# HSC +2 Maths - Sample 5 Mark Questions - Part 4

Find Plus2 Important Maths 5 Mark Questions Part 4 in this post. For more practice questions on Maths, refer to the other parts in the 'Related Post' section.

1) If $A=\begin{bmatrix} 5\;\;\;\;\; 2 \\ 7\;\;\;\;\; 3 \\ \end{bmatrix}$ and $B=\begin{bmatrix} 2\;\;\;\;\; -1 \\ -1\;\;\;\;\; 1 \\ \end{bmatrix}$ verify that

(i) $(AB)^{-1}=B^{-1}A^{-1}$

(ii) $(AB)^T=B^TA^T$

2) Solve: x + y+ 2x = 0; 3x + 2y +z = 0; 2x + y –z =0

3) Solve the following homogenous linear equations: x + 2y – 5z = 0; 3x + 4y + 6z = 0; x + y + z = 0

4)         (i) For any vector $\overrightarrow{r}$, prove that $\overrightarrow{r}=(\overrightarrow{r}.\overrightarrow{i})\overrightarrow{i}+(\overrightarrow{r}.\overrightarrow{j})\overrightarrow{j}+(\overrightarrow{r}.\overrightarrow{k})\overrightarrow{k}$
(ii) Find the projection of the vector $7\overrightarrow{i}+\overrightarrow{j}-4\overrightarrow{k}$ on $2\overrightarrow{i}+6\overrightarrow{j}+3\overrightarrow{k}$

5) Angle in a semi-circle is a right angle. Prove by vector method.

6) Find the vectors of magnitude 6 which are perpendicular to both the vector $4\overrightarrow{i}-\overrightarrow{j}+3\overrightarrow{k}$ and $-2\overrightarrow{i}+\overrightarrow{j}-2\overrightarrow{k}$

7) Show that the torque above the point A(3, –1, 3) of a force $4\overrightarrow{i}+2\overrightarrow{j}+\overrightarrow{k}$ through the point B(5, 2, 4) is $\overrightarrow{i}+2\overrightarrow{j}-8\overrightarrow{k}$

8) If $arg(z-1) = \frac{\pi}{6}$ and $arg(z+1) = 2\frac{\pi}{3}$, then prove |z|=1.

9) If $cos\; \alpha+cos \;\beta+cos\; \gamma=0=sin\; \alpha+sin\; \beta+sin\; \gamma$, prove that
(i) $cos\; 3\alpha+cos \; 3\beta+cos\; 3\gamma=3cos(\alpha+\beta+\gamma)$
(ii) $sin\; 3\alpha+sin \; 3\beta+sin\; 3\gamma=3sin(\alpha+\beta+\gamma)$

10) Find all the values: $(-\sqrt 3-i)^{\frac{2}{3}}$

11) Prove that if $\omega^3=1$, then $\frac{1}{1+2\omega}-\frac{1}{1+\omega}+\frac{1}{2+\omega}=0$

12) A reflecting telescope has a parabolic mirror for which the distance from the vertex to the focus is 9 mts. If the distance across (diameter) the top of the mirror is 160 cm, how deep is the mirror at the middle?

13) Find the equation of the ellipse whose vertices are (-1, 4) and (-7, 4) and eccentricity is $\frac{1}{3}$

14) Find the equations of the tangent and normal to the hyperbola $\frac{x^2}{9}-\frac{y^2}{12}=1$ at $\Theta=\frac{\pi}{6}$

15)    Find the angle between the asymptotes of the hyperbola $4x^2-5y^2-16x+10y+31=0$
OR
Show that the tangent to a rectangular hyperbola terminated by its asymptotes is bisected at the point of contact.