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

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The equation of the circle passing through the foci of the ellipse `x^2/16 +  y^2/9` = 1 having centre at (0, 3) is

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Consider an ellipse whose centre is of the origin and its major axis is a long x-axis. If its eccentricity is `3/5` and the distance between its foci is 6, then the area of the quadrilateral’ inscribed in the ellipse with diagonals as major and minor axis, of the ellipse is

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Area of the greatest rectangle inscribed in the ellipse `x^2/"a"^2 + y^2/"b"^2` = 1 is

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If the two tangents drawn from a point P to the parabola y² = 4r are at right angles then the locus of P is 

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)

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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"`