Condition for Coplanarity of Four Points

Q.

Let $\stackrel{\rightharpoonup }{a}=2i-3j+4k,\overrightarrow{b}=7i+j-6k$. If $\overrightarrow{r}\times \overrightarrow{a}=\overrightarrow{r}\times \overrightarrow{b},\overrightarrow{r}·(i+2j+k)=-3,$ then $\overrightarrow{r}·(2i-3j+k)$ is equal to

1. $10$

2. $13$

3. $12$

4. $8$

Q. The lines $x=ay–1=z–2$ and $x=3y–2=bz–2,$ $(ab\ne 0)$ are coplanar, if

1. $a=2,b=2$
2. $a=2,b=3$
3. $a=1,b\in R-\left\{0\right\}$
4. $b=1,a\in R-\left\{0\right\}$

Q.

If the least and the largest real values of $\alpha$, for which the equation $z+\alpha \left|z-1\right|+2\iota =0$, where $z\in C$ and $\iota =\sqrt{-1}$, has a solution, are $p$ and $q$ respectively, then $4\left(p^2+q^2\right)$ is equal to

Q. Let $S$ be the set of all real values of $\lambda$ such that plane passing through the points $(-{\lambda }^2,1,1),(1,-{\lambda }^2,1)$ and $(1,1,-{\lambda }^2)$ also passes through the point $(-1,-1,1)$. Then $S$ is equal to :

1. $\left\{\sqrt{3}\right\}$
2. $\{1,-1\}$
3. $\{3,-3\}$
4. $\{\sqrt{3},-\sqrt{3}\}$

Q.

Joint equation of pair of lines through$(3,-2)$ and parallel to $x^2-4xy+3y^2=0$ is

1. $x^2+3y^2-4xy-14x+24y+45=0$

2. $x^2+3y^2+4xy-14x+24y+45=0$

3. $x^2+3y^2+4xy-14x+24y-45=0$

4. $x^2+3y^2+4xy-14x-24y-45=0$

Q.

Nonzero vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ satisfy $\overrightarrow{a}.\overrightarrow{b}=0,(\overrightarrow{b}-\overrightarrow{a}).(\overrightarrow{b}+\overrightarrow{c})=0and2|\overrightarrow{b}+\overrightarrow{c}|=|\overrightarrow{b}-\overrightarrow{a}|$. If $\overrightarrow{a}=\mu \overrightarrow{b}+4\overrightarrow{c}then\mu =$ ___

Q. If $(1,5,35),(7,5,5),(1,\lambda ,7)$ and $(2\lambda ,1,2)$ are coplanar, then the sum of all possible values of $\lambda$ is :

1. $-\frac{44}{5}$
2. $\frac{39}{5}$
3. $-\frac{39}{5}$
4. $\frac{44}{5}$

Q.

The equation of the plane passing through the points (3, 2, 2) and (1, 0, -1) and parallel to the line $\frac{x-1}{2}=\frac{y-1}{-2}=\frac{z-2}{3}$, is

1. None of these

2. 

3. 

4. 

Q. Three lines through origin having direction ratios $(1,a,a^2),$ $(1,b,b^2)$ and $(1,c,c^2)$ are non- coplanar. But the lines with direction ratios $(a,a^2,1+a^3),$ $(b,b^2,1+b^3)$ and $(c,c^2,1+c^3)$ are coplanar. Then the value of $abc$ is

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

Q. The plane passing through the point (5, 1, 2) perpendicular to the line 2(x - 2) = y – 4 = z – 5 will meet the line in the point

1. (1, 2, 3)
2. (2, 3, 1)
3. (1, 3, 2)
4. (3, 2, 1)

Q. The points A(4, 5, 1), B(0, -1, -1), C(3, 9, 4) and D(-4, 4, 4) are

1. Collinear
2. Coplanar
3. Non- coplanar
4. Non- Collinear and non-coplanar

Q. Consider the matrix equation X² = I, I being a 2x2 unit matrix, ‘X’ is a real matrix (all the elements being real). Which of the following is/are correct?

1. The equation has exactly two solutions
2. The equation has infinitely many Solutions
3. $x=\left[\begin{array}{cc}\sqrt{2}-\sqrt{3}& 4\sqrt{2}\\ \frac{\sqrt{3}}{2}-\frac{1}{\sqrt{2}}& \sqrt{3}-\sqrt{2}\end{array}\right]$
4. $x=\left[\begin{array}{cc}\sqrt{3}-\sqrt{2}& \frac{3}{2}-1\\ 4& \sqrt{2}-\sqrt{3}\end{array}\right]$

Q.

Let $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$ be three unit vectors such that $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=0$. If $\lambda =\overrightarrow{a}.\overrightarrow{b}+\overrightarrow{b}.\overrightarrow{c}+\overrightarrow{c}.\overrightarrow{a}$ and $\overrightarrow{d}=\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}$, then the ordered pair $\left(\lambda ,\overrightarrow{d}\right)$ is

1. $\left(\frac{3}{2},3\overrightarrow{a}\times \overrightarrow{c}\right)$

2. $\left(-\frac{3}{2},3\overrightarrow{c}\times \overrightarrow{b}\right)$

3. $\left(-\frac{3}{2},3\overrightarrow{a}\times \overrightarrow{b}\right)$

4. $\left(\frac{3}{2},3\overrightarrow{b}\times \overrightarrow{c}\right)$

Q.

The vector having initial and terminal points are (2, 5, 0) and (-3, 7, 4), respectively is

(a) $-\hat{i}+12\hat{j}+4\hat{k}$

(b) $5\hat{i}+2\hat{j}-4\hat{k}$

(c) $-5\hat{i}+2\hat{j}+4\hat{k}$

(d) $\hat{i}+\hat{j}+\hat{k}$

Q. The three lines drawn from O with direction ratios, (1, –1, k), (2, –3, 0) and (1, 0, 3) are coplanar. Then k =

1. no such k exists
2. 1
3. 0
4. 7

Q.

Find the length and the foot of perpendicular from the point $(1,\frac{3}{2},2)$ to the plane 2x-2y+4z+5=0.

Q.

Which of the following are APs?

If they form an A.P. find the common difference $d$ and write three more terms.

$a,a^2,a^3,a^4\dots$

Q. The lines $\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{-k}$ and $\frac{x-1}{k}=\frac{y-4}{2}=\frac{z-5}{1}$ are coplanar if

1. $k=1\text{or}-1$
2. $k=0\text{or}-3$
3. $k=3\text{or}-3$
4. $k=0\text{or}-1$

Q. If $\overrightarrow{c}=x\hat{i}+y\hat{j}+z\hat{k}$ is the internal angle bisector between $\overrightarrow{a}=2\hat{i}-\hat{j}+\hat{k}$ and $\overrightarrow{b}=\hat{i}+2\hat{j}-\hat{k}$ and $|\overrightarrow{c}|=\sqrt{40}.$ Then the value of $(x+y+z)=$

Q. $\text{There are}2\text{indian couples,}2\text{american couple and}1\text{unmarried person}$
$\begin{array}{cccc}& \text{List- I}& & \text{List-II}\\ & & & \\ \left(I\right)& \text{The total mumber of ways in which they can sit}& \left(P\right)& 5760\\ & \text{in a row such that an indian wife and an american}& & \\ & \text{wife are always on either side of the unmarried person}& & \\ & & \left(Q\right)& 24230\\ \left(II\right)& \text{The total mumber of ways in which they can sit}& & \\ & \text{in a row such that the unmarried man always}& & \\ & \text{occupy the midddle position}& & \\ & & \left(R\right)& 40320\\ \left(III\right)& \text{The total mumber of ways in which they can sit}& & \\ & \text{around a circular table such that indian wife and}& & \\ & \text{american wife are on either side of unmarried person}& & \\ & & \left(S\right)& 1920\\ \left(IV\right)& \text{The total number of ways in which they can sit in}& & \\ & \text{a row such that all couples sit together}& & \\ & & \left(T\right)& 28410\end{array}$

$2)$ Which of the following is only INCORRECT combination?

1. $I\to \left(R\right)$
2. $II\to \left(P\right)$
3. $III\to \left(P\right)$
4. $IV\to \left(S\right)$

Q.

The points A(4, 5, 1), B(0, -1, -1), C(3, 9, 4) and D(-4, 4, 4) are
[Kurukshetra CEE 2002]

1. Coplanar

2. Non- coplanar

3. Collinear

4. Non- Collinear and non-coplanar

Q. If the vector area of the triangle whose adjacent sides are $2\overrightarrow{i}+3\overrightarrow{j}$ and $-2i+4\overrightarrow{j}$ is $\lambda \hat{k},$ then the value of $\lambda$ is

Q. Find the value of $\overrightarrow{a}$ if the vectors $2\hat{i}-\hat{j}+\hat{k},\hat{i}+2\hat{j}-3\hat{k}$ and $3\hat{i}+a\hat{j}+5\hat{k}$ are coplanar?

Q. The lines $\frac{x+3}{-3}=\frac{y-1}{1}=\frac{z-5}{5}$ and $\frac{x+1}{-1}=\frac{y-2}{\alpha }=\frac{z-5}{5}$ are coplanar. Then the value of $\alpha$ is

1. $-1$
2. $2$
3. $5$
4. $1$

Q.

Curve $16x^2+8xy+y^2-74x-78y+212=0$ represents

1. Parabola

2. Hyperbola

3. Ellipse

4. None of these

Q. If vectors $2\hat{i}-\hat{j}+\hat{k},\hat{i}+2\hat{j}-3\hat{k}$ and $3\hat{i}+a\hat{j}+5\hat{k}$ are coplanar, then find the value of $a$.

Q. If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are non coplanar vectors such that $\overrightarrow{b}\times \overrightarrow{c}=\overrightarrow{a},\overrightarrow{c}\times \overrightarrow{a}=\overrightarrow{b}$ and $\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{c}$, then

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

Q. If $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ are three non-coplanar vectors, prove that
$[\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}\overrightarrow{a}+\overrightarrow{b}\overrightarrow{a}+\overrightarrow{c}]=-\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]$.

Q.

The equation of the line passing through (1, 2, 3) and parallel to the planes x - y + 2z = 5 and 3x + y + z = 6, is
[DSSE 1986]

1. 
   

2. 
   

3. 
   

4. None of these

Q. The number of distinct solutions of ${\int }_0^x(4t^2+1)e^{2t^2}dt=5x$ for $x\ge 0$ is

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