**(I) Select the appropriate option from the given alternatives.**

**Question 1.**

Hint:

**Question 2.**

Hint:

**Question 3.**

Hint:

**Question 4.**

Hint:

**Question 5.Suppose f(x) is the derivative of g(x) and g(x) is the derivative of h(x).If h(x) = a sin x + b cos x + c, then f(x) + h(x) =(A) 0(B) c(C) -c(D) -2(a sin x + b cos x)Answer:**

(B) c

Hint:

h(x) = a sin x + b cos x + c

Differentiating w.r.t. x, we get

h'(x) = a cos x – b sin x = g(x) …..[given]

Differentiating w.r.t. x, we get

g'(x) = -a sin x – b cos x = f(x) …..[given]

∴ f(x) + h(x) = -a sin x – b cos x + a sin x + b cos x + c

∴ f(x) + h(x) = c

**Question 6.If f(x) = 2x + 6, for 0 ≤ x ≤ 2= ax**

^{2}+ bx, for 2 < x ≤ 4is differentiable at x = 2, then the values of a and b are

Hint:

f(x) = 2x + 6, 0 ≤ x ≤ 2

= ax

^{2}+ bx, 2 < x ≤ 4

Lf'(2) = 2, Rf'(2) = 4a + b

Since f is differentiable at x = 2,

Lf'(2) = Rf'(2)

∴ 2 = 4a + b …..(i)

f is continuous at x = 2.

∴ 4a + 2b = 2(2) + 6

∴ 4a + 2b = 10

∴ 2a + b = 5 …..(ii)

Solving (i) and (ii), we get

**Question 7.If f(x) = x ^{2} + sin x + 1, for x ≤ 0= x^{2} – 2x + 1, for x ≤ 0, then(A) f is continuous at x = 0, but not differentiable at x = 0(B) f is neither continuous nor differentiable at x = 0**

**(C) f is not continuous at x = 0, but differentiable at x = 0(D) f is both continuous and differentiable at x = 0Answer:**(A) f is continuous at x = 0, but not differentiable at x = 0

Hint:

**Question 8.**

**(A) 48(B) 49(C) 50(D) 51Answer:**(C) 50

Hint:

**(II).**

**Question 1.Determine whether the following function is differentiable at x = 3 where,f(x) = x**

^{2}+ 2, for x ≥ 3= 6x – 7, for x < 3.Solution:

f(x) = x

^{2}+ 2, x ≥ 3

= 6x – 7, x < 3

Differentiability at x = 3

Here, Lf'(3) = Rf'(3)

∴ f is differentiable at x = 3.

**Question 2.Find the values of p and q that make function f(x) differentiable everywhere on R.f(x) = 3 – x, for x < 1= px**

^{2}+ qx, for x ≥ 1.Solution:

f(x) is differentiable everywhere on R.

∴ f(x) is differentiable at x = 1.

∴ f(x) is continuous at x = 1.

f(x) is differentiable at x = 1.

∴ Lf'(1) = Rf'(1)

∴ -1 = 2p + q …..(ii)

Subtracting (i) from (ii), we get

p = -3

Substituting p = -3 in (i), we get

p + q = 2

∴ -3 + q = 2

∴ q = 5

**Question 3.Determine the values of p and q that make the function f(x) differentiable on R wheref(x) = px**

^{3}, for x < 2= x

^{2}+ q, for x ≥ 2Solution:

f(x) is differentiable on R.

∴ f(x) is differentiable at x = 2.

∴ f(x) is continuous at x = 2.

Continuity at x = 2:

f(x) is continuous at x = 2.

f(x) is differentiable at x = 2.

∴ Lf'(2) = Rf'(2)

∴ 12p = 4

**Question 4.Determine all real values of p and q that ensure the functionf(x) = px + q, for x ≤ 1is differentiable at x = 1.Solution:**

f(x) is differentiable at x = 1.

∴ f(x) is continuous at x = 1.

Continuity at x= 1:

f(x) is continuous at x = 1.

**Question 5.Discuss whether the function f(x) = |x + 1| + |x – 1| is differentiable ∀ x ∈ R.Solution:**

Here, Lf'(1) ≠ Rf'(1)

∴ f is not differentiable at x = 1.

∴ f is not differentiable at x = -1 and x = 1.

∴ f is not differentiable ∀ x ∈ R.

**Question 6.Test whether the functionf(x) = 2x – 3, for x ≥ 2= x – 1, for x < 2is differentiable at x = 2.Solution:**

**Question 7.Test whether the functionf(x) = x**

^{2}+ 1, for x ≥ 2= 2x + 1, for x < 2is differentiable at x = 2.Solution:

**Question 8.Test whether the functionf(x) = 5x – 3x**

^{2}, for x ≥ 1= 3 – x, for x < 1is differentiable at x = 1.Solution:

Here, Lf'(1) = Rf'(1)

∴ f(x) is differentiable at x = 1.

**Question 9.Solution:**

**Question 10.Solution:**