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NCERT solutions for Mathematics Exemplar Class 12 chapter 5 - Continuity And Differentiability [Latest edition]

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Mathematics Exemplar Class 12 - Shaalaa.com
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Chapter 5: Continuity And Differentiability

Solved ExamplesExercise
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Solved Examples [Pages 91 - 107]

NCERT solutions for Mathematics Exemplar Class 12 Chapter 5 Continuity And Differentiability Solved Examples [Pages 91 - 107]

Short Answer

Solved Examples | Q 1 | Page 91

Find the value of the constant k so that the function f defined below is continuous at x = 0, where f(x) = `{{:((1 - cos4x)/(8x^2)",", x ≠ 0),("k"",", x = 0):}`

Solved Examples | Q 2 | Page 91

Discuss the continuity of the function f(x) = sin x . cos x.

Solved Examples | Q 3 | Page 92

If f(x) = `{{:((x^3 + x^2 - 16x + 20)/(x - 2)^2",", x ≠ 2),("k"",", x = 2):}` is continuous at x = 2, find the value of k.

Solved Examples | Q 4 | Page 92

Show that the function f defined by f(x) = `{{:(x sin  1/x",", x ≠ 0),(0",", x = 0):}` is continuous at x = 0.

Solved Examples | Q 5 | Page 92

Given f(x) = `1/(x - 1)`. Find the points of discontinuity of the composite function y = f[f(x)]

Solved Examples | Q 6 | Page 93

Let f(x) = x|x|, for all x ∈ R. Discuss the derivability of f(x) at x = 0

Solved Examples | Q 7 | Page 93

Differentiate `sqrt(tansqrt(x))` w.r.t. x

Solved Examples | Q 8 | Page 93

If y = tan(x + y), find `("d"y)/("d"x)`

Solved Examples | Q 9 | Page 94

If ex + ey = ex+y , prove that `("d"y)/("d"x) = -"e"^(y - x)`

Solved Examples | Q 10 | Page 94

Find `("d"y)/("d"x)`, if y = `tan^-1 ((3x - x^3)/(1 - 3x^2)), -1/sqrt(3) < x < 1/sqrt(3)`

Solved Examples | Q 11 | Page 95

If y = `sin^-1 {xsqrt(1 - x) - sqrt(x) sqrt(1 - x^2)}` and 0 < x < 1, then find `("d"y)/(dx)`

Solved Examples | Q 12 | Page 95

If x = a sec3θ and y = a tan3θ, find `("d"y)/("d"x)` at θ = `pi/3`

Solved Examples | Q 13 | Page 96

If xy = ex–y, prove that `("d"y)/("d"x) = logx/(1 + logx)^2`

Solved Examples | Q 14 | Page 96

If y = tanx + secx, prove that `("d"^2y)/("d"x^2) = cosx/(1 - sinx)^2`

Solved Examples | Q 15 | Page 96

If f(x) = |cos x|, find f'`((3pi)/4)`

Solved Examples | Q 16 | Page 97

If f(x) = |cos x – sinx|, find `"f'"(pi/6)`

Solved Examples | Q 17 | Page 97

Verify Rolle’s theorem for the function, f(x) = sin 2x in `[0, pi/2]`.

Solved Examples | Q 18 | Page 98

Verify mean value theorem for the function f(x) = (x – 3)(x – 6)(x – 9) in [3, 5].

Long Answer

Solved Examples | Q 19 | Page 98

If f(x) = `(sqrt(2) cos x - 1)/(cot x - 1), x ≠ pi/4` find the value of `"f"(pi/4)`  so that f (x) becomes continuous at x = `pi/4`

Solved Examples | Q 20 | Page 99

Show that the function f given by f(x) = `{{:(("e"^(1/x) - 1)/("e"^(1/x) + 1)",", "if"  x ≠ 0),(0",",  "if"  x = 0):}` is discontinuous at x = 0.

Solved Examples | Q 21 | Page 100

Let f(x) = `{{:((1 - cos 4x)/x^2",",  "if"  x < 0),("a"",",  "if"  x = 0),(sqrt(x)/(sqrt(16) + sqrt(x) - 4)",", "if"  x > 0):}`. For what value of a, f is continuous at x = 0?

Solved Examples | Q 22 | Page 101

Examine the differentiability of the function f defined by
f(x) = `{{:(2x + 3",",  "if"  -3 ≤ x < - 2),(x + 1",",  "if"  -2 ≤ x < 0),(x + 2",",  "if"  0 ≤ x ≤ 1):}`

Solved Examples | Q 23 | Page 102

Differentiate `tan^-1 (sqrt(1 - x^2)/x)` with respect to`cos^-1(2xsqrt(1 - x^2))`, where `x ∈ (1/sqrt(2), 1)`

Objective Type Questions from 24 to 25

Solved Examples | Q 24 | Page 103

The function f(x) = `{{:(sinx/x + cosx",",  "if" x ≠ 0),("k"",",  "if" x = 0):}` is continuous at x = 0, then the value of k is ______.

  • 3

  • 2

  • 1

  • 1.5

Solved Examples | Q 25 | Page 103

The function f(x) = [x], where [x] denotes the greatest integer function, is continuous at ______.

  • 4

  • – 2

  • 1

  • 1.5

Solved Examples | Q 26 | Page 103

The number of points at which the function f(x) = `1/(x - [x])` is not continuous is ______.

  • 1

  • 2

  • 3

  • None of these

Solved Examples | Q 27 | Page 104

The function given by f (x) = tanx is discontinuous on the set ______.

  • `{"n"pi: "n" ∈ "Z"}`

  • `{2"n"pi: "n" ∈ "Z"}`

  • `{(2"n" + 1) pi/2 : "n" ∈ "Z"}`

  • `{("n"pi)/2 : "n" ∈ "Z"}`

Solved Examples | Q 28 | Page 104

Let f(x)= |cosx|. Then, ______.

  • f is everywhere differentiable

  • f is everywhere continuous but not differentiable at n = nπ, n ∈ Z

  • f is everywhere continuous but not differentiable at x = `(2"n" + 1) pi/2, "n" ∈ "Z"`

  • None of these

Solved Examples | Q 29 | Page 104

The function f(x) = |x| + |x – 1| is ______.

  • Continuous at x = 0 as well as at x = 1

  • Continuous at x = 1 but not at x = 0

  • Discontinuous at x = 0 as well as at x = 1

  • Continuous at x = 0 but not at x = 1

Solved Examples | Q 30 | Page 104

The value of k which makes the function defined by f(x) = `{{:(sin  1/x",",  "if"  x ≠ 0),("k"",",  "if"  x = 0):}`, continuous at x = 0 is ______.

  • 8

  • 1

  • –1

  • None of these

Solved Examples | Q 31 | Page 104

The set of points where the functions f given by f(x) = |x – 3| cosx is differentiable is ______.

  • R

  • R – {3}

  • `(0, oo)`

  • None of these

Solved Examples | Q 32 | Page 105

Differential coefficient of sec (tan–1x) w.r.t. x is ______.

  • `x/sqrt(1 + x^2)`

  • `x/(1 + x^2)`

  • `xsqrt(1 + x^2)`

  • `1/sqrt(1 + x^2)`

Solved Examples | Q 33 | Page 105

If u = `sin^-1 ((2x)/(1 + x^2))` and v = `tan^-1 ((2x)/(1 - x^2))`, then `"du"/"dv"` is ______.

  • `1/2`

  • x

  • `(1 - x^2)/(1 + x^2)`

  • 1

Solved Examples | Q 34 | Page 105

The value of c in Rolle’s Theorem for the function f(x) = e x sinx, x ∈ π [0, π] is ______.

  • `pi/6`

  • `pi/4`

  • `pi/2`

  • `(3pi)/4`

Solved Examples | Q 35 | Page 105

The value of c in Mean value theorem for the function f(x) = x(x – 2), x ∈ [1, 2] is ______.

  • `3/2`

  • `2/3`

  • `1/2`

  • `3/2`

Match the column

Solved Examples | Q 36 | Page 105
COLUMN-I COLUMN-II
(A) If a function
f(x) = `{((sin3x)/x, "if"  x = 0),("k"/2",",  "if"  x = 0):}`
is continuous at x = 0, then k is equal to
(a) |x|
(B) Every continuous function is differentiable (b) True
(C) An example of a function which is continuous
everywhere but not differentiable at exactly one point
(c) 6
(D) The identity function i.e. f (x) = x ∀ ∈x R
is a continuous function
(d) False

Fill in the blanks 37 to 41

Solved Examples | Q 37 | Page 106

The number of points at which the function f(x) = `1/(log|x|)` is discontinuous is ______.

Solved Examples | Q 38 | Page 106

If f(x) = `{{:("a"x + 1,  "if"  x ≥ 1),(x + 2,  "if"  x < 1):}` is continuous, then a should be equal to ______.

Solved Examples | Q 39 | Page 106

The derivative of log10x w.r.t. x is ______.

Solved Examples | Q 40 | Page 106

If y = `sec^-1 ((sqrt(x) + 1)/(sqrt(x + 1))) + sin^-1((sqrt(x) - 1)/(sqrt(x) + 1))`, then `"dy"/"dx"` is equal to ______.

Solved Examples | Q 41 | Page 106

The derivative of sin x w.r.t. cos x is ______.

State whether the following is True or False: 42 to 46

Solved Examples | Q 42 | Page 106

For continuity, at x = a, each of `lim_(x -> "a"^+) "f"(x)` and `lim_(x -> "a"^-) "f"(x)` is equal to f(a).

  • True

  • False

Solved Examples | Q 43 | Page 106

y = |x – 1| is a continuous function.

  • True

  • False

Solved Examples | Q 44 | Page 106

A continuous function can have some points where limit does not exist.

  • True

  • False

Solved Examples | Q 45 | Page 106

|sinx| is a differentiable function for every value of x.

  • True

  • False

Solved Examples | Q 46 | Page 107

cos |x| is differentiable everywhere.

  • True

  • False

Exercise [Pages 107 - 116]

NCERT solutions for Mathematics Exemplar Class 12 Chapter 5 Continuity And Differentiability Exercise [Pages 107 - 116]

Short Answer

Exercise | Q 1 | Page 107

Examine the continuity of the function f(x) = x3 + 2x2 – 1 at x = 1

Find which of the functions in 2 to 10 is continuous or discontinuous at the indicated points:

Exercise | Q 2 | Page 107

f(x) = `{{:(3x + 5",", "if"  x ≥ 2),(x^2",", "if"  x < 2):}` at x = 2

Exercise | Q 3 | Page 107

f(x) = `{{:((1 - cos 2x)/x^2",", "if"  x ≠ 0),(5",", "if"  x = 0):}` at x = 0

Exercise | Q 4 | Page 107

f(x) = `{{:((2x^2 - 3x - 2)/(x - 2)",", "if"  x ≠ 2),(5",", "if"  x = 2):}` at x = 2

Exercise | Q 5 | Page 107

f(x) = `{{:(|x - 4|/(2(x - 4))",", "if"  x ≠ 4),(0",", "if"  x = 4):}` at x = 4

Exercise | Q 6 | Page 107

f(x) = `{{:(|x|cos  1/x",", "if"  x ≠ 0),(0",", "if"  x = 0):}` at x = 0

Exercise | Q 7 | Page 107

f(x) = `{{:(|x - "a"| sin  1/(x - "a")",",  "if"  x ≠ 0),(0",",  "if"  x = "a"):}` at x = a

Exercise | Q 8 | Page 107

f(x) = `{{:(("e"^(1/x))/(1 + "e"^(1/x))",", "if"  x ≠ 0),(0",", "if"  x = 0):}` at x = 0 

Exercise | Q 9 | Page 107

f(x) = `{{:(x^2/2",",  "if"  0 ≤ x ≤ 1),(2x^2 - 3x + 3/2",",  "if"  1 < x ≤ 2):}` at x = 1

Exercise | Q 10 | Page 107

f(x) = |x| + |x − 1| at x = 1

Find the value of k in the 11 to 14 so that the function f is continuous at the indicated point:

Exercise | Q 11 | Page 108

f(x) = `{{:(3x - 8",",  "if"  x ≤ 5),(2"k"",",  "if"  x > 5):}` at x = 5

Exercise | Q 12 | Page 108

f(x) = `{{:((2^(x + 2) - 16)/(4^x - 16)",",  "if"  x ≠ 2),("k"",",  "if"  x = 2):}` at x = 2

Exercise | Q 13 | Page 108

f(x) = `{{:((sqrt(1 + "k"x) - sqrt(1 - "k"x))/x",",  "if" -1 ≤ x < 0),((2x + 1)/(x - 1)",",  "if"  0 ≤ x ≤ 1):}` at x = 0

Exercise | Q 14 | Page 108

f(x) = `{{:((1 - cos "k"x)/(xsinx)",",   "if"  x ≠ 0),(1/2",",  "if"  x = 0):}` at x = 0

Exercise | Q 15 | Page 108

Prove that the function f defined by 
f(x) = `{{:(x/(|x| + 2x^2)",",  x ≠ 0),("k",  x = 0):}`
remains discontinuous at x = 0, regardless the choice of k.

Exercise | Q 16 | Page 108

Find the values of a and b such that the function f defined by
f(x) = `{{:((x - 4)/(|x - 4|) + "a"",",  "if"  x < 4),("a" + "b"",",  "if"  x = 4),((x - 4)/(|x - 4|) + "b"",", "if"  x > 4):}`
is a continuous function at x = 4.

Exercise | Q 17 | Page 108

Given the function f(x) = `1/(x + 2)`. Find the points of discontinuity of the composite function y = f(f(x))

Exercise | Q 18 | Page 109

Find all points of discontinuity of the function f(t) = `1/("t"^2 + "t" - 2)`, where t = `1/(x - 1)`

Exercise | Q 19 | Page 109

Show that the function f(x) = |sin x + cos x| is continuous at x = π.

Exercise | Q 20 | Page 109

Examine the differentiability of f, where f is defined by
f(x) = `{{:(x[x]",",  "if"  0 ≤ x < 2),((x - 1)x",",  "if"  2 ≤ x < 3):}` at x = 2

Exercise | Q 21 | Page 109

Examine the differentiability of f, where f is defined by
f(x) = `{{:(x^2 sin  1/x",",  "if"  x ≠ 0),(0",", "if"  x = 0):}` at x = 0

Exercise | Q 22 | Page 109

Examine the differentiability of f, where f is defined by
f(x) = `{{:(1 + x",",  "if"  x ≤ 2),(5 - x",",  "if"  x > 2):}` at x = 2

Exercise | Q 23 | Page 109

Show that f(x) = |x – 5| is continuous but not differentiable at x = 5.

Exercise | Q 24 | Page 109

A function f: R → R satisfies the equation f( x + y) = f(x) f(y) for all x, y ∈ R, f(x) ≠ 0. Suppose that the function is differentiable at x = 0 and f′(0) = 2. Prove that f′(x) = 2f(x).

Differentiate the following w.r.t. x 25 to 43:

Exercise | Q 25 | Page 109

`2^(cos^(2_x)`

Exercise | Q 26 | Page 109

`8^x/x^8`

Exercise | Q 27 | Page 109

`log (x + sqrt(x^2 + "a"))`

Exercise | Q 28 | Page 109

`log [log(logx^5)]`

Exercise | Q 29 | Page 109

`sin sqrt(x) + cos^2 sqrt(x)`

Exercise | Q 30 | Page 109

sinn (ax2 + bx + c)

Exercise | Q 31 | Page 109

`cos(tan sqrt(x + 1))`

Exercise | Q 32 | Page 109

sinx2 + sin2x + sin2(x2)

Exercise | Q 33 | Page 109

`sin^-1  1/sqrt(x + 1)`

Exercise | Q 34 | Page 109

(sin x)cosx 

Exercise | Q 35 | Page 109

sinmx . cosnx

Exercise | Q 36 | Page 109

(x + 1)2(x + 2)3(x + 3)4

Exercise | Q 37 | Page 110

`cos^-1 ((sinx + cosx)/sqrt(2)), (-pi)/4 < x < pi/4`

Exercise | Q 38 | Page 110

`tan^-1 (sqrt((1 - cosx)/(1 + cosx))), - pi/4 < x < pi/4`

Exercise | Q 39 | Page 110

`tan^-1 (secx + tanx), - pi/2 < x < pi/2`

Exercise | Q 40 | Page 110

`tan^-1 (("a"cosx - "b"sinx)/("b"cosx - "a"sinx)), - pi/2 < x < pi/2` and `"a"/"b" tan x > -1`

Exercise | Q 41 | Page 110

`sec^-1 (1/(4x^3 - 3x)), 0 < x < 1/sqrt(2)`

Exercise | Q 42 | Page 110

`tan^-1 ((3"a"^2x - x^3)/("a"^3 - 3"a"x^2)), (-1)/sqrt(3) < x/"a" < 1/sqrt(3)`

Exercise | Q 43 | Page 110

`tan^-1 ((sqrt(1 + x^2) + sqrt(1 - x^2))/(sqrt(1 + x^2) - sqrt(1 - x^2))), -1 < x < 1, x ≠ 0`

Find dy/dx of the functions expressed in parametric form in 44 to 48.

Exercise | Q 44 | Page 110

x = `"t" + 1/"t"`, y = `"t" - 1/"t"`

Exercise | Q 45 | Page 110

x = `"e"^theta (theta + 1/theta)`, y= `"e"^-theta (theta - 1/theta)`

Exercise | Q 46 | Page 110

x = 3cosθ – 2cos3θ, y = 3sinθ – 2sin3θ

Exercise | Q 47 | Page 110

sin x = `(2"t")/(1 + "t"^2)`, tan y = `(2"t")/(1 - "t"^2)`

Exercise | Q 48 | Page 110

x = `(1 + log "t")/"t"^2`, y = `(3 + 2 log "t")/"t"`

Exercise | Q 49 | Page 110

If x = ecos2t and y = esin2t, prove that `"dy"/"dx" = (-y log x)/(xlogy)`

Exercise | Q 50 | Page 110

If x = asin2t (1 + cos2t) and y = b cos2t (1–cos2t), show that `("dy"/"dx")_("at  t" = pi/4) = "b"/"a"`

Exercise | Q 51 | Page 110

If x = 3sint – sin 3t, y = 3cost – cos 3t, find `"dy"/"dx"` at t = `pi/3`

Exercise | Q 52 | Page 111

Differentiate `x/sinx` w.r.t. sin x

Exercise | Q 53 | Page 111

Differentiate `tan^-1 ((sqrt(1 + x^2) - 1)/x)` w.r.t. tan–1x, when x ≠ 0

Find dy/dx when x and y are connected by the relation given in 54 to 57

Exercise | Q 54 | Page 111

`sin xy + x/y` = x2 – y

Exercise | Q 55 | Page 111

sec(x + y) = xy

Exercise | Q 56 | Page 111

tan–1(x2 + y2) = a

Exercise | Q 57 | Page 111

(x2 + y2)2 = xy

Exercise | Q 58 | Page 111

If ax2 + 2hxy + by2 + 2gx + 2fy + c = 0, then show that `"dy"/"dx" * "dx"/"dy"` = 1 

Exercise | Q 59 | Page 111

If x = `"e"^(x/y)`, prove that `"dy"/"dx" = (x - y)/(xlogx)`

Exercise | Q 60 | Page 111

If yx = ey – x, prove that `"dy"/"dx" = (1 + log y)^2/logy`

Exercise | Q 61 | Page 111

If y = `(cos x)^((cos x)^((cosx)....oo)`, show that `"dy"/"dx" = (y^2 tanx)/(y log cos x - 1)`

Exercise | Q 62 | Page 111

If x sin (a + y) + sin a cos (a + y) = 0, prove that `"dy"/"dx" = (sin^2("a" + y))/sin"a"`

Exercise | Q 63 | Page 111

If `sqrt(1 - x^2) + sqrt(1 - y^2) = "a"(x - y)`, proe that `"dy"/"dx" = sqrt((1 - y^2)/(1 - x^2)`

Exercise | Q 64 | Page 111

If y = tan–1x, find `("d"^2y)/("dx"^2)` in terms of y alone.

Verify the Rolle’s theorem for the functions in 65 to 69.

Exercise | Q 65 | Page 112

f(x) = x(x – 1)2 in [0, 1]

Exercise | Q 66 | Page 112

f(x) = `sin^4x + cos^4x` in `[0, pi/2]`

Exercise | Q 67 | Page 112

f(x) = log(x2 + 2) – log3 in [–1, 1]

Exercise | Q 68 | Page 112

f(x) = `x(x + 3)e^((–x)/2)` in [–3, 0]

Exercise | Q 69 | Page 112

f(x) = `sqrt(4 - x^2)` in [– 2, 2]

Exercise | Q 70 | Page 112

Discuss the applicability of Rolle’s theorem on the function given by f(x) = `{{:(x^2 + 1",",  "if"  0 ≤ x ≤ 1),(3 - x",",  "if"  1 ≤ x ≤ 2):}`

Exercise | Q 71 | Page 112

Find the points on the curve y = (cosx – 1) in [0, 2π], where the tangent is parallel to x-axis

Exercise | Q 72 | Page 112

Using Rolle’s theorem, find the point on the curve y = x(x – 4), x ∈ [0, 4], where the tangent is parallel to x-axis

Verify mean value theorem for the functions given 73 to 76

Exercise | Q 73 | Page 112

f(x) = `1/(4x - 1)` in [1, 4]

Exercise | Q 74 | Page 112

f(x) = x3 – 2x2 – x + 3 in [0, 1]

Exercise | Q 75 | Page 112

f(x) = sinx – sin2x in [0, π]

Exercise | Q 76 | Page 112

f(x) = `sqrt(25 - x^2)` in [1, 5]

Exercise | Q 77 | Page 112

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

Exercise | Q 78 | Page 112

Using mean value theorem, prove that there is a point on the curve y = 2x2 – 5x + 3 between the points A(1, 0) and B(2, 1), where tangent is parallel to the chord AB. Also, find that point

Long Answer

Exercise | Q 79 | Page 112

Find the values of p and q so that f(x) = `{{:(x^2 + 3x + "p"",",  "if"  x ≤ 1),("q"x + 2",",  "if"  x > 1):}` is differentiable at x = 1

Exercise | Q 80. (i) | Page 113

If xm . yn = (x + y)m+n, prove that `"dy"/"dx" = y/x`

Exercise | Q 80. (ii) | Page 113

If xm . yn = (x + y)m+n, prove that `("d"^2"y")/("dx"^2)` = 0

Exercise | Q 81 | Page 113

If x = sint and y = sin pt, prove that `(1 - x^2) ("d"^2"y")/("dx"^2) - x "dy"/"dx" + "p"^2y` = 0

Exercise | Q 82 | Page 113

Find `"dy"/"dx"`, if y = `x^tanx + sqrt((x^2 + 1)/2)`

Objective Type Questions from 83 to 96

Exercise | Q 83 | Page 113

If f(x) = 2x and g(x) = `x^2/2 + 1`, then which of the following can be a discontinuous function ______.

  • f(x) + g(x)

  • f(x) – g(x)

  • f(x) . g(x)

  • `("g"(x))/("f"(x))`

Exercise | Q 84 | Page 113

The function f(x) = `(4 - x^2)/(4x - x^3)` is ______.

  • Discontinuous at only one point

  • Discontinuous at exactly two points

  • Discontinuous at exactly three points

  • None of these

Exercise | Q 85 | Page 113

The set of points where the function f given by f(x) = |2x − 1| sinx is differentiable is ______.

  • R

  • `"R" - {1/2}`

  • `(0, oo)`

  • None of these

Exercise | Q 86 | Page 114

The function f(x) = cot x is discontinuous on the set ______.

  • {x = nπ : n ∈ Z}

  • {x = 2nπ : n ∈ Z}

  • `{x = (2"n" + 1)pi/2 ; "n" ∈ "Z"}`

  • `{x = ("n"pi)/2 ; "n" ∈ "Z"}`

Exercise | Q 87 | Page 114

The function f(x) = `"e"^|x|` is ______.

  • Continuous everywhere but not differentiable at x = 0

  • Continuous and differentiable everywhere

  • Not continuous at x = 0

  • None of these

Exercise | Q 88 | Page 114

If f(x) = `x^2 sin  1/x` where x ≠ 0, then the value of the function f at x = 0, so that the function is continuous at x = 0, is ______.

  • 0

  • – 1

  • 1

  • None of these

Exercise | Q 89 | Page 114

If f(x) = `{{:("m"x + 1",",  "if"  x ≤ pi/2),(sin x + "n"",",  "If"  x > pi/2):}`, is continuous at x = `pi/2`, then ______.

  • m = 1, n = 0

  • m = `("n"pi)/2 + 1`

  • n = `("m"pi)/2`

  • m = n = `pi/2`

Exercise | Q 90 | Page 114

Let f(x) = |sin x|. Then ______.

  • f is everywhere differentiable

  • f is everywhere continuous but not differentiable at x = nπ, n ∈ Z

  • f is everywhere continuous but not differentiable at x = `(2"n" + 1)  pi/2`, n ∈ Z

  • None of these

Exercise | Q 91 | Page 114

If y = `log ((1 - x^2)/(1 + x^2))`, then `"dy"/"dx"` is equal to ______.

  • `(4x^3)/(1 - x^4)`

  • `(-4x)/(1 - x^4)`

  • `1/(4 - x^4)`

  • `(-4x^3)/(1 - x^4)`

Exercise | Q 92 | Page 115

If y = `sqrt(sinx + y)`, then `"dy"/"dx"` is equal to ______.

  • `cos/(2y - 1)`

  • `cosx/(1 - 2y)`

  • `sinx/(1 - 2y)`

  • `sinx/(2y - 1)`

Exercise | Q 93 | Page 115

The derivative of cos–1(2x2 – 1) w.r.t. cos–1x is ______.

  • 2

  • `(-1)/(2sqrt(1 - x^2)`

  • `2/x`

  • 1 – x2 

Exercise | Q 94 | Page 115

If x = t2, y = t3, then `("d"^2"y")/("dx"^2)` is ______.

  • `3/2`

  • `3/(4"t")`

  • `3/(2"t")`

  • `3/4`

Exercise | Q 95 | Page 115

The value of c in Rolle’s theorem for the function f(x) = x3 – 3x in the interval `[0, sqrt(3)]` is ______.

  • 1

  • – 1

  • `3/2`

  • `1/3`

Exercise | Q 96 | Page 116

For the function f(x) = `x + 1/x`, x ∈ [1, 3], the value of c for mean value theorem is ______.

  • 1

  • `sqrt(3)`

  • 2

  • None of these

Fill in the blanks 97 to 101:

Exercise | Q 97 | Page 116

An example of a function which is continuous everywhere but fails to be differentiable exactly at two points is ______.

Exercise | Q 98 | Page 116

Derivative of x2 w.r.t. x3 is ______.

Exercise | Q 99 | Page 116

If f(x) = |cosx|, then `"f'"(pi/4)` = ______.

Exercise | Q 100 | Page 116

If f(x) = |cosx – sinx| , then `"f'"(pi/4)` = ______.

Exercise | Q 101 | Page 116

For the curve `sqrt(x) + sqrt(y)` = 1, `"dy"/"dx"` at `(1/4, 1/4)` is ______.

State whether the following is True or False: 102 to 106

Exercise | Q 102 | Page 116

Rolle’s theorem is applicable for the function f(x) = |x – 1| in [0, 2].

  • True

  • False

Exercise | Q 103 | Page 116

If f is continuous on its domain D, then |f| is also continuous on D.

  • True

  • False

Exercise | Q 104 | Page 116

The composition of two continuous function is a continuous function.

  • True

  • False

Exercise | Q 105 | Page 116

Trigonometric and inverse-trigonometric functions are differentiable in their respective domain.

  • True

  • False

Exercise | Q 106 | Page 116

If f.g is continuous at x = a, then f and g are separately continuous at x = a.

  • True

  • False

Chapter 5: Continuity And Differentiability

Solved ExamplesExercise
Mathematics Exemplar Class 12 - Shaalaa.com

NCERT solutions for Mathematics Exemplar Class 12 chapter 5 - Continuity And Differentiability

NCERT solutions for Mathematics Exemplar Class 12 chapter 5 (Continuity And Differentiability) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics Exemplar Class 12 solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. NCERT textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Mathematics Exemplar Class 12 chapter 5 Continuity And Differentiability are Higher Order Derivative, Algebra of Continuous Functions, Derivative - Exponential and Log, Concept of Differentiability, Proof Derivative X^n Sin Cos Tan, Infinite Series, Continuous Function of Point, Mean Value Theorem, Second Order Derivative, Derivatives of Functions in Parametric Forms, Logarithmic Differentiation, Exponential and Logarithmic Functions, Derivatives of Implicit Functions, Derivatives of Inverse Trigonometric Functions, Derivatives of Composite Functions - Chain Rule, Concept of Continuity.

Using NCERT Class 12 solutions Continuity And Differentiability exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in NCERT Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 12 prefer NCERT Textbook Solutions to score more in exam.

Get the free view of chapter 5 Continuity And Differentiability Class 12 extra questions for Mathematics Exemplar Class 12 and can use Shaalaa.com to keep it handy for your exam preparation

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