NCERT Solutions Class 12 Mathematics Chapter 5 Continuity And Differentiability Download In Pdf

Chapter 5 Continuity and Differentiability Download in pdf

**Question 1.** Prove that the function f (x) = 5x â€“ 3 is continuous at x = 0, at x = â€“ 3 and at x = 5.

**QuestioQuestion ** Examine the continuity of the function f (x) = 2x2 â€“ 1 at x = 3.

**Question 3.** Examine the following functions for continuity.

(a) f (x) = x â€“ 5

(b) f (x) =
1
x âˆ’ 5

(c) f (x) =
2 25
5
x
x
âˆ’
+

(d) f (x) = | x â€“ 5 |

**Question 4. **Prove that the function f (x) = xn is continuous at x = n, where n is a positive
integer.

**Question 5.** Is the function f defined by
, if 1
( )
5, if > 1
x x
f x
x
â‰¤
=
continuous at x = 0? At x = 1? At x = 2?
Find all points of discontinuity of f, where f is defined by

**
Question 18.** For what value of Î» is the function defined by
( 2 2 ), if 0 ( )
4 1, if 0
f x x x x
x x
âŽ§âŽªÎ» âˆ’ â‰¤ = + >
continuous at x = 0? What about continuity at x = 1?

**Question 19.** Show that the function defined by g (x) = x â€“ [x] is discontinuous at all integral
points. Here [x] denotes the greatest integer less than or equal to x.

**Question 20. **Is the function defined by f (x) = x2 â€“ sin x + 5 continuous at x = Ï€?

**Question 21. **Discuss the continuity of the following functions:

(a) f (x) = sin x + cos x

(b) f (x) = sin x â€“ cos x

(c) f (x) = sin x . cos x

**Question 22.** Discuss the continuity of the cosine, cosecant, secant and cotangent functions.

**Question 23. **Find all points of discontinuity of f, where
sin , if 0
( )
1, if 0
x x
f x x
x x < = + â‰¥

**Question 24. **Determine if f defined by
2 sin 1 , if 0
( )
0, if 0
x x
f x x
x
â‰ = =
is a continuous function?

is a continuous function.

**
Question 31. **Show that the function defined by f (x) = cos (x2) is a continuous function.

**Question 32.** Show that the function defined by f (x) = | cos x | is a continuous function.

**Question 33. **Examine that sin | x | is a continuous function.

**Question 34.** Find all the points of discontinuity of f defined by f (x) = | x | â€“ | x + 1 |.

EXERCISE 5.2

**Differentiate the functions with respect to x in Exercises 1 to 8.**

**Question 1.** sin (x2 + 5)

**Question 2. **cos (sin x)

**Question 3.** sin (ax + b)

**Question 4. **sec (tan ( x ))

**Question 5. **sin ( )
cos ( )
ax b
cx d
+
+

**Question 6. **cos x3 . sin2 (x5)

**Question 7. **2 cot ( x2 )

**Question 8.** cos( x )

**Question 9. **Prove that the function f given by
f (x) = | x â€“ 1 |, x âˆˆ R
is not differentiable at x = 1.

**Question 10. **Prove that the greatest integer function defined by
f (x) = [x], 0 < x < 3
is not differentiable at x = 1 and x = 2.

EXERCISE 5.4

**Differentiate the following w.r.t. x:**

**Question 1.** sin
ex
x

**Question 2.** esin 1 x âˆ’

**Question 3.** ex3

**Question 4. **sin (tanâ€“1 eâ€“x)

**Question 5. **log (cos ex)

**Question 6. **2 5 ex + ex +... + ex

**Question 7.** e x , x > 0

**Question 8.** log (log x), x > 1

**Question 9.** cos , 0
log
x x
x
>

**Question 10.** cos (log x + ex), x > 0

EXERCISE 5.5

**Differentiate the functions given in Exercises 1 to 11 w.r.t. x.**

**Question 1.** cos x . cos 2x . cos 3x

**Question 2.** ( 1)( 2)
( 3)( 4)( 5)
x x
x x x
âˆ’ âˆ’
âˆ’ âˆ’ âˆ’

**Question 3. **(log x)cos x

**Question 4.** xx â€“ 2sin x

**Question 5.** (x + 3)2 . (x + 4)3 . (x + 5)4

**Question 6.** 1 1 1

**Question 7.** (log x)x + xlog x

**Question 8. **(sin x)x + sinâ€“1 x

**Question 9. **xsin x + (sin x)cos x

**Question 10. **2
cos
2
1
1
xx x x
x
+
+
âˆ’

**Question 11.** (x cos x)x +
1
(xsin x)x
Find
dy
dx of the functions given in Exercises 12 to 15.

**Question 12.** xy + yx = 1

**Question 13.** yx = xy

**Question 14. **(cos x)y = (cos y)x

**Question 15. **xy = e(x â€“ y)

**Question 16.** Find the derivative of the function given by f (x) = (1 + x) (1 + x2) (1 + x4) (1 + x8)
and hence find f â€²(1).

**Question 17. **Differentiate (x2 â€“ 5x + 8) (x3 + 7x + 9) in three ways mentioned below:

(i) by using product rule

(ii) by expanding the product to obtain a single polynomial.

(iii) by logarithmic differentiation.
Do they all give the same answer?

**Question 18. **If u, v and w are functions of x, then show that
d
dx (u. v. w) =
du
dx v. w + u .
dv
dx . w + u . v
dw
dx
in two ways - first by repeated application of product rule, second by logarithmic
differentiation.

EXERCISE 5.6

If x and y are connected parametrically by the equations given in Exercises 1 to 10,
without eliminating the parameter, Find dy
dx
.

**
Question 1.** x = 2at2, y = at4

**Question 2. **x = a cos Î¸, y = b cos Î¸

**Question 3.** x = sin t, y = cos 2t 4. x = 4t, y =
4
t

**Question 5.** x = cos Î¸ â€“ cos 2Î¸, y = sin Î¸ â€“ sin 2Î¸
6. x = a (Î¸ â€“ sin Î¸), y = a (1 + cos Î¸)

**Question 7. **x =
sin3
cos 2
t
t
,
cos3
cos2
y t
t
=

**Question 8.** cos log tan
2
x = a = a sin t

**Question 9.** x = a sec Î¸, y = b tan Î¸

**Question 10.** x = a (cos Î¸ + Î¸ sin Î¸), y = a (sin Î¸ â€“ Î¸ cos Î¸)

**Question 11.** If x asin 1 t , y acos 1 t , show that dy y
dx x

EXERCISE 5.7

** Find the second order derivatives of the functions given in Exercises 1 to 10.**

**
Question1.** x2 + 3x +

**Question ** ** 2.** x20

**Question 3.** x . cos x

**Question 4.** log x

**Question 5.** x3 log x

**Question 6.** ex sin 5x

**Question 7.** e6x cos 3x

**Question 8.** tanâ€“1 x

**Question 9.** log (log x)

**Question 10.** sin (log x)

**Question 11.** If y = 5 cos x â€“ 3 sin x, prove that
2
2 d y y 0
dx
+ =

**Question 12.** If y = cosâ€“1 x, Find
2
2
d y
dx
in terms of y alone.

**Question 13.** If y = 3 cos (log x) + 4 sin (log x), show that x2 y2 + xy1 + y = 0

**Question 14.** If y = Aemx + Benx, show that
2
2 d y (m n) dy mny 0
dx dx
âˆ’ + + =

**Question 15.** If y = 500e7x + 600eâ€“ 7x, show that
2
2 d y 49y
dx
=

**Question 16.** If ey (x + 1) = 1, show that
2 2
2
d y dy
dx dx
=

**Question 17.** If y = (tanâ€“1 x)2, show that (x2 + 1)2 y2 + 2x (x2 + 1) y1 = 2

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- Chapter 1 Relations and Functions
- Chapter 2 Inverse Trigonometric Functions
- Chapter 3 Matrices
- Chapter 4 Determinants
- Chapter 5 Continuity and Differentiability
- Chapter 6 Application of Derivatives
- Chapter 7 Integral
- Chapter 8 Application of Integrals
- Chapter 9 Differential Equations
- Chapter 10 Vector Algebra
- Chapter 11 Three Dimensional Geometry
- Chapter 12 Linear Programming
- Chapter 13 Probability

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