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

Prove that f(x, y) = x³ – 2x²y + 3xy² + y³ is homogeneous. What is the degree? Verify Euler’s Theorem for f

Prove that g(x, y) = `x log(y/x)` is homogeneous What is the degree? Verify Eulers Theorem for g

If `"u"(x , y) = (x^2 + y^2)/sqrt(x + y)`, prove that `x (del"v")/(delx) + y (del"u")/(dely) = 3/2 "u"`

If v(x, y) = `log((x^2 + y^2)/(x + y))`, prove that `x (del"v")/(delx) + y (del"u")/(dely) = 1`

If w(x, y, z) = `log((5x^3y^4 + 7y^2xz^4 - 75y^3zz^4)/(x^2 + y^2))`, find `x (del"w")/(delx) + y (del"w")/(dely) + z (del"w")/(delz)`

Choose the correct alternative:

If v(x, y) = log(e^x + e^y), then `(del"v")/(delx) + (del"u")/(dely)` is equal to

Choose the correct alternative:

If w(x, y) = x^y, x > 0, then `(del"w")/(delx)` is equal to

Choose the correct alternative:

If f(x, y) = e^xy, then `(del^2"f")/(delxdely)` is equal to

Choose the correct alternative:

f u(x, y) = x² + 3xy + y – 2019, then `(delu)/(delx) "|"_(((4 , - 5)))` is equal to

Choose the correct alternative:

If w(x, y, z) = x²(y – z) + y²(z – x)+ z²(x – y) then `(del"w")/(delz) + (del"w")/(dely) + (del"w")/(delz)` is 

Choose the correct alternative:

If f(x, y, z) = xy + yz + zx, then f_x – f_z is equal to

Evaluate the following:

`int_0^(pi/2) sin^10 x  "d"x`

Evaluate the following:

`int_0^(pi/2) cos^7 x  "d"x`

Evaluate the following:

`int_0^(pi/4) sin^6 2x  "d"x`

Evaluate the following:

`int_0^(pi/6) sin^5 3x  "d"x`

Evaluate the following:

`int_0^(pi/2) sin^2x cos^4 x  "d"x`

Evaluate the following:

`int_0^(2pi) sin^7  x/4  "d"x`

Evaluate the following:

`int_0^(pi/2) sin^3theta cos^5theta  "d"theta`

Evaluate the following:

`int_1^0 x^2 (1 - x)^3  "d"x`

Choose the correct alternative:

The value of `int_0^(pi/6) cos^3 3x  "d"x` is