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] $
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] $
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