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Q1. Multiple choice questions (Attempt all · 12 × 1 = 12)

For the matrices $A = \begin{bmatrix} 2 & 3 \\ -5 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 4 & -1 \\ 1 & 3 \end{bmatrix}$ , the value of matrix X in the matrix equations 2X - A = B will be :
1. $\begin{bmatrix} 12 & 4 \\ -8 & 8 \end{bmatrix}$
2. $\begin{bmatrix} 6 & 2 \\ -4 & 4 \end{bmatrix}$
3. $\begin{bmatrix} 3 & 1 \\ -2 & 2 \end{bmatrix}$
4. $\begin{bmatrix} 9 & 3 \\ -3 & 3 \end{bmatrix}$
What is the position of the particle from mean position when amplitude is 0.0004mm and angle is $\frac{\pi}{3}$
1. $\frac{1}{500}(1 +\sqrt{3i})$
2. $\frac{1}{5000}(1 +\sqrt{3i})$
3. $\frac{1}{5000}(\sqrt{3} + i)$
4. $\frac{1}{500}(\sqrt{3} + i)$
The projection of $\underline{i} - 2\underline{j} + a\underline{k}$ along $2\underline{i} - 3\underline{j} + 2\sqrt{3}\underline{k}$ is 4 then $\alpha$ is :
1. $\sqrt{3}$
2. $2\sqrt{3}$
3. - $\sqrt{3}$
4. $3\sqrt{3}$
The matrix in reduced echelon form is :
1. $\begin{bmatrix} 0 & 1 & -\frac{1}{2} & 0 & 6 \\ 1 & 0 & \frac{2}{3} & 4 & -5 \\ 0 & 0 & 0 & 1 & 0.4 \\ 0 & 1 & 1 & 0 & 0 \end{bmatrix}$
2. $\begin{bmatrix} 1 & 3 & -6 & 10 & 0 \\ 0 & 0 & 1 & 4 & -3 \\ 0 & 0 & 0 & 1 & \frac{1}{2} \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}$
3. $\begin{bmatrix} 1 & 3 & 0 & 0 & 0\\ 0 & 0 & 1 & 4 & \frac{1}{2} \\ 0 & 0 & 1 & 0 & -3 \\ 0 & 0 & 0 & 0 & 0\end{bmatrix}$
4. $\begin{bmatrix} 1 & 3 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & -3 \\ 0 & 0 & 0 & 1 & \frac{1}{2} \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}$
Part of complex number which is along y-axis is called
1. Real part
2. Imaginary part
3. Complex part
4. Both a and c
The line segment joining the midpoints of the diagonals of a trapezium is parallel to the parallel sides and its length is half the ____________ lengths of the parallel sides.
1. Sum
2. Difference
3. Product
4. Quotient
The principal diagonal / main diagonal elements in the matrix $A=\begin{bmatrix} 1 & -3 & 1 \\ 2 & 4 & -1 \\ 3 & 5 & 6 \end{bmatrix}$ are:
1. 1 , - 3 , 1
2. 1 , 4 , 3
3. 1 , 4 , 6
4. 3 , 5 , 6
Algebraic form of $z = \sqrt{2} (\cos 315^\circ + i \sin 315^\circ)$ is
1. $1 + i$
2. $1 - i$
3. $i$
4. $2 + i$
If $\underline{a} = \underline{i} - 2\underline{k}$ and $\underline{b} = 3\underline{i} - 2\underline{j}$ , then $\underline{a} \times \underline{b} = ?$
1. $2\left( \underline{i} - 3\underline{j} +\underline{k}\right)$
2. $-2\left(2 \underline{i}+ 3\underline{j} +\underline{k}\right)$
3. $2\left( \underline{i} + 3\underline{j} -\underline{k}\right)$
4. $2\underline{i} + 3\underline{j} +4\underline{k}$
If $A = \begin{bmatrix} 1 & 2 \\ 3 & - 3 \end{bmatrix}$ and $B= \begin{bmatrix} - 1 \\ 6 \end{bmatrix}$ , given that AX =B , then X =
1. $\begin{bmatrix} 1 \\ 1 \end{bmatrix}$
2. $\begin{bmatrix} -1 \\ 1 \end{bmatrix}$
3. $\begin{bmatrix} 1 \\ -1 \end{bmatrix}$
4. $\begin{bmatrix} -1 \\ -1 \end{bmatrix}$
If $z = ( 3 + 4i ) ( 2 - 3i)$ then $|{z}|$
1. $7\sqrt{2}$
2. $5\sqrt{13}$
3. $13\sqrt{5}$
4. $2\sqrt{7}$
If $\overrightarrow{AB}=\overrightarrow{CD}$ , then coordinates of point A is when B ( 1 , 2 ) , C ( - 2 , 5 ) and D ( 4 , 11 ) are.
1. A ( 5 , 4 )
2. A ( - 5 , 4 )
3. A ( - 5 , - 4 )
4. A ( 5 , - 4 )

Q2. Short questions (Attempt any 8 of 12 · 8 × 2 = 16)

Find the value of λ, so that the given matrix are singular. $\begin{bmatrix} \lambda & 2 & 0 \\ 2 & 1 & 3 \\ \lambda & 2 & 1 \end{bmatrix}$
Find the moment of the force $\vec{F} = 3\hat{i} = 2\hat{j} - 5\hat{k}$ about the point ( 2 , 1 , -1 ) when it is applied at point ( 3 , 0 , 2 ).
Write the following equation and inequation in term of x and y by taking z = x + 𝑖y. $| 2z - 𝑖 | = 4$
If $A = \begin{bmatrix} 1 & 2 & 0 \\ 0 & 3 & 0 \\ 2 & 1 & 6 \end{bmatrix}$ then find $|A| = |A^t|$
Find the angle between the vectors $\hat{i} -2\hat{j} + \hat{k}$ and $2\hat{i} -3\hat{j} + \hat{k}$ .
Find the magnitude of the following complex number. $\left( \sqrt{3} - \sqrt{-8} \right) \left( \sqrt{3} + \sqrt{-8} \right)$
If $A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 1 & 0 \\ 0 & 2 & 0 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 2 & 0 \\ 2 & 1 & 0 \end{bmatrix}$ then find $|B| = -|A|$
If $\vec{a} = 2\hat{i} - 3\hat{j} + \hat{k}$ and $\vec{b} = \hat{i} - 3\hat{j} + 5\hat{k}$ then find: A vector of magnitude $\frac{3}{7}$ in the direction $3\vec{a} + \vec{b}$ .
Factorize the following: $4x^2 + 40$
Determine which of the following matrix are singular and which are non - singular. $\begin{bmatrix} 3 & 1 & 2 \\ 2 & 3 & 1 \\ -4 & 1 & -3 \end{bmatrix}$
If $\vec{a} = 2\hat{i} + 5\hat{j} - 2\hat{k} \quad \text{and} \quad \vec{b} = \hat{i} - \hat{j} + 3\hat{k}$ are two adjacent sides of a parallelogram, then find its area.
Write the following equation and inequation in term of x and y by taking z = x + 𝑖y. $\frac{1}{2}Re (𝑖\bar{z}) = 4$

Q3. Long questions (Attempt any 3 of 4 · 3 × 4 = 12)

OPQR is a trapezium made from three equilateral triangles with $\overrightarrow{OP} = \vec{r}$ , $\overrightarrow{OS} = \vec{s}$ and M is the mid point of $\overline{QR}$ . (1) Write $\overrightarrow{PS}$ in terms of $\vec{r}$ and $\vec{s}$ . (2) Show that $\overrightarrow{OQ}$ is parallel to $\overrightarrow{SM}$ .
Verify that $\sqrt{2} |z| \geq |\text{Re } (z)| + |\text{Im } (z)| \quad \text{Hint: (Start with } (|x| - |y|)^2 \geq 0)$
Using the properties of the determinant prove the following. $\begin{vmatrix} 3a & 1 & 2a+1 \\ 2a+1 & 1 & a+2 \\ 3 & 1 & 2 \end{vmatrix} = (a-1)(a-2)$
Find the volume of the tetrahedron with given vertices. A ( 0, 1, 0) ; B (2, 0, 1) ; C (3, 1, 2) ; D (5, 6, -1)