1) Solve the matrix inversion method of the following system of linear equation:
2x – y + z = 7, 3X + y – 5z = 13, x + y + z = 5
2) Forces $ 2\overrightarrow{i}+7\overrightarrow{j},\; 2\overrightarrow{i}-5\overrightarrow{j}+6\overrightarrow{k}, \;-\overrightarrow{i}+2\overrightarrow{j}-\overrightarrow{k} $ act at a point P whose position vector is $ 4\overrightarrow{i}-3\overrightarrow{j}-2\overrightarrow{k} $. Find the moment of the resultant of three forces acting at P about the point Q whose position vector is $ 6\overrightarrow{i}+\overrightarrow{j}-3\overrightarrow{k} $.
3) If $ \overrightarrow{a}=2\overrightarrow{i}+3\overrightarrow{j}-\overrightarrow{k}, \;\overrightarrow{b}=-2\overrightarrow{i}+5\overrightarrow{k},\; \overrightarrow{c}=\overrightarrow{j}-3\overrightarrow{k} $, Verify that $ \overrightarrow{a}\times(\overrightarrow{b}\times\overrightarrow{c})=(\overrightarrow{a}.\overrightarrow{c})\overrightarrow{b}-(\overrightarrow{a}.\overrightarrow{b})\overrightarrow{c} $
4) Find the modulus and argument of the following complex numbers:
a) (i) i
(ii) $ -\sqrt2+i\sqrt2 $
b) If $ z^{2}=(0, 1) $ find ‘z’
5) If a parabolic reflector is 20cm in diameter and 5cm deep, find the distance of the focus from the centre of the reflector.
6) Find the equation of the ellipse whose vertices are (-1, 4) and (-7, 4) and eccentricity is 1/3.
7) A water tank has the shape of an inverted circular cone with base radius 2 metres and height 4 metres. If water is being pumped into the tank at a rate of $ 2m^{3}/min $, find the rate at which the water level is rising when the water is 3m deep.
8) Find the intervals of concavity and the points of inflection of the function $ f(x)=(x-1)^{\frac{1}{3}} $
9) (i) Using Euler’s theorem prove that $ u=xy^{2}\;sin(\frac{x}{y}) $, show that $ x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}=3u $.
(ii) If $ U=(x-y)(y-z)(z-x) $ then show that $ U_{x}+U_{y}+U_{z}=0 $
10) Evaluate:
(i) $ \int_{0}^{\frac{\pi}{2}}cos^{9}\;xdx $
(ii) $ \int_{0}^{1}xe^{-2x}\;dx $
11) Evaluate the problem using properties of integration $ \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}x^{3}\;cos^{3}\;xdx $
12) Solve: $ (x^{2}-yx^{2})dy+(y^{2}+xy^{2})dx=0 $
13) Solve: $ \frac{\mathrm{d} y}{\mathrm{d} x}+\frac{y}{x}=sin(x^{2}) $
14) State and prove reversal law of the group (G ; *)
15) Show that the set of G of all positive rationals forms a group under the composition * defined by $ a\ast b=\frac{ab}{3} $, for all $ a,b\;\epsilon\; G $
OR
The probability density function of a random variable ‘x’ is: $ f(x)=\left\{\begin{matrix} kx^{\alpha -1}e^{\beta x^{\alpha}}\;\;\;\;\; x, \alpha, \beta>0 \\ 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; elsewhere \\ \end{matrix}\right. $.
Find (i) k; (ii) P(X>10)
2x – y + z = 7, 3X + y – 5z = 13, x + y + z = 5
2) Forces $ 2\overrightarrow{i}+7\overrightarrow{j},\; 2\overrightarrow{i}-5\overrightarrow{j}+6\overrightarrow{k}, \;-\overrightarrow{i}+2\overrightarrow{j}-\overrightarrow{k} $ act at a point P whose position vector is $ 4\overrightarrow{i}-3\overrightarrow{j}-2\overrightarrow{k} $. Find the moment of the resultant of three forces acting at P about the point Q whose position vector is $ 6\overrightarrow{i}+\overrightarrow{j}-3\overrightarrow{k} $.
3) If $ \overrightarrow{a}=2\overrightarrow{i}+3\overrightarrow{j}-\overrightarrow{k}, \;\overrightarrow{b}=-2\overrightarrow{i}+5\overrightarrow{k},\; \overrightarrow{c}=\overrightarrow{j}-3\overrightarrow{k} $, Verify that $ \overrightarrow{a}\times(\overrightarrow{b}\times\overrightarrow{c})=(\overrightarrow{a}.\overrightarrow{c})\overrightarrow{b}-(\overrightarrow{a}.\overrightarrow{b})\overrightarrow{c} $
4) Find the modulus and argument of the following complex numbers:
a) (i) i
(ii) $ -\sqrt2+i\sqrt2 $
b) If $ z^{2}=(0, 1) $ find ‘z’
5) If a parabolic reflector is 20cm in diameter and 5cm deep, find the distance of the focus from the centre of the reflector.
6) Find the equation of the ellipse whose vertices are (-1, 4) and (-7, 4) and eccentricity is 1/3.
7) A water tank has the shape of an inverted circular cone with base radius 2 metres and height 4 metres. If water is being pumped into the tank at a rate of $ 2m^{3}/min $, find the rate at which the water level is rising when the water is 3m deep.
8) Find the intervals of concavity and the points of inflection of the function $ f(x)=(x-1)^{\frac{1}{3}} $
9) (i) Using Euler’s theorem prove that $ u=xy^{2}\;sin(\frac{x}{y}) $, show that $ x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}=3u $.
(ii) If $ U=(x-y)(y-z)(z-x) $ then show that $ U_{x}+U_{y}+U_{z}=0 $
10) Evaluate:
(i) $ \int_{0}^{\frac{\pi}{2}}cos^{9}\;xdx $
(ii) $ \int_{0}^{1}xe^{-2x}\;dx $
11) Evaluate the problem using properties of integration $ \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}x^{3}\;cos^{3}\;xdx $
12) Solve: $ (x^{2}-yx^{2})dy+(y^{2}+xy^{2})dx=0 $
13) Solve: $ \frac{\mathrm{d} y}{\mathrm{d} x}+\frac{y}{x}=sin(x^{2}) $
14) State and prove reversal law of the group (G ; *)
15) Show that the set of G of all positive rationals forms a group under the composition * defined by $ a\ast b=\frac{ab}{3} $, for all $ a,b\;\epsilon\; G $
OR
The probability density function of a random variable ‘x’ is: $ f(x)=\left\{\begin{matrix} kx^{\alpha -1}e^{\beta x^{\alpha}}\;\;\;\;\; x, \alpha, \beta>0 \\ 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; elsewhere \\ \end{matrix}\right. $.
Find (i) k; (ii) P(X>10)
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