CBSE Class 12th Mathematics Chapter 5 - Continuity and Differentiability Important Questions with Answers

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Differentiate \(e^{\sqrt{3 x}}\), with respect to x. (All India 2019)

Question 2.

If y = cos (?3x), then find \(\frac{d y}{d x}\). (All India 2019)

Question 3.

If f(x) = x + 1, find \(\frac{d}{d x}\) (fof) (x). (Delhi 2019)

Question 4.

If f(x) = x + 7 and g(x) = x – 7, x ? R, then find the values of \(\frac{d}{d x}\) (fog) x. (Delhi 2019)

Question 5.

If y = x|x|, find \(\frac{d y}{d x}\) for x < 0. (All India 2019)

Question 6.

If (cos x)^y = (cos y)^x, then find \(\frac{d y}{d x}\). (All India 2019; Delhi 2012)

Question 7.

If \(\sqrt{1+y}+y \sqrt{1+x}=0\) = 0,(x ? y), then prove that \(\frac{d y}{d x}=-\frac{1}{(1+x)^{2}}\). (All India 2019: Foreign 2012; Delhi 2011C)

Question 8.

If y = (sin^-1 x)² prove that
(1 – x²)\(\frac{d^{2} y}{d x^{2}}\) – x \(\frac{d y}{d x}\) – 2 = 0 (Delhi 2019)

Question 9.

If (x – a)² + (y – b)² = c², for some c > 0,
prove that
Continuity and Differentiability Class 12 Maths Important Questions Chapter 5 51

independent of a and b. (All India 2019)

Question 10.

If x = ae^t(sin t + cos t) and y = ae^t(sin t – cos t), then prove that \(\frac{d y}{d x}=\frac{x+y}{x-y}\) (All India 2019)

Question 1.

Differentiate x^{sin x} + (sin x)^{cos x} with respect to x. (All IndIa 2019)

Question 2.

If log (x² + y²) = 2 tan^-1\(\left(\frac{y}{x}\right)\) show that \(\frac{d y}{d x}=\frac{x+y}{x-y}\) (Delhi 2019)

Question 3.

If x^y – y^x = ab, find \(\frac{d y}{d x}\). (Delhi 2019)

Question 4.

If x = cos t + log tan\(\left(\frac{t}{2}\right)\), y = sin t, then find the values of \(\frac{d^{2} y}{d t^{2}}\) and \(\frac{d^{2} y}{d x^{2}}\) at t = \(\frac{\pi}{4}\). (Delhi 2019; All IndIa 2012 C)

Question 5.

Differentiate tan^-1 \(\left(\frac{1+\cos x}{\sin x}\right)\) with respect to x. (CBSE 2018)

Question 6.

Differentiate tan^-1 \(\left(\frac{\cos x-\sin x}{\cos x+\sin x}\right)\) with respect to x. (CBSE 2018 C)

Question 7.

If y = sin (sin x), prove that
\(\frac{d^{2} y}{d x^{2}}\) + tan x \(\frac{d y}{d x}\) + y cos x = 0. (CBSE 2018)

Question 8.

If (x² + y²)² = xy, find \(\frac{d y}{d x}\). (CBSE 2018)

Question 9.

If x = a(2? – sin 2?) and y = a(1 – cos 2?), find \(\frac{d y}{d x}\) when ? = \(\frac{\pi}{3}\). (CBSE 2018)

Question 10.

If sin y = x cos(a + y), then show that
\(\frac{d y}{d x}=\frac{\cos ^{2}(a+y)}{\cos a}\).
Also, show that \(\frac{d y}{d x}\) = cos a, when x = 0. (CBSE 2018 C)