HSC Science कक्षा १२ - Tamil Nadu Board of Secondary Education Question Bank Solutions for Mathematics

Evaluate the following limits, if necessary use l’Hôpital Rule:

`lim_(x -> 0^+) (cos x)^(1/x^2)`

Evaluate the following limits, if necessary use l’Hôpital Rule:

If an initial amount A₀ of money is invested at an interest rate r compounded n times a year, the value of the investment after t years is A = `"A"_0 (1 + "r"/"n")^"nt"`. If the interest is compounded continuously, (that is as n → ∞), show that the amount after t years is A = A₀e^rt 

Choose the correct alternative:

The value of the limit `lim_(x -> 0) (cot x - 1/x)` is

Find the partial dervatives of the following functions at indicated points.

f(x, y) = 3x² – 2xy + y² + 5x + 2, (2, – 5)

Find the partial dervatives of the following functions at indicated points.

g(x, y) = 3x² + y² + 5x + 2, (2, – 5)

Find the partial derivatives of the following functions at indicated points.

 h(x, y, z) = x sin (xy) + z²x, `(2, pi/4, 1)`

Find the partial derivatives of the following functions at the indicated points.

`"G"(x, y) = "e"^(x + 3y)  log(x^2 + y^2), (- 1, 1)`

For the following functions find the f_x, and f_y and show that f_xy = f_yx 

f(x, y) = `(3x)/(y + sinx)`

For the following functions find the f_x, and f_y and show that f_xy = f_yx 

f(x, y) = `tan^-1 (x/y)`

For the following functions find the f_x, and f_y and show that f_xy = f_yx 

f(x, y) = `cos(x^2 - 3xy)`

If U(x, y, z) = `(x^2 + y^2)/(xy) + 3z^2y`, find `(del"U")/(delx), (del"U")/(dely)` and `(del"U")/(del"z)`

If U(x, y, z) = `log(x^3 + y^3 + z^3)`,  find `(del"U")/(delx) + (del"U")/(dely) + (del"U")/(del"z)`

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

g(x, y) = xe^y + 3x²y

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)