lim ₓ→₄ (f(x)- f(4))/(x-4) has the value
A) 5/4 B) -4/5 C) 4/5D) none
2) If f(x)= log|x|, x ≠ 0 then f'(x) is
A)1/|x| B) 1/x C) -1/x D) none
3) If f(x)= |logx|, then for x≠1, f'(x) is..
A) 1/x B) 1|x| C) -1/x D) none
4) If f(9)= 9 and f'(9)= 4 then lim ₓ→₉ {√f(x) -3}/(√x -3) is..
A) 9 B) 4 C) 36 D) none
5) If f(x)= |log|x||, then f'(x) is
A) 1/|x|, x≠0 B) 1/x for |x|> 1 and-1/x for |x|<1. C) -1/x for |x|> 1 and 1/x for |x|< 1. D) 1/x for x> 0 and -1/x for x< 0
6) if f(x)= (cosx+ i sinx)(cos 3x + i sin3x).... (Cos(2n-1)x + i sin(2n -1)x, then f"(x)=
A) n² f(x) B)- n⁴f(x). C) -n² f(x) D) n⁴ f(x)
7) If f(x)= xⁿ, then the value of f(1)+ f'(1)/1! + f"(1)/2! + f"'(1)/3! + ...+ fⁿ(1)/n! is
A) n B) 2ⁿ C) 2ⁿ⁻¹ D) n(n+1)/2
8) Let f(x) be a polynomial in x, Then the second order of derivative of f(eˣ) is
A) f"(eˣ). eˣ+ f'(eˣ)
B) f"(eˣ). e²ˣ+ f'(eˣ).e²ˣ
C) f"(eˣ). e²ˣ
D) f"(eˣ). e²ˣ+ f'(eˣ) eˣ.
9) Let f(x)= xⁿ, n being a non- negative integer. The value of n for which the equality f'(x+y)= f'(y) is
valid for all x, y > 0 is
A) 0 B) 1 C) 2 D) none
10) If f(x)= sinx, g(x)= x² and h(x)= logx . if F(x)= (hog of)(x), then F"(x) is equals to
A) 2 cosec²x
B) 2 cotx² - 4x² cosec²x²
C) 2x cotx² D) - 2 cosec²x
11) If y= sin⁻¹{sink sinx)/(1- cosk sinx)}, then y'(0) is..
A) 1 B) 2 tank C) 1/2 tank D) sink
12) If y= logₓ2₊₄ (7x²-5x+1) then dy/dx is equal to
A) log(x²+4){(14x-5)/(7x²-5x+1) - 2xy/(x²+4)}
B) 1/{log(x²+4)}{(14x-5)/(7x²-5x+1) - 2xy/(x²+4)}
C) log(7x²-5x+1){(2x/(x²+4) - (14x-5)y/(7x²-5x+1)}
D) 1/{log(7x²-5x+1)}{(2x/(x²+4) - (14x-5)y/(7x²-5x+1)}
13) If f(x)= sin{π/3 [x] -x²} for 2<x<3
and [x] denotes the greatest integer less than or equal to x, then f'(√(π/3)) is equal to
A) √(π/3). B) - √(π/3) C) -√(π) D) n
14) If f(x)= cot⁻¹{ xˣ - x⁻ˣ}/2 then f'(1) is
A) -1 B) 1 C) log2 D) - log 2
15) The function satisfy u=eˣ sinx ; v= eˣ cosx satisfy the equation
A) v du/dx - u dv/dx= u² + v²
B) d²u/dx² = 2v C) d²v/dx² = -2u D)n
16) If f(x)= |x- 2| and g(x)= f(f(x)), then for x> 20, g'(x) equals
A) -1 B) 1 C) 0 D) none
17) If f(x) = |x- 2| and g(x)= f(f(x)), then for 2<x <4, g'(x) equals
A) -1 B) 1 C) 0 D) none
18) If a curve is given by x= a cost + b/2 cos 2t and y= a sint + b/2 sin2t , then the points for which d²y/dx²= 0 are given by
A) sint= (2a²+b²)/3ab
B) cost=- (a²+ 2b²)/3ab
C) tan t= a/b. D) none
19) if f(x)= logₓ |ln (x)|, then f'(x)= e is
A) e. B) -e c) e² D) 1/e
20) Let f(t)= ln(t), then {d/dx ∫f(t) dt} at (x³, x²)
A) has a value zero when x= 0
B) has a value zero when x=1 and x= 4/9
C) has a value 9e² - 4e when x= e
D) has a the differential coefficient 27e - 8 for x= e
21) If g is inverse of f and f'(x)= 1/(1+xⁿ) , then f'(x) equals
A) 1+ xⁿ B) 1+[f(x)]ⁿ C) 1+[g(x)]ⁿ D)n
22) If f(x)= |x| ^|sinx|, then f'(-π/4) =
A) (π/4)^(1/√2){√2/2 ln(4/π -2√2/π}
B) (π/4)^(1/√2){√2/2 ln(4/π+2√2/π}
C) (π/4)^(1/√2){√2/2 ln(π/4 -2√2/π}
D)(π/4)^(1/√2){√2/2 ln(4/π +2√2/π}
23) If y= cos⁻¹{2x/(1+x²)}, then y' is
A) -2/(1+x²) for all x
B) -2/(1+x²) for all |x|> 1
C) 2/(1+x²) for |x|< 1 D) none
24) If ʸ₀∫ 1/√(1+4t²) dt, then d²y/dx²
A) 2y B) 4y C) 8y D) 6y
25) If f(x)=√(x² -2x +1), then
A) f'(x)= 1 for all x
B) f'(x)= -1 for all x ≤ 1
C) f'(x)= 1 for all x ≥ 1. D) none
26) If f(x)= √(1- sin2x), then f'(x)=?
A) -(cosx+ sinx), for x ∈ (π/4,π/2)
B) (cosx+ sinx), for x ∈ (0,π/4)
C) -(cosx+ sinx), for x ∈ (0,π/4)
D) (cosx- sinx), for x ∈ (π/4,π/2)
27) If f(x)=|x²-5x+6|, then f'(x)=?
A) 2x-5 for 2<x<3
B) 5- 2x for 2<x<3
C) 2x-5 for 2≤x≤3
D) 5 - 2x for 2≤x≤3
28) If x² + y²= a² and k -1/a, then k is equal to
A) y"/(√(1+y') B) |y"|/(√(1+y' ²)³
C) 2y"/(√(1+y' ²) D) y"/(2√(1+y' ²)³
29) If f(x)= sinx and g(x)=s g n sinx, then g'(1) equal to
A) 0 B) - cos 1 C) cos 1 D) none
30) Let f(x) and g(x) be two functions having finite non-zero 3rd order derivatives f"'(x) and g"'(x) for all x ∈ R. If f(x) g(x)= 1 for all x ∈ R, then f"/f' - g"'/g' is equal to
A) 3(f"/g - g"/f) B) 3(f"/f - g"/g)
C) 3(g"/g - f"/g) D) 3(f"/g - g"/f)
31) If y= cos⁻¹{(2 cosx - 3sinx)/√13} then dy/dx is.
A) zero B) constant=1
C) constant≠1 D) none
32) If y= x + eˣ, then y" is
A) eˣ B) - eˣ/(1+ eˣ)³
C) - eˣ/(1+ eˣ)² C) 1/(1+ eˣ)²
33) If f(x)= 1/x² ˣ₀ ∫ (4t² - 2 F'(t) dt, then F'(4) equal to
A) 32/9 B) 64/3 C) 64/9 D) n
34) If y²= P(x) is a polynomial of degree 3, then 2 dy/dx(y³ d²y/dx²) is
A) P(x)+ P"(x). B) P(x)
C) P(x) P"'(x). D) constant
35) 2ˣ+ 2ʸ = 2ˣ⁺ʸ, then the value of dy/dx at x=y=1 is
A) 0 B) -1 C) 1 D) 2
36) tan⁻¹[{√(1+x²) - 1}/x] w.r.t. tan⁻¹[{2x √(1-x²)}/(1-2x²)] at x= 0, is
A)1/8 B) 1/4 C) 1/2 D) 1
37) If y= tan⁻¹{log(e/x²)/log(ex²)}] + tan⁻¹{(3+ 2log x)/(1- 6 logx)} then y"
A) 2 B) 1 C) 0 D) -1
38) The expression of dy/dx of the function y= aˣ·······ᵅ is
A) y²/{x(1- y logx)}
B) y²log y/{x(1- y logx)}
C) y²logy /{x(1- y logx logy)}
D) y²logy/{x(1+ y logx logy)}
39) If √(1-x²)+ √(1-y²)= a(x-y), then dy/dx is..
A) √{(1-x²)(1-y²). B) √{(1-y²)/(1-x²)}
C) √{(1-x²)/(1-y²)} D) none
40) if y= log꜀ₒₛₓsinx , then dy/dx is
A) (cotx log cosx+ tanx log sinx)/(log cosx)²
B) (tanx log cosx+ cotx log sinx)/(log cosx)²
C) (cotx log cosx+ tanx log sinx)/(log sinx)². D) none
41) xʸ = eˣ⁻ʸ, then dy/dx is equal to
A) (1+ logx)⁻¹ B) (1+ logx)⁻²
C) logx (1+ logx)⁻². D) none
42) Let f(x)= x²/(1-x²) , x ≠ 0, ±1, then derivative of f(x) with respect to x is
A) 2x/(1+x²)² B) 1/(2+x²)³
C) 1/(1- x²)². D) 1/(2- x²)²
43) If y= ₑsin⁻¹x & u= logx, then dy/du is
A) ₑsin⁻¹x/√(1-x²) B) x ₑsin⁻¹x
C) (xₑsin⁻¹x)/√(1-x²) D) none
44) The differential coefficient of log(logx) w.r.t. x is
A) x/(logx) B) (logx)/x
C) 1/(xlogx). D) x logx
45) The derivative of log |x| is
A) 1/x, x> 0 B) 1/|x|, x≠ 0
C) 1/x, x≠0 D) none
46) If y= sin⁻¹[{√(x+1)+(1-x)})2] w.r.t.x
A) -1/{2√(1-x²)} B) 1/{2√(1-x²)}
C) 2/{√(1-x²)} D) -2/{√(1-x²)}
47) sec t tan²t 1
t secx tanx x
1 tanx - tan t 0 then f'(t) is
A) 0 B) -1 C) independent t D) none
48) The differential coefficient of f(logx) where f(x)= logx is
A) x/(logx) B) 1/x/(xlogx)
C) (logx)/x D) none
49) If xᵖ yᑫ = (x+y)ᵖ⁺ᑫ , then dy/dx is..
A) y/x B) py/qx. C) x/y D) qy/px
50) If sec⁻¹{1/(2x²-1)} w.r.t. √(1-x²) at x=1/2.
A) 2 B) 4 C) 1 D) -2
51) y= sec⁻¹{(x+1)/(x-1)}+ sin⁻¹{(x-1)/(x+1), y' is
A) 1 B) (x-1)/(x+1( C) 0 D) (x+1)/(x-1)
52) f(x)= sin(logx) and y= f{(2x+3)/(3-2x)} then f'(x) is.
A) sin(logx).1/(xlogx)
B) 12/(3-2x)² sin[log{(2x+3)/(3-2x)}]
C) sin[log{(2x+3)/(3-2x). D) none
53) If f(x)=( log꜀ₒₜₓ tanx) (logₜₐₙₓcotx)⁻¹+ tan⁻¹{4x/√(4-x²)} then f'(0) is
A) 2 B) 0 C) 1/2 D) -2
54) If f(x)= eˣ g(x), g(0)=2, g'(0)= 1, then f'(0) is.
A) 1 B) 3 C) 2 D) 0
55) If sin⁻¹{(x²-y²)/(x²+ y²)}= log a, then dy/dx is
A) x/y B) y/x² C) (x²-y²)/(x²+y²) D) y/x
56) y= sec⁻¹{(√x+1)/(√x-1)}+ sin⁻¹{(√x-1)/(√x+1)} then dy/dx
A) 1 B) 0 C) (√x+1)/(√x-1)}
D) (√x-1)/(√x+1)}
57) If x²+ y²=t- 1/t and x⁴ + y⁴= t²+ 1/t² then x³y dy/dx is
A) 0 B) 1 C) -1 D) none
58) If y= tan⁻¹[{√(1+x²)+ √(1-x²)}/{√(1+x²) - √(1-x²)}] then y' is
A) 1/√(1-x⁴) B) -1/√(1-x⁴)
C) x/√(1-x⁴). D) -x/√(1-x⁴)
59) ˣₒ∫ f(t) sin {k(x-t)} dt, then d²y/dx² + k²y is..
A) 0 B) y C) kf(x) D) k²f(x)
60) If fₙ(x), gₙ(x), hₙ(x), n= 1, 2, 3 are polynomial in x such that fₙ(a) = gₙ(a)= hₙ(a), n= 1, 2, 3 and
f₁(x) f₂(x) f₃(x)
F(x)= g₁(x) g₂(x) g₃(x)
h₁(x) h₂(x) h₃(x) then F'(a) is equal to
A) 0 B) f₁(a) g₂(a) h₃(a)
C) 1 D) none
61) If x= g(t), y= h(t), then d²y/dx² is
A) (g'h" - h' g")/(g')²
B) (g'h" - h' g")/(g')³
C) g'/h" D) h"/g"
62) If y² = ax² + bx+ c where a, b are constants, then y³ d²y/dx² is equal
A) a constant
B) a function of x
C) a function of y
D) a function of x and y both
63) If x= a cost, y= b sint , then d³y/dx³ is equal to
A) - 3b/a³ cosec⁴t cot⁴t
B) 3b/a³ cosec⁴t cott
C) - 3b/a³ cosec⁴t cott
D) none
64) If y= sinx + eˣ, then d²y/dx² is..
A) 1/(-sinx + eˣ)
B) (sinx - eˣ)/(cosx + eˣ)²
C) (sinx - eˣ)/(cosx + eˣ)³
D) (sinx + eˣ)/(cosx + eˣ)³
65) If variable x and y are related by the equation x ʸ₀ ∫ du/√(1+9y²) then dy/dx is..
A) 1/√(1+9y²). B) √(1+9y²)
C) (1+9y²). D) 1/(1+9y²)
66) The differential coefficient of ₐ log₁₀cosec⁻¹x is
A) ₐ log₁₀cosec⁻¹x /cosec⁻¹x .1/{x√(x²-1)} log₁₀a
B) - ₐ log₁₀cosec⁻¹x /cosec⁻¹x .1/{|x|√(x²-1)} log₁₀a
C) - ₐ log₁₀cosec⁻¹x /cosec⁻¹x .1/{|x|√(x²-1)} logₐ10
D) ₐ log₁₀cosec⁻¹x /cosec⁻¹x .1/{x√(x²-1)} logₐ10
67) If f(x)= tan⁻¹[{log(e/x²)}/{log(ex²)}] + tan⁻¹{(3+ 2 logx)/(1-6 logx)}, then dⁿy/dxⁿ
A) tan⁻¹{log xⁿ)} B) 0 C) 1/2 D) n
68) If y= sin²a+ cos²(a+b)+ 2 sina sin b cos (a-b), then d³y/dx³ is
A) sin²(a+b)/cosa
B) cos(a+3b). C) 0 D) none
69) If y= cos2x cos3x, then yₙ is
equals to
A) 6ⁿ cos(2x + nπ/2) cos(3x + nπ/2)
B)6ⁿ sin(2x + nπ/2) cos(3x + nπ)/2
C) 1/2[5ⁿsin(5x +nπ/2) +sin(x+π/2)]
D) none
70) If f(x)= (x+1)tan⁻¹(e⁻²ˣ), then f'(0) is...
A) π/2+ 1 B) π/4 - 1 C) π/6+ 5 D) n
71) If f(x)= 3. ₑx², then f'(x)= 2x f(x) + 1/3 f(0)- f'(0) is equals to
A) 0 B) 1 C) 7/3 ₑx² D) none
72) If y= c. ₑx/(x-a), then dy/dx is
A) a(x-a)². B) - ay/(x-a)²
C) a²(x-a)². D) none
73) A curve is given by the equations x= a cost + 1/2 b cos2t, y= a sint + 1/2 b sin2t. Then the points for which d²y/dx²= 0 are given by
A) sint= (2a²+b²)/5ab
B) tan t= (3a²+ 2b²)/4ab
C) cos t= (a²+ 2b²)/3ab. D) none
74) If ¹⁾ᵐ = [x + √(1+x²)] then (1+x²) y₂ + xy₁ is equals to
A) m²y B) my² C) m²y² D) none
75) If eʸ + xy= e , then d²y/dx² at x= 0
A) 1/e B) 1/e². C) 1/e³. D) none
76) If √(x +y)√(y - x)= c, then d²y/dx²
A) 2/c B) -2/c² C) 2/c². D) none
77) If ax² + 2hxy + by²= 1 then d²y/dx² is
A) (h²+ab)/(hx+ by)²
B) (h²- ab)/(hx+ by)²
C) (h²+ab)/(hx+ by)³
D) (h²- ab)/(hx+ by)³
78) If f(x)= ₓ² (logx), then f'(x) at x= e is
A) 0 B) 1 C) 1/e D) 1/2e
79) The differential coefficient of f(logx) w.r.t.x where f(x)= logx is
A) x/(logx) B) (logx)/x
C) 1/(x logx) D) none
80) The derivative of the function cos⁻¹[cos 2x)¹⁾²] at x=π/6 is
A) (2/3)¹⁾² B) (1/3)¹⁾²
C) 3¹⁾² D) 6¹⁾²
81) Differential coefficient of sec(tan⁻¹x) is
A) x/(1+x²). B) x √(1+x²)
C) 1/√(1+x²). D) x/√(1+x²)
82) If f(x)= tan⁻¹{(1+sinx)/(1-sinx)}, 0≤ x ≤π/2, then f'(π/6) is
A)-1/4 B) -1/2 C) 1/4 D) 1/2
83) If y= (1+ 1/x)ˣ then dy/dx is
A) (1+ 1/x)ˣ[log(1+ 1/x) - 1/(x+1)]
B) (1+ 1/x)ˣ log(1+ 1/x)
C) (x+ 1/x)ˣ[log(x+ 1) - x/(x+1)]
D) (x+ 1/x)ˣ[log(1+ 1/x) - 1/(x+1)]
84) If xʸ = eˣ⁻ʸ, dy/dx is
A) (1+x)/(1+ logx)
B) (1- logx)/(1+ logx)
C) not defined
D) logx)/(1+ logx)²
85) lim ₓ→₁₊ {√(x+1)+ √(x-1)}/√x²-1)=
A) 1/2 B) √2 C) 1 D) 1/√2
86) If x= a cos³t,.y= a sin³t, √(1+ (dy/dx)²)=
A) tan²t B) sec²t C) sect D) |sect|
87) If y= sin⁻¹{(1-x²)/(1+x²), then dy/dx=
A) -2/(1+x²) B) 2/(1+x²)
C) 1/(2- x²) D) 2/(2 -x²)
88) sec{1/(2x²+1)} w.4.t. √(1+3x) at x=-1/3
A) doesn't exist B) 0 C) 1/2 D) 1/3
89) for the curve√x +√y= 1, dy/dx at (1/4,1/4) is
A) 1/2 B) 1 C) -1 D) 2
90) If sin(x+y)= log(x+y), then dy/dx =
A) 2 B)) -2 .C) 1 D) -1
91) Let U= sin⁻¹{2x/(1+x²)} and V= tan ⁻¹{2x/(1-x²)}, then dU/dV is
A) 1/2 B) x C) (1-x²)(1+ x²) D) 1
92) If x= a cos nt - b sin nt, then d²y/dx² is
A)n²x B) -n²x C) -nx D) nx
93) d/dx [tan⁻¹{cosx/(1+sinx)}=
A)1/2 B) -1/2 C) 1 D) -1
94) d/dx[log{eˣ.{(x-2)/(x+2)}³⁾⁴}]=
A) (x² -1)/(x² -4)
B) 1 C) (x² +1)/(x² -4)
D) eˣ{(x² +1)/(x² -4)}
95) If y= √(sinx + y), then dy/dx
A) sinx/(2y-1). B) sinx/(1-2y)
C) cosx/(1-2y). D) cosx/(2y-1)
96) x= at² ,.y= 2at, then d²y/dx²=
A) -1/t² B) 1/2at³ C) -1/t³ D)-1/2at³
97) if y= axⁿ⁺¹ + bx⁻ⁿ , then x² d²y/dx²=
A) n(n-1)y. B) n(n+1)y.
C) ny D) n²y
98) d²⁰/dx²⁰ (2 cosx cos3x)=
A) 2²⁰(cos2x - 2²⁰cos3x)
B) 2²⁰(cos2x + 2²⁰cos 4x)
C) 2²⁰(sin2x + 2²⁰ sin 4x)
D) 2²⁰(sin 2x - 2²⁰ sin 4x)
99) If y= a + bx², a, b arbitrary constants, then
A) d²y/dx²= 2xy
B) xd²y/dx²= dy/dx
C) xd²y/dx² - dy/dx+ y= 0
D) xd²y/dx²= 2xy
100) If 3 sin(xy) + 4 cos(xy)= 5, then dy/dx=
A) -y/x
B) (3sin(xy)+4 cos(xy))/(3cos(xy) - 4 sin(xy))
C) (3cos(xy)+4 sin(xy))/(4cos(xy) - 3 sin(xy)). D) none
101) If siny = x sin(a+y), then dy/dx
A) sina/{sina sin²(a+y)}
B) sin²(a+y)/sin a
C) sina sin²(a-y)}
D) sin²(a+y)/sina
102) Cos⁻¹(2x² -1) w.r.t. cos⁻¹x is
A) 2 B) 1/{2√(1-x²)} C) 2/x D) 1- x²
FILL IN THE BLANKS:
1) If y= f{(2x-1)/(x²+1)} and f'(x)= sin x², then dy/dx= ____
2) If f(x)= logₓ(logₑx(, then f'(x) at x= e is _____
3) If f(x)= |x -2| and g(x)= f o f(x), then for x> 20 g'(x)= ____
4) If f(x)=|x - a| and g(x)= = f(f(f(x))), then g'(x)= ___ for x> 4a.
5) If f(x)= log tan{(2x+1)/4}, then f'(0)= _____
6) If y= (e²ˣ -1)/(e²ˣ +1), then y² + dy/dx= _____
7) If y=√[x+ √{y+ √(x+ √(y+...∞))}] then dy/dx is equal to____
8) If d/dx{(1+x²+ x⁴)/(1+x+ x²)}= ax + b, then a = ____, b= ____.
9) If y= sin⁻¹x satisfies the equation (1-x²) y₂ = f(x) y₁ , then f(x)=_____
10) If y= log{x+√(1+x²)}, then y₂(0)=______
11) If f(x)= sin(log x) and y= f{(2x+3)/(3-2x)}, then dy/dx=-----
TRUE/FALSE
1) if xy = c² where c is constant and if u is any function of x, then x dy/dx + y du/dc= 0
2) If x=f(t) and y= g(t) and if d²y/dx²= 0 , then dx/dy . d²y/dt²= dy/dx. d²x/dt².
3) If y= tan⁻¹{4x/(1+5x²)} + tan⁻¹{(2+3x)/(3-2x)}, then dy/dx= 5x/(1+25x²)
4) If u= ax+ b, then dⁿ/dxⁿ [f(ax+b)] is equal to aⁿ. dⁿ/duⁿ[f(u)].
EXERCISE -2
1) If x= √t + 1/√t, then 2t dx/dt + x
A) √t B) 2√t C) 3√t D) none
2) If (x+4)y= x, then x dy/dx + y(y-1) is
A) 1 B) 2 C) 1/2 D) none
3) If y= xⁿ⁻¹ logx, then x dy/dx +(1-n)y is
A) xⁿ⁻¹ B) xⁿ⁻² C) nxⁿ⁻¹ D) none
4) If y= 6x⁴ + 4x³ - 24x² - 36x+ 21 and dy/dx at x= 2 is 12 then a is..
A) 1 B) 2 C) 3 D) 4
5) if y= 1/(x ᑫ⁻ᵖ + x ʳ⁻ᵖ) + 1/(1+ x ᵖ⁻ᑫ + x ʳ⁻ᑫ) + 1/(1+ x ᵖ⁻ʳ +x ᑫ⁻ʳ) then dy/dx is
A) x ᵖ⁺ᑫ⁺ʳ B) 1/(p+q+r) C) 1 D) 0
6) If y= (1+sinx)/cosx, then cosx dy/dx is..
A) sin²x B)1+y C) y D) none
7) If x²+ y²= 4, then y dy/dx + x is
A) 4 B) 0 C) -1 D) none
8) y= x ˣ (x >0), then dy/dx is..
A) y(1+logx). B) (1+logx)
C) y logx D) x(1+logx)
9) If y= 2x tan⁻¹x - log(1+x²) then dy/dx = k tan⁻¹x where the value of k is
A) 1 B) 1/2 C) 1/3. D) 2
10) y= sin x°, then dy/dx = π/180 cost where t is equal to
A) x B) x ᶜ C) x/180 D) πx/180
11) If y= log(logx), then dy/dx= k/(logx) where k is..
A) x B) 1 C) 1/x. D) none
12) Let y= sin⁻¹(cosx). Then
A) dy/dx= -1 when sinx < 0
B) dy/dx= 1 when sinx > 0
C) dy/dx does not exist at x=π
D) dy/dx does not exist anywhere
13) cosx ={1-t²)/(1+t²)}, tany ={(3t-t³)/(1-3t³)} (0< t <1/√3), then dy/dx is
A) 1 B)1/2 C) 3/2. D) 2
14) If x⁶y²= (x + y)⁸, then dy/dx is
A) x/y. B) y/x C) - y/x. D) 1
15) If x²+ y²= t+ 1/t and x⁴ +y⁴=t²+ 1/t², then dy/dx is
A) -1/x³y B) 1/x³y C) x³/y D) -y/x³
16) If y= cos[2 sin⁻¹(cosx), then dy/dx is
A) 4sinx cosx. B) sin2x
C) 2 cos2x D) none
17) If y= log(secx + tanx), then dy/dx is..
A)sinx B) cosx C) secx D) Cosecx
18) If y= log tan(π/4+ x/2), then dy/dx is
A) secx B) tanx C) Cosecx D) n
19) If y= log(log(logx)), then dy/dx is..
A) (log(logx)). B) logx (log(logx))
C) 1/(x log(log(logx)))
D) 1/(logx(log(logx)))
20) If y= (tanx)ˢᶦⁿˣ then dy/dx ₓ₌π/4 is equal to
A) √2 B) 1/√2. C) 2. D) none
21) e ˣʸ - 4xy = 4, then dy/dx is
A) x/y. B) y/x C) - x/y D) -y/x
22) If y= eʸ + xy = e, then dy/dxₓ₌₀ is equal to.
A) e B) -1/e C) 1/e. D) -e
23) A differentiable function f is defined for all x> 0 and f(x²)= x³ for a x> 0. Then f'(16) is
A) 1 B) 4 C) 6 D) 16
24) If x= log(1+t²), y= t - tan⁻¹ t, then dy/dx=
A) 1 B) 1/2 C) t/2 D) t
25) If x⁷ y³ = (x+y)¹⁰, then d²y/dx²
A)0 B) 1 C) 10 D) x²y²
26) If y= cot⁻¹x, then d²y/dx²|ₓ₌₁is
A) 1/3 B) 1/2 C) 1 D) 5/2
27) If y= (cos⁻¹x)², then (1- x²) d²y/dx² is..
A) 0 B) 1 C) 2 D) 4
28) If y= sin(2 sin⁻¹x), then (1-x²) d²y/dx² - x dy/dx + 4y is
A) 0 B) 2y C) -2y D) none
29) If y= f(x) and x= 1/z, then d²f/dx² = 2z³ dy/dz + k d²y/dz² where k is equal to
A) -2 B) 2 C) 2z⁴ D) z⁴
30) If f(x)=x³ cos(1/x), when x ≠0
= 0, when x= 0, then
A) f'(0)=0= f"(0)
B) f"(x)|ₓ₌₀ does not exist
C) f'(x)ₓ₌₀ does not exist D) N
31) If x= e ᵗ and y= sint, then d²y/dx²|ₜ₌π/4
A) does not exist B) equal e²
C) equals - e^(-π) D) equal e^(π)
32) If y= ₑ m sin⁻¹x, then (1-x²) d²x/dx² - x dy/dx - ky = 0 where k is equal to
A) m² B) 2 C) -1 D) D) - m².
33) If y= sin⁻¹x, then (1-x²)y₂/y₁
A) x B) x² C) 2x D) none
34) If y⁴ + 5xy + y= 2, then d²y/dx²| ₓ₌₀ ,ᵧ₌₀is equal to
A) 2/5 B) -2/5 C) 1/5 D) -1/5
35) If y²= 4ax, then d²y/dx² . d²x/dy² is equal to
A) 1 B) a C) -2a/y³ D) -a/y²
36) Let f(x-y)= f(x)/f(y) for all reals x, y and f'(0)= p, f'(5)= q. Then f'(-5) is
A) q B) -q C) 0 D) none
37) If ₑsin⁻¹x, z= ₑ - cos⁻¹x, then dy/dz
A) doesn't exist finitely
B) is equal to eˣ
C) exists and is independent of x
D) exists and is a function of x
38) If f(a)=2, f'(a)=1, g(a)= -1, g'(a)= 2, then the value of
lim ₓ→ₐ (g(x)f(a) - g(a)f(x))/(x -a) is equal to
A) -1 B) 5 C) 1/2 D) 2a
39) If y= 4 cos 5x, then d²y/dx² = ky , where k is equal to
A) 5 B)-5 c) 25 D) -25
40) The derivative of f(logx) with respect to x, where f(x)= logx, is equal to
A) 1/x B)1/(x logx) C) 1/logx D) x logx
41) If g(x)= log₅ log₃ x, then g'(e) is equal to
A) e log 5 B) e log 3
C) 1/(e log 5) D) 1/(e log 3)
42) If g is the inverse of f and f'(x)=1/(1+x³), then g'(x) is ..
A) (1+3x²) B) 1/{1+[g(x)]³}
C) 1+[g(x)]³ D) [g(x)]³ - 1
43) If f(x)= | cosx |, then f'(3π/4) is equal to
A) 1/√2 B) -1/√2 C) √2. D) 1
44) Given that (f(x))ⁿ = f(nx) for all x. Then f'(x) f(nx) is equal to
A) xⁿ B) nxⁿ⁻¹ C) f(x) D) f(x)f'(nx).
45) If xʸ = eˣ⁻ʸ, then dy/dx is
A) x/(logx)². B) logx/(1+ logx)²
C) (1+x)/(1-x) D) none
46) If xᵐ yⁿ = (x+y)ᵐ⁺ⁿ , then dy/dx is equal to
A) xy B) x/y C) y/x D) (x+y)/xy
47) Let f(x + y)= f(x) f(y) for all x, y. If f'(0) exists and f(0) ≠ 0, then for all x, f'(x) exists and equal to
A) f(x)f'(0). B) f'(0)/f(x)
C) f(x)/f'(0) D) none
48) If f'(a) and g'(a) exists for two functions f(x) and g(x) at x= a, then lim ₓ→ₐ {g(x)f(a)- f(x)g(a)}/{g(x) - g(a)} equal
A){g(a)f'(a)- f(a)g'(a)}/g'(a)
B) {f(a)g'(a)- g(a)f'(a)}/g'(a)
C) {g(a)f'(a)- f(a)g'(a)}/f(a). D) n
49) If y= 2 sin⁻¹√(1-x) + sin⁻¹{2√(x(1-x))} for 0 < x< 1/2, then dy/dx is equal to
A) 2/√(x(1-x)) B) √{(1-x)/x}
C) -1/√{x(1-x)} D) 0
50) Let f(x)= |sin³x| and g(x)= sin³x then in (-π/2, π/2) which of the following is true?
A) g'(x)= [f'(x)], for all x belongs to (-π/2, π/2)
B) f'(x)= g'(x), for all x belongs to (-π/2, π/2)
C) f'(x)= g'(x), for all x belongs to (-π/2, π/2). D) none
51) If f(x)= sinx, g(x) = x², h(x)= logx and p(x) = h(g(f(x))), then d²p/dx² is equal to
A) 2x cot²x B) 2x cot x²
C) - 2 cosec²x D) 2 cosecx²
52) For x> 0, if g(x)= x ˡᵒᵍ and f(x)=ₑg(x) , then f'(x) is equal to
A) x ²ˡᵒᵍ ˣ ⁻¹ B) (1+x)e
C) 2xˡᵒᵍ ˣ ⁻¹ logx f(x) D) none
53) The function f(x)= 1/{1+|x|}
A) is differential at all real x.
B) is differential at all real except ± x.
C) is differential at all real 0
D) satisfies none of these
54) f"(x)= - f(x) and g(x)= f'(x) and F(x) = f²(x/2) + g²(x/2) and given that F(5)= 5, then F(10) is equal to
A) 5 B) 10 C) 0 D) 15
55) If f(x) = min {1, x,x²}, then
A) f(x) is continuous for all reals
B) f'(x) > 0 for all x > 1
C) f(x) is continuous for all real x but is not differentiable for atleast one x
D) f(x) is not differentiable for two values of x
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