Dot Product

Q.

The number of points at which $f\left(x\right)=\frac{1}{log|x|}$ is discontinuous is.

Q.

Find the ratio in which the line segment joining $A(1,-5)$ and $B(-4,5)$ is divided by the $x$-axis. Also, find the coordinates of the point of division.

Q.

If a is a non-zero vector of magnitude a and $\lambda$ is a non-zero scalar, then $\lambda$ a is unit vector if

a) $\lambda =1$

b) $\lambda =-1$

c) $a=|\lambda |$

d) $a=\frac{1}{|\lambda |}$

Q. If $\overrightarrow{a}$ and $\overrightarrow{b}$ are two non-parallel unit vectors and the vector $\alpha \overrightarrow{a}+\overrightarrow{b}$ bisects the internal angle between $\overrightarrow{a}$ and $\overrightarrow{b},$ then $\alpha$ is

1. $\frac{1}{2}$
2. $1$
3. $2$
4. $\frac{5}{2}$

Q. A typist charges Rs.145 for typing 10 English and 3 Hindi pages, while charges for typing 3 English and 10 Hindi pages are Rs.180. Using matrices, find the charges of typing one English and one Hindi page separately. However, typist charged only Rs.2 per page from a poor student Shyam for 5 Hindi pages. How much less was charged from this poor boy ? Which values are reflected in this problem ?

Q.

The points $(1,1),(-5,5)$ and $(13,\lambda )$ lie on the same straight line if $\lambda$ is equal to

1. $7$

2. $-7$

3. $\pm 7$

4. $0$

Q. Find the vector and cartersian equation of the line passing through the point (2, 1, 3) and perpendicular to the lines $\frac{x-1}{1}=\frac{y-2}{2}=\frac{z-3}{3}$ and $\frac{x}{-3}=\frac{y}{2}=\frac{z}{5}$

Q.

If the points $(a,0)$, $(0,b)$ and $(1,1)$ are collinear then which of the following is true?

1. $\frac{1}{a}+\frac{1}{b}=1$

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

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

4. $\frac{1}{a}+\frac{1}{b}=2$

Q. In quadrilateral $ABCD,\overrightarrow{AB}=\overrightarrow{a},\overrightarrow{BC}=\overrightarrow{b},\overrightarrow{AD}=\overrightarrow{b}-\overrightarrow{a}.$ If $M$ is the mid point of $\overrightarrow{BC}$ and $N$ is a point on $\overrightarrow{DM}$ such that $\overrightarrow{DN}=\frac{4}{5}\overrightarrow{DM,}$ then $\overrightarrow{AN}=$

1. $\frac{1}{5}\overrightarrow{AC}$
2. $\frac{2}{5}\overrightarrow{AC}$
3. $\frac{3}{5}\overrightarrow{AC}$
4. $\frac{4}{5}\overrightarrow{AC}$

Q. The complex number satisfying arg (z + i) = $\frac{\pi }{4}$ and arg (2z + 3 - 2i) = $\frac{3\pi }{4}$ simultaneously is

1. 
2. 
3. None
4. 

Q.

Show that the points $A\left(1,2\right)$, $B\left(5,4\right)$, $C\left(3,8\right)$ and $D\left(-1,6\right)$ are the vertices of a square.

Q.

Show that the points A(1, -2, -8), B(5, 0, -2) and C(11, 3, 7) are collinear and find the ratio in which B divides AC.

Q.

If $\mathbf{C}$ be the centroid of the triangle having vertices$\mathbf{}\mathbf{(}\mathbf{3}\mathbf{,}\mathbf{}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{,}\mathbf{}\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{}\mathbf{3}\mathbf{)}$ and $\mathbf{(}\mathbf{2}\mathbf{,}\mathbf{}\mathbf{4}\mathbf{)}$. Let $\mathit{P}$ be the point of intersection of the lines $\mathit{x}\mathbf{+}\mathbf{3}\mathit{y}\mathbf{-}\mathbf{1}\mathbf{=}\mathbf{}\mathbf{0}\mathbf{}$and$\mathbf{3}\mathit{x}\mathbf{-}\mathit{y}\mathbf{+}\mathbf{1}\mathbf{=}\mathbf{0}$, then the line passing through the points$CandP$ also passes through the point:

1. $\mathbf{(}\mathbf{-}\mathbf{9}\mathbf{,}\mathbf{}\mathbf{-}\mathbf{7}\mathbf{)}$

2. $\mathbf{(}\mathbf{-}\mathbf{9}\mathbf{,}\mathbf{}\mathbf{-}\mathbf{6}\mathbf{)}$

3. $\mathbf{(}\mathbf{7}\mathbf{,}\mathbf{}\mathbf{6}\mathbf{)}$

4. $\mathbf{(}\mathbf{9}\mathbf{,}\mathbf{}\mathbf{7}\mathbf{)}$

Q. If vector $\overrightarrow{a}$ is collinear with vector $\overrightarrow{b}=3\hat{i}+6\hat{j}+6\hat{k}$ and $\overrightarrow{a}\cdot \overrightarrow{b}=27.$ Then $\overrightarrow{a}$ is equal to

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

Q.

The line segment joining the points $P(3,3),Q(6,-6)$ is trisected at the points $A$ and $B$ such that $A$ is nearer to $P$. If $A$ also lies on the line given by $2x+y+k=0$, find the value of $k$.

Q. A straight line through A(6, 8) meets the curve $2x^2+y^2=2$ at B and C. P is a point on BC such that AB, AP, AC are in H.P, then the minimum distance of the origin from the locus of ‘P’ is

1. $\frac{67}{\sqrt{52}}$
2. $\frac{5}{\sqrt{52}}$
3. $\frac{10}{\sqrt{52}}$
4. $\frac{15}{\sqrt{52}}$

Q. In $△OAB,$ if $\overrightarrow{OA}=\overrightarrow{a},\overrightarrow{OB}=\overrightarrow{b},L$ is mid point of $\overrightarrow{OA}$ and $M$ is a point on $\overrightarrow{OB}$ such that $\overrightarrow{OM}:\overrightarrow{MB}=2:1.$ If $P$ is the mid point of $\overrightarrow{LM},$ then $\overrightarrow{AP}=$

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

Q. A line drawn through the point P (-1, 2) meets the hyperbola $xy=c^2$ at the points A and B (points A and B lie on same side of P) . Q is a point on AB such that PA, PQ and PB are in H.P then locus of Q is

1. $x-2y=2c^2$
2. $2x-y=2c^2$
3. $x+2y+2c^2$
4. $2x+y+2c^2=0$

Q. If $\overrightarrow{p},\overrightarrow{q}$ are two non collinear and non zero vectors such that $(b-c)\overrightarrow{p}\times \overrightarrow{q}+(c-a)\overrightarrow{p}+(a-b)\overrightarrow{q}=0$ where $a,b,c$ are the lengths of the sides of triangle, then the triangle is:

1. Right angled
2. Obtuse angled
3. Equilateral
4. Isosceles

Q. The point $Q$ is the image of the point $P(1,5)$ about the line $y=x$ and $R$ is the image of the point $Q$ about the line $y=–x$. The circumcenter of the $\Delta PQR$ is

1. $(5,1)$
2. $(-5,1)$
3. $(1,-5)$
4. $(0,0)$

Q. Let the vectors $\overrightarrow{PQ},\overrightarrow{QR},\overrightarrow{RS},\overrightarrow{ST},\overrightarrow{TU}$ and $\overrightarrow{UP}$ represent the sides of a regular hexagon.
Statement - 1: $\overrightarrow{PQ}\times (\overrightarrow{RS}+\overrightarrow{ST})\ne 0$
Statement - 2: $\overrightarrow{PQ}\times \overrightarrow{RS}=0$ and $\overrightarrow{PQ}\times \overrightarrow{ST}\ne 0$

1. Statement $-1$ is True, Statement $-2$ is True; Statement $-2$ is the correct explanation of Statement $-1$.
2. Statement $-1$ is True, Statement $-2$ is True; Statement $-2$ is the incorrect explanation of Statement $-1$.
3. Statement $-1$ is True, Statement $-2$ is False.
4. Statement $-1$ is False, Statement $-2$ is True.

Q. Let $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ be three non zero vectors which are pairwise noncollinear. If $\overrightarrow{a}+3\overrightarrow{b}$ is collinear with $\overrightarrow{c}$ and $\overrightarrow{b}+2\overrightarrow{c}$ is collinear with $\overrightarrow{a}$, then $\overrightarrow{a}+3\overrightarrow{b}+6\overrightarrow{c}$ is:

1. $\overrightarrow{a}+\overrightarrow{c}$
2. $\overrightarrow{a}$
3. $\overrightarrow{c}$
4. $0$

Q. The length of the perpendicular from the origin to the plane passing through the point $\overline{a}$ and containing the line $\overline{r}=\overline{b}+\lambda \overline{c}$ is :

1. $=\frac{\left[\overline{a}\overline{b}\overline{c}\right]}{|\overline{a}\times \overline{b}+\overline{b}\times \overline{c}|}$
2. $=\frac{\left[\overline{a}\overline{b}\overline{c}\right]}{|\overline{c}\times \overline{a}+\overline{a}\times \overline{b}|}$
3. $=\frac{\left[\overline{a}\overline{b}\overline{c}\right]}{|\overline{c}\times \overline{a}-\overline{a}\times \overline{b}|}$
4. $=\frac{\left[\overline{a}\overline{b}\overline{c}\right]}{|\overline{b}\times \overline{c}+\overline{c}\times \overline{a}|}$

Q. If $P,Q$ and $R$ are foot of perpendiculars drawn from point $A(1,1,1)$ to planes $P_1:x+2y+2z=2$, $P_2:2x-2y+z=-8$ and to line of intersection of $P_1$ and $P_2$ respectively, then area of $\Delta PQR$ is -

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

Q. The vectors $\overrightarrow{a}+\overrightarrow{b},\overrightarrow{a}-k\overrightarrow{b},$ where $k$ is scalar, are collinear for

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

Q. Let $\overrightarrow{a}=\hat{i}+\hat{j}$ and $\overrightarrow{b}=2\hat{i}-\hat{k}$ then the point of intersection of the line $\overrightarrow{r}\times \overrightarrow{a}=\overrightarrow{b}\times \overrightarrow{a}$ and $\overrightarrow{r}\times \overrightarrow{b}=\overrightarrow{a}\times \overrightarrow{b}$

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

Q. Let $\overrightarrow{p}$ and $\overrightarrow{q}$ be the position vectors of $P$ and $Q$ respectively with respect to $O$ and $|\overrightarrow{p}|=p,|\overrightarrow{q}|=q.$ The points $R$ and $S$ divides $PQ$ internally and externally in the ratio $3:2$ respectively. If $\overrightarrow{OR}$ and $\overrightarrow{OS}$ are perpendicular, then

1. $9p^2=4q^2$
2. $4p^2=9q^2$
3. $9p=4q$
4. $4p=9q$

Q. If the points $(p,0)$, $(0,q)$ and $(1,1)$ are collinear, then $\frac{1}{p}+\frac{1}{q}$ is equal to:

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

Q. The point collinear with $(1,-2,-3)$ and $(2,0,0)$ among the following is

1. $(0,-4,-5)$
2. $(0,-4,-6)$
3. $(0,4,6)$
4. $(0,-4,6)$

Q. Show that the points having position vectors, $\overline{a},\overline{b}$ and $3\overline{a}-2\overline{b}$ are colliera, where $\overline{a},\overline{b},\overline{c}$ any vectors.