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

Find the equations of tangents to the hyperbola `x^2/16 - y^2/64` = 1 which are parallel to10x − 3y + 9 = 0

Show that the line x – y + 4 = 0 is a tangent to the ellipse x² + 3y² = 12. Also find the coordinates of the point of contact

Find the equation of the tangent to the parabola y² = 16x perpendicular to 2x + 2y + 3 = 0

Prove that the point of intersection of the tangents at ‘t₁‘ and t₂’ on the parabola y² = 4ax is [at₁ t₂, a (t₁ + t₂)]

If the normal at the point ‘t₁‘ on the parabola y² = 4ax meets the parabola again at the point ‘t₂‘, then prove that t₂ = `- ("t"_1 + 2/"t"_1)`

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The ellipse E₁ : `x^2/9 + y^2/4` = 1 is inscribed in a rectangle R whose sides are parallel to the co-ordinate axes. Another ellipse E₂ passing through the point (0, 4) circumscribes the rectangle R. The eccentricity of the ellipse is

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Tangents are drawn to, the, hyperbola `x^2/9 - y^2/4` = 1 parallel to the straight line 2x – y – 1. One of the points of contact of tangents on the hyperbola is

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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)