Plus Two Maths Chapter Wise Questions and Answers Chapter 5 Continuity and Differentiability are part of Plus Two Maths Chapter Wise Questions and Answers. Here we have given Plus Two Maths Chapter Wise Questions and Answers Chapter 5 Continuity and Differentiability.

Board | SCERT, Kerala |

Text Book | NCERT Based |

Class | Plus Two |

Subject | Maths Chapter Wise Questions |

Chapter | Chapter 5 |

Chapter Name | Continuity and Differentiability |

Number of Questions Solved | 56 |

Category | Plus Two Kerala |

## Kerala Plus Two Maths Chapter Wise Questions and Answers Chapter 5 Continuity and Differentiability

**Short Answer Type Questions**

**(Score 3)**

Question 1.

Consider the function

i. Show that f(x) is discontinuous at x = 0.

ii. Redefine the function in such a way that it becomes continuous at x = 0.

Answer.

Question 2.

Consider the function

i. Evaluate Right hand derivative and Left hand derivative of f(x) at x = 2.

ii. Check the differentiability of f(x) at x = 2.

Answer.

Question 3.

If

i. Put x = tanθ and prove that y = 2tan^{-1}x

ii. Find

Answer.

Question 4.

sin y = x sin(a + y)

i. Express x as a function of y.

ii. Show that

Answer.

i. sin y = x sin(a + y)

Question 5.

Consider f:[3, ∞] → [1, ∞] given by f(x) = x² – 6x + 10.Find f^{-1}.

Answer.

f(x) = (x-3)² + 1

y = (x-3 )² + 1

Question 6.

Find the relationship between a and b so that the function f defined by

is continuous at x = 3

Answer.

f(3) = 3a + 1

since f is continuous at 3

3a+ 1 = 3b+3

⇒3 (a – b) = 2

Question 7.

Find if

where a is a constant.

Answer

differentiating both sides w.r.t.x we get

Question 8.

Find the derivative of log(tan e^{x}) w.r.t.x.

Answer.

Let y = log(tane^{x})

Differentiating w.r.t.x,

Question 9.

Find of 2x + 3y = sinx

Answer.

Given, 2x+3y = sin x

Differentiating both sides w.r.t.x, we get

Question 10.

Consider the parametric function

x = acos^{3}θ and y = bsin^{3}θ.

i. Find .

ii. Show that

Answer.

i. x = acos^{3}θ

**Long Answer Type Question**

**(Score 4)**

Question 1.

Differentiate the following.

Answer.

Question 2.

If

i. Prove that log y = x^{x} (logx)

ii. Find

Answer.

i. Let = x^{u}, where u = x^{x} then

logy = ulogx = x^{x} logx

ii. Differentiating,

Question 3.

Consider

Answer.

Question 4.

sin(xy)+ = x² – y.

i. Differentiate throughout w.r.t x.

ii. Evaluate dy/dx.

Answer.

Question 5.

Given x^{y }= y^{x}

i. Take logarithm on both sides.

ii. Find dy/dx.

Answer.

i. If x^{y} = y^{x}; ylogx = x logy

Question 6.

Answer.

Question 7.

If x = acos^{3}θ, y = asin^{3}θ

Answer.

Question 8.

If x^{3} + y^{3} = 3axy

i. Differentiate through out with respect to x.

ii. Show that

Answer.

x^{3} + y^{3} = 3axy

Differentiating

Question 9.

Answer.

Question 10.

y = tanx + secx

Answer.

i. y = tanx + secx

**Very Long Answer Type Questions**

**(Score 6)**

Question 1.

Answer.

Question 2.

u =(sin x)^{tanx} ,v = (cos x )^{secx}

i. Find and

ii. Find dy/dx if

y = (sin x )^{tanx} + (cos x)^{secx}

Answer.

i. Let u = (sin x)^{tanx} Then

log u = tan x log sin x.

Differentiating both sides with respect to x, we get

Question 3.

Given that y = a sinnx + b cosnx

Answer.

i. y = a sinnx + b cosnx

= acos(nx)n + b. – sin(nx).n

= an cos(nx) – bn sin(nx)

Question 4.

i. Differentiate

with respect to x

ii. Differentiate

with respect to x.

iii. Differentiate

with respect to x.

Answer.

Question 5.

Let

i. Evaluate

ii. Find the values of a and b , so that f(x) is continuous.

Answer.

Question 6.

Consider

i. Differentiate x.sin^{-1}x with respect to x.

ii. Find using quotient rule.

Answer.

Question 7.

i. Verify mean value theorem for the function f(x) = (x – 2)² in [1, 4].

ii. Find a point on the curve y = (x – 2)² at which the tangent is parallel to the chord joining the points (1, 1) and (4, 4).

iii. Find a point on the above curve at which the tangent is parallel to the x-axis.

Answer.

1. f(x) = (x – 2)², x ∈ [1, 4]

f(x) is continuous in [1, 4]

f(x) is differentiable in (2, 4)

There exists c ∈ (1, 4) so that

Question 8.

If y sin(m sin^{-1}x),

i. Find

ii. Show that

iii. Hence prove that

Answer.

i. y = sin (m sin^{-1}x)

Differentiating w.r.t.x

Question 9.

Verify Mean Value Theorem, if f(x) = x^{3} – 5x^{2} – 3x in the interval [a, b] where a = 1 and b = 3. Find all c ∈ (1,3) for which f ‘(c) = 0. 6

Answer.

f(x) = x^{3} – 5x^{2} – 3x

f ‘(x) = 3x^{2} – 10x – 3

Since f ‘(x) exists, f(x) is continuous on [1, 3]

f(x) is differentiable on (1, 3)

f ‘(c) = 3c^{2} – 10c – 3

f(b) = f(3) = -27

f(a) = f(1) = -7

Hence Mean Value Theorem is verified for f(x) on (1,3).

Question 10.

Using mathematical induction, prove that

for all positive integers n.

Answer.

Hence P(k + 1) is true.

i.e., P(k + 1) is true whenever P(k) is true. Hence by the principle of mathematical induction,

is true for all positive integer n.

**Edumate Questions & Answers**

Question 1.

i. Choose the correct answer from the bracket. If x = sint and y = cost, then is equal to …..

{ a. tant b. cot t

c. -tant d. -cot t}

ii. Find the value of k such that the function f(x) defined by

is continuous function at

iii. If yx = e^{y-x}, prove that

Answer.

i. c – tant

Question 2.

i. Choose the correct answer from the bracket If y = logcosx, then the value of at x = is

{a.∞ b.1 c. 0 d.-1}

ii. Find the value of a such that the function f(x) defined by

is continuous at x = 2.

iii. If x = acos^{3}θ and y = asin^{3}θ. Prove that

Answer

i. d. -1

ii. 2a + 3 = 2a^{2 }– 1

a = 2, a = -1

Question 3.

i. Choose the correct answer from the bracket.

If x = at², y = 2at, then the value of at t = 1 is

{a.0 b.1 c.2 d.∞}

ii. Is the function defined by

a continuous function ? Justify.

iii. Find

Answer.

i. a. 1

ii. Left limit ≠ Right limit

It is not continuous

Question 4.

i. Choose the correct answer from the bracket.

If x² + 2xy + 2y² = 1 then at the point where y = 1 is equal to

{a.0 b.1 c. 2 d.-1}

ii. Find the relationship between a and b so that the functions f defined by

a continuous at x = 2

iii. If x^{y}y^{x} = 16, then find at (2,2)

Answer.

i. a,0

ii. 2a+1 = 2b-3

a – b = -2

iii. ylogx + xlogy = log 16

Question 5.

Find the values of a and b such that the function f defined by

is a continuous function.

ii. Verify Langrange’s Mean Value theorem for the function f(x) = 2x² – 10x + 29 in [2, 9].

Answer.

i. a+b = 2

5a+b = 14

a = 3, b = -1

f(2) = 17, f(9) = 101

ii. f ‘(c) = 4c – 10 = 12

c = ∈[2,9]

Question 6.

i. If

then find f ‘(1).

ii. Verify Rolles theorem for the function f(x) = sin3x on [0,]

Answer.

i. f(x) = loge^{x} + [log(3 - x) - log(3 + x)]

Question 7.

Answer.

Question 8.

i. Find if sinx + cosy = xy

ii. Find if y = (sinx)^{logx}

Answer.

Question 9.

Differentiate the following

Answer.

Question 10.

Let siny = xsin (a + y)

i. Express x as a function of y.

ii. Show that

Answer.

i. siny = xsin (a + y)

Question 11.

i. Let f: R → R given by f(x) = [x], where [x] denotes greatest integer less than or equal to x. Is f(x) continuous at x = 0?

ii. Let f: R → R given by f(x) = |x|. Is f(x) differentiable at x = 0?

iii. Let f: R → R given by f(x) = |x| . Is f(x) differentiable at x ≠ 0 ?

Answer.

i. f(x) is not continuous at x = 0, since it breaks at integral values.

f '(0+) ≠ f '(0-) so not differentiable at x = 0

At x > 0, f(x) = x, a straight line graph.

At x < 0, f(x) = -x, a straight line graph.

So differentiable at x ≠ 0.

**NCERT Questions & Answers**

Question 1.

Differentiate the following function w.r.t x.

(logx)^{x} + x^{logx}

Answer.

Let u = (logx)^{x} ,v = x^{logx}, and y = u+v

Differentiating w.r.t. x,

Question 2.

If y^{x} = e^{y-x},

Prove that

Answer.

We have, y^{x} = e^{y-x},

Taking log on both sides,

x log y = (y-x) log e

Question 3.

If

Find .

Answer.

y = (sin x - cos x)^{(sinx-cosx)}

log y = (sinx-cosx) log(sinx-cosx)

Question 4.

Consider the function

i. Find Rf '(0) and Lf '(0)

ii. Hence prove that f (x) is differentiable at x = 0.

Answer.

ii. Since Rf '(0) = Lf '(0) f is differentiable at x = 0

Question 5.

Given that y = 3cos(logx) + 4sin(logx)

Answer.

i. y = 3cos(logx) + 4sin(logx)

Question 6.

Given that log y = tan^{-1} x

i. Find y_{1}.

ii. Show that (1+x²)y_{1} = y

iii. Show that (1+x²) y_{2} + (2x-1) y_{1} = 0.

Answer.

i. logy = tan^{-1 }x

Question 7.

y = e^{ax} sin bx:

i. Find y_{1}

ii. Prove that y_{2} - 2a y_{1}+ (a^{2}+ b^{2}) y = 0

Answer.

i. y_{1} = e^{ax} sinbx

y_{1} = e^{ax} cosbx . b + sinbx . e^{ax} . a

y_{1} = e^{ax} (b cosbx + a sinbx)

ii. Differentiating,

y_{2} = e^{ax} (-b^{2}sinbx + ab cosbx) + (b cosbx + a sinbx)ae^{ax}

y_{2} = e^{ax} (-b^{2}sìnbx+2ab cosbx+a^{2} sinbx)

Now LHS = y_{2} - 2ay_{1} + (a^{2} + b^{2})y

e^{ax}(-b^{2}sinbx+2ab cosbx+a^{2}sinbx) 2a^{ax}

(bcosbx+asinbx)+(a^{2} + b^{2})e^{ax}sinbx = 0

Question 8.

If y = sin(logx) prove that

Answer.

Question 9.

Consider the function f (x)=(x - a)^{m} (x - b)^{n} where m and n are integers greater than 1.

a. Does f satisfy conditions of Rolle’s theorem?

b. If so verify Rolle’s theorem.

Answer.

a. f(x) = (x - a)^{m} (x - b)^{n}

on expanding (x-a)^{m} and (x-a)^{n}, we find that f(x) is a polynomial of degree m+n.

∴f(x) is continuous

f '(x) = m(x-a)^{m-1} (x-b)^{n} +(x-a)^{m} n(x-b)^{n-1}

= (x-a)^{m-1} (x-b)^{n-1}[m(x-b)+n(x-a)]

which exists for all x ∈ (a, b)

Thus all the conditions of Rolle’s theorem are satisfied.

b.f '(c) = 0 ⇒ m(c-b)+n(c-a) = 0

in the ratio m: n]. Hence Rolle’s theorem is verified.

Question 10.

If x^{y} = e^{x-y}

i. Express y ¡n terms of x.

ii. Find

Answer.

i. x^{y} = e^{x-y}

Taking logarithm,

y logx = (x - y) loge

y Iogx = x - y

ylogx + y = x ; y(logx+1) = x

Question 11.

Find if y^{x}+x^{y}+x^{x} = a

Answer.

Let u = y^{x}, v = x^{y }and w = x^{x} then

u + v + w = a^{b} Differentiating w.r.t. x,

Question 12.

Differentiate

w.r.t. x

Answer.

Question 13.

i. Let u = sin² x & v = e^{cosx}

ii. Differentiate sin² x w.r.t. e^{cosx}

Answer

i. Let u = sin² x and v = e^{cosx}

Question 14.

If for - 1 < x < 1,

i. Express y as function of x.

ii. Prove that

Answer.

Question 15.

i. find &

ii. find

Answer.

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