Math Answers

Sin^8g/a^3+cos^8y/b^3=1/(a+b)^3

How many solutions does the equation sin(x) sin(2x) sin(3x)...sin(11x) sin(12x) = 0
have in the interval (0,π ] ?

Let O be the origin of coordinate plane, A, B lie on the upper half plane satisfying
OA=OB . If line OA has slope 1, line OB has slope −7 , what is the slope of line AB?

Calculate the work done by a force F vector = (x − y)i cap + xy j cap in moving a particle counter- clockwise along the circle x2 + y2 = 4 from the point (2, 0) to the point (0, −2

Suppose that a bike rents for $4 plus $ 1.50 per hours. write an equation in slope - intercept form that models this situation

A room is 10m long and 6m breadth it has one door measuring 2mx1m and two windows each measuring 1.5mx1m. Find the cost of painting the walls of the room at the rate of Rs. 15.50 per m. Sq

1. Compute the divergence and curl of each of the following vector fields:
a) hx
2 + y
2
, x2 − y
2
, z2
i
b) hx + y, x − y, zi
2. For any two vector fields F, G show that:
∇.(F × G) = G.(∇ × F) − F.(∇ × G).
3. Given scalar functions u(x, y, z), v(x, y, z), w(x, y, z), φ(x, y, z) and a vector field F(x, y, z) =
hu(x, y, z), v(x, y, z), w(x, y, z)i, prove that
div(φF) = φ(div F) + (∇φ) · F.
4. Find a function f such that F = ∇f where F = hyz, xz, xy + 2zi
5. Let r = hx, y, zi and r = |r|, verify that
a) ∇ · r = 3
b) div (r
n
r) = (n + 3)r
n
6. Given u = xy2
z
2 and v = yz − 3x
2
, find ∇ · [(∇u) × (∇v)]
7. Given the vector field F = he
x
sin y, ex
cos y, zi, show that F is irrotational, hence or
otherwise, find the functions f such that ∇f = F.

Obtain all the first and second order partial derivatives of the function: f (x, y)=x^2 sin y+y^2 cos x

Obtain the directional derivative for a scalar field 2 3 2
φ(x, y,z) = 3x y − y z at the
point (1, −2, −1) in the direction .

Amir is older than Nadeem by 4 years and Danish is younger than Saleem by 4 year. But Saleem's age is 1/4th of Amir. If Danish's Age is 8 years .
(a): How many time is Nadeem in age to saleems age?
(b): What is the age of Amir Nadeem Saleem And Danish?