Class12 ISC Maths Chapter Continuity and Differentiability Exercise5-14

Class XII ISC Chapter Continuity and Differentiability

Explore our extensive collection of Continuity and Differentiability questions, tailored for class 12 students following the ISC curriculum. This topic is vital for understanding how to optimize a particular outcome given certain constraints. Our practice questions cover various aspects of Linear Programming, including formulating linear inequalities, graphical methods, and finding optimal solutions. Whether you're looking to strengthen your problem-solving skills or prepare for your exams, these questions provide the perfect resource to master Continuity and Differentiability concepts with ease.

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Continuity & Differentiability

Class XII ISC Chapter Continuity and Differentiability

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Q1 (i) f(x) = x2 - 5x + 6 in [1, 4] (ii) f(x) = x2 + 2 on [-2, 2] (iii) f(x) = x2 + 2x - 8, x € [-4, 2]

Q2(i) f(x) = x3 + 3x2 - 24x - 80 in [-4, 5] (ii) f (x)= x(x - 1)2 in [0, 1] (iii) f(x) = (x-1)(x-2)(x-3) in [1 , 3] (iv) f(x) = √4-x2 in [-2, 2]

Q3(i) f(x) = cos 2x in [-π/4 , π/4] (ii) f(x) = sinx - 1 in [π/2 , 5π/2]

Q4(i) f(x) = sin 2x in [0 , π/2] (ii) f(x) = sin 3x in [0, π]

Q5(i) f(x) = ex sin x on [0 , π] (ii) f(x) = ex cos x on [-π/2 , π/2] (iii) f (x) = e2x (sin 2x - cos 2x) in [π/8 , 5π/8]

Q6 Verify the conditions of Rolle’s Theorem for the following function: f (x) = log (x2 + 2) - log 3 on [-1, 1]. Find a point in the given interval where the tangent to the curve is parallel to x-axis.

Q7 Apply Rolle’s theorem to find point (or points) on the following curves where the tangent is parallel to x-axis:

(i) y= x2 in [-2, 2]

(ii) y = -1 + cos x on [0, 2π]

Q8 If Rolle’s theorem holds for the function f (x) = x3 + ax2 + bx in [1, 2] at the point x = 4/3 then find the values of a and b.

Q9 Examine if Rolle’s theorem is applicable to the function f(x) = [x] for x € [-2, 2]. What can you say about the converse of Rolle’s theorem?

Q10 What can you say about the applicability of Rolle’s theorem for the following functions in the indicated intervals?

(i) f(x) = x1/3 in [-1, 1] (ii) f(x) = x2/3 in [-2, 2]

(iii) f(x) = 2 + (x - 1)2/3 in [0, 2]

(iv) f(x) = 1 +|x - 2| in [0, 4]

(v) f(x) = tan x in [0, π]

(vi) f(x) = sec x in [0, 2π]

(vii) f(x) = x(x-2) / x-1 in [0,2]

(viii) f(x) = x2 + 1 in [-1, 2].

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Q1 (i) f(x) = x² - 5x + 6 in [1, 4]
(ii) f(x) = x² + 2 on [-2, 2]
(iii) f(x) = x² + 2x - 8, x € [-4, 2]

Q2(i) f(x) = x³ + 3x² - 24x - 80 in [-4, 5]
(ii) f (x)= x(x - 1)² in [0, 1]
(iii) f(x) = (x-1)(x-2)(x-3) in [1 , 3]
(iv) f(x) = √4-x² in [-2, 2]

Q3(i) f(x) = cos 2x in [-π/4 , π/4]
(ii) f(x) = sinx - 1 in [π/2 , 5π/2]

Q4(i) f(x) = sin 2x in [0 , π/2]
(ii) f(x) = sin 3x in [0, π]

Q5(i) f(x) = e^x sin x on [0 , π]
(ii) f(x) = e^x cos x on [-π/2 , π/2]
(iii) f (x) = e^2x (sin 2x - cos 2x) in [π/8 , 5π/8]

Q6 Verify the conditions of Rolle’s Theorem for the following function:
f (x) = log (x² + 2) - log 3 on [-1, 1].
Find a point in the given interval where the tangent to the curve is parallel to x-axis.

(i) y= x² in [-2, 2]

Q8 If Rolle’s theorem holds for the function f (x) = x³ + ax² + bx in [1, 2] at the point x = 4/3 then find the values of a and b.

(i) f(x) = x^1/3 in [-1, 1]
(ii) f(x) = x^2/3 in [-2, 2]

(iii) f(x) = 2 + (x - 1)^2/3 in [0, 2]

(vii) f(x) =
x(x-2) / x-1
in [0,2]

(viii) f(x) = x² + 1 in [-1, 2].