PUC Science 2nd PUC Class 12 - Karnataka Board PUC Question Bank Solutions for Mathematics

Using Lagrange's mean value theorem, prove that (b − a) sec² a < tan b − tan a < (b − a) sec² b
where 0 < a < b < \[\frac{\pi}{2}\] ?

State Rolle's theorem ?

State Lagrange's mean value theorem ?

If the value of c prescribed in Rolle's theorem for the function f (x) = 2x (x − 3)ⁿ on the interval \[[0, 2\sqrt{3}] \text { is } \frac{3}{4},\] write the value of n (a positive integer) ?

Find the value of c prescribed by Lagrange's mean value theorem for the function \[f\left( x \right) = \sqrt{x^2 - 4}\] defined on [2, 3] ?

If the polynomial equation \[a_0 x^n + a_{n - 1} x^{n - 1} + a_{n - 2} x^{n - 2} + . . . + a_2 x^2 + a_1 x + a_0 = 0\] n positive integer, has two different real roots α and β, then between α and β, the equation \[n \ a_n x^{n - 1} + \left( n - 1 \right) a_{n - 1} x^{n - 2} + . . . + a_1 = 0 \text { has }\].

If 4a + 2b + c = 0, then the equation 3ax² + 2bx + c = 0 has at least one real root lying in the interval

For the function f (x) = x + \[\frac{1}{x}\] ∈ [1, 3], the value of c for the Lagrange's mean value theorem is 

If from Lagrange's mean value theorem, we have \[f' \left( x_1 \right) = \frac{f' \left( b \right) - f \left( a \right)}{b - a}, \text { then }\]

Rolle's theorem is applicable in case of ϕ (x) = a^sin x, a > a in

The value of c in Rolle's theorem when
f (x) = 2x³ − 5x² − 4x + 3, x ∈ [1/3, 3] is

When the tangent to the curve y = x log x is parallel to the chord joining the points (1, 0) and (e, e), the value of x is ______.

The value of c in Rolle's theorem for the function \[f\left( x \right) = \frac{x\left( x + 1 \right)}{e^x}\] defined on [−1, 0] is

The value of c in Lagrange's mean value theorem for the function f (x) = x (x − 2) when x ∈ [1, 2] is

The value of c in Rolle's theorem for the function f (x) = x³ − 3x in the interval [0,\[\sqrt{3}\]] is 

If f (x) = e^x sin x in [0, π], then c in Rolle's theorem is

Find the points on the curve x² + y² − 2x − 3 = 0 at which the tangents are parallel to the x-axis ?

Find the shortest distance between the lines

Find the shortest distance between the lines