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

If p and q are the roots of the equation lx² + nx + n = 0, show that `sqrt("p"/"q") + sqrt("q"/"p") + sqrt("n"/l)` = 0

If the equations x² + px + q = 0 and x² + p’x + q’ = 0 have a common root, show that it must be equal to `("pq'" - "p'q")/("q" - "q")` or `("q" - "q'")/("p'" - "P")`

A 12 metre tall tree was broken into two parts. It was found that the height of the part which was left standing was the cube root of the length of the part that was cut away. Formulate this into a mathematical problem to find the height of the part which was left standing

Find the value of `sin^-1(sin((2pi)/3))`

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`