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

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

The Slope of the tangent to the curve at any point is the reciprocal of four times the ordinate at that point. The curve passes through (2, 5). Find the equation of the curve

Show that y = e^–x + mx + n is a solution of the differential equation `"e"^x(("d"^2y)/("d"x^2)) - 1` = 0

Show that y = `"a"x + "b"/x ≠ 0` is a solution of the differential equation x²yⁿ + xy’ – y = 0

Show that y = ae^–3x + b, where a and b are arbitrary constants, is a solution of the differential equation `("d"^2y)/("d"x^2)  + 3("d"y)/("d"x)` = 0

Show that the differential equation representing the family of curves y² = `2"a"(x + "a"^(2/3))`, where a is a postive parameter, is `(y^2 - 2xy ("d"y)/("d"x))^3 = 8(y ("d"y)/("d"x))^5`

Show that y = a cos bx is a solution of the! differential equation `("d"^2y)/("d"x^2) + "b"^2y` = 0

Choose the correct alternative:

The general solution of the differential equation `("d"y)/("d"x) = y/x` is