2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability

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Karnataka 2nd PUC Maths Question Bank Chapter 5 Continuity and Differentiability

2nd PUC Maths Continuity and Differentiability One Mark Questions and Answers

Question 1.
Find the derivative of cos (x²) with respect to x.
Answer:
Let y = cos (x²)
\(\frac{d y}{d x}\) = -sin(x²).2x.

Question 2.
If tan (2x + 3), find \(\frac{d y}{d x}\)
Answer:
Let y = tan (2x + 3)
\(\frac{d y}{d x}\) = 2.sec²(2x + 3).

Question 3.
The function f(x) = \(\frac{1}{x-5}\) is not continuous at x = 5. Justify the statement.
Answer:
The function is not defined at x = 5.

Question 4.
Sin (x² + 5).
Answer:
Let y = Sin (x² + 5)
Differentiate both sides w.r.t. x, we get

= cos (x² + 5) (2x + 0) = 2x cos (x² + 5).

Question 5.
Cos (sin x).
Answer:
Let y = cos (sin x) .
Differentiate both sides w.r.t. x, we get

= – sin (sin x) cos x = – cos x sin (sin x).

Question 6.
Sin (ax + b).
Answer:
Let y = sin (ax + b)
Differentiate both sides w.r.t. x, we get

= cos (ax + b) {a × 1 + 0} = a cos (ax + b).

Question 7.
Sec (tan √x).
Answer:
Let y = sec (tan √x)
Differentiate both sides w.r.t. x, we get

Question 8.
cos (√x)
Answer:
Let y = cos(√x)
Differentiate both sides w.r.t. x

Question 9.
Prove that the function f(x) = 5x – 3 is continuous at x = 0, at x = – 3 and at x = 5.
Answer:
Here, f(x) = 5x – 3

Question 10.
Examine the continutiy of the function f(x) = 2x² – 1 at x = 3.
Answer:
Here, f(x) = 2x² – 1

Question 11.
Examine the following functions for continuity:
(a) f(x) = x – 5
(b) f(x) = \(\frac{1}{x-1}\) ≠ 0
(c) f(x) = \(\frac{x^{2}-25}{x+5}\), x ≠ 5
(d) f(x) = |x – 5|.
Answer:
(a) f(x) = x – 5 is a polynomial function, so f(x) is continuous for all values of x.

(b) f(x) = \(\frac{1}{x-5}\) is a quotient function of two polynomial functions, so ,
f(x) is continuous for all values of x provided x ≠ 5.

(c) f(x) = \(\frac{x^{2}-25}{x+5}=\frac{(x+5)(x-5)}{x+5}=x-5\)
∴ f(x) = x – 5 is a polynomial function, so f(x) is continuous at all values of x.

(d)

Also, f(5) = 5 – 5 = 0
∵ LHL = RHL = f(5). therefore, the function is continuous at x = 5.

Question 12.
Prove that the function f(x) = xⁿ is continuous at x = n, where n is a positive integer.
Answer:
Here, f(x) = xⁿ

Thus, f(x) is continuous at x = n, where n is a positive integer.

Question 13.
Differentiate sin (cos (x²)) with respect to x.
Answer:
Let y = sin (cos(x²))
\(\frac{d y}{d x}\) = cos(cos(x²)). – sin(x²).2x

Question 14.
Differentiate sin (x²) with respect to x.
Answer:
Let y = sin(x²)
\(\frac{d y}{d x}\) = cos(x²).2x

Question 15.
Differentiate cos(x³).sin² (x⁵) with respect tox.
Answer:
Let y = cos(x³).sin² (x⁵)

Question 16.
If 2x + 3y = sin x find \(\frac{d y}{d x}\)
Answer:
Let 2x + 3y = sinx
Differentiating both sides w.r. to x.
2(1) + 3\(\frac{d y}{d x}\) = cos x
\(\frac{d y}{d x}\) = \(\frac{\cos x-2}{3}\)

Question 17.
If 2x + 3y = sin y find \(\frac{d y}{d x}\)
Answer:
Differentiate both sides w.r. t. x.

Question 18.
If ax + by² = cos y find \(\frac{d y}{d x}\)
Answer:

Question 19.
If xy + y² = tan x + y find \(\frac{d y}{d x}\)
Answer:

Question 20.
Differentiate x² + xy + y² = 100 with respect to X.
Answer:
x² + xy + y² = 100
Differentiating with respect to x.

Question 21.
If x³ + x²y + xy² + y³ = 81 find \(\frac{d y}{d x}\)
Answer:
x³ + x²y + xy² + y³ = 81
Differentiating w.r. to. x.

Question 22.
If sin²x + cos xy = k find \(\frac{d y}{d x}\)
Answer:

Question 23.
If sin²x + cos²y = 1 find \(\frac{d y}{d x}\).
Answer:

Question 24.
Differentiate w.r. to x : x^x + a^x + x^a + a^a
Answer:
Let y = x^x + a^x + x^a + a^a
\(\frac{d y}{d x}\) = x^x(1 + log x) + a^x log_ea + ax^n-1 + 0

Question 25.
Differentiate x^x w.r. to x
Answer:
Let y = x^x
log y = log x^x
log y = x log x.
\(\frac{1}{y} \frac{d y}{d x}\) = x.\(\frac{1}{x}\) + logx(1)
\(\frac{d y}{d x}\) = y(1 + log x) = x^x (1 + log x)

Question 26.
Differentiate a^x w.r. to x
Answer:
Let y = a^x
= x log a
log y = log a^x
\(\frac{1}{y} \frac{d y}{d x}\) = x(0) + log a(1)
\(\frac{d y}{d x}\) = y log a = a^x log_ea
Note : \(\frac{d }{d x}\)(log_ex) = \(\frac{1}{x}\)
\(\frac{d }{d x}\)(log_ea) = 0 ∵ a is a constant.

Question 27.
If y = x⁵5^x find \(\frac{d y}{d x}\)
Soin.
y = x⁵ .5^x
\(\frac{d y}{d x}\) = x⁵ .5^x log_e5 + 5^x .5x⁴

Question 28.
If y = x³.2^x find \(\frac{d y}{d x}\)
Answer:
y = x³.2^x
\(\frac{d y}{d x}\) = x³.2^x log_e2 + 2^x(3x²)

Question 29.
If y = (log x)^{cos x} find \(\frac{d y}{d x}\)
Answer:
y = (log x)^{cos x}
log y = log(log x)

Question 30.
If y = cosx.cos2x.cos3x find \(\frac{d y}{d x}\).
Answer:
logy = log(cosx.cos2x.cos3x)
= logcosx + logcos2x + logcos3x

cos x cos 2x cos 3x [-tan x – 2 tan 2x – 3 tan 3x]

2nd PUC Maths Continuity and Differentiability Two Marks Questions and Answers

Question 1.
Check the continuity of the function f given by f(x) = 2x + 3 at x = 1.
Answer:

Question 2.
Find the derivative of (3x² – 7x + 3)^5/2 with respect to x.
Answer:
Let y = (3x² – 7x + 3)^5/2
\(\frac{d y}{d x}=\frac{5}{2}\) (3x² – 7x + 3)^5/2 (6x – 7).

Question 3.
If y = (sin^-1 x)^x find \(\frac{d y}{d x}\)
Answer:

Question 4.

Answer:

Question 5.
If y = sin (Iog_e x) prove that \(\frac{d y}{d x}=-\sqrt{\frac{y}{x}}\)
Answer:

Question 6.
Find the derivative of x^x – 2^{sin x} with respect to x.
Answer:
Let y = x^x – 2^{sin x}
Let u = x^x and v = 2^{sin x}
∴ y = u – v
Differentiating both sides w.r.t. x, we get

Now, u = x^x
Taking log on both sides.
⇒ log u = log x^x ⇒ log u = x log x

Question 7.
Find the derivative of \(\frac{e^{x}}{\sin x}\) w.r.t x.
Answer:

Question 8.
Find the derivative of e^sinx-1 w.r.t. x.
Answer:
y = e^sinx-1
Differentiate both sides wor.t. x, we get

Question 9.
Differentiate \(\left(x+\frac{1}{x}\right)^{x}\) W. r. to x. (M. 2015)
Answer:

Question 10.

Answer:

Question 11.
Differentiate (x + 3)² (x + 4)³ (x + 5)⁴ w.r. to x.
Answer:
Let y = (x + 3)² (x + 4)³ (x + 5)⁴
log y = 2 log (x + 3) + 3 log (x + 4) + 4 log (x + 5)

Question 12.
If y = log₇ (log x) find \(\frac{d y}{d x}\)
Answer:

Question 13.
Differentiate w.r. to x: \(\sqrt{3 x+2}+\frac{1}{\sqrt{2 x^{2}+4}}\)
Answer:

Question 14.
If y = cos ^-1(sin x) find \(\frac{d y}{d x}\).
Answer:

Question 15.

Answer:

Question 16.

Answer:

Question 17.
Differentiate sin² x w.r. t e^cosx
Answer:

Question 18.
If y = (3x² – 9x + 5)⁹ find \(\frac{d y}{d x}\)
Answer:
\(\frac{d y}{d x}\) = 9(3x² – 9x + 5)⁸ (6x – 9)

Question 19.
If y = sin³x + cos⁶x find \(\frac{d y}{d x}\)
Answer:
y = sin³x + cos⁶x
\(\frac{d y}{d x}\) = 3 sin²x cosx + 6cos⁵x(-sin x)

Question 20.
If y = (5x)^{3 cos 2x} find \(\frac{d y}{d x}\)
Answer:
y = y = (5x)^{3 cos 2x}

Question 21.
Differentiate w.r. to x. (log x)^x
Answer:
Let y = (log x)^x
log y = x log (log x)

Question 22.
Differentiate w.r. to x. x^logx.
Answer:
Let y = x^logx
log y = log x ^{log x}
= log x log x = (log x)²

Question 23.
Differentiate (log x)^{log x} w.r. to x.
Answer:
Let y = (log x)^{log x}
log y = log (log x)

Question 24.
If y = sin^-1 (x√x) find \(\frac{d y}{d x}\).
Answer:

Question 25.
If x^y = y^x \(\frac{d y}{d x}\)
Answer:
x^y = y^x
log x^y = log y^x

Question 26.
Differentiate : x^{sin x} with respect to x.
Answer:
Let y = x^{sin x}

2nd PUC Maths Continuity and Differentiability Three Marks Questions and Answers

Question 1.

Answer:

Question 2.
y = sin^-1\(\left(\frac{2 x}{1+x^{2}}\right)\)
Answer:
Given y = sin^-1\(\left(\frac{2 x}{1+x^{2}}\right)\)
Putting θ = tan^-1 x i.e., x = tan θ

Question 3.

Answer:
Substitute tan^-1 x = θ i.e., x = tan θ

⇒ y = tan^-1 (tan 3θ) = 3θ = 3 tan^-1 x
Differentiating both sides w.r.t. x, we get
⇒ 3\(\frac{d}{d x}\)(tan^-1) = \(\frac{3}{1+x^{2}}\)

Question 4.
y = cos^-1\(\left(\frac{1-x^{2}}{1+x^{2}}\right)\), 0 < x < 1.
Answer:
Let tan^-1 x = θ i.e., x = tan θ

⇒ y = cos^-1 (cos 2θ) = 2θ = 2 tan^-1 x
Differentiating both sides w.r.t. x, we get

Question 5.
y = sin^-1\(\left(\frac{1-x^{2}}{1+x^{2}}\right)\), 0 < x < 1.
Answer:
Substitute x = tan θ ⇒ tan^-1 x = θ

Question 6.
y = cos^-1\(\left(\frac{2 x}{1+x^{2}}\right)\), -1 < x < 1.
Answer:
Substitute x = tan θ ⇒ tan^-1 x = θ

Question 7.

Answer:
Let sin^-1 x = θ , then x = sin θ

⇒ y = sin^-1 (2 sin θ cos θ) = sin^-1(sin 2θ)
⇒ y = 2θ ⇒ 2 sin^-1 x
Differentiating both sides w.r.t. x, we get
⇒ \(\frac{d y}{d x}=\frac{2}{\sqrt{1-x^{2}}}\)

Question 8.

Answer:
Let cos^-1 x = θ, i.e., x = cos θ,

⇒ y sec^-1 (sec 2θ) ⇒ y = 2θ ⇒ y = 2 cos^-1 x
Differentiating both sides w.r.t. x, we get

Question 9.
Find \(\frac{d y}{d x}\), if x = at², y = 2at.
Answer:
Given that x = at², y = 2at

Question 10.
x = 2at², y = at⁴.
Answer:
Given, x = 2at², y = at⁴
Differentiating w.r.t. t, we get

Question 11.
x = a cos θ, y = b sin θ.
Answer:
Given, x = a cos θ, y = b sin θ

Question 12.
x = sin t, y = cos 2t.
Answer:
Given, x = sin t, y = cos 2t.
Differentiating w.r.t. t, we get

Question 13.
x = 4t, y = \(\frac{4}{t}\)
Answer:
Given x = 4t, y = \(\frac{4}{t}\)

Question 14.
x = cos θ – cos 2θ, y = sin θ – sin 2θ.
Answer:
Given x = cos θ – cos 2θ, y = sin θ – sin 2θ
Differentiating w.r.t. θ, we get

Question 15.
x = a (θ – sin θ), y = a (1 + cos θ).
Answer:
Given, x = a (θ – sin θ), y = a (1 + cos θ)
Differentiating w.r.t. 0, we get

Question 16.

Answer:

Question 17.
x = a (cos t + log tan\(\frac{t}{2}\)), y = a sint.
Answer:
Given, x = a (cos t + log tan\(\frac{t}{2}\))
Differentiating w.r.t. t, we get

Question 18.
x = a sec θ, y = b tan θ.
Answer:
Given, x = a sec θ, y = b tan θ
Differentiating w.r.t. θ, we get

Question 19.
x = a (cos θ + θ sin θ), y = a (sin θ – θ cos θ).
Answer:
Given, x = a (cos θ + θ sin ), y = a (sin θ – θ cos θ)
Differentiating w.r.t. θ, we get
∴ = a {- sin θ + (θ cos θ + sin θ.1)} = a θ cos θ
(Using product rule in \(\frac{d}{d \theta}\) (θ sin θ)

Question 20.
If x = a (θ + sin θ) and y = a (1 – cos θ) prove that \(\frac{d y}{d x}\) = tan\(\left(\frac{\theta}{2}\right)\)
Answer:
We have \(\frac{d x}{d \theta}\) = a(1 + cosθ), \(\frac{d y}{d \theta}\) = a(sin θ)

Question 21.
If a function f(x) is differentiable at x = c prove that it is continuous at x = c.
Answer:
Since f is differentiable at c, we have

Question 22.
Differentiate with respect to x : (sin x)^x + sin^-1 √x
Answer:
Let y =(sin x)^x + sin^-1 √x
y = u + v

Question 23.
Find \(\) given x^y + y^x = 1
Answer:

Question 24.

Answer:

x² (1 + y) = y² (1 + x)
x² + x²y = y² + y²x
x² – y² = y²x – x²y
(x – y)(x + y) = xy (y – x) ⇒ (x – y)(x + y) = -xy (x – y)
∵ x ≠ y
⇒ x + y = -xy
x + y + xy = 0
y + xy = -x

Question 25.

Answer:

Question 26.
If cos y = x cos (a + y) with cos a ≠ ±1
Prove that \(\frac{d y}{d x}=\frac{\cos ^{2}(a+y)}{\sin a}\)
Answer:

Question 27.
If x = a(cos t + t sin t) y = a(sin t – t cos t) find \(\frac{d^{2} y}{d x^{2}}\)
Answer:
x = a(cos t + t sin t)

Question 28.
If x^y = e^x-y prove that \(\frac{d y}{d x}=\frac{\log _{e} x}{\left(1+\log _{e} e\right)^{2}}\)
Answer:
x^y = e^x-y
log_ex^y = log_ee^x-y
y log_ex = (x – y)log_ee
y log_ex = x – y
y + y log_ex = x

2nd PUC Maths Continuity and Differentiability Four Marks Questions and Answers

Question 1.

Answer:

Question 2.

Answer:

Question 3.

Answer:

Question 4.

Answer:

Question 5.

Answer:

Question 6.
Define a continuity of a function at a point. Find all the points of discontinuity of f defined by f(x) = |x| – |x + 1|.
Answer:
Let f(x) be a real valued function on a subset of the real numbers and let c be a point in the domain off. Then f (x) is continuous at x = c if.

Let g (x) = |x| and h (x) = |x + 1|
Now, g (x) = |x| is the absolute valued function, so it is a continuous function for all x ∈ R,
H (x) = |x + 1| is the absolute valued function, so it is a continuous function for all x ∈ R.
Since g (x) and h (x) are both continuous functions for all x ∈ R, so difference of two continuous function is a continuous function for all x ∈ R. Thus f(x) = |x| – |x + 1| is continuous at all points. Hence there is no point at which f(x) is discontinuous.

Question 7.

Answer:

3a + 1 = 3 b+ 3 = 3a + 1
⇒ 3a + 1 = 3b + 3
⇒ 3a = 3b + 3 – 1
⇒ 3a = 3b + 2
⇒ a = b + \(\frac { 2 }{ 3 }\)

Question 8.
Find the points of discontinuity of the function f(x) = x – [x] where [x] indicates the greatest integer not greater than x. Also write the set of values of x where the function is continuous.
Answer:
Let a ∈ Z

∴ f(x) = x – [x] ¡s discontinuous at integral points.
∴ f(x) is continuous at all points.
x ∈ R – Z

Question 9.

Answer:
f(x) is defined at all points on the real line.
Let C be a real number.
Case (1) : Let c < 2.

Question 10.
Define continuity of a function at a point. Find all the points of discontinuity of f defined by f(x) = |x| – |x +1|
Answer:
Let f(x) be a real valued function on a subset of the real numbers and let c be a point in the domain off. Then f(x) is continuous at x = c if.

Let g (x) = |x| and h (x) = |x + 1|
Now, g (x) = |x| is the absolute valued function, so it is a continuous function for all x∈R.
H (x) = |x + 1| is the absolute valued function, so it is a continuous function for all x ∈ R.
Since g (x) and h (x) are both continuous functions for all X ∈ R, so difference of two continuous function is a continuous function for all x ∈ R. Thus f(x) = |x| – |x + 1| is continuous at all points. Hence there is no point at which/(x) is discontinuous.

2nd PUC Maths Continuity and Differentiability Five Marks Questions and Answers

Question 1.
If y = A sin x + B cos x, then prove that \(\frac{d^{2} y}{d x^{2}}\) + y = 0.
Answer:

Question 2.

Answer:

Question 3.

Answer:

Question 4.
If y = 3cos(log x) + 4 sin(log x) show that x²y₂ + xy₁ + y = 0.
Answer:
Given, y = 3 cos (log x) + 4 sin (log x)
Differentiating w.r.t. s, we get ……… (i)

Multiplying by x, we get
xy₁ = -3 sin (log x) + 4 cos (log x) ……….. (ii)
Again differentiating w.r.t. x, we obtain

Multiplying through out by x, we have
x²y₂ + xy₁ = – (3 cos (log x) + 4 sin (log x)) [from Eq. (i)]
⇒ x²y₂ + xy₁ = -y ⇒ x²y₂ + xy₁ + y = 0

Question 5.
If y = 5 cos x – 3 sin x, prove that \(\frac{d^{2} y}{d x}\) + y = 0.
Answer:
Given, y = 5 cos x – 3 sin x
Differentiating twicely w.r.t. x, we get
\(\frac{d y}{d x}\) = -5 sin x – 3 cos x

Question 6.

Answer:

Question 7.

Answer:

Question 8.
If e^y (x + 1) = 1, show that \(\frac{d^{2} y}{d x^{2}}=\left(\frac{d y}{d x}\right)^{2}\)
Answer:

Question 9.
y = 500e^7x + 600e^-7x prove that \(\frac{d^{2} y}{d x^{2}}\) = 49 y.
Answer:
y = 500e^7x + 600e^-7x

Question 10.

Answer:

Question 11.

Answer:

Question 12.

Answer:

Question 13.
If y^x = e^y-x, prove that \(\frac{d y}{d x}=\frac{(1+\log y)^{2}}{\log y}\)
Answer:
y^x = e^y-x
log y^x = log e^y-x
x log y = (y – x)log e
x + x log y = y
x(1 + log y) = y

Question 14.

Answer:
y = (cos x)^y
log y = log (cos x)^y
log y = y log cos x

Question 15.
If y = tan^-1x find \(\frac{d^{2} y}{d x^{2}}\) im terms of y alone.
Answer:
y = tan^-1x
x = tan y