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

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The function sin⁴x + cos⁴x is increasing in the interval

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The minimum value of the function `|3 - x| + 9` is

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The maximum value of the function x² e^-2x, x > 0 is

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The maximum value of the product of two positive numbers, when their sum of the squares is 200, is

If w(x, y) = x³ – 3xy + 2y², x, y ∈ R, find the linear approximation for w at (1, –1)

If u(x, y) = x²y + 3xy⁴, x = e^t and y = sin t, find `"du"/"dt"` and evaluate if at t = 0

Let u(x, y, z) = xy²z³ x = sin t, y = cos t, z = 1 + e^2t, Find `"du"/"dt"`

If w(x, y, z) = x² + y² + z², x = e^t, y = e^t sin t and z = e^t cos t, find `("d"w)/"dt"`

Let U(x, y, z) = xyz, x = e^–t, y = e^–t cos t, z – sin t, t ∈ R, find `"dU"/"dt"`

Let w(x, y) = 6x³ – 3xy + 2y², x = e^s, y = cos s, s ∈ R. Find `("d"w)/"ds"` and evaluate at s = 0

Let z(x, y) = x tan^–1(xy), x = t², y = s e^t, s, t ∈ R. Find `(delz)/(del"s")` and `(delz)/(del"t")` at s = t = 1

Let U(x, y) = e^x sin y where x = st², y = s²t, s, t ∈ R. Find `(del"U")/(del"s"), (del"u")/(del"t")` and evaluate them at s = t = 1

Let z(x, y) = x³ – 3x²y³ where x = se^t, y = se^–t, s, t ∈ R. Find `(delz)/(del"s")` and `(delz)/(delt)`

W(x, y, z) = xy + yz + zx, x = u – v, y = uv, z = u + v, u, v ∈ R. Find `(del"W")/(del"u"), (del"W")/(del"v")` and evaluate them at `(1/2, 1)`

In the following, determine whether the following function is homogeneous or not. If it is so, find the degree.

f(x, y) = x²y + 6x³ + 7

In the following, determine whether the following function is homogeneous or not. If it is so, find the degree.

h(x, y) = `(6x^3y^2 - piy^5 + 9x^4y)/(2020x^2 + 2019y^2)` 

In the following, determine whether the following function is homogeneous or not. If it is so, find the degree.

g(x, y, z) = `sqrt(3x^2+ 5y^2+z^2)/(4x + 7y)`

In the following, determine whether the following function is homogeneous or not. If it is so, find the degree.

U(x, y, z) = `xy + sin((y^2 - 2z^2)/(xy))`

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