MCA Entrance Previous Year Question Bank

Let $\alpha ,\beta ,\gamma$ be distinct real numbers. The points with position vectors $\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}$, $\beta \hat{i}+\gamma \hat{j}+\alpha \hat{k}$, and $\gamma \hat{i}+\alpha \hat{j}+\beta \hat{k}$:

If $\vec{A}+\vec{B}+\vec{C}=0$, $|\vec{A}|=3$, $|\vec{B}|=5$, and $|\vec{C}|=7$, then the angle between $\vec{A}$ and $\vec{B}$ is:

Let $\vec{A}=2\hat{i}+\hat{j}-2\hat{k}$ and $\vec{B}=\hat{i}+\hat{j}$. If $\vec{C}$ is a vector such that $\vec{A}\cdot \vec{C}=|\vec{C}|,|\vec{C}-\vec{A}|=2\sqrt{2}$ and the angle between $\vec{A}\times \vec{B}$ and $\vec{C}$ is $30^{\circ }$, then $|(\vec{A}\times \vec{B})\times \vec{C}|$ is equal to:

The value of $\lambda$ such that the four points whose position vectors are $3\hat{i}-2\hat{j}+\lambda \hat{k}$, $6\hat{i}+3\hat{j}+\hat{k}$, $5\hat{i}+7\hat{j}+3\hat{k}$ and $2\hat{i}+2\hat{j}+6\hat{k}$ are coplanar is:

The value of $\lambda$ for which the volume of parallelopiped formed by the vectors $\hat{i}+\lambda \hat{j}+\hat{k}$, $\hat{j}+\lambda \hat{k}$ and $\lambda \hat{i}+\hat{k}$ is minimum is given by:

$\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ are non-coplanar unit vectors such that $\mathbf{a}\times (\mathbf{b}\times \mathbf{c})=\frac{\mathbf{b}+\mathbf{c}}{\sqrt{2}}$, then the angle between $\mathbf{a}$ and $\mathbf{b}$ is:

If $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ are unit vectors, then $|\mathbf{a}-\mathbf{b}{|}^2+|\mathbf{b}-\mathbf{c}{|}^2+|\mathbf{c}-\mathbf{a}{|}^2$ does not exceed:

The vector $\vec{B}=3\hat{i}+4\hat{k}$ is to be written as the sum of a vector ${\vec{B}}_1$ parallel to $\vec{A}=\hat{i}+\hat{j}$ and a vector ${\vec{B}}_2$ perpendicular to $\vec{A}$, then ${\vec{B}}_1$ is:

If $\int e^x(f(x)-f^{\prime }(x))dx=\varphi (x)$, then the value of $\int e^{x}f(x)dx$ is:

If $\vec{a},\vec{b}$ and $\vec{c}$ are the position vectors of the vertices $A,B,C$ of a triangle $ABC$, then the area of the triangle $ABC$ is: