Notes and Study Materials
- Concepts of Continuity and Differentiability
- Master File for Continuity and Differentiability
- Continuity and Differentiability Note
- NCERT Solutions for – Continuity and Differentiability
- NCERT Exemplar Solutions for – Continuity and Differentiability
- R D Sharma Solution of Algebra of Differentiability
- R D Sharma Solution of Algebra of Continuity
- Past Many Years CBSE Questions and Answer Of Relation and Function
- Continuity and Differentiability Mind Map
Examples and Exercise
CBSE Revision Notes for CBSE Class 12 Mathematics Continuity and Differentiability Continuity and differentiability, derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit functions. Concept of exponential and logarithmic functions. Derivatives of logarithmic and exponential functions. Logarithmic differentiation, derivative of functions expressed in parametric forms. Second order derivatives. Rolle’s and Lagrange’s Mean Value Theorems (without proof) and their geometric interpretation.
CONTINUITY
DEFINITION A function f (x) is said to be continuous at x = a; where a ∈ domain of f(x), if
i.e., LHL=RHL = value of a function at x= a
Reasons of discontinuity
If f (x) is not continuous at x = a, we say that f (x) is discontinuous at x = a.
There are following possibilities of discontinuity
PROPERTIES OF CONTINUOUS FUNCTIONS
THE INTERMEDIATE VALUE THEOREM
Suppose f(x) is continuous on an interval I, and a and b are any two points of I. Then if y₀ is a number between f(a) and f(b), their exits a number c between a and b such that f(c)=y₀.
Note :-
That a function f which is continuous in [a, b] possesses the following properties:
( i ) If f (a) and f(b) possess opposite signs, then there exists at least one solution of the equation f(x) =0 in the open interval (a, b).
(ii) If K is any real number between f(a) and f(b), then there exists at least one solution of the equation f (x) = K in the open interval (a, b).
CONTINUITY IN AN INTERVAL
(a) A function f is said to be continuous in (a, b) if f is continuous at each and every point ∈ (a, b).
(b) A function f is said to be continuous in a closed interval [a, b] if:
(1) f is continuous in the open interval (a, b) and
(2) f is right continuous at ‘a’ i.e. limit ₓ→ₐ⁺ f(x)=f(a) = a finite quantity.
(3) f is left continuous at ‘b’; i.e. limit ₓ→b⁻f(x) = f(b) a finite quantity.
A LIST OF CONTINUOUS FUNCTIONS
TYPES OF DISCONTINUITIES
Type-1: (Removable type of discontinuities)
In case, limit ₓ→c f(x) exists but is not equal to f (c) then the function is said to have a removable discontnuity or discontinuity of the first kind. In this case, we can redefine the function such that limit ₓ→c f(x) = f (c) and make it continuous at x=c. Removable type of discontinuity can be further classified as:
(a) Missing Point Discontinuity :
Where limit ₓ→ₐ f(x) exists finitely but f(a) is not defined.
From the adjacent graph note that
– f is continuous at x =-1
– f has isolated discontinuity at x = 1
– f has missing point discontinuity at x = 2
– f has non-removable (finite type) discontinity at the origin.
(a) In case of dis-continuity of the second kind the non- negative difference between the value of the RHL at x=a and LHL at x=a is called the jump of discontinuity. A function having a finite number of jumps in a given interval I is called a piece wise continuous or sectionally continuous function in this interval.
(b) All Polynomials, Trigonometrical functions, exponential and Logarithmic functions are continuous in their domains.
(c) If f (x) is continuous and g (x) is discontinuous at x = a then the product function ∅ (x)=f(x). g (x) is not necessarily be discontinuous at x = a. e.g.
DIFFERENTIABILITY
DIFFERENTIABILITY IN A SET
1. A function f (x) defined on an open interval (a, b) is said to be differentiable or derivable in open interval (a, b), if it is differentiable at each point of (a, b).
2. A function f (x) defined on closed interval [a, b] is said to be differentiable or derivable. “If f is derivable in the open interval (a, b) and also the end points a and b, then f is said to be derivable in the closed interval [a, b]”.
A function f is said to be a differentiable function if it is differentiable at every point of its domain.
Note :-
1. If f (x) and g (x) are derivable at x = a then the functions f(x)+g (x), f(x)-g(x), f(x). g (x) will also be derivable at x=a and if g (a) ≠ 0 then the function f (x)/g(x) will also be derivable at x=a.
2. If f (x) is differentiable at x = a and g (x) is not differentiable at x = a, then the product function F (x) = f (x). g (x) can still be differentiable at x=a.
E.g. f (x) =x and g (x) = |x|.
3. If f (x) and g (x) both are not differentiable at x = a then the product function; F (x) = f(x). g (x) can still be differentiable at x = a.
E.g., f(x) = |x| and g (x) = |x|.
4. If f (x) and g (x) both are not differentiable at x=a then the sum function F (x) =f (x)+g (x) may be a differentiable function.
E.g., f (x) = |x| and g (x)=-|xl.
5. If f(x) is derivable at x= a
• F’ (x) is continuous at x=a.
RELATION B/W CONTINUITY & DIFFERENNINBILINY
In the previous section we have discussed that if a function is differentiable at a point, then it should be continuous at that point and a discontinuous function cannot be differentiable. This fact is proved in the following theorem.
Theorem : If a function is differentiable at a point, it is necessarily continuous at that point. But the converse is not necessarily true,
Or f(x) is differentiable at x=c
f(x) is continuous at x = c.
Note.
Converse : The converse of the above theorem is not necessarily true i.e., a function may be continuous at a point but may not be differentiable at that point.
E.g., The function f (x) = |x| is continuous at x = 0 but it is not differentiable at x = 0, as shown in the figure.
The figure shows that sharp edge at x = 0 hence, function is not differentiable but continuous at x = 0.
Note:-
(a) Let f´⁺(a) = p & f´⁻(a)=q where p & q are finite then
(b) If a function f is not differentiable but is continuous at x = a it geometrically implies a sharp corner at X=a.
Theorem 2 : Let fand g be real functions such that fog is defined if g is continuous at x = a and f is continuous at g (a), show that fog is continuous at x = a.
DIFFERENTIATION
DEFINITION
(a) Let us consider a function y=f(x) defined in a certain interval. It has a definite value for each value of the independent variable x in this interval.
Now, the ratio of the increment of the function to the increment in the independent variable
DERIVATIVE OF STANDARD FUNCTION
THEOREMS ON DERIVATIVES
METHODS OF DIFFERENTIATION
4.1 Derivative by using Trigonometrical Substitution
4.2 Logarithmic Differentiatio
DERIVATIVE OF ORDER TWO & THREE
Let a function y =f (x) be defined on an open interval (a, b). It’s derivative, if it exists on (a, b), is a certain function f'(x) [or (dy/dx) or y’] & is called the first derivative of y w.r.t. x. If it happens that the first derivative has a derivative on (a, b) then this derivative is called the second derivative of y w.r.t. x & is denoted by f”(x) or (d²y/dx²) or y”.
Similarly, the 3rd order derivative of y w.r.t. x, if it exists, is
Important Questions for CBSE Class 12 Maths Continuity
Previous Year Examination Questions
4 Marks Questions
CBSE Class 12 Maths Differntiability
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