# HSC Maths Sample Question Paper- Part 1 - AA2

AA2 Sample Question Paper Series for State Board / HSC Mathematics Examination. Take these questions as practice questions only and prepare well for your exams. If you know the answer to any of these questions, share your answers in the comments section.
1) The rank of the matrix $\begin{bmatrix} 2\;\;\;\;\; -4 \\ -1\;\;\;\;\; 2 \\ \end{bmatrix}$ is:

a) 1                     b) 2                     c) 0                     d) 8

2) $(A^T)^{-1}$ is equal to:

a) $A^{-1}$             b) $A^{T}$            c) A             d) $(A^{-1})^T$

3) If $\rho (A)$, then which of the following is correct?

a) All the minors of order r which do not vanish                         b) A has at-least one minor of r which does not vanish

c) A has at-least one (r+1) order minor which vanishes              d) All (r+1) and higher order minors should not vanish

4) In echelon form, which of the following is incorrect?

a) Every row of A which has all its entries O occurs below every row which has a non-zero entry

b) The first non-zero entry in each non-zero row is 1

c) The number of zeros before the first non-zero element in a row is less than the number of such zeros in the next row

d) Two rows can have same number of zeros before the first non-zero entry

5) In the system of 3 linear equations with three unknowns, if $\Delta=0$ and one of $\Delta_x, \;\Delta_y$ or $\Delta_z$ is non-zero then the system is:

a) Consistent                                               b) consistent and the system reduces to two equations

c) Inconsistent                                             d) consistent and the system reduces to a single equations

6) In the system of 3 linear equations with three unknowns, if $\Delta=0$ and all 2x2 minors of $\Delta,\;\Delta_x, \;\Delta_y,\;\Delta_z$ are zeros and at-least one non-zero element is in $\Delta$ then the system is:

a) Consistent                                               b) consistent and the system reduces to two equations

c) Inconsistent                                             d) consistent and the system reduces to a single equations

7) If $\rho (A)=\rho (AB)$, then the system is:

a) Consistent and has infinitely many solution                       b) consistent and has a unique solution

c) consistent                                                                       d) inconsistent

8) Cramer’s rule is applicable only (with three unknowns) when:

a) $\Delta \neq 0$           b) $\Delta = 0$           c) $\Delta = 0,\;\Delta \neq 0$           d) $\Delta x=\Delta y=\Delta z= 0$

9) The value of $\overrightarrow{a}.\overrightarrow{b}$, when $\overrightarrow{a}=\overrightarrow{i}-2\overrightarrow{j}+\overrightarrow{k}$ and $\overrightarrow{b}=4\overrightarrow{i}-4\overrightarrow{j}+7\overrightarrow{k}$ is:

a) 19                        b) 3                        c) -19                        d) 14

10) If $m\overrightarrow{i}+2\overrightarrow{j}+\overrightarrow{k}$ and $4\overrightarrow{i}-9\overrightarrow{j}+2\overrightarrow{k}$ are perpendicular then m is:

a) -4                        b) 8                        c) 4                        d) 12

11) The angle between the vectors $3\overrightarrow{i}-2\overrightarrow{j}-6\overrightarrow{k}$ and $4\overrightarrow{i}-\overrightarrow{j}+8\overrightarrow{k}$ is:

a) $cos^{-1}(\frac{34}{63})$           b) $sin^{-1}(-\frac{34}{63})$           c) $sin^{-1}(\frac{34}{63})$           d) $cos^{-1}(-\frac{34}{63})$

12) The projection of the vector $7\overrightarrow{i}+\overrightarrow{j}-4\overrightarrow{k}$ on $2\overrightarrow{i}+6\overrightarrow{j}+3\overrightarrow{k}$ is:

a) $\frac{7}{8}$           b) $\frac{8}{\sqrt{66}}$           c) $\frac{8}{7}$           d) $\frac{\sqrt{66}}{8}$

13) If the vectors $\overrightarrow{a}=3\overrightarrow{i}+2\overrightarrow{j}+9\overrightarrow{k}$ and $\overrightarrow{i}+m\overrightarrow{j}+3\overrightarrow{k}$ are parallel then m is:

a) $\frac{3}{2}$           b) $\frac{2}{3}$           c) $\frac{-3}{2}$           d) $\frac{-2}{3}$

14) Let $\overrightarrow{u}, \; \overrightarrow{v}$ and $\overrightarrow{w}$ be vector such that $\overrightarrow{u}+\overrightarrow{v}+\overrightarrow{w}=\overrightarrow{0}$. If $|\overrightarrow{u}|=3$, $|\overrightarrow{v}|=4$ and $|\overrightarrow{w}|=5$, then $\overrightarrow{u}.\overrightarrow{v}+\overrightarrow{v}.\overrightarrow{w}+\overrightarrow{w}.\overrightarrow{u}$ is :

a) 25                        b) -25                        c) 5                        d) $\sqrt{5}$

15) The projection of $\overrightarrow{i}-\overrightarrow{j}$ on z-axis is:

a) 0                        b) 1                        c) -1                        d) 2

16) The work done by the force $\overrightarrow{F}=a\overrightarrow{i}+\overrightarrow{j}+\overrightarrow{k}$ in moving the point of application form (1, 1, 1) to (2, 2, 2) along a straight line is given to 5 units. The value of a is:

a) -3                        b) 3                        c) 8                        d) –8

17) The unit normal vectors to the plane 2x-y+2z=5 are

a) $2\overrightarrow{i}-\overrightarrow{j}+2\overrightarrow{k}$

b) $\frac{1}{3}(2\overrightarrow{i}-\overrightarrow{j}+2\overrightarrow{k})$

c) $\frac{-1}{3}(2\overrightarrow{i}-\overrightarrow{j}+2\overrightarrow{k})$

d) $\pm\frac{1}{3}(2\overrightarrow{i}-\overrightarrow{j}+2\overrightarrow{k})$

18) The centre and radius of the sphere $|2\overrightarrow{r}+(3\overrightarrow{i}-\overrightarrow{j}+4\overrightarrow{k})|=4$ are:

a) $(\frac{-3}{2},\frac{1}{2}, -2),4$                              b) $(\frac{-3}{2},\frac{1}{2}, -2),2$

c) $(\frac{-3}{2},\frac{1}{2}, -2),6$                              d) $(\frac{-3}{2},\frac{1}{2}, -2),5$

19) The vector equation of a plane whose distance from the origin is p and perpendicular to a vector n is:

a) $\overrightarrow{r}.\overrightarrow{n}=p$                              b) $\overrightarrow{r}.\hat{n}=p$

c) $\overrightarrow{r} \times \overrightarrow{n}=p$                    d) $\overrightarrow{r}.\hat{n}=p$

20) The vector equation of a sphere whose centre is origin and radius ‘a’ is:

a) $|\overrightarrow{r}|=|\overrightarrow{a}|$

b) $\overrightarrow{r}-\overrightarrow{c}=\overrightarrow{a}$

c) $\overrightarrow{r}=\overrightarrow{a}$

d) $\overrightarrow{r}=a$