Continuity and Differentiability

The function \(f(x) = |x|\) is:

1. adifferentiable at \(x = 0\)
2. bcontinuous but not differentiable at \(x = 0\)
3. cnot continuous at \(x = 0\)
4. dboth continuous and differentiable at \(x = 0\)

The derivative of \(\sin(2x)\) with respect to \(x\) is:

1. a\(\cos(2x)\)
2. b\(-2\cos(2x)\)
3. c\(2\cos(2x)\)
4. d\(\cos x\)

If \(y = e^{3x}\), then \(\dfrac{dy}{dx}\) equals:

1. a\(e^{3x-1}\)
2. b\(e^{3x}\)
3. c\(3xe^{3x}\)
4. d\(3e^{3x}\)

If \(y = \sin^{-1}\left(2x\sqrt{1-x^{2}}\right)\), then \(\dfrac{dy}{dx}\) equals (for \(|x| < \dfrac{1}{\sqrt{2}}\)):

1. a\(\dfrac{2}{1-x^{2}}\)
2. b\(\dfrac{1}{\sqrt{1-x^{2}}}\)
3. c\(\dfrac{2}{\sqrt{1-x^{2}}}\)
4. d\(\dfrac{2x}{\sqrt{1-x^{2}}}\)

\(\dfrac{d}{dx}\left(\log(\sin x)\right)\) equals:

1. a\(\csc x\)
2. b\(\tan x\)
3. c\(\sec x\)
4. d\(\cot x\)

If \(f(x) = x^{2}\) and \(g(x) = \sqrt{x}\), then \(\dfrac{d}{dx}(f \circ g)\) at \(x = 4\) is:

1. a\(2\)
2. b\(1\)
3. c\(\dfrac{1}{2}\)
4. d\(\dfrac{1}{4}\)

If \(y = x^{x}\), then \(\dfrac{dy}{dx}\) equals:

1. a\(x^{x}\)
2. b\(x^{x}\log x\)
3. c\(x \cdot x^{x-1}\)
4. d\(x^{x}(1 + \log x)\)

If \(f(x) = \begin{cases} 3x - 2, & \text{if } x \le 1 \\ kx + 5, & \text{if } x > 1 \end{cases}\) is continuous at \(x = 1\), then \(k\) is:

1. a\(-2\)
2. b\(-4\)
3. c\(1\)
4. d\(2\)

If \(x = a\cos\theta\), \(y = a\sin\theta\), then \(\dfrac{dy}{dx}\) equals:

1. a\(\cot\theta\)
2. b\(-\tan\theta\)
3. c\(-\cot\theta\)
4. d\(\tan\theta\)

If \(y = x\cos x\), then \(\dfrac{d^{2}y}{dx^{2}}\) equals:

1. a\(-x\cos x\)
2. b\(-2\sin x - x\cos x\)
3. c\(-2\sin x + x\cos x\)
4. d\(2\sin x - x\cos x\)

Assertion (A): The function \(f(x) = x^{2}\sin\left(\dfrac{1}{x}\right)\) for \(x \ne 0\) and \(f(0) = 0\) is differentiable at \(x = 0\).

Reason (R): \(\left|h\sin\dfrac{1}{h}\right| \le |h|\), and the squeeze theorem gives \(\lim_{h \to 0} h\sin\dfrac{1}{h} = 0\), so \(f'(0)\) exists.

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

Assertion (A): If \(f\) is differentiable at \(x = a\), then \(f\) is continuous at \(x = a\).

Reason (R): Differentiability at \(a\) requires \(\lim_{h \to 0}\dfrac{f(a+h)-f(a)}{h}\) to exist, which in turn forces \(\lim_{x \to a} f(x) = f(a)\).

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

Assertion (A): The function \(f(x) = |x| + |x-1|\) is continuous everywhere but not differentiable at \(x = 0\) and \(x = 1\).

Reason (R): \(|x-c|\) is continuous everywhere on \(\mathbb{R}\) but not differentiable at \(x = c\) (a corner).

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

1. I. The product rule: \((uv)' = u'v + uv'\).
2. II. The quotient rule: \(\left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^{2}}\) for \(v \ne 0\).

Which statements about derivatives are correct?

1. aBoth
2. bII only
3. cI only
4. dNeither

1. I. Every differentiable function is continuous.
2. II. Every continuous function is differentiable.

Which statements are correct?

1. aBoth
2. bII only
3. cI only
4. dNeither

1. I. \(\dfrac{d}{dx}(\sin^{-1} x) = \sqrt{1-x^{2}}\).
2. II. Implicit differentiation of \(x^{2} + y^{2} = r^{2}\) gives \(\dfrac{dy}{dx} = -\dfrac{x}{y}\).

Which statements are correct?

1. aI only
2. bNeither
3. cBoth
4. dII only

Match each function with its derivative:

1. a(i)-R, (ii)-S, (iii)-P, (iv)-Q
2. b(i)-Q, (ii)-P, (iii)-S, (iv)-R
3. c(i)-S, (ii)-R, (iii)-Q, (iv)-P
4. d(i)-P, (ii)-Q, (iii)-R, (iv)-S

Match each function with its derivative:

1. a(i)-P, (ii)-Q, (iii)-R, (iv)-S
2. b(i)-Q, (ii)-P, (iii)-S, (iv)-R
3. c(i)-R, (ii)-S, (iii)-P, (iv)-Q
4. d(i)-S, (ii)-R, (iii)-Q, (iv)-P

A particle's position is \(s(t) = t^{3} - 6t^{2} + 9t\) m. Its velocity at \(t = 2\) s is:

1. a0 m/s
2. b3 m/s
3. c\(-3\) m/s
4. d6 m/s

A taxi company in Dubai models the fare (in dirhams) for a ride of \(x\) km by \(f(x) = 2x^{2} + 10\) for \(0 \le x \le 4\) and \(f(x) = ax + b\) for \(x > 4\). To make the fare change smoothly at \(x = 4\) (the fare is continuous and has no sudden change in its rate), the value of \(a + b\) is:

1. a\(6\)
2. b\(-6\)
3. c\(22\)
4. d\(38\)