MCA Entrance Previous Year Question Bank

Let $\vec{a}=\hat{i}-2\hat{j}+3\hat{k}$, $\vec{b}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{c}$ be a vector such that $\vec{a}+(\vec{b}\times \vec{c})=\vec{0}$ and $\vec{b}\cdot \vec{c}=5$. Then, the value of $3(\vec{c}\cdot \vec{a})$ is equal to ________.

Let $A,B,C$ be three points whose position vectors respectively are

$\vec{a}=\hat{i}+4\hat{j}+3\hat{k}$

$\vec{b}=2\hat{i}+\alpha \hat{j}+4\hat{k},\alpha \in R$

$\vec{c}=3\hat{i}-2\hat{j}+5\hat{k}$ If $\alpha$

is the smallest positive integer for which $\vec{a},\vec{b},\vec{c}$ are noncollinear, then the length of the median, in $\Delta ABC$, through $A$ is:

Let $\vec{a}=\alpha \hat{i}+3\hat{j}-\hat{k}$, $\vec{b}=3\hat{i}-\beta \hat{j}+4\hat{k}$ and $\vec{c}=\hat{i}+2\hat{j}-2\hat{k}$ where $\alpha ,\beta \in R$, be three vectors. If the projection of $\vec{a}$ on $\vec{c}$ is $\frac{10}{3}$ and $\vec{b}\times \vec{c}=-6\hat{i}+10\hat{j}+7\hat{k}$, then the value of $\alpha +\beta$ is equal to:

Let $\vec{a}=2\hat{i}-7\hat{j}+5\hat{k}$, $\vec{b}=\hat{i}+\hat{k}$ and $\vec{c}=\hat{i}+2\hat{j}-3\hat{k}$ be three given vectors. If $\vec{r}$ is a vector such that $\vec{r}\times \vec{a}=\vec{c}\times \vec{a}$ and $\vec{r}\cdot \vec{b}=0$, then $|\vec{r}|$ is equal to:

Let $\vec{a}=5\hat{i}-\hat{j}-3\hat{k}$ and $\vec{b}=\hat{i}+3\hat{j}+5\hat{k}$ be two vectors. Then which one of the following statements is TRUE?

$A(2,6,2),B(-4,0,\lambda ),C(2,3,-1)$ and $D(4,5,0)$, $|\lambda |\le 5$ are the vertices of a quadrilateral $ABCD$. If its area is $18$ square units, then $5-6\lambda$ is equal to:

Let $\vec{v}=\alpha \hat{i}+2\hat{j}-3\hat{k}$, $\vec{w}=2\alpha \hat{i}+\hat{j}-\hat{k}$ and $\vec{u}$ be a vector such that |\vec{u}| = \alpha > 0. If the minimum value of the scalar triple product $[\vec{u}\vec{v}\vec{w}]$ is $-\alpha \sqrt{3401}$ and $|\vec{u}\cdot \hat{i}{|}^2=\frac{m}{n}$ where $m$ and $n$ are coprime natural numbers, then $m+n$ is equal to:

Let $\vec{a},\vec{b},\vec{c}$ be three vectors such that $|\vec{a}|=\sqrt{31},4|\vec{b}|=|\vec{c}|=2$ and $2(\vec{a}\times \vec{b})=3(\vec{c}\times \vec{a})$. If the angle between $\vec{b}$ and $\vec{c}$ is $\frac{2\pi }{3}$, then ${\left(\frac{\vec{a}\times \vec{c}}{\vec{a}\cdot \vec{b}}\right)}^2$ is equal to ________.

Let $\vec{a}=\hat{i}+2\hat{j}+3\hat{k}$, $\vec{b}=\hat{i}-\hat{j}+2\hat{k}$ and $\vec{c}=5\hat{i}-3\hat{j}+3\hat{k}$ be three vectors. If $\vec{r}$ is a vector such that $\vec{r}\times \vec{b}=\vec{c}\times \vec{b}$ and $\vec{r}\cdot \vec{a}=0$, then $25|\vec{r}{|}^2$ is equal to:

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}|=\sqrt{14}$, $|\vec{b}|=\sqrt{6}$ and $|\vec{a}\times \vec{b}|=\sqrt{48}$. Then $(\vec{a}\cdot \vec{b}{)}^2$ is equal to: