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.
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.
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