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

Find the absolute extrema of the following functions on the given closed interval.

f(x) = x² – 12x + 10; [1, 2]

Find the absolute extrema of the following functions on the given closed interval.

f(x) = 3x⁴ – 4x³ ; [– 1, 2]

Find the absolute extrema of the following functions on the given closed interval.

f(x) = `6x^(4/3) - 3x^(1/3) ; [-1, 1]`

Find the absolute extrema of the following functions on the given closed interval.

f(x) = `2 cos x + sin 2x; [0, pi/2]`

Find the intervals of monotonicities and hence find the local extremum for the following functions:

f(x) = 2x³ + 3x² – 12x

Find the intervals of monotonicities and hence find the local extremum for the following functions:

f(x) = `x/(x - 5)`

Find the intervals of monotonicities and hence find the local extremum for the following functions:

f(x) = `"e"^x/(1 - "e"^x)`

Find the intervals of monotonicities and hence find the local extremum for the following functions:

f(x) = `x^3/3 - log x`

Find the intervals of monotonicities and hence find the local extremum for the following functions:

f(x) = sin x cos x + 5, x ∈ (0, 2π)

Choose the correct alternative:

The function sin⁴x + cos⁴x is increasing in the interval

Choose the correct alternative:

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

Choose the correct alternative:

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

Choose the correct alternative:

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