Equal Vectors

Q.

Let the plane $ax+by+cz+d=0$ bisect the line joining the points $\left(4,-3,1\right)$ and $\left(2,3,-5\right)$ at the right angles. If $a$, $b$, $c$, $d$ are integers, then the minimum value of $\left(a^2+b^2+c^2+d^2\right)$ is

Q.

The value of $a$, for which the points $A,B$ and $C$ with position vectors $2i-j+k$ and $i-3j-5k$ and $ai-3j+k$ respectively are the vertices of a right-angled triangle with $C=\frac{\mathrm{\pi }}{2}$ are?

1. $-2$ and $-1$

2. $-2$ and $1$

3. $2$ and $-1$

4. $2$ and $1$

Q.

The area bounded athe curves $y=e^x,y=e^{-x}$ and the straight line $x=1$ is given by

1. $e+e^{-1}–2$

2. $e-e^{-1}$

3. $e+e^{-1}$

4. $e+e^{-1}+2$

Q.

How do you factor completely $a^4-b^4$?

Q. If vector $\overrightarrow{PQ}$ with coordinates of the end points (1, -2) and (4-1) is equal to vector is with coordinates of initial and final points (a, -1) and (0, 0) respectively, then $\left|a\right|=$ ___

Q. If |a+b| = |a-b| then (a, b) =

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

Q.

The shortest distance between the line$y-x=1$ and the $x=y^2$ is

1. $\frac{3\sqrt{2}}{8}$

2. $\frac{2\sqrt{3}}{8}$

3. $\frac{3\sqrt{2}}{5}$

4. $\frac{\sqrt{3}}{4}$

Q. If |a+b| = |a-b| then (a, b) =

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

Q. Which of the following is/are true?
1) Equal vectors have same direction but not necessarily the same magnitude.
2) Equal vectors have same magnitude and same direction.
3) Equal vectors cannot start from different points.

1. 1, 2 and 3 are true
2. None of them is true
3. Only 2 is true
4. 2 and 3 are true

Q. There are two vectors. The first one has initial and terminal points as A(–1, 3), B(2, 4) respectively. The second one has initial and terminal points as. C(1, -2), D(4, -1) respectively. The two vectors are not equal vectors.

1. True
2. False