CBSE Class 12 Maths –Chapter 5 Continuity and Differentiability- Important Questions

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) =MCQ Questions for Class 12 Maths Chapter 5 Continuity and Differentiability with Answersis 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) = MCQ Questions for Class 12 Maths Chapter 5 Continuity and Differentiability with Answers
(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 \(\frac{dy}{dx}\)
MCQ Questions for Class 12 Maths Chapter 5 Continuity and Differentiability with Answers

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
MCQ Questions for Class 12 Maths Chapter 5 Continuity and Differentiability with Answers

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


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