1) Examine the consistency of the system:
x + y + z = 7
x + 2y + 3z = 18
y + 2z = 6
If it is consistent then solve by using rank method.
2) Verify that $ (A^{-1})^{T} = (A^{T})^{-1} $ for the matrix $ A=\begin{bmatrix} -2\;\;\; -3 \\ 5\;\;\; -6\\ \end{bmatrix} $
3) Find the vector and Cartesian equations of the sphere whose centre is (1, 2, 3) and which passes through the point (5, 5, 3)
4) Show that the points representing the complex number (7 + 9i), (-3 + 7i) and (3 + 3i) form a right-angled triangle on the Argand diagram.
5) Prove that $ (1+i)^{n}+(1-i)^{n}=2^{\frac{n+2}{2}}\; cos\frac{n\pi}{4}(n \;\varepsilon \;N) $.
6) The headlight of a motor vehicle is a parabolic reflector of diameter 12 cm and depth 4 cm. Find the position of bulb on the axis of the reflector for effective functioning of the headlight.
7) Verify Lagrange’s law of mean for the function $ f(x)=2x^{3}+x^{2}-x-1 $, on [0, 2].
8) (1) Obtain the Maclaurin’s series for $ e^{x} $.
(2) Find the critical numbers of $ x^{\frac{3}{5}}(4-x) $.
9) Find $ \frac{\partial \omega }{\partial u} $ and $ \frac{\partial \omega }{\partial v} $, if $ \omega=x^{2}+y^{2} $, where $ x=u^{2}-v^{2} $ and y = 2uv by using chain rule for partial derivatives.
10) Solve $ (D^{2}+14D+49)y=e^{-7x}+4 $.
11) Show that $ p\leftrightarrow q \; \equiv \; ((\sim p)\; \vee \; q) \; \wedge \; ((\sim q)\vee \; p) $
12) State and prove reversal law on inverses of a group.
13) (1) If $ F(x)=\frac{1}{\pi}(\frac{\pi}{2}+tan^{-1}x), -\infty < x < \infty $ is a distribution function of a continuous variable X, find $ p(0\leq x\leq1) $.
(2) The difference between the mean and the variance of a binomial distribution is 1 and the difference between their squares is 11. Find n.
14) Alpha particles are emitted by a radioactive source at an average rate of 5 in 20 minutes interval. Using Poisson distribution, find the probability that there will be:
a) 2 emissions
b) at least 2 emissions
in a particular 20 minute interval. $ (e^{-5}=0.0067) $
15) (1) If $ \overrightarrow{a}=\overrightarrow{i}+\overrightarrow{j}+2\overrightarrow{k} $ and $ \overrightarrow{b}=3\overrightarrow{i}+2\overrightarrow{j}-\overrightarrow{k} $, find $ (\overrightarrow{a}+3\overrightarrow{b}).(2\overrightarrow{a}-\overrightarrow{b}) $.
(2) The volume of parallelopiped whose edges are represented by $ -12\overrightarrow{i}+\lambda\overrightarrow{k}, \;3\overrightarrow{j}-\overrightarrow{k}, \;2\overrightarrow{i}+\overrightarrow{j}-15\overrightarrow{k} $ is 546. Find the value of $ \lambda $
(3) Evaluate: $ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}x \;sin\;x \;dx $, using properties of definite integrals.
16) Two sides of a triangle have length 12m and 15m. The angle between them is increasing at a rate of $ 2^{\circ} \;/min $. How fast is the length of third side increasing, when the angle between the sides of fixed length is $ 6-^{\circ} $?
17) If the curve $ y^{2}=x $ and $ xy=k $ are orthogonal then prove that $ 8k^{2}=1 $.
18) (1) At noon, ship A is 100 km west of ship B. Ship A is sailing east at 35 km/hr and Ship B is sailing north at 25 km/hr. How fast is the distance between the ships changing at 4.00 pm.
(2) Use differentials to find an approximate value for the given number: $ \sqrt36.1 $
19) (1) Find $ \frac{\partial w}{\partial u} $ and $ \frac{\partial w}{\partial v} $, if $ w=sin^{-1}\; xy $ where $ x=u+v, y=u-v $
(2) Evaluate: $ lim\; x\rightarrow 0 \; \frac{cot\;x}{cot\;2x} $
20) Show that $ f(x) = tan^{-1}(sin \;x+ cos\;x), x>0 $ is a strictly increasing function in the interval $ [0,\frac{\pi}{4}] $.
21) Find the local maximum and minimum values of $ sin^{2} \; \Theta, \; [0,\pi] $
22) Find the intervals of concavity and the points of inflection of the following function: $ y=12x^{2}-2x^{3}-x^{4} $.
x + y + z = 7
x + 2y + 3z = 18
y + 2z = 6
If it is consistent then solve by using rank method.
2) Verify that $ (A^{-1})^{T} = (A^{T})^{-1} $ for the matrix $ A=\begin{bmatrix} -2\;\;\; -3 \\ 5\;\;\; -6\\ \end{bmatrix} $
3) Find the vector and Cartesian equations of the sphere whose centre is (1, 2, 3) and which passes through the point (5, 5, 3)
4) Show that the points representing the complex number (7 + 9i), (-3 + 7i) and (3 + 3i) form a right-angled triangle on the Argand diagram.
5) Prove that $ (1+i)^{n}+(1-i)^{n}=2^{\frac{n+2}{2}}\; cos\frac{n\pi}{4}(n \;\varepsilon \;N) $.
6) The headlight of a motor vehicle is a parabolic reflector of diameter 12 cm and depth 4 cm. Find the position of bulb on the axis of the reflector for effective functioning of the headlight.
7) Verify Lagrange’s law of mean for the function $ f(x)=2x^{3}+x^{2}-x-1 $, on [0, 2].
8) (1) Obtain the Maclaurin’s series for $ e^{x} $.
(2) Find the critical numbers of $ x^{\frac{3}{5}}(4-x) $.
9) Find $ \frac{\partial \omega }{\partial u} $ and $ \frac{\partial \omega }{\partial v} $, if $ \omega=x^{2}+y^{2} $, where $ x=u^{2}-v^{2} $ and y = 2uv by using chain rule for partial derivatives.
10) Solve $ (D^{2}+14D+49)y=e^{-7x}+4 $.
11) Show that $ p\leftrightarrow q \; \equiv \; ((\sim p)\; \vee \; q) \; \wedge \; ((\sim q)\vee \; p) $
12) State and prove reversal law on inverses of a group.
13) (1) If $ F(x)=\frac{1}{\pi}(\frac{\pi}{2}+tan^{-1}x), -\infty < x < \infty $ is a distribution function of a continuous variable X, find $ p(0\leq x\leq1) $.
(2) The difference between the mean and the variance of a binomial distribution is 1 and the difference between their squares is 11. Find n.
14) Alpha particles are emitted by a radioactive source at an average rate of 5 in 20 minutes interval. Using Poisson distribution, find the probability that there will be:
a) 2 emissions
b) at least 2 emissions
in a particular 20 minute interval. $ (e^{-5}=0.0067) $
15) (1) If $ \overrightarrow{a}=\overrightarrow{i}+\overrightarrow{j}+2\overrightarrow{k} $ and $ \overrightarrow{b}=3\overrightarrow{i}+2\overrightarrow{j}-\overrightarrow{k} $, find $ (\overrightarrow{a}+3\overrightarrow{b}).(2\overrightarrow{a}-\overrightarrow{b}) $.
(2) The volume of parallelopiped whose edges are represented by $ -12\overrightarrow{i}+\lambda\overrightarrow{k}, \;3\overrightarrow{j}-\overrightarrow{k}, \;2\overrightarrow{i}+\overrightarrow{j}-15\overrightarrow{k} $ is 546. Find the value of $ \lambda $
(3) Evaluate: $ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}x \;sin\;x \;dx $, using properties of definite integrals.
16) Two sides of a triangle have length 12m and 15m. The angle between them is increasing at a rate of $ 2^{\circ} \;/min $. How fast is the length of third side increasing, when the angle between the sides of fixed length is $ 6-^{\circ} $?
17) If the curve $ y^{2}=x $ and $ xy=k $ are orthogonal then prove that $ 8k^{2}=1 $.
18) (1) At noon, ship A is 100 km west of ship B. Ship A is sailing east at 35 km/hr and Ship B is sailing north at 25 km/hr. How fast is the distance between the ships changing at 4.00 pm.
(2) Use differentials to find an approximate value for the given number: $ \sqrt36.1 $
19) (1) Find $ \frac{\partial w}{\partial u} $ and $ \frac{\partial w}{\partial v} $, if $ w=sin^{-1}\; xy $ where $ x=u+v, y=u-v $
(2) Evaluate: $ lim\; x\rightarrow 0 \; \frac{cot\;x}{cot\;2x} $
20) Show that $ f(x) = tan^{-1}(sin \;x+ cos\;x), x>0 $ is a strictly increasing function in the interval $ [0,\frac{\pi}{4}] $.
21) Find the local maximum and minimum values of $ sin^{2} \; \Theta, \; [0,\pi] $
22) Find the intervals of concavity and the points of inflection of the following function: $ y=12x^{2}-2x^{3}-x^{4} $.
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