Science (English Medium) इयत्ता १२ - CBSE Question Bank Solutions for Mathematics

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem  f(x) = x² + x − 1 on [0, 4] ?

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem  f(x) = sin x − sin 2x − x on [0, π] ?

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem f(x) = x³ − 5x² − 3x on [1, 3] ?

Discuss the applicability of Lagrange's mean value theorem for the function
f(x) = | x | on [−1, 1] ?

Show that the lagrange's mean value theorem is not applicable to the function
f(x) = \[\frac{1}{x}\] on [−1, 1] ?

Verify the  hypothesis and conclusion of Lagrange's man value theorem for the function
f(x) = \[\frac{1}{4x - 1},\] 1≤ x ≤ 4 ?

Find a point on the parabola y = (x − 4)², where the tangent is parallel to the chord joining (4, 0) and (5, 1) ?

Find a point on the parabola y = (x − 3)², where the tangent is parallel to the chord joining (3, 0) and (4, 1) ?

Find the points on the curve y = x³ − 3x, where the tangent to the curve is parallel to the chord joining (1, −2) and (2, 2) ?

Find a point on the curve y = x³ + 1 where the tangent is parallel to the chord joining (1, 2) and (3, 28) ?

Let C be a curve defined parametrically as \[x = a \cos^3 \theta, y = a \sin^3 \theta, 0 \leq \theta \leq \frac{\pi}{2}\] . Determine a point P on C, where the tangent to C is parallel to the chord joining the points (a, 0) and (0, a).

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