# HSC +2 Maths - Important 5 Mark Questions - Part2

Visit the previous page for Part1 Maths model 5 mark questions.

1) State and prove reversal law for inverses of matrices.

2) Solve the following non-homogenous equations of 3 unknowns:

x + y + 2z = 4
2x + 2y + 4z = 8
3x + 3y + 6z = 10
by determinant method.

3)      (1) The planes $\overrightarrow{r}.(2\overrightarrow{i}+\lambda\overrightarrow{j}-3\overrightarrow{k}) = 10$ and $\overrightarrow{r}.(\lambda\overrightarrow{i}+3\overrightarrow{j}+\overrightarrow{k}) = 5$ are perpendicular. Find $\lambda$.

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

4) Prove that $(\overrightarrow{a}\times \overrightarrow{b}).(\overrightarrow{c}\times \overrightarrow{d}) \;+ \; (\overrightarrow{b}\times \overrightarrow{c}).(\overrightarrow{a}\times \overrightarrow{d}) \;+ \;(\overrightarrow{c}\times \overrightarrow{a}).(\overrightarrow{b}\times \overrightarrow{d}) =0$

5) State and prove the triangle inequality of complex numbers.

6) Prove that $(1+cos\Theta +isin\Theta)^{n}-(1+cos\Theta -isin\Theta)^{n}=2^{n+1}\;cos^{n}(\frac{\Theta}{2})cos(\frac{n\Theta}{2})$

7) The tangent at any point of the R.H $xy=c^{2}$ makes intercepts a,b and the normal at the point makes intercepts p,q on the axes. Prove that ap+bq=0.

8) Evaluate: limit $limit\;x\rightarrow \frac{\pi}{2} \;\; (tan x)^{cosx}$

9) Find $\frac{\partial w}{\partial r}$ if $\frac{\partial w}{\partial \Theta}$ if $w=log(x^{2}+y^{2})$, when $x=r \;cos\Theta$, $y=r \;sin\Theta$.

10) Show that of all the rectangles with a given perimeter the one with the greater area is a square.

11) Find the area of the circle whose radius is $\frac{a}{2}$ using integration.

12) Solve $\frac{\mathrm{dy} }{\mathrm{d} x}=sin(x+y)$

13) Show that $p\leftrightarrow q \; \equiv \; ((\sim p)\; \vee \; q) \; \wedge \; ((\sim q)\vee \; p)$

14) Two cards are drawn without replacement from a well shuffled deck of 52 cards. Find the mean and variance for the number of aces.

15)    (1) Show that the set of all $2\times2$ non singular matrices form a non ableian infinite group under matrix multiplication (where the entries belong to R).

(2) The life of army shoes is normally distributed with mean 8 months and standard deviation 2 months. If 5000 pairs are issued, how many pairs would be expected to need replacement after 12 months. $[P(0\leqZ\leq-2)=0.4772, \; P(0\leqZ\leq1.2)=0.3849]$