Mathematics

Q. If the ratio in which the XY plane divides the line joining the points (2, 4, 5) and (–4, 3, –2) is k: 1, then find the value of 10k

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Q. Let $\overrightarrow{a}=\hat{i}+\hat{j};\overrightarrow{b}=2\hat{i}-\hat{k}$. Then, vector $\overrightarrow{r}$ satisfying the equations $\overrightarrow{r}\times \overrightarrow{a}=\overrightarrow{b}\times \overrightarrow{a}$ and $\overrightarrow{r}\times \overrightarrow{b}=\overrightarrow{a}\times \overrightarrow{b}$ is

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

Q. Let ABCD be a parallelogram whose diagonals intersect at P and let O be the origin, then $\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC}+\overrightarrow{OD}$ equals

1. $2\overrightarrow{OP}$
2. $\overrightarrow{OA}$
3. $3\overrightarrow{OP}$
4. $4\overrightarrow{OP}$

Q. Let $\overrightarrow{a}=2\hat{i}+\hat{j}-\hat{k}$ and $\overrightarrow{b}=\hat{i}+2\hat{j}+\hat{k}$ be two vectors. Consider a vector $\overrightarrow{c}=\alpha \overrightarrow{a}+\beta \overrightarrow{b},\alpha ,\beta \in R.$ If the projection of $\overrightarrow{c}$ on the vector $(\overrightarrow{a}+\overrightarrow{b})$ is $3\sqrt{2},$ then the minimum value of $(\overrightarrow{c}-(\overrightarrow{a}\times \overrightarrow{b}\left)\right).\overrightarrow{c}$ equals

Q. If the vectors AB=$-3\hat{i}+4\hat{k}$ and AC=$5\hat{i}-2\hat{j}+4\hat{k}$ are the sides of a triangle ABC, then the length of median through the point A is

1. $\sqrt{14}$
2. $\sqrt{18}$
3. $\sqrt{29}$
4. $\sqrt{25}$

Q. If $\overrightarrow{v_1}and\overrightarrow{v_2}$ are 2 vectors, then $c_1\overrightarrow{v_1}+c_2\overrightarrow{v_2}.\overrightarrow{v_2}$ is the linear combination of these 2 vectors where $c_1,c_2\stackrel{`}{\epsilon }R$.

1. True
2. False

Q. Given orthocentre $\overline{H}$ and circumcentre $\overline{C}$ for a triangle as $2\hat{i}+3\hat{j}and4\hat{i}+5\hat{k}$.Then the centroid of triangle can be given by,

1. 
2. 
3. 
4. 

Q. The ratio in which $\hat{i}+2\hat{j}+3\hat{k}$ divides the join of $-2\hat{i}+3\hat{j}+5\hat{k}$ and $7\hat{i}-\hat{k}$ is

1. -3 : 2
2. 1 : 2
3. 2 : 3
4. -4 : 3

Q. If $\overrightarrow{p}=\hat{i}+\hat{j}+\hat{k}$ and $\overrightarrow{q}=a\hat{i}+b\hat{j}+c\hat{k},$ where $a,b,c\in \{-2,-1,0,1,2\},$ then the number of vectors $\overrightarrow{q}$ perpendicular to $\overrightarrow{p}$ is

Q. If $\overrightarrow{a}$ and $\overrightarrow{b}$ are the position vectors of points A and B respectively, find the position vector of point C on AB produced such that $\overrightarrow{AC}=3\overrightarrow{AB}$

1. 
2. 
3. 
4.