Important Questions for CBSE Class 12 Maths Continuity
CBSE Class 12 Mathematics Important Questions Chapter 5 – Continuity and Differentiability
4 Marks Questions
1. Find the values of K so that the function f is continues at the given value of x.
Ans.
K = 6
2. Differentiate the function
Ans. Let y = u + v
When u = x sinx, v = (sinx)cosx
Taking log both side
log u = log xsinx
log u = sinx . logx
diff. both side w.r. to x
Taking log both side
log v = log (sinx)cosx
Differentiation both side w.r. to x
Hence
3. If show that
Ans.
Square both side
Differentiation
Dividing (2) and (1)
4. If y = (tan-1x)2 show that (x2 + 1)2 y2 + 2x (x2 + 1)y1 = 2
Ans. y = (tan-1 x)2 (given)
Differentiation both side w.r. to x
Again differentiation both side w.r. to
5. Verify Rolle’s Theorem for the function y = x2 +2 , [ -2 , 2]
Ans. y = x2 + 2 is continuous in [-2, 2] and differentiable in (-2, 2). Also f (-2) = f(2) = 6
Hence all the condition of Rolle’s Theorem are verified hence their exist value c such that
(c) = 0
0 = 2c.
C = 0
Hence prove.
6. Differentiate
Ans.
7. Differentiate sin2x w.r. to ecosx
Ans.
8. If prove that
Ans.
Square both side
9. If cosy = x cos (a + y) prove that
Ans.
10. If x = a (cos t + t sin t)
y = a (sin t – t cos t )
find
Ans.
11. Find all points of discontinuity if
Ans. At x = -3
f(-3) = |-3| + 3 = 3 + 3 = 6
Hence continuous at x = -3
At x = 3
Hence it is continuous
12. Differentiate
Ans.
13. Find if
Ans. Differentiate both side w.r.t. to x, x3 + x2y + xy2 + y3 = 81
14. Differentiate xy = e(x-y)
Ans.
Taking log both side
Diff. both side w.r.t. to x
15. Find if
Ans.
16. If y = 3 cos (log x) + 4 sin (log x). Show that x2y2 + xy1 + y = 0
Ans.
Diff. both side w.r.t. to x
Again diff.
17. Verify Rolle’s Theorem for the function f(x) = x2 + 2x – 8, x [-4, 2]
Ans. The function
Continuous in [-4, 2] and differentiable in (-4, 2)
Also
Hence all the condition of all Rolle ’s Theorem, is verified
Their exist a value C
Such that (c) = 0
(c) = 2c +2
0 = 2C+2
C = -1
18. Find
Ans.
19. If x = a (cos t + t sin t) and y = a (sin t – t cos t), find
Ans.
20. If Prove that
Ans.
21. Find the value of K so that function is continuous at the given value.
Ans.
22. Differentiate
Ans.
23. Find
Ans.
24. Find
Ans. Let
Therefore — (1)
Taking log both side
Differentiate both side w.r.t. to x
— (2)
Taking log both side
— (3)
Taking log both side
— (4)
(by putting 2,3 and 4 in 1)
25. Find when
Ans.
26. If Prove that
Ans.
=
LHS
27. If Show that
Ans.
28. If
Prove is a constant independent of a & b.
Ans.
Diff. both side w.r.t. to x
Again diff. both side
Put (y-b) in equation (1)
Put the value of (x-a) and (y-b) in equation (1)
Hence prove
29. Find if
Ans.
Differentiate both side w.r.t. x
30. Find
Ans.
Taking log both side
Differentiate both side w.r.t. x
31. Discuss the continuity of the function
Ans. At x = -1
f(-1) = -2
Hence continuous at x = -1
Continuous
32. Find if
Ans.
33. Find if
Ans.
Diff.
34. Find , if y=
Ans. Let
Where
Taking log both side
Differentiate
Taking log both side
Differentiate
35. , find
Ans.
36. If show that
Ans.
Differentiate
37. Find
Ans.
38.
Ans.
39. If Prove that
Ans. Let
Squaring both side
Differentiate
40. Show that
Ans.
,
hence
41. For what value of K is the following function continuous at x = 2?
Ans.
A T
42. Differentiate the following w.r.t. to x
Ans.
43. If find
Ans.
Squaring and adding
44. Discuss the continuity of the following function at x = 0
Ans.
Hence continuous
45. Verify L.M.V theorem for the following function f(x) = x2 + 2x + 3, for [4, 6]
Ans. Since f(x) is polynomial hence continuous in the interval [4, 6] thus f(x) is differentiable in (4, 6) both condition of L.M.V theorem are satisfied.
46. If find also find
Ans.
47. If prove that
Ans.
Taking log both side
Differentiate both side w.r.t. to x
48. If find the value of at t = 0
Ans.
49. If prove that
Ans.
50. If
prove that OR
If prove that
Ans. Let
Squaring both side
Differentiate both side w.r.t. to x
OR
Differentiate both side w.r.t. to x
Continuity and Differentiability Class 12 MCQs Questions with Answers
Question 1.
If f (x) = 2x and g (x) = \(\frac{x^2}{2}\) + 1, then’which of the following can be a discontinuous function
(a) f(x) + g(x)
(b) f(x) – g(x)
(c) f(x).g(x)
(d) \(\frac{g(x)}{f(x)}\)
Answer
Answer: (d) \(\frac{g(x)}{f(x)}\)
Question 2.
The function f(x) = \(\frac{4-x^2}{4x-x^3}\) is
(a) discontinuous at only one point at x = 0
(b) discontinuous at exactly two points
(c) discontinuous at exactly three points
(d) None of these
Answer
Answer: (a) discontinuous at only one point at x = 0
Question 3.
The set of points where the function f given by f (x) =| 2x – 1| sin x is differentiable is
(a) R
(b) R = {\(\frac{1}{2}\)}
(c) (0, ∞)
(d) None of these
Answer
Answer: (b) R = {\(\frac{1}{2}\)}
Question 4.
The function f(x) = cot x is discontinuous on the set
(a) {x = nπ, n ∈ Z}
(b) {x = 2nπ, n ∈ Z}
(c) {x = (2n + 1) \(\frac{π}{2}\) n ∈ Z}
(d) {x – \(\frac{nπ}{2}\) n ∈ Z}
Answer
Answer: (a) {x = nπ, n ∈ Z}
Question 5.
The function f(x) = e|x| is
(a) continuous everywhere but not differentiable at x = 0
(b) continuous and differentiable everywhere
(c) not continuous at x = 0
(d) None of these
Answer
Answer: (a) continuous everywhere but not differentiable at x = 0
Question 6.
If f(x) = x² sin\(\frac{1}{x}\), where x ≠ 0, then the value of the function f(x) at x = 0, so that the function is continuous at x = 0 is
(a) 0
(b) -1
(c) 1
(d) None of these
Answer
Answer: (a) 0
Question 7.
If f(x) =is continuous at x = \(\frac{π}{2}\), then
(a) m = 1, n = 0
(b) m = \(\frac{nπ}{2}\) + 1
(c) n = \(\frac{mπ}{2}\)
(d) m = n = \(\frac{π}{2}\)
Answer
Answer: (c) n = \(\frac{mπ}{2}\)
Question 8.
If y = log(\(\frac{1-x^2}{1+x^2}\)), then \(\frac{dy}{dx}\) is equal to
(a) \(\frac{4x^3}{1-x^4}\)
(b) \(\frac{-4x}{1-x^4}\)
(c) \(\frac{1}{4-x^4}\)
(d) \(\frac{-4x^3}{1-x^4}\)
Answer
Answer: (b) \(\frac{-4x}{1-x^4}\)
Question 9.
Let f(x) = |sin x| Then
(a) f is everywhere differentiable
(b) f is everywhere continuous but not differentiable at x = nπ, n ∈ Z
(c) f is everywhere continuous but no differentiable at x = (2n + 1) \(\frac{π}{2}\) n ∈ Z
(d) None of these
Answer
Answer: (b) f is everywhere continuous but not differentiable at x = nπ, n ∈ Z
Question 10.
If y = \(\sqrt{sin x+y}\) then \(\frac{dy}{dx}\) is equal to
(a) \(\frac{cosx}{2y-1}\)
(b) \(\frac{cosx}{1-2y}\)
(c) \(\frac{sinx}{1-xy}\)
(d) \(\frac{sinx}{2y-1}\)
Answer
Answer: (a) \(\frac{cosx}{2y-1}\)
Question 11.
The derivative of cos-1 (2x² – 1) w.r.t cos-1 x is
(a) 2
(b) \(\frac{-1}{2\sqrt{1-x^2}}\)
(c) \(\frac{2}{x}\)
(d) 1 – x²
Answer
Answer: (a) 2
Question 12.
If x = t², y = t³, then \(\frac{d^2y}{dx^2}\)
(a) \(\frac{3}{2}\)
(b) \(\frac{3}{4t}\)
(c) \(\frac{3}{2t}\)
(d) \(\frac{3}{4t}\)
Answer
Answer: (b) \(\frac{3}{4t}\)
Question 13.
The value of c in Rolle’s theorem for the function f(x) = x³ – 3x in the interval [o, √3] is
(a) 1
(b) -1
(c) \(\frac{3}{2}\)
(d) \(\frac{1}{3}\)
Answer
Answer: (a) 1
Question 14.
For the function f(x) = x + \(\frac{1}{x}\), x ∈ [1, 3] the value of c for mean value theorem is
(a) 1
(b) √3
(c) 2
(d) None of these
Answer
Answer: (b) √3
Question 15.
Let f be defined on [-5, 5] as
f(x) = {\(_{-x, if x is irrational}^{x, if x is rational}\) Then f(x) is
(a) continuous at every x except x = 0
(b) discontinuous at everyx except x = 0
(c) continuous everywhere
(d) discontinuous everywhere
Answer
Answer: (b) discontinuous at everyx except x = 0
Question 16.
Let function f (x) =
(a) continuous at x = 1
(b) differentiable at x = 1
(c) continuous at x = -3
(d) All of these
Answer
Answer: (d) All of these
Question 17.
If f(x) = \(\frac{\sqrt{4+x}-2}{x}\) x ≠ 0 be continuous at x = 0, then f(o) =
(a) \(\frac{1}{2}\)
(b) \(\frac{1}{4}\)
(c) 2
(d) \(\frac{3}{2}\)
Answer
Answer: (b) \(\frac{1}{4}\)
Question 18.
let f(2) = 4 then f”(2) = 4 then \(_{x→2}^{lim}\) \(\frac{xf(2)-2f(x)}{x-2}\) is given by
(a) 2
(b) -2
(c) -4
(d) 3
Answer
Answer: (c) -4
Question 19.
It is given that f'(a) exists, then \(_{x→2}^{lim}\) [/latex] \(\frac{xf(a)-af(x)}{(x-a)}\) is equal to
(a) f(a) – af'(a)
(b) f'(a)
(c) -f’(a)
(d) f (a) + af'(a)
Answer
Answer: (a) f(a) – af'(a)
Question 20.
If f(x) = \(\sqrt{25-x^2}\), then \(_{x→2}^{lim}\)\(\frac{f(x)-f(1)}{x-1}\) is equal to
(a) \(\frac{1}{24}\)
(b) \(\frac{1}{5}\)
(c) –\(\sqrt{24}\)
(d) \(\frac{1}{\sqrt{24}}\)
Answer
Answer: (d) \(\frac{1}{\sqrt{24}}\)
Question 21.
If y = ax² + b, then \(\frac{dy}{dx}\) at x = 2 is equal to ax
(a) 4a
(b) 3a
(c) 2a
(d) None of these
Answer
Answer: (a) 4a
Question 22.
If x sin (a + y) = sin y, then \(\frac{dy}{dx}\) is equal to
(a) \(\frac{sin^2(a+y)}{sin a}\)
(b) \(\frac{sin a}{sin^2(a+y)}\)
(c) \(\frac{sin(a+y)}{sin a}\)
(d) \(\frac{sin a}{sin(a+y)}\)
Answer
Answer: (a) \(\frac{sin^2(a+y)}{sin a}\)
Question 23.
If x \(\sqrt{1+y}+y\sqrt{1+x}\) = 0, then \(\frac{dy}{dx}\) =
(a) \(\frac{x+1}{x}\)
(b) \(\frac{1}{1+x}\)
(c) \(\frac{-1}{(1+x)^2}\)
(d) \(\frac{x}{1+x}\)
Answer
Answer: (c) \(\frac{-1}{(1+x)^2}\)
Question 24.
If y = x tan y, then \(\frac{dy}{dx}\) =
(a) \(\frac{tan x}{x-x^2-y^2}\)
(b) \(\frac{y}{x-x^2-y^2}\)
(c) \(\frac{tan y}{y-x}\)
(d) \(\frac{tan x}{x-y^2}\)
Answer
Answer: (b) \(\frac{y}{x-x^2-y^2}\)
Question 25.
If y = (1 + x) (1 + x²) (1 + x4) …….. (1 + x2n), then the value of \(\frac{dy}{dx}\) at x = 0 is
(a) 0
(b) -1
(c) 1
(d) None of these
Answer
Answer: (c) 1
Question 26.
If f(x) = \(\frac{5x}{(1-x)^{2/3}}\) + cos² (2x + 1), then f'(0) =
(a) 5 + 2 sin 2
(b) 5 + 2 cos 2
(c) 5 – 2 sin 2
(d) 5 – 2 cos 2
Answer
Answer: (c) 5 – 2 sin 2
Question 27.
If sec(\(\frac{x^2-2x}{x^2+1}\)) – y then \(\frac{dy}{dx}\) is equal to
(a) \(\frac{y*2}{x^2}\)
(b) \(\frac{2y\sqrt{y^2-1}(x^2+x-1)}{(x^2+1)^2}\)
(c) \(\frac{(x^2+x-1)}{y\sqrt{y^2-1}}\)
(d) \(\frac{x^2-y^2}{x^2+y^2}\)
Answer
Answer: (b) \(\frac{2y\sqrt{y^2-1}(x^2+x-1)}{(x^2+1)^2}\)
Question 28.
If f(x) = \(\sqrt{1+cos^2(x^2)}\), then the value of f’ (\(\frac{√π}{2}\)) is
(a) \(\frac{√π}{6}\)
(b) –\(\frac{√π}{6}\)
(c) \(\frac{1}{√6}\)
(d) \(\frac{π}{√6}\)
Answer
Answer: (b) –\(\frac{√π}{6}\)
Question 29.
Differential coefficient of \(\sqrt{sec√x}\) is
(a) \(\frac{1}{4√x}\) = sec √x sin √x
(b) \(\frac{1}{4√x}\) = (sec√x)3/2 sin√x
(c) \(\frac{1}{2}\) √x sec√x sin √x.
(d) \(\frac{1}{2}\)√x (sec√x)3/2 sin√x
Answer
Answer: (b) \(\frac{1}{4√x}\) = (sec√x)3/2 sin√x
Question 30.
Let f(x)={\(_{1-cos x, for x ≤ 0}^{sin x, for x > 0}\) and g (x) = ex. Then the value of (g o f)’ (0) is
(a) 1
(b) -1
(c) 0
(d) None of these
Answer
Answer: (c) 0
Question 31.
If xmyn = (x + y)m+n, then \(\frac{dy}{dx}\) is equal to
(a) \(\frac{x+y}{xy}\)
(b) xy
(c) \(\frac{x}{y}\)
(d) \(\frac{y}{x}\)
Answer
Answer: (d) \(\frac{y}{x}\)
Question 32.
If \(\sqrt{(x+y)}\) + \(\sqrt{(y-x)}\) = a, then \(\frac{dy}{dx}\)
Answer
Answer: (a) \(\frac{\sqrt{(x+y)}-\sqrt{(y-x)}}{\sqrt{y-x}+\sqrt{x+y}}\)
Question 33.
If ax² + 2hxy + by² = 1, then \(\frac{dy}{dx}\)equals
(a) \(\frac{hx+by}{ax+by}\)
(b) \(\frac{ax+by}{hx+by}\)
(c) \(\frac{ax+hy}{hx+hy}\)
(d) \(\frac{-(ax+hy)}{hx+by}\)
Answer
Answer: (d) \(\frac{-(ax+hy)}{hx+by}\)
Question 34.
If sec (\(\frac{x-y}{x+y}\)) = a then \(\frac{dy}{dx}\) is
(a) –\(\frac{y}{x}\)
(b) \(\frac{x}{y}\)
(c) –\(\frac{x}{y}\)
(d) \(\frac{y}{x}\)
Answer
Answer: (d) \(\frac{y}{x}\)
Question 35.
If y = tan-1(\(\frac{sinx+cosx}{cox-sinx}\)) then \(\frac{dy}{dx}\) is equal to
(a) \(\frac{1}{2}\)
(b) \(\frac{π}{4}\)
(c) 0
(d) 1
Answer
Answer: (d) 1
Question 36.
If y = tan-1(\(\frac{√x-x}{1+x^{3/2}}\)), then y'(1) is equal to
(a) 0
(b) (\(\frac{√x-x}{1+x^{3/2}}\))
(c) -1
(d) –\(\frac{1}{4}\)
Answer
Answer: (d) –\(\frac{1}{4}\)
Question 37.
The differential coefficient of tan-1(\(\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}\)) is
(a) \(\sqrt{1-x^2}\)
(b) \(\frac{1}{\sqrt{1-x^2}}\)
(c) \(\frac{1}{2\sqrt{1-x^2}}\)
(d) x
Answer
Answer: (c) \(\frac{1}{2\sqrt{1-x^2}}\)
Question 38.
\(\frac{d}{dx}\)[tan-1(\(\frac{a-x}{1+ax}\))] is equal to
Answer
Answer: (a) –\(\frac{1}{1+x^2}\)
Question 39.
\(\frac{d}{dx}\)(x\(\sqrt{a^2-x^2}+a^2 sin^{-1}(\frac{x}{a})\)) is equal to
(a) \(\sqrt{a^2-x^2}\)
(b) 2\(\sqrt{a^2-x^2}\)
(c) \(\frac{1}{\sqrt{a^2-x^2}}\)
(d) None of these
Answer
Answer: (b) 2\(\sqrt{a^2-x^2}\)
Question 40.
If f(x) = tan-1(\(\sqrt{\frac{1+sinx}{1-sinx}}\)), 0 ≤ x ≤ \(\frac{π}{2}\), then f'(\(\frac{π}{6}\)) is
(a) –\(\frac{1}{4}\)
(b) –\(\frac{1}{2}\)
(c) \(\frac{1}{4}\)
(d) \(\frac{1}{2}\)
Answer
Answer: (d) \(\frac{1}{2}\)
Question 41.
If y = sin-1(\(\frac{√x-1}{√x+1}\)) + sec-1(\(\frac{√x+1}{√x-1}\)), x > 0, then \(\frac{dy}{dx}\) is equal to
(a) 1
(b) 0
(c) \(\frac{π}{2}\)
(d) None of these
Answer
Answer: (b) 0
Question 42.
If x = exp {tan-1(\(\frac{y-x^2}{x^2}\))}, then \(\frac{dy}{dx}\) equals
(a) 2x [1 + tan (log x)] + x sec² (log x)
(b) x [1 + tan (log x)] + sec² (log x)
(c) 2x [1 + tan (logx)] + x² sec² (log x)
(d) 2x [1 + tan (log x)] + sec² (log x)
Answer
Answer: (a) 2x [1 + tan (log x)] + x sec² (log x)
Question 43.
If y = e3x+n, then the value of \(\frac{dy}{dx}\)|x=0 is
(a) 1
(b) 0
(c) -1
(d) 3e7
Answer
Answer: (d) 3e7
Question 44.
Let f (x) = ex, g (x) = sin-1 x and h (x) = f |g(x)|, then \(\frac{h'(x)}{h(x)}\) is equal to
(a) esin-1x
(b) \(\frac{1}{\sqrt{1-x^2}}\)
(c) sin-1x
(d) \(\frac{1}{(1-x^2)}\)
Answer
Answer: (b) \(\frac{1}{\sqrt{1-x^2}}\)
Question 45.
If y = aex+ be-x + c Where a, b, c are parameters, they y’ is equal to
(a) aex – be-x
(b) aex + be-x
(c) -(aex + be-x)
(d) aex – bex
Answer
Answer: (a) aex – be-x
Question 46.
If sin y + e-xcos y = e, then \(\frac{dy}{dx}\) at (1, π) is equal to
(a) sin y
(b) -x cos y
(c) e
(d) sin y – x cos y
Answer
Answer: (c) e
Question 47.
Derivative of the function f (x) = log5 (Iog,x), x > 7 is
(a) \(\frac{1}{x(log5)(log7)(log7-x)}\)
(b) \(\frac{1}{x(log5)(log7)}\)
(c) \(\frac{1}{x(logx)}\)
(d) None of these
Answer
Answer: (a) \(\frac{1}{x(log5)(log7)(log7-x)}\)
Question 48.
If y = log10x + log y, then \(\frac{dy}{dx}\) is equal to
(a) \(\frac{y}{y-1}\)
(b) \(\frac{y}{x}\)
(c) \(\frac{log_{10}e}{x}\)(\(\frac{y}{y-1}\))
(d) None of these
Answer
Answer: (c) \(\frac{log_{10}e}{x}\)(\(\frac{y}{y-1}\))
Question 49.
If y = log [ex(\(\frac{x-1}{x-2}\))\(^{1/2}\)], then \(\frac{dy}{dx}\) is equal to
(a) 7
(b) \(\frac{3}{x-2}\)
(c) \(\frac{3}{(x-1)}\)
(d) None of these
Answer
Answer: (d) None of these
Question 50.
If y = e\(\frac{1}{2}\) log(1+tan²x), then \(\frac{dy}{dx}\) is equal to
(a) \(\frac{1}{2}\) sec² x
(b) sec² x
(c) sec x tan x
(d) e\(\frac{1}{2}\) log(1+tan²x)
Answer
Answer: (c) sec x tan x
Question 51.
If y = 2x32x-1 then \(\frac{dy}{dx}\) is equal to dx
(a) (log 2) (log 3)
(b) (log lg)
(c) (log 18²) y²
(d) y (log 18)
Answer
Answer: (d) y (log 18)
Question 52.
If xx = yy, then \(\frac{dy}{dx}\) is equal to
(a) –\(\frac{y}{x}\)
(b) –\(\frac{x}{y}\)
(c) 1 + log (\(\frac{x}{y}\) )
(d) \(\frac{1+logx}{1+logy}\)
Answer
Answer: (d) \(\frac{1+logx}{1+logy}\)
Question 53.
If y = (tan x)sin x, then \(\frac{dy}{dx}\) is equal to
(a) sec x + cos x
(b) sec x+ log tan x
(c) (tan x)sin x
(d) None of these
Answer
Answer: (d) None of these
Question 54.
If xy = ex-y then \(\frac{dy}{dx}\) is
(a) \(\frac{1+x}{1+log x}\)
(b) \(\frac{1-log x}{1+log y}\)
(c) not defined
(d) \(\frac{-y}{(1+log x)^2}\)
Answer
Answer: (d) \(\frac{-y}{(1+log x)^2}\)
Question 55.
The derivative of y = (1 – x) (2 – x)…. (n – x) at x = 1 is equal to
(a) 0
(b) (-1) (n – 1)!
(c) n ! – 1
(d) (-1)n-1 (n – 1)!
Answer
Answer: (b) (-1) (n – 1)!
Question 56.
If f(x) = cos x, cos 2 x, cos 4 x, cos 8 x, cos 16 x, then the value of'(\(\frac{π}{4}\)) is
(a) 1
(b) √2
(c) \(\frac{1}{√2}\)
(d) 0
Answer
Answer: (b) (-1) (n – 1)!
Question 57.
xy. yx = 16, then the value of \(\frac{dy}{dx}\) at (2, 2) is
(a) -1
(b) 0
(c) -1
(d) None of these
Answer
Answer: (a) -1
Question 58.
If y = ex+ex+ex+….to∞ find \(\frac{dy}{dx}\) =
(a) \(\frac{y^2}{1-y}\)
(b) \(\frac{y^2}{y-1}\)
(c) \(\frac{y}{y-1}\)
(d) \(\frac{-y}{y-1}\)
Answer
Answer: (c) \(\frac{y}{y-1}\)
Question 59.
If x = \(\frac{1-t^2}{1+t^2}\) and y = \(\frac{2t}{1+t^2}\) then \(\frac{dy}{dx}\) is equal to dx
(a) –\(\frac{y}{x}\)
(b) \(\frac{y}{x}\)
(c) –\(\frac{x}{y}\)
(d) \(\frac{x}{y}\)
Answer
Answer: (c) –\(\frac{x}{y}\)
Question 60.
If x = a cos4 θ, y = a sin4 θ. then \(\frac{dy}{dx}\) at θ = \(\frac{3π}{4}\) is
(a) -1
(b) 1
(c) -a²
(d) a²
Answer
Answer: (a) -1
Question 61.
If x = sin-1 (3t – 4t³) and y = cos-1 (\(\sqrt{1-t^2}\)) then \(\frac{dy}{dx}\) is equal to
(a) \(\frac{1}{2}\)
(b) \(\frac{2}{5}\)
(c) \(\frac{3}{2}\)
(d) \(\frac{1}{3}\)
Answer
Answer: (d) \(\frac{1}{3}\)
Question 62.
Let y = t10 + 1 and x = t8 + 1, then \(\frac{d^2y}{dx^2}\), is equal to
(a) \(\frac{d^2y}{dx^2}\)
(b) 20t8
(c) \(\frac{5}{16t^6}\)
(d) None of these
Answer
Answer: (d) \(\frac{1}{3}\)
Question 63.
The derivative of ex3 with respect to log x is
(a) ee3
(b) 3x22ex3
(c) 3x3ex3
(d) 3x2ex3+ 3x2
Answer
Answer: (c) 3x3ex3
Question 64.
If x = et sin t, y = etcos t, t is a parameter, then \(\frac{dy}{dx}\) at (1, 1) is equal to
(a) –\(\frac{1}{2}\)
(b) –\(\frac{1}{4}\)
(c) 0
(d) \(\frac{1}{2}\)
Answer
Answer: (c) 0
Question 65.
The derivative of sin-1 (\(\frac{2x}{1+x^2}\)) with respect to cos-1 (\(\frac{1-x^2}{1+x^2}\)) is
(a) -1
(b) 1
(c) 2
(d) 4
Answer
Answer: (b) 1