MTH282 --------------------------------------------------------------------------------------------------- Question:

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Question: If

r = (t^{3} + 2 t) i - 3 e^{- 2 t} j + 2 \sin 5 t k,

find

\frac{d r}{d t}

Answer:

(3 t^{2} + 2) i + 6 e^{- 2 t} j + 10 \cos 5 t k

Question: Find the gradient of the scalar field

ϕ = x y^{2} z^{3}

Answer:

y^{2} z^{3} i + 2 x y z^{3} j + 3 x y^{2} z^{2} k

Question: If function

R (x, y, z) = x z^{3} - 2 x^{2} y z j + 2 y z^{4} k

. Find the curl

R

Answer:

(2 z^{4} + 2 x^{2} y) i + 3 x z^{2} j - 4 x y z k

Question: Find the polar form of Complex number

2 + 2 i \sqrt{3}

Answer:

4 (\cos 60^{o} + i \sin 60^{o}

) Question: Given that

z_{1} = r_{1} (\cos θ_{1} + i \sin θ_{1})

and

z_{2} = r_{2} (\cos θ_{2} + i \sin θ_{2})

, evaluate

z_{1} z_{2}

Answer:

r_{1} r_{2} \{\cos (θ_{1} + θ_{2}) - i \sin (θ_{1} + θ_{2})\}

Question: Find the fifth roots of

- 1

Answer:

\cos \frac{9 π}{5} + i \sin \frac{9 π}{5}

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Answer: Question: Answer: Question: Answer: Question: Answer: Question: Answer: Question: Answer: Question: Answer: 0 Question: Answer: Question: Answer: 195 Question: Answer: -10 Question: Given A=1+3j-2k and B=4i-2j+4k, find |3A+2B| Answer: Question: Answer: -14 Question: Answer: 2, 1 Question: Answer: -i+7j+5k Question: Answer: 24i+7j-5k Question: Answer: 10i+3j+11k Question: Given that $A = (t^{3} + 2 t) i - 3 e^{- 2 t} j + 2 \sin 5 t k$ , the ${\frac{d^{2} A}{d t^{2}}|}_{t = 0}$ is _______ Answer: 12 Answer: twelve Question: Given function $ϕ (x, y, z) = x^{2} y + y z$ at the point (1, 2,-1),then its rate of change is __________ Answer: 4i+2k Question: The modulus of the complex number $z = \frac{(1 + i) (2 + i)}{(3 - i)}$ is _______ Answer: 1 Answer: one Question: The real number $x$ and $y$ such that $3 x + 2 i y - i x + 5 y = 7 + 5 i$ is x= ____ and y=_______ Answer: -1, 2 Question: the angle between $A = 2 i + 2 j - k$ and $B = 6 i - 3 j + 2 k$ is ____ degress Answer: 79 Question: The projection of the vector $A = i - 2 j + k$ on the vector $B = 4 i - 4 j + 7 k$ is ___ Answer: 19/9 Answer: 2.11 Question: The work done in moving an object along a vector $r = 3 i + 2 j - 5 k$ , if the applied force is $F = 2 i - j - k$ is _________ Answer: 9 Question: If $A = 2 i - 3 j - k$ and $A = i + 4 j - 2 k$ , then $(A + B) \times (A - B)$ is _____________ Answer: -20i-6j-22k Question: The solution of $(2 i - 3 j) ∙ [(i + j - k) \times (3 i - k)]$ is ____________ Answer: 4 Question: Given $A = i \sin t - j \sin t + k$ , the value of $|\frac{d^{2} A}{d t^{2}}|$ is _______________ Answer: 1 Question: The solution of $\frac{\partial^{2}}{\partial x \partial y} (ϕ A)$ at the point (2,-1,1) of the functions $ϕ (x, y, z) = x y^{2} z$ and $A = x z i - x y^{2} j + x z^{2} k$ is ________ Answer: 4i-2j Question: The magnitude of the velocity of a particle which moves along the curve $x = 2 \sin 3 t, y = 2 \cos 3 t$ and $z = 8 t$ at any time $t > 0$ is ____________ Answer: 18 Question: If $ϕ = 2 x z^{4} - x^{2},$ the solution of $\nabla ϕ$ at point (2, -2, -1) is ______________ Answer: 10i-4j-16k Question: The solution of $A ∙ \nabla ϕ$ at point (1, - 1, 1)of the functions $A = x^{2} i - 3 y z j + x z^{2}$ and ${ϕ = 2 z - x}^{3} y$ is ____________ Answer: 5 Question: The solution of $A \times \nabla ϕ$ at point (1, - 1, 1)of the functions $A = x^{2} i - 3 y z j + x z^{2}$ and ${ϕ = 2 z - x}^{3} y$ is ____________ Answer: 7i-j-11k Question: The work done in moving an object along a straight line from (3, 2, -1) to (2, -1, 4) in a force field by $F = 4 i - 3 j + 2 k$ Answer: 15 Question: If $A = x^{2} y z i - 2 x z^{3} j + x z^{2} k$ and $B = 2 z i + y j - x^{3} k$ ,then $\frac{\partial^{2}}{\partial x \partial y} (A \times B)$ at point (1, 0, -2) is __________ Answer: -4i-8j Question: The solution of the complex number (3+2i)(2-i) is ___________ Answer: 8+i Question: The solution of the complex number (3+2i)(2-i) is ___________ Answer: 3-i Question: Answer: 6 Question: Answer: -2 Question: Answer: 7 Question: Answer: -6 Question: Answer: -14 Question: Answer: 4x Question: Answer: 2.24 Question: Answer: -74 Question: Answer: 6 Question: Answer: 4-5i Question: Answer: 48, 72 Question: Answer: -1 -2i Question: Answer: -6 Question: Answer: -4 Question: Answer: 2, 3/2 Answer: 2, 1.5 Question: Answer: 2i Question: Answer: 16 Question: Answer: 47.83 Question: Answer: 0 Answer: 0 Question:

Two vectors are said to be perpendicular if their scalar product is ______ Answer: 0 Question:

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Answer: 1 Question: The density is an example _______ quantity Answer: scalar Question: at the points (0, 0, 0) is __________ Answer: 5 Question: at the points (1, -2, 2) is __________ Answer: 19 Question: The projection of the vector 2i-3j+6k on the vector i+2j+k is _______ Answer: 8/3 Answer: 2. 67 Question: The projection of the vector 4i-3j+k on the line passing through the points (2,3, -1) and (-2, -4, 3) is _______ Answer: 1 Question: The work done in moving an object along a straight line from (3, 2, -1) to (2, -1, 4) in a force field given by F=4i-3j+2k is _______ Answer: 15 Question: The work done in moving an object along a vector r=3i+2j-5k, if the applied force is F=2i-j-k is ______ Answer: 9 Question: is ____________ Answer: 4 Question: The projection of the vector 4i-3j+k on the line passing through the points (2,3, -1) and (-2, -4, 3) is _______ Answer: 1