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

Find the local extrema for the following functions using second derivative test:

f(x) = x log x

Find the local extrema for the following functions using second derivative test:

f(x) = x² e^–2x 

For the function f(x) = 4x³ + 3x² – 6x + 1 find the intervals of monotonicity, local extrema, intervals of concavity and points of inflection

Choose the correct alternative:

The curve y = ax⁴ + bx² with ab > 0

Choose the correct alternative:

The point of inflection of the curve y = (x – 1)³ is

Evaluate the following:

`int_0^oo x^5 "e"^(-3x)  "d"x`

Evaluate the following:

`int_0^(pi/2) ("e"^(-tanx))/(cos^6x)  "d"x`

Evaluate the following:

If `int_0^oo "e"^(-"a"x^2) x^3  "d"x` = 32, `alpha > 0`, find `alpha`

Choose the correct alternative:

The value of `int_0^1 x(1 - x)^99  "d"x` is

Choose the correct alternative:

If `("I'"("n" + 2))/("I'n")` = 90 then n is

Choose the correct alternative:

The value of `int_0^oo "e"^(-3x) x^2  "d"x` is

Solve the following Linear differential equation:

`cos x  ("d"y)/("d"x) + y sin x ` = 1

Solve the following Linear differential equation:

`(1 - x^2) ("d"y)/("d"x) - xy` = 1

Solve the following Linear differential equation:

`("d"y)/("d"x) + y/x = sin x`

Solve the following Linear differential equation:

`(x^2 + 1) ("d"y)/("d"x) + 2xy = sqrt(x^2 + 4)`

Solve the following Linear differential equation:

(2x – 10y³)dy + y dx = 0

Solve the following Linear differential equation:

`x sin x ("d"y)/("d"x) + (x cos x + sin x)y = sinx`

Solve the following Linear differential equation:

`(y - "e"^(sin^-1)x) ("d"x)/("d"y) + sqrt(1 - x^2)` = 0

Solve the following Linear differential equation:

`("d"y)/("d"x) + y/((1 - x)sqrt(x)) = 1 - sqrt(x)`

Solve the following Linear differential equation:

`(1 + x + xy^2) ("d"y)/("d"x) + (y + y^3)` = 0