Addition of Vectors

Q.

Use the sign of $>,<or=$in the box to make the statement true:$39+(-24)-\left(15\right)[]36+(-52)-(-36)$

Q.

What happens to the force between two charges if the distance between them is cut in half?

Q. The resultant of $P$ and $Q$ is $R$. If $Q$ is doubled, $R$ is also doubled and if $Q$ is reversed, $R$ is again doubled. Then, $P^2:Q^2:R^2$ given by

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

Q.

Two charges each equal to $q$, are kept at $x=a$and $x=-a$ on the $x$-axis. A particle of mass $m$ and charge $q_0=\frac{q}{2}$ is placed at the origin. If a charge $q_0$ is given, a small displacement ($y<<a$) along the $y$-axis, the net force acting on the particle is proportional to

1. $y$

2. $-y$

3. $\frac{1}{y}$

4. $-\frac{1}{y}$

Q.

Find the missing number $0,7,26,63,\_\_\_$.

Q. Let three vectors $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ be such that $\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{c},\overrightarrow{b}\times \overrightarrow{c}=\overrightarrow{a}$ and $|\overrightarrow{a}|=2.$ Then which one of the following is not true:

1. $\overrightarrow{a}\times ((\overrightarrow{b}+\overrightarrow{c})\times (\overrightarrow{b}-\overrightarrow{c}))=\overrightarrow{0}$
2. Projection of $\overrightarrow{a}$ on $(\overrightarrow{b}\times \overrightarrow{c})$ is $2$
3. $\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]+\left[\overrightarrow{c}\overrightarrow{a}\overrightarrow{b}\right]=8$
4. $|3\overrightarrow{a}+\overrightarrow{b}-2\overrightarrow{c}{|}^2=51$

Q.

The vectors vector $AB=3\hat{i}+4\hat{k}$ and vector $AC=5\hat{i}-2\hat{j}+4\hat{k}$ are the sides of triangle $ABC.$The length of the median through $A$ is

1. $7\sqrt{2}$

2. $\sqrt{33}$

3. $\sqrt{288}$

4. $\sqrt{18}$

Q.

$Givenvector\overrightarrow{A}=2\hat{i}+3\hat{j},theanglebetween\overrightarrow{A}andy-axisis$

1. ${\tan }^{-1}(3/2)$

2. ${\tan }^{-1}(2/3)$

3. ${\sin }^{-1}(2/3)$

4. ${\cos }^{-1}(2/3)$

Q. If ABCDEF is a regular hexagon with $\overrightarrow{AB}=\overrightarrow{a}$ and $\overrightarrow{BC}=\overrightarrow{b}$, then $\overrightarrow{CE}$

1. $Noneofthese$
2. $-\overrightarrow{b}$
3. $\overrightarrow{v}-\overrightarrow{b}$
4. $\overrightarrow{b}-2\overrightarrow{a}$

Q.

The product of the roots of the equation $9x^2-18\left|x\right|+5=0$ is

1. $\frac{25}{81}$

2. $\frac{5}{9}$

3. $\frac{5}{27}$

4. $\frac{25}{9}$

Q.

The sum of the series $1+(1+2)+(1+2+3)+.......$upto $n$ terms will be

1. $n^2-2n+6$

2. $\frac{n(n+1)(2n-1)}{6}$

3. $n^2-2n-6$

4. $\frac{n(n+1)(n+2)}{6}$

Q. If the diagonals of a parallelogram are given $3\hat{i}+\hat{j}-2\hat{k}$ and $\hat{i}-3\hat{j}+4\hat{k}$, then the lengths of its sides are

1. $\sqrt{8},\sqrt{10}$
2. $\sqrt{6},\sqrt{14}$
3. $None$
4. $\sqrt{5},\sqrt{12}$

Q. A car goes 5 km east, 3km south, 2km west and 1km north. The magnitude of the resultant displacement will be ___ km at an angle of $ta{n}^{-1}$(- / -) with the positive x-axis (where x axis is in the east direction) in the clock wise direction.

1. 
2. 
3. 
4. None of hese

Q.

The two adjacent sides of a parallelogram are $2\hat{i}-4\hat{j}+5\hat{k}$ and $\hat{i}-2\hat{j}-3\hat{k}$. Find the unit vector parallel to its diagonal. Also, find its area.

Q.

The sum of $n$ terms of the following series $1.2+2.3+3.4+4.5+...$shall be

1. $n^3$

2. $\frac{n(n+1)(n+2)}{3}$

3. $\frac{n(n+1)(n+2)}{6}$

4. $\frac{n(n+1)(2n+1)}{3}$

Q.

The product of three geometric means between $4$ and $\frac{1}{4}$ will be

1. $4$

2. $2$

3. $-1$

4. $1$

Q.

The heights of 50 students, measured to the nearest centimeters, have been found to be as follows: $161150154165168161154162150151162164171165158154156172160170153159161170162165166168165164154152153156158162160161173166161159162167168159158153154159$Represent the data given above by a grouped frequency distribution table, taking the class intervals as $160-165,165-170$, etc.

Q.

$\overrightarrow{AB}\times \overrightarrow{AC}=2\hat{i}-4\hat{j}+4\hat{k}$ then the area of triangle $ABC$ is

1. $3squnits$

2. $4squnits$

3. $16squnits$

4. $9squnits$

Q.

The value of $\frac{1}{8}(3-4\cos 2\mathrm{\theta }+\cos 4\mathrm{\theta })$ is

1. $\cos 4\mathrm{\theta }$

2. $\sin 4\mathrm{\theta }$

3. ${\sin }^4\mathrm{\theta }$

4. ${\cos }^4\mathrm{\theta }$

5. ${\sin }^4\left(\raisebox{1ex}{$\mathrm{\theta }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)$

Q. Four point charges $q_A=2\mu C$, $q_B=5\mu C$, $q_C=2\mu C$, $q_D=5\mu C$ are located at the four corners of a square ABCD of side 10 cm. The force on a charge of $1\mu C$ placed at the centre of the square is

1. Towards AD
2. Zero
3. Towards AB
4. Towards BC

Q. The area of triangle with vertices $(K,0),(4,0),(0,2)$ is $4$ square units, then the value of $K$ is

1. $0$ or $8$
2. $0$ or $-8$
3. $0$
4. $8$

Q. The two vectors $\hat{i}+\hat{j}+\hat{k}$ and $\hat{i}+3\hat{j}+5\hat{k}$ represent the two sides $\overrightarrow{AB}$ and $\overrightarrow{AC}$ respectively of a triangle $ABC.$ The length of the median through $A$ is

1. $7$
2. $14$
3. $\sqrt{14}$
4. $\frac{\sqrt{14}}{2}$

Q.

For any two statements $p$ and $q$, $~(p\vee q)\vee (~p\wedge q)$ is logically equivalent to

1. $p$

2. $~p$

3. $q$

4. $~q$

Q. If $\overrightarrow{x}$ is a vector in the direction of $(2,-2,1)$ of magnitude $6$ and $\overrightarrow{y}$ is a vector in the direction of $(1,1,-1)$ of magnitude $\sqrt{3}$, then $|\overrightarrow{x}+2\overrightarrow{y}|$ is equal to

1. $40$
2. $\sqrt{35}$
3. $\sqrt{17}$
4. $2\sqrt{10}$

Q.

$\Delta ABC$ is an isosceles triangle in which$\mathit{}AB\mathit{=}AC$. Side $BA$is produced to $D$ such that $AD=AB$ (see Fig.) . Show that $\angle BCD$is a right angle.

Q.

The point in the interval $[0,2\pi ],$ where $f\left(x\right)=e^x\sin x$ has maximum slope, is

1. $\frac{\mathrm{\pi }}{4}$

2. $\frac{\mathrm{\pi }}{2}$

3. $\mathrm{\pi }$

4. $\frac{3\mathrm{\pi }}{2}$

Q.

The equation of the ellipse whose distance between the foci is equal to $8$ and distance between the directrix is $18$, is:

1. $5x^2-9y^2=180$

2. $9x^2+5y^2=180$

3. $x^2+9y^2=180$

4. $5x^2+9y^2=180$

Q.

Two vectors are given by $\text{a=-2i+j-3k}$ and $\text{b=5i+3j-2k}$. if $\text{3a+2b-c=0}$ then third vector $\text{c}$ is

Q.

A vector of magnitude $\sqrt{2}$ coplanar with the vectors $\overrightarrow{a}=\hat{i}+\hat{j}+2\hat{k}\text{and}\overrightarrow{b}=\hat{i}+2\hat{j}+\hat{k}$ and perpendicular to the vector $\overrightarrow{c}=\hat{i}+\hat{j}+\hat{k},$ is

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

Q. The volume of the tetrahedron whose co-terminous edges are $(-12\hat{i}+\lambda \hat{k}),(3\hat{j}-\hat{k})$ and $(2\hat{i}+\hat{j}-15\hat{k})$ is $546$ cubic units.Then the value of $\lambda$ is

1. $458$
2. $-458$
3. $300$
4. $-3$