Class12 ISC Maths Chapter Continuity and Differentiability Exercise5-7

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 Find the domain of differentiability of the following functions : (i) sin-1x (ii) cos-1x (iii) sec-1x (iv) cosec-1x (v) tan-1x (vi) cot-1x

Q2 Find the derivatives of the following functions : (i) tan-1(x√x) (ii) cos-1 3x (iii) √(sin-12x) (iv) sin(tan-1x) (v) cot-1√x (vi) (tan-1 x)2

Q3 Find the derivatives of the following functions : (i) sec-1 (cosec x) (ii) tan-1 (cot x).

Q4 (i) If f(x) = cot-1 (tan x), show that f’(x) = -1. (ii) If f(x) = sin-1 x + sec-1 1/x, find f’(x).

Q5 (i) If y = tan-1 x + cot-1 1/x , x > 0, find dy/dx (ii) If y = sin-1 (cos x) + 3 cosec-1 (sec x), find dy/dx

Differentiate the following (6 to 10) functions w.r.t. x :

Q6 (i) x tan-1 x (ii) x cos-1√x (iii) sin-1x / x

Q7 (i) cos-1 ( 1 - x / 1 + x ) (ii) sec-1 ( 1 / x - 1 ) (iii) sin-1( 1 / √(x + 1) )

Q8 (i) sin-1 x + sin-1√(1 - x2) (ii) x (sin-1 x)2+ 2√(1-x2) sin-1x - 2x

Q9 (i) tan-1 (sin x + cos x) (ii) tan-1 ( sinx / √(1 + cosx) ) (iii) cot-1 ( 1 + cosx / sinx ) (iv) tan-1 ( cosx / √(1 + sinx) )

Q10 (i) cot-1 (cosec x + cot x) (ii) tan-1 (√ 1 - cosx / 1 + cosx ) (iii) cot-1 (√ 1 - sinx / 1 + sinx )

Q11 (i) If y = sin-1x / √1 - x2 , prove that (1 - x2) dy / dx - xy = 1. (ii) If y = sin-1( 2x / 1 + x2 ) , prove that (1 + x2) dy / dx = 2.

Differentiate the following (12 to 18) functions w.r.t. x :

Q12 (i) tan-1 ( 2x / 1-x2 ) (ii) cos-1 ( 1-x2 / 1+x2 ) (iii) cos-1 (1 - 2x2) (iv) sin-1 (2x√ 1 - x2)

Q13 (i) sin-1 (3x - 4x3) (ii) tan-1 ( 3x - x3 / 1-3x2 ) (iii) cos-1 ( 2x / 1 + x2 ) (iv) sec-1 ( 1 / 4x3 - 3x )

Q14 (i) sec-1 ( x2 + 1 / x2-1 ) (ii) sin-1 ( 1 - x2 / 1 + x2 ) (iii) tan-1 √(1 + x2) - x (iv) cos-1 √(1 + x2) + x

Q15 (i) tan-1 ( x / √a2- x2 ) (ii) sin-1 ( 1 / √1+x2 ) (iii) cos-1 (√ 1+x / 2 ) (iv) tan-1 ( x / √x2+ a2 ) (v) tan-1 (√ a - x / a + x )

Q16 (i) sin-1 (√ 1 + x2 / 2 ) (ii)tan-1 ( √(1+x) - √(1-x) / √(1+x) + √(1-x) ) (iii) sin2(cot-1 √ 1 + x / 1 - x ) (iv) sin-1 ( 2 secx / 1 + sec2x )

Q17 (i) cos-1 ( sinx + cosx / √2 ) (ii) cos-1 ( 3 cosx - 4 sinx / 5 ) (iii) sin-1 ( x + √1- x2 / √2 ) (iv) cos-1 ( 3x + 4√1-x2 / 5 )

Q18 (i) sin-1 ( 6x - 4 √1 - x2 / 5 ) (ii) tan-1 ( x1/3 + a1/3 / 1 - x1/3 a1/3 ) (iii) sin(2 sin-1x) (iv) cot-1 ( √1 + sinx + √1 - sinx / √1 + sinx - √1 - sinx )

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Q1 Find the domain of differentiability of the following functions :
(i) sin^-1x
(ii) cos^-1x
(iii) sec^-1x
(iv) cosec^-1x
(v) tan^-1x
(vi) cot^-1x

Q2 Find the derivatives of the following functions :
(i) tan^-1(x√x)
(ii) cos^-1 3x
(iii) √(sin^-12x)
(iv) sin(tan^-1x)
(v) cot^-1√x
(vi) (tan^-1 x)²

Q3 Find the derivatives of the following functions :
(i) sec^-1 (cosec x) (ii) tan^-1 (cot x).

Q4 (i) If f(x) = cot^-1 (tan x), show that f’(x) = -1.

(ii) If f(x) = sin^-1 x + sec^-1 1/x, find f’(x).

Q5 (i) If y = tan^-1 x + cot^-1 1/x , x > 0, find dy/dx

(ii) If y = sin^-1 (cos x) + 3 cosec^-1 (sec x), find dy/dx

Q6 (i) x tan^-1 x (ii) x cos^-1√x (iii)
sin^-1x / x

Q7 (i) cos^-1 (
1 - x / 1 + x
) (ii) sec^-1 (
1 / x - 1
) (iii) sin^-1(
1 / √(x + 1)
)

Q8 (i) sin^-1 x + sin^-1√(1 - x²)

(ii) x (sin^-1 x)²+ 2√(1-x²) sin^-1x - 2x

Q9 (i) tan^-1 (sin x + cos x)

(ii) tan^-1 (
sinx / √(1 + cosx)
)
(iii) cot^-1 (
1 + cosx / sinx
)
(iv) tan^-1 (
cosx / √(1 + sinx)
)

Q10 (i) cot^-1 (cosec x + cot x)
(ii) tan^-1 (√
1 - cosx / 1 + cosx
)
(iii) cot^-1 (√
1 - sinx / 1 + sinx
)

Q11 (i) If y =
sin^-1x / √1 - x²
, prove that (1 - x²)
dy / dx
- xy = 1.
(ii) If y = sin^-1(
2x / 1 + x²
) , prove that (1 + x²)
dy / dx
= 2.

Q12 (i) tan^-1 (
2x / 1-x²
)
(ii) cos^-1 (
1-x² / 1+x²
)
(iii) cos^-1 (1 - 2x²)
(iv) sin^-1 (2x√ 1 - x²)

Q13 (i) sin^-1 (3x - 4x³)
(ii) tan^-1 (
3x - x³ / 1-3x²
)
(iii) cos^-1 (
2x / 1 + x²
)
(iv) sec^-1 (
1 / 4x³ - 3x
)

Q14 (i) sec^-1 (
x² + 1 / x²-1
)
(ii) sin^-1 (
1 - x² / 1 + x²
)
(iii) tan^-1 √(1 + x²) - x
(iv) cos^-1 √(1 + x²) + x

Q15 (i) tan^-1 (
x / √a²- x²
)
(ii) sin^-1 (
1 / √1+x²
)
(iii) cos^-1 (√
1+x / 2
)
(iv) tan^-1 (
x / √x²+ a²
)
(v) tan^-1 (√
a - x / a + x
)

Q16 (i) sin^-1 (√
1 + x² / 2
)
(ii)tan^-1 (
√(1+x) - √(1-x) / √(1+x) + √(1-x)
)
(iii) sin²(cot^-1 √
1 + x / 1 - x
)
(iv) sin^-1 (
2 secx / 1 + sec²x
)

Q17 (i) cos^-1 (
sinx + cosx / √2
)
(ii) cos^-1 (
3 cosx - 4 sinx / 5
)
(iii) sin^-1 (
x + √1- x² / √2
)
(iv) cos^-1 (
3x + 4√1-x² / 5
)

Q18 (i) sin^-1 (
6x - 4 √1 - x² / 5
)
(ii) tan^-1 (
x^1/3 + a^1/3 / 1 - x^1/3 a^1/3
)
(iii) sin(2 sin^-1x)
(iv) cot^-1 (
√1 + sinx + √1 - sinx / √1 + sinx - √1 - sinx
)