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

Find the value of `sin^-1 (sin((5pi)/4))`

For what value of x does sin x = sin^–1x?

Find the value of `sin^-1(sin  (5pi)/9 cos  pi/9 + cos  (5pi)/9 sin  pi/9)`

Choose the correct alternative:

If the function `f(x) = sin^-1 (x^2 - 3)`, then x belongs to

Explain why Rolle’s theorem is not applicable to the following functions in the respective intervals

`f(x) = |1/x|, x ∈ [- 1, 1]`

Explain why Rolle’s theorem is not applicable to the following functions in the respective intervals

`f(x)` = tan x, x ∈ [0, π]

Explain why Rolle’s theorem is not applicable to the following functions in the respective intervals

`f(x)` = x – 2 log x, x ∈ [2, 7]

Using the Rolle’s theorem, determine the values of x at which the tangent is parallel to the x-axis for the following functions:

`f(x)` = x² – x, x ∈ [0, 1]

Using the Rolle’s theorem, determine the values of x at which the tangent is parallel to the x-axis for the following functions:

`f(x) = (x^2 - 2x)/(x + 2), x ∈ [-1, 6]`

Using the Rolle’s theorem, determine the values of x at which the tangent is parallel to the x-axis for the following functions:

`f(x) = sqrt(x) - x/3, x ∈ [0, 9]`

Explain why Lagrange’s mean value theorem is not applicable to the following functions in the respective intervals:

`f(x) = (x + 1)/x, x ∈ [-1, 2]`

Explain why Lagrange’s mean value theorem is not applicable to the following functions in the respective intervals:

`f(x) = |3x + 1|, x ∈ [-1, 3]`

Using the Lagrange’s mean value theorem determine the values of x at which the tangent is parallel to the secant line at the end points of the given interval:

`f(x) = x^3 - 3x + 2, x ∈ [-2, 2]`

Using the Lagrange’s mean value theorem determine the values of x at which the tangent is parallel to the secant line at the end points of the given interval:

`f(x) = (x - 2)(x - 7), x ∈ [3, 11]`

Show that the value in the conclusion of the mean value theorem for `f(x) = 1/x` on a closed interval of positive numbers [a, b] is `sqrt("ab")`

Show that the value in the conclusion of the mean value theorem for `f(x) = "A"x^2 + "B"x + "C"` on any interval [a, b] is `("a" + "b")/2`

A race car driver is kilometer stone 20. If his speed never exceeds 150 km/hr, what is the maximum kilometer he can reach in the next two hours

Suppose that for a function f(x), f'(x) ≤ 1 for all 1 ≤ x ≤ 4. Show that f(4) – f(1) ≤ 3

Does there exist a differentiable function f(x) such that f(0) = – 1, f(2) = 4 and f(x) ≤ 2 for all x. Justify your answer

Show that there lies a point on the curve `f(x) = x(x + 3)e^(pi/2), -3 ≤ x ≤ 0` where tangent drawn is parallel to the x-axis