# HSC +2 Maths - Practice 5 Mark Questions - Part4

Visit the previous pages for Plus2 Maths Part1 , HSC Maths Sample Part2 and +2 Mathematics practice Part3 Important 5 Mark questions.

1) Solve the non-homogenous system of linear equations by determinant method
2x + y - z = 4,
x + y – 2z = 0,
3x + 2y – 3z = 4.

2) Examine the consistency of the equations:
2x + 3y + 7z = 5,
3x + y – 3z = 13,
2x + 19y – 47z = 32.

3) Solve by matrix inversion method for the system of linear equations: 7x + 3y = –1; 2x + y =0.

4) Show the adjoint of A is $3A^{T}$, where $A=\begin{bmatrix} -1\;\;\;\; -2\;\;\;\; -2\\2\;\;\;\;\;\;\;\; 1 \;\;\;\;\;\;-2\\2\;\;\;\;\; -2 \;\;\;\;\;\;\;1 \\ \end{bmatrix}$

5) Find the rank of the matrix: $A=\begin{bmatrix} 1\;\;\;\; 2\;\;\;\; -1\;\;\;\; 3\\2\;\;\;\; 4 \;\;\;\;\; 1\;\; -2\\3\;\;\;\; 6\;\;\;\;\; 3\;\; -7\\ \end{bmatrix}$

6) Solve the non-homogenous system of three unknowns by determinant method.
x + y + 2z = 4,
2x + 2y + 4z = 8,
3x + 3y + 6z = 10.

7) Prove that $[\overrightarrow{a}\times\overrightarrow{b}, \overrightarrow{b}\times\overrightarrow{c},\overrightarrow{c}\times\overrightarrow{a}]=[\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}]^{2}$

8) Solve the equation $x^{4}-4x^{3}+11x^{2}-14x+10=0$, if one root is 1+2i.

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

10) A standard rectangular hyperbola has its vertices at (5, 7) and (-3, –1). Find its equation and asymptotes.

11) Find the equation of the tangent and normal to the curves $y=2\;sin^{2}\;3x$ at $x=\frac{\pi}{6}$.

12) Obtain the Maclaurin’s series expansion for: $\frac{1}{1+x}$

13)     (i) The radius of a sphere was measured and found to be 21 cm with a possible error in measurement of atmost 0.05 cm. What is the maximum error in using this value of the radius to compute the volume of the sphere?
(ii) Determine: $\frac{\partial u}{\partial x}, \frac{\partial u}{\partial y},\frac{\partial^2 u}{\partial x^2}$ if $u(x,y)=x^{4}+y^{3}+3x^{2}y^{2}+3x^{2}y$

14) If $k, \mu , \sigma$ of the normal distribution whose probability function is given by $f(x)=ke^{-2x^{2}+4x-2}$.

15)    (i) Solve: $(D^{2}-2D-3)y=sinx\;cosx$
(or)
(ii) State and prove the cancellation laws.

16) Of $U=e^{x^{3}+y^{3}}$, prove that $x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}=3U\; logU$

17) Evaluate: $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{dx}{1+\sqrt cot\;x}$

18) Solve: $(1+x^{2})\frac{\mathrm{d}y }{\mathrm{d} x}+2xy=cos\;x$

19) The life of army shoes is normally distributed with mean 8 months and standard deviation 2 months. If 5000 pairs are given, how many pairs would be expected to need replacement within 12 months.

20) Obtain the Maclarin’s expansion for $tan\;x, \frac{-\pi}{2}<x<\frac{\pi}{2}$
(OR)
In a Poisson distribution. Prove that the total probability is one.