of the determinant $\left|\begin{array}{ccc}15!& 16!& 17!\\ 16!& 17!& 18!\\ 17!& 18!& 19!\end{array}\right|=$

1. $15!+16!$

2. $2\left(15!\right)\left(16!\right)\left(17!\right)$

3. $15!+16!+17!$

4. $16!+17!$

Q. Let $\alpha \in R$ and the three vectors $\overrightarrow{a}=\alpha \hat{i}+\hat{j}+3\hat{k},\overrightarrow{b}=2\hat{i}+\hat{j}-\alpha \hat{k}$ and $\overrightarrow{c}=\alpha \hat{i}-2\hat{j}+3\hat{k}$. Then the set $S=\{\alpha :\overrightarrow{a},\overrightarrow{b}\text{and}\overrightarrow{c}\text{are coplanar}\}$

1. is empty
2. is singleton
3. contains exactly two positive numbers
4. contains exactly two numbers only one of which is positive

Q. The points with position vectors $60\hat{i}+3\hat{j},40\hat{i}-8\hat{j}$ and $a\hat{i}-52\hat{j}$ are collinear, then the value of $a$ is

1. $-40$
2. $10$
3. $20$
4. $-30$

Q.

If $a,b,candu,v,w$ are complex numbers representing the vertices of two triangles such that $c=(1-r)a+rbandw=(1-r)u+rv$, where $r$ is a complex number, then the two triangles.

1. Have the same area

2. Are similar

3. Are congruent

4. None of these

Q. If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c},\overrightarrow{d}$ are non-zero vectors satisfying $\overrightarrow{a}=\overrightarrow{b}+\overrightarrow{c}$, $\overrightarrow{b}\times \overrightarrow{d}=\overrightarrow{0}$ and $\overrightarrow{c}\cdot \overrightarrow{d}=0$ , then $\frac{\overrightarrow{d}\times (\overrightarrow{a}\times \overrightarrow{d})}{|\overrightarrow{d}{|}^2}$ is always equal to

1. $\overrightarrow{a}$
2. $\overrightarrow{d}$
3. $\overrightarrow{b}$
4. $\overrightarrow{c}$

Q. A vector $\left(\overrightarrow{d}\right)$ is equally inclined to three vectors $\overrightarrow{a}=\hat{i}-\hat{j}+\hat{k},\overrightarrow{b}=2\hat{i}+\hat{j}$ and $\overrightarrow{c}=3\hat{j}-2\hat{k}$. Let $\overrightarrow{x},\overrightarrow{y},\overrightarrow{z}$ be three vectors in the plane of $\overrightarrow{a},\overrightarrow{b};\overrightarrow{b},\overrightarrow{c};\overrightarrow{c},\overrightarrow{a}$, respectively. Then which of the following is INCORRECT?

1. $\overrightarrow{x}.\overrightarrow{d}=0$
2. $\overrightarrow{y}.\overrightarrow{d}=0$
3. $\overrightarrow{z}.\overrightarrow{d}=0$
4. none of these

Q. $Let\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k},\overrightarrow{b}=\hat{i}-\hat{j}+2\hat{k}and\overrightarrow{c}=x\hat{i}+\left(x-2\right)\hat{j}-\hat{k}.If\overrightarrow{c}liesintheplaneof\overrightarrow{a}and\overrightarrow{b},thenx=$

1. \N
2. -2
3. -1

Q.

The vectors $\overrightarrow{a}=\hat{i}+\hat{j}+m\hat{k},\overrightarrow{b}=\hat{i}+\hat{j}+(m+1)\hat{k}$ and $\overrightarrow{c}=\hat{i}-\hat{j}+m\hat{k}$ are coplanar, if $m$ is equal to?

1. $1$

2. $4$

3. $3$

4. no value of $m$ for which vectors are coplanar

Q. The value of $x\in R$ for which vectors $\overrightarrow{a}=(1,-2,1),\overrightarrow{b}=(-2,3,-4),\overrightarrow{c}=(1,-1,x)$ form a linearly dependent system, is equal to

Q. Find the value of $\lambda$ so that the points $P,Q,R$ and $S$ on the sides $OA,OB,OC$ and $AB$ respectively of a regular tetrahedron $OABC$ are coplanar. It is given that $\frac{OP}{OA}=\frac{1}{3},\frac{OQ}{OB}=\frac{1}{2}$ and $\frac{OS}{AB}=\lambda$.

1. $\lambda =\frac{1}{2}$
2. For no value of $\lambda$
3. $\lambda =0$
4. $\lambda =-1$

Q. Let four vectors $\overrightarrow{r}=3\hat{i}+2\hat{j}-5\hat{k},\overrightarrow{a}=2\hat{i}-\hat{j}+\hat{k},\overrightarrow{b}=\hat{i}+3\hat{j}-2\hat{k}$ and $\overrightarrow{c}=-2\hat{i}+\hat{j}-3\hat{k}$ are such that $\overrightarrow{r}=\lambda \overrightarrow{a}+\mu \overrightarrow{b}+v\overrightarrow{c},$ then $\lambda +\mu +v$ is

1. $2$
2. $3$
3. $4$
4. $6$

Q. If non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are equally inclined to coplanar vector $\overrightarrow{c}$, then $\overrightarrow{c}$ can be

1. $\frac{|\overrightarrow{a}|}{|\overrightarrow{a}|+2|\overrightarrow{b}|}\overrightarrow{a}+\frac{|\overrightarrow{b}|}{|\overrightarrow{a}|+|\overrightarrow{b}|}\overrightarrow{b}$
2. $\frac{|\overrightarrow{b}|}{|\overrightarrow{a}|+|\overrightarrow{b}|}\overrightarrow{a}+\frac{|\overrightarrow{a}|}{|\overrightarrow{a}|+|\overrightarrow{b}|}\overrightarrow{b}$
3. $\frac{|\overrightarrow{a}|}{|\overrightarrow{a}|+2|\overrightarrow{b}|}\overrightarrow{a}+\frac{|\overrightarrow{b}|}{|\overrightarrow{a}|+2|\overrightarrow{b}|}\overrightarrow{b}$
4. $\frac{|\overrightarrow{b}|}{2|\overrightarrow{a}|+|\overrightarrow{b}|}\overrightarrow{a}+\frac{|\overrightarrow{a}|}{2|\overrightarrow{a}|+|\overrightarrow{b}|}\overrightarrow{b}$

Q. $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are non-coplanar vectors such that $\overrightarrow{P}=\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}$, $\overrightarrow{Q}=\overrightarrow{4a}+\overrightarrow{3b}+\overrightarrow{4c}$ and $\overrightarrow{R}=\overrightarrow{a}+\alpha \overrightarrow{b}+\beta \overrightarrow{c}$ are linearly dependent vectors, then number of possible values of $\alpha$ is -

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

Q.

The vectors $\lambda \hat{i}+\hat{j}+2\hat{k},\hat{i}+\lambda \hat{j}-\hat{k}and2\hat{i}-\hat{j}+\lambda \hat{k}arecoplanar,if$

(a) $\lambda =-2$ (b) $\lambda =0$

(a) $\lambda =1$ (b) $\lambda =-1$

Q. Assertion :If $a\hat{i}+\hat{j}+\hat{k},\hat{i}+b\hat{j}+\hat{k},\hat{i}+\hat{j}+c\hat{k}$ are coplanar, then $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=1$ provided $a\ne 1,b\ne 1,c\ne 1$. Reason: Vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are coplanar, then $\overrightarrow{a}\cdot (\overrightarrow{b}\times \overrightarrow{c})=0$.

1. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion
2. Both Assertion & Reason are individually true but Reason is not the , correct (proper) explanation of Assertion
3. Assertion is true but Reason is false
4. Assertion is false but Reason is true

Q. Find the values of $x$ and $y$ so that the vectors $2\hat{i}+3\hat{j}$ and $x\hat{i}+y\hat{j}$ are equal

Q. The three points A, B, C with position vectors $-2a+3b+5c,a+2b+3c,7a-c$

1. Coinitial
2. Colinear
3. equal
4. None of these

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. Let $\hat{i},\hat{j},\hat{k}$ be the orthonormal system of vectors, and $\overrightarrow{a}$ be any vector. If $\overrightarrow{a}\times \hat{i}+2\overrightarrow{a}-5\hat{j}=\overline{0}$ , then $\overrightarrow{a}$ is

1. $2\hat{j}+\hat{k}$
2. $2\hat{i}-\hat{k}$
3. $2\hat{i}-\hat{j}$
4. $2\hat{i}+\hat{k}$

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. \N
3. 1
4. None of these

Q.

Find two solutions the equation $(3x-2y=5)$

Q. Let $\overrightarrow{a}=\hat{i}+2\hat{j}+4\hat{k},\overrightarrow{b}=\hat{i}+\lambda \hat{j}+4\hat{k}$ and $\overrightarrow{c}=2\hat{i}+4\hat{j}+({\lambda }^2-1)\hat{k}$ be coplanar vectors. Then the non-zero vector $\overrightarrow{a}\times \overrightarrow{c}$ is :

1. $-10\hat{i}-5\hat{j}$
2. $-14\hat{i}-5\hat{j}$
3. $-10\hat{i}+5\hat{j}$
4. $-14\hat{i}+5\hat{j}$

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

Q.

If $A=\left|\begin{array}{ccc}2& \lambda & -3\\ 0& 2& 5\\ 1& 1& 3\end{array}\right|$, then $A^{-1}$ exists, if

(a) $\lambda =2$
(b) $\lambda \ne 2$
(c) $\lambda \ne -2$
(d) None of these

Q. The set of value(s) of $\alpha \in R$ for which vectors $\overrightarrow{a}=x\hat{i}+(x-1)\hat{j}+\hat{k}$ and $\overrightarrow{b}=(x+1)\hat{i}+\hat{j}+\alpha \hat{k}$ make an acute angle for $\forall x\in R,$ is

1. $\alpha \in [0,\infty )$
2. $\alpha \in (-\infty ,0)$
3. $\alpha \in (-2,2)$
4. $\alpha \in (2,\infty )$

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