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 $\vec{a}=2\hat{i}-7\hat{j}+5\hat{k}$, $\vec{b}=\hat{i}+\hat{k}$ 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?

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 ________.

If the four points, whose position vectors are $3\hat{i}-4\hat{j}+2\hat{k}$, $\hat{i}+2\hat{j}-\hat{k}$, $-2\hat{i}-\hat{j}+3\hat{k}$, and $5\hat{i}-2\alpha \hat{j}+4\hat{k}$ are coplanar, then $\alpha$ is equal to:

Let $\vec{a}$ be a non-zero vector parallel to the line of intersection of the two planes described by $\hat{i}+\hat{j},\hat{i}+\hat{k}$ and $\hat{i}-\hat{j},\hat{i}-\hat{k}$. If $\theta$ is the angle between the vector $\vec{a}$ and the vector $\vec{b}=2\hat{i}-2\hat{j}+\hat{k}$ and $\vec{a}\cdot \vec{b}=6$, then the ordered pair $(\theta ,|\vec{a}\times \vec{b}|)$ is equal to:

Let $\vec{a}=2\hat{i}+7\hat{j}-\hat{k}$, $\vec{b}=3\hat{i}+5\hat{k}$ and $\vec{c}=\hat{i}-\hat{j}+2\hat{k}$. Let $\vec{d}$ be a vector which is perpendicular to both $\vec{a}$ and $\vec{b}$, and $\vec{c}\cdot \vec{d}=12$. The $(-\hat{i}+\hat{j}-\hat{k})\cdot (\vec{c}\times \vec{d})$ is equal to:

The area of the quadrilateral ABCD with vertices A(2,1,1), B (1,2, 5), C(–2,–3, 5) and D (1, –6, –7) is equal to