HSC Arts Class 12 - Tamil Nadu Board of Secondary Education Question Bank Solutions

For the following functions find the g_xy, g_xx, g_yy and g_yx 

g(x, y) = log(5x + 3y)

For the following functions find the g_xy, g_xx, g_yy and g_yx 

g(x, y) = x² + 3xy – 7y + cos(5x)

Let w(x, y, z) = `1/sqrt(x^2 + y^2 + z^2)` = 1, (x, y, z) ≠ (0, 0, 0), show that `(del^2w)/(delx^2) + (del^2w)/(dely^2) + (del^2w)/(delz^2)` = 0

If V(x, y) = e^x (x cosy – y siny), then Prove that `(del^2"V")/(delx^2) + (del^2"V")/(dely^2)` = 0

If w(x, y) = xy + sin(xy), then Prove that `(del^2w)/(delydelx) = (del^2w)/(delxdely)`

If v(x, y, z) = x³ + y³ + z³ + 3xyz, Show that `(del^2"v")/(delydelz) = (del^2"v")/(delzdely)`

A from produces two types of calculates each week, x number of type A and y number of type B. The weekly revenue and cost functions = (in rupees) are R(x, y) = 80x + 90y + 0.04xy – 0.05x² – 0.05y² and C (x, y) = 8x + 6y + 2000 respectively. Find the profit function P(x, y)

A from produces two types of calculates each week, x number of type A and y number of type B. The weekly revenue and cost functions = (in rupees) are R(x, y) = 80x + 90y + 0.04xy – 0.05x² – 0.05y² and C(x, y) = 8x + 6y + 2000 respectively. Find `(del"P")/(delx)` (1200, 1800) and `(del"P")/(dely)` (1200, 1800) and interpret these results

Let z(x, y) = x²y + 3xy⁴, x, y ∈ R, Find the linear approximation for z at (2, –1)

If v(x, y) = `x^2 - xy + 1/4  y^2 + 7, x, y ∈ "R"`, find the differential dv

Let V (x, y, z) = xy + yz + zx, x, y, z ∈ R. Find the differential dV

Choose the correct alternative:

If g(x, y) = 3x² – 5y + 2y², x(t) = e^t and y(t) = cos t then `"dg"/"dt"` is equal to

Evaluate the following:

`int_0^(pi/2) ("d"x)/(1 + 5cos^2x)`

Evaluate the following:

`int_0^(pi/2) ("d"x)/(5 + 4sin^2x)`

Choose the correct alternative:

The value of  `int_10^pi sin^4x  "d"x`

Choose the correct alternative:

If `int_0^x f("t")  "dt" = x + int_x^1 "t" f("t")  "dt"`, then the value of `f(1)` is

Show the following expressions is a solution of the corresponding given differential equation.

y = 2x² ; xy’ = 2y

Show the following expressions is a solution of the corresponding given differential equation.

y = ae^x + be^–x ; y'' – y = 0

Find the value of m so that the function y = e^mx solution of the given differential equation.

y’ + 2y = 0

Find the value of m so that the function y = e^mx solution of the given differential equation.

y” – 5y’ + 6y = 0