In the new HSC +2 Maths questions set AA3, we provide a collected set of 40 questions for solving, that are of objective type. You may wish to post the answers to us in the comments section to know the correct answers. This is part 1 of this set which contains questions from 1 to 20.
1) The rank of the matrix $ \begin{bmatrix} 7\;\;\;\;\; -1 \\ 2\;\;\;\;\;\;\;\;\; 1 \\ \end{bmatrix} $ is:
a) 9 b) 2 c) 1 d) 5
2) Which of the following is not elementary transformation?
a) $ R_{i}\leftrightarrow R_{j} $ b) $ R_{i}\rightarrow 2R_{i}R_{j} $
c) $ C_{i}\rightarrow C_{j}+C_{i} $ d) $ R_{i}\rightarrow R_{i}+C_{j} $
3) If $ \Delta \neq 0 $, then the system is:
a) Consistent and has unique solution b) consistent and has infinitely many solutions
c) inconsistent d) either consistent or inconsistent
4) In the system of 3 linear equations with three unknowns, if $ \Delta = 0 $ and $ \Delta_{x}, \;\Delta_{y}, \;\Delta_{z} $ are zeros and at least one 2x2 minors of $ \Delta $ is non-zero then the system is:
a) consistent b) inconsistent c) consistent and the system reduces to two equations d) consistent and the system reduces to a single equation
5) $ \rho(A) \neq \rho(A,B) $, then the system is:
a) consistent and has infinitely many solution b) consistent and has a unique solution c) consistent d) inconsistent
6) In the homogenous system with three unknowns, $ \rho(A) = $ number of unknowns then the system has:
a) only trivial solution b) reduces to 2 equations and has infinitely many solution c) reduces to a single equation and has infinitely many solution d) is inconsistent
7) In the homogenous system $ \rho(A) < $ the number of unknowns then the system has:
a) only trivial solution b) trivial solution and infinitely many non-trivial solutions c) only non-trivial solutions d) no solution
8) Which of the following statement is correct regarding homogenous system:
a) always inconsistent b) has only trivial solution c) has only non-trivial solution d) has only trivial solution only if rank of the coefficient matrix equal to the number of unknowns
9) The value of $ \overrightarrow{a}.\overrightarrow{b} $ when $ \overrightarrow{a}=\overrightarrow{j}-2\overrightarrow{k} $ and $ \overrightarrow{b}=2\overrightarrow{i}+3\overrightarrow{j}-2\overrightarrow{k} $ is:
a) 7 b) -7 c) 5 d) 6
10) The angle between the vectors $ \overrightarrow{i}-\overrightarrow{j} $ and $ \overrightarrow{j}-\overrightarrow{k} $ is:
a) $ \frac{\pi}{3} $ b) $ \frac{-2\pi}{3} $ c) $ \frac{-\pi}{3} $ d) $ \frac{2\pi}{3} $
11) If $ \overrightarrow{a}, \;\overrightarrow{b}, \;\overrightarrow{c} $ are three mutually perpendicular unit vectors, then $ |\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}|= $
a) 3 b) 9 c) $ 3\sqrt{3} $ d) $ \sqrt{3} $
12) The projection of $ 3\overrightarrow{i}+\overrightarrow{j}-\overrightarrow{k} $ on $ 4\overrightarrow{i}-\overrightarrow{j}+2\overrightarrow{k} $ is:
a) $ \frac{9}{\sqrt{21}} $ b) $ \frac{-9}{\sqrt{21}} $ c) $ \frac{81}{\sqrt{21}} $ d) $ \frac{-81}{\sqrt{21}} $
13) If $ |\overrightarrow{a}|=3, \; |\overrightarrow{b}|=4 $ and $ \overrightarrow{a}.\overrightarrow{b}=9 $, then $ |\overrightarrow{a} \times \overrightarrow{b}| $ is:
a) $ 3\sqrt{7} $ b) 63 c) 69 d) $ \sqrt{69} $
14) The d.c.s of a vector whose direction ratios are 2, 3, –6 are:
a) $ (\frac{2}{7},\frac{3}{7},\frac{-6}{7}) $ b) $ (\frac{2}{7},\frac{3}{7},\frac{6}{7}) $
c) $ (\frac{2}{49},\frac{3}{49},\frac{-6}{49}) $ d) $ (\frac{\sqrt{2}}{7},\frac{\sqrt{3}}{7},\frac{-\sqrt{6}}{7}) $
15) Chord AB is a diameter of the sphere $ |\overrightarrow{r}-(2\overrightarrow{i}+\overrightarrow{j}-6\overrightarrow{k})|= \sqrt{18} $ with the coordinate of A as (3, 2, –2). The coordinates of B is:
a) (1, 0, 10) b) (-1,0, –10) c) (-1, 0, –10) d) (1, 0, –10)
16) The non parametric vector equation of a plane passing through the points whose, P.Vs are $ \overrightarrow{a}, \overrightarrow{b} $ and parallel to $ \overrightarrow{v}, $ is:
a) $ [\overrightarrow{r}-\overrightarrow{a} \;\;\;\;\; \overrightarrow{b}-\overrightarrow{a} \;\;\;\;\; \overrightarrow{v}]=0 $
b) $ [\overrightarrow{r} \;\;\;\;\; \overrightarrow{b}-\overrightarrow{a} \;\;\;\;\; \overrightarrow{v}]=0 $
c) $ [\overrightarrow{a} \;\;\;\;\; \overrightarrow{b} \;\;\;\;\; \overrightarrow{v}]=0 $
d) $ [\overrightarrow{r} \;\;\;\;\; \overrightarrow{a} \;\;\;\;\; \overrightarrow{b}]=0 $
17) The non-parametric vector equation of a plane passing through three points whose P.Vs are $ \overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c} $ is:
a) $ [\overrightarrow{r}-\overrightarrow{a} \;\;\;\;\; \overrightarrow{b}-\overrightarrow{a} \;\;\;\;\; \overrightarrow{c}-\overrightarrow{a}]=0 $
b) $ [\overrightarrow{r} \;\;\;\;\; \overrightarrow{a} \;\;\;\;\; \overrightarrow{b}]=0 $
c) $ [\overrightarrow{r} \;\;\;\;\; \overrightarrow{b} \;\;\;\;\; \overrightarrow{c}]=0 $
d) $ [\overrightarrow{a} \;\;\;\;\; \overrightarrow{b} \;\;\;\;\; \overrightarrow{c}]=0 $
18) The vector equation of a plane passing through the line of intersection of the planes $ \overrightarrow{r}.\overrightarrow{n_{1}}=\overrightarrow{q_{1}} $ and $ \overrightarrow{r}.\overrightarrow{n_{2}}=\overrightarrow{q_{2}} $ is:
a) $ (\overrightarrow{r}.\overrightarrow{n_{1}} - \overrightarrow{q_{1}})+ \lambda (\overrightarrow{r}.\overrightarrow{n_{2}} - \overrightarrow{q_{2}})=0 $
b) $ \overrightarrow{r}.\overrightarrow{n_{1}} + \overrightarrow{r}.\overrightarrow{n_{2}} = q_{1}+ \lambda q_{2} $
c) $ \overrightarrow{r} \times \overrightarrow{n_{1}} + \overrightarrow{r} \times \overrightarrow{n_{2}} = q_{1}+ q_{2} $
d) $ \overrightarrow{r} \times \overrightarrow{n_{1}} - \overrightarrow{r} \times \overrightarrow{n_{2}} = q_{1}+ q_{2} $
19) The angle between the line $ \overrightarrow{r}=\overrightarrow{a}+t\overrightarrow{b} $ and the plane $ \overrightarrow{r}.\overrightarrow{n}=\overrightarrow{q} $ is connected by the relation:
a) $ cos\; \theta= \frac{\overrightarrow{a}.\overrightarrow{n}}{q} $
b) $ cos\; \theta= \frac{\overrightarrow{b}.\overrightarrow{n}}{|\overrightarrow{b}||\overrightarrow{n}|} $
c) $ sin\; \theta= \frac{\overrightarrow{a}.\overrightarrow{b}}{|\overrightarrow{n}|} $
d) $ sin\; \theta= \frac{\overrightarrow{b}.\overrightarrow{n}}{|\overrightarrow{b}||\overrightarrow{n}|} $
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 $
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