A) discontinuous at only one point
B) discontinuously exactly at two points
C) discontinuously exactly three points D) none
2) Let f(x)=|x| and g(x)=|x³|, then
A) f(x) and g(x) both are continuously at x= 0
B) f(x) and g(x) both are differentiable at x= 0
C) f(x) is differentiable but g(x) is not differentiable at x= 0
D) f(x) and g(x) both are not differentiable at x= 0
3) the function f(x) = sin⁻¹(cosx) is
A) discontinuous at x= 0
B) continuous at x= 0
C) differentiable at x= 0 D) none
4) the set of points where the function f(x) =x |x| is differentiable is
A) (-∞,∞) B) (-∞,0) ∪ (0,∞)
C) (0,∞) D) [0,∞]
5) On interval I=[-2,2], the function
f(x)= (x+1) ₑ-(1|x| + 1/x), x≠ 0
0 , x= 0
A) is continuous for all x∈ I
B) is continuous for all x∈ I -{0}
C) assume all intermediate values from f(-2) to f(2)
D) has a maximum value equal to 3/e
6) If f(x)= |x+2|/{tan⁻¹(x+2)}, x≠-2
2 , x =-2 then f(x) is
A) continuous at x=-2
B) not continue at x=-2
C) differentiable at x=-2
D) continuous but not derivable at x=-2
7) Let f(x)= (x+|x|(|x|), then for all x
A) f is continuous
B) f is differenciable for some x.
C) f' is continuous
D) f" is continuous
8) the set of points where the function f(x)= √(1- ₑ-x²) is differentiable is
A) (-∞,∞) B) (-∞,0) ∪ (0,∞)
C) (-1,∞) D) none
9) 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) n
10) The function f(x)=|cosx| is
A) everywhere continuous and differentiable
B) everywhere continuous but not differentiable at (2n+1)π/2, n ∈ Z
C) neither continuous nor differentiable on (2n+1)π/2, n ∈ Z
D) none
11) If f(x)= √{1- √(1-x²)}, then f(x) is
A) continuous on [-1,1] and differentiable on (-1,1)
B) continuous on [-1,1] and differentiable on (-1,0)U(0,1)
C) continuous and differentiable on [-1,1] D) none
12) If f f(x)= sin⁻¹{2x/(1+x²)}, then f(x) is differenciable on
A)[-1,1] B) R-[-1,1] C) R-(-1,1) D)n
13) If f(x)= a |sin x|+ b ₑ|x| + c |x³| and f(x) is differenciable at x= 0, then
A) a=b=c= 0 B) a=0,b= 0, c∈R
C) b=c= 0, a∈R D) c=0,a= 0, b∈R
14) If f(x)= | x - a| g(x), where g(x) is continuous function, then
A) f'(a')= g(a) B) f'(a⁻)= - g(a)
C) f'(a⁺)= f'(a⁻) D) none
15) If f(x)= x² + x²/(1+x²) + x²(1+x²)² + .....+ x²/(1+x²)ⁿ +...., then at x= 0, f(x)
A) has no limit
B) is continuous
C) is continuous but not differentiable
D) is differenciable
16) If f(x)= | log₁₀ x|, then at x=1
A) f(x) is continuous and f'(1⁺)= log₁₀e
B) f(x)= is continuous and f'(1⁺)= logₑ10
C) f(x) is continuous and f'(1⁻)=logₑ10
D) f(x) is continuous and f(x) f'(1⁻)= - log₁₀e
17) If f(x)= | logₑ x|, then
A) f'(1⁺)= 1 B) f'(1⁻)= -1
C) f'(1)= 1 D) f'(1⁻)= -1
18) If f(x)= | log|x||, then
A) f(x) is continuous and differentiable for all x in its domain
B) f(x) is continuous for all for all x in its Domain but not differentiable at x=±1.
C) f(x) is neither continuous nor differentiable at x=±1 D) none
19) Let f(x)= 1/|x| for |x|≥ 1
ax²+b for |x| < 1
If f(x) is continuous and differentiable at any point, then
A) a=1/2, b= -3/2
B) a= -1/2, b= 3/2
C) a=1, b= -1 D) none
20) Let h(x)= min (x,x²}, for every real numbers x, then
A) h is continuous for all x
B) h is differenciable for all x
C) h'(x)= 1, for all x > 1
D) h is not differenciable at two values of x
21) If f(x)= {(36ˣ - 9ˣ -4ˣ+1)/(√2-√(1+cosx)}, x≠ 0
k , x= 0 is continuous at x= 0, then k equals
A) 16√2 log 2 log 3 B) 16√2 ln 6
C) 16√2 ln 2 ln 3 D) none
22) If f(x)= |x -4|, for x≥ 1
(x³/2)- x²+ 3x+1/2, x< 1 then
A) f(x) is continuous at x= 1 and at x= 4
B) f(x) is differentiable at x= 4
C) f(x) is continuous and differentiable at x=1
D) f(x) is only continuous at x = 1
23) Let f(x)= sin 2x, 0<x≤ π/6
ax+b, π/6<x <1 if f(x) and f'(x) are continuous, then
A) a=1, b= 1/√2 + π/6
B) a=1/√2, b= 1/√2
C) a=1 b= √3/2-π/6 D) none
24) Let f(x)= ˣ₀∫{5+ |1- t|}dt, if x> 2
5x+1, if x≤ 2
Then
A) f(x) is continuous at x= 2
B) f(x) is continuous but not differentiable at x= 2
C) f(x) is everywhere differentiable
D) the right derivative of f(x) at x= 2 does not exist.
25) The function f defined by
f(x) = (sin x²)/x , x ≠ 0
0 , x= 0 is
A) continuous and available at x= 0
B) neither continuous nor derivable at x= 0
C) continuous but not derivable at x= 0 D) none
26) If f(x) is continuous at x= 0 and f(0) = 2, then
lim ₓ→₀ ˣ₀∫ f(x) du /x is
A) 0 B) 2 C) f(3) D) none
27) If f(x) defined by
f(x)= {|x²- x|}/(x²-x) , x ≠ 0, 1
1, x= 0
-1, x= 1 then f(x) is continuous for all
A) x B) x except at x= 0
C) x except at x= 1
D) x except at x= 0 and x= 1
28) If f(x)={(1-sinx)/(π-2x)²}. {(log sinx)/(log(1+π²- 4πx+ 4x²) is
K. x=π/2 continuous at x=π/2, then k is
A) -1/16 B) -1/32 C) -1/64 D)-1/28
29) The set of points of differentiability of the function
f(x)={√(x+1)-1}/√x, for x≠ 0
0,. for x= 0 is
A) R B) [0,∞) C) (0,∞) D) R-{0}
30) The set of the points where the function f(x)= |x-1| eˣ is
A) R B) R-{1} C) R-{-1} D) R-{0}
31) If f(x)=(x+1)ᶜᵒᵗ ˣ be continuous at x= 0, f(0)=
A) 0 B) 1/e C) e D) none
32) If f(x)= {log(1+ax)- log(1-bx)}/x, x≠ 0
K, x= 0 and f(x) is continuous at x= 0, then the value of k is
A) a - b B) a + b
C) log a + log b D) none
33) the function
f(x)={e¹⁾ˣ -1)/(e¹⁾ˣ+ 1)}, x≠ 0
0, x= 0
A) is continuous at x= 0
B) is not continuous at x= 0
C) is not continuous at x= 0, but can be made continuous at x= 0
D) none
34) Let f(x)={(x-4)/|x-4|} +a, x < 4
a+b, x= 4
{(x-4)/|x-4|} +b, x> 4
Then f(x) is continuous at x=4 when
A) a=0, b=0 B) a=1, b= 1
C) a=-1, b= 1 D) a=1, b= -1
35) if the function
f(x)= (cosx)¹⁾ˣ, x≠ 0
K, x= 0 is continuous at x=0, then the value of k is
A) 0 B) 1 C) -1 D) e
36) let f(x)=|x| + |x -1|, then
A) f(x) is continuous at x= 0 as well as at x=1
B) f(x) is continuous at x= 0, but not at x=1
C) f(x) continuous at x= 1, but not at x= 0 D) none
37) let f(x)= {(x²-5x²+4)/|(x-1)(x-2)|}, x≠ 1,2
6, x= 1
12 x= 3 then f(x) is continuous on the set
A) R B) R -{1} C) R -{2} D) R -{1,2}
38) f(x)= {sin(a+1)x+ sinx}/x,x<0
c, x= 0
{√(x+bx²)- √x}/(bx√x) x> 0
is continuous at x= 0, then
A) a=-3/2, b=0,c=1/2
B) a=-3/2, b=1 ,c= -1/2
C)a=-3/2, b belongs to R ,c=1/2 D) none
39) if f(x)= my+1, x≤π/2
Sin x+ n, x>π/2
is continuous at x= π/2, then
A) m= 1, n= 0 B) m= n π/2+1
C) n= m π/2 D) m= n= π/2
40) The value of f(0), so that the function
f(x)={√(a²-ax+x²)- √(a²+ax+x²)}/{√(a+x) - √(a-x)} becomes continuous for all x, given by
A) a³⁾² B) a¹⁾² C) - a¹⁾² D) -a³⁾²
41) The function
f(x)= 1, |x|>1
1/n², 1/n < |x| <1/(n-1), n=2,3..
0, x= 0
A) is continuous at finitely many points
B) is continuous everywhere
C) is continuous only at x=1/n, n belongs Z -{0} and x= 0 D) none
42) The value of f(0), so that function
f(x)={(27-2x)¹⁾³ -3}/{9 -3(243+5x)¹⁾⁵ (x≠0) is continuous is given by
A) 2/3 B) 6 C) 2 D) 4
43) The value of f(0) so that the function
f(x)={2 - (256-7x)¹⁾⁸}/{(32+5x)¹⁾⁵ -2} (x≠ 0) is continuous everywhere, is given by
A) -1 B) 1 C) 2⁶ D) none
44) The following functions are continuous on (0,π)
A) tanx B) ˣ₀ ∫ t sin(1/t) dt
C) f(x)= 1, 0<x ≤ 3π/4
2 sin(2x/9), 3π/4<x ≤ π
D) f(x)= x sinx 0<x ≤ π/2
π/2 sin(π+x), π/2<x<π
45) If f(x)= x sin(1/x), x≠ 0, then the value of the function at x= 0, so that the function is continuous at x= 0 is
A) 1 B) -1 C) 0 D) indeterminate
46) Let f(x)= [x] and
g(x)= 0, x belongs to Z
x², x belongs to R - Z then
A) lim ₓ→₁ g(x) exists, but g(x) is not continuous at x= 1
B) lim ₓ→₁ f (x) does not exist and f(x) is not continuous at x=1
C) g o f is continuous for all x
D) fog is continuous for all x
47) Let f(x)= lim ₓ→∞ (sinx)²ⁿ, then f is
A) continuous at x=π/2
B) discontinuous at x=π/2
C) discontinuous at x= -π/2
D) discontinuous at infinite number of points.
48) let f(x) be a function differentiable at x= c. Then
lim ₓ→꜀ f(x) equals
A) f'(c) B) f''(c) C) 1/f'(c) D) none
49) If lim ₓ→꜀ {f(x)- f(c)}/+x-c) exists finitely, then
A) lim ₓ→꜀ f(x) = f(c)
B) lim ₓ→꜀ f'(x) = f'(c)
C) lim ₓ→꜀ f(x) does not exist
D) lim ₓ→꜀ f(x) may or may not exist.
50) If f(x)= (x log cosx)/(log(1+x²) x≠0
0, x= 0
A) f(x) is not continuous
B) f(x) is continuous at x= 0
C) f(x) is continuous at x= 0 but not differentiable at x= 0
D) f(x) is differenciable at x= 0
51) the function f(x)=|x|+|x-1is
A) continuous at x= 1, but not differentiable
B) both continuous and differenciable at x= 1
C) not differentiable at x= 1 D) n
52) the function
f(x)= |x - 3|, x≥ 1
x²/4 - 3x/2 + 13/4, x< 1 is
A) continuous at x=1
B) derivable at x= 1
C) continuous at x= 3
D) derivable at x= 3
53) let f(x)= xⁿ sin(1/x), x≠ 0
0, x= 0
then f(x) is continuous but not differentiable at x= 0 if
A) n belongs (0,1]
B) n belongs [1,∞]
C) n belongs (-∞,0) D) n= 0
54) if x+ 4 |y| = 6y, then y as a function of x is
A) continuous at x= 0
B) derivable at x= 0
C) dy/dx= 1/2 for all x D) none
55) If f(x)=x³ sin x, then
A) f is derivable at x= 0
B) f is continuous but not derivable at x= 0
C) LHD at x= 0
D) RHD at x= 0 is 0
56) the function
f(x)= (tan{π[x-π]})/(1+[x]²) where [x] denotes the greatest integer less than or equal to x, is
A) discontinue at some x
B) continuous at all x, but f'(x) does not exist for some x
D) f'(x) exist exist for all x
57) If f(x)= x² sin(1/x), x≠ 0
0, x= 0
Then
A) f and f' are continuous at x= 0
B) f is derivable at x= 0
C) f is derivable at x= 0 and f' is not continuous at x=0
D) f' is derivable at x= 0
58) The following functions are differentiable on (-1,2)
A) ²ˣₓ∫ (log t)² dt B) ²ˣ₀∫ (sint)/t dt
C) ˣ₀∫ (1-t+t²)/(1+t+t²) D) none
59) if f(x)= √{x+2)√(2x-4)} +√{x-2)√(2x-4)}, then f(x) is differentiable on
A) (-∞,∞) B) (2,∞) -{4}
C) [2,∞) D) none
60) the derivative of f(x)=|x|³ at x= 0
A) -1 B) 0 C) does not exist D) n
61) f(x)= x{√x -√(x+1)} then
A) f is continuous but not differentiable at x= 0
B) f is differentiable at x=0
C) f is differentiable but not continuous at x= 0
D) f is not differentiable at x=0
62) The value of the derivative of |x -1| + |x -3| at x=2 is
A) continuous at x= zero
B) continuous in (-1,0)
C) differentiable at x= one
D) differentiable in (-1,1) E) n
63) if function f(x)= [x sin πx] , then f(x) is
A) continuous nowhere
B) continuous everywhere
C) differentiable nowhere
D) not differentiable at an infinite number of points
E) not differentiable at x=zero
65) If f(x)= 1 , x< 0
1+ sinx, 0≤ x <π/2 then derivative of f(x) at x= 0.
A) is equals to 1
B) is equal to zero
C) is equal to -1
D) does not exist
66) let [x] denotes the greatest integer less than or equal to x and f(x)= [tan²x]. Then,
A) lim ₓ→₀ f(x) does not exist
B) f(x) is continuous at x= 0
C) f(x) is not differentiable at x=0
D) f'(0)= 1
67) A function f: R--> R satisfies the equation f(x+y)=f(x)f(y) for all x,y belongs to R and f(x)≠ 0 for all x belongs to R. If f(x) is differentiable at x= 0 and f'(0)= 2, then f'(x)=
A) f(x) B) -f(x) C) 2f(x) D) none
68) let f(x) be defined on R such f(1)=2, f(2)= 8 and f(u+v)= f(u) + kuv - 2v² for all u, v belongs to R (k is fixed constant). then
A) f'(x)= 8x B) f(x)=8x C) f'(x)=x D) n
69) let f(x) be a function satisfying f(x+y)=f(x)+ f(y) and f(x)= x g(x) for all x,y belongs to R, where g(x) is continuous. then
A) f'(x)=g'(x)
B) f'(x)=g(x)
C) f'(x)=g(0) D) none
70) if f(x)= ax² - b , |x|< 1
1/|x|, |x|≥ 1
A) a= 1/2, b= -1/2
B) a= -1/2, b= -3/2
C) a= 1/2, b= 1/2
D) a= -1/2, b= -1/2
71) If f(x)= (x - x₀) ∅(x) and ∅(x) is continuous at x= x₀, then f(x₀) is equals to:
A) ∅'(x₀) B) ∅(x₀) C) x₀∅(x₀) D) n
72) If f(x+y)= f(x) f(y) for all x,y belongs to R, f(5)= 2, f'(0)=3. Then f(5) equal to
A) 6 B) 3 C) 5 D) none
73) If f(xy)= f(x) f(y) for all x,y belongs to R, f'(1)= 2, f(4)=4, Then f'(4) equal to
A) 4 B) 1 C) 1/2 D) 8
74) If f(x+y)= f(x) f(y) for all x,y belongs to R, f(3)= 3, f'(0)=11, Then f'(3) equal to
A) 22 B) 44 C) 28 D) none
75) If f(x+y)= f(x)+ f(y) and f(x)= x² g(x) for all x,y belongs to R, where g(x) is continuous function then f'(x) is equal to
A) g'(x) B) g(0) C) g'(x) +g(0) D) 0
76) If f(x+y)= f(x) f(y) and f(x)= 1+ x g(x) for all x,y belongs to R, where lim ₓ→₀ g(x) = 1, then f'(x) is equal to
A) g'(x) B) g(x) C) f(x) D) none
77) If f(x+y)= f(x) f(y) and f(x)= 1+ x g(x) G(x), for all x,y belongs to R, where lim ₓ→₀ g(x) = a and
lim ₓ→₀ G(x) = b, then f'(x) is equal to
A) 1+ ab B) ab C) a/b D) none
78) If f(x+y)= f(x) f(y) and f(x)= 1+(sin 2x) where g(x) is continuous. then f'(x) is equal to
A) f(x)+ g(0) B) 2f(x) g(0) C) 2g(0) D) none
79) Let g(x) be the inverse of an invertible function f(x) which is differentiable at x= c, then g'(f(c)) equal
A) f'(c) B) 1/f'(c) C) f(c) D) none
80) Let g(x) be the inverse of the function f(x) and f'(x)=1/(1+x³), then g'(x) is equal to
A) 1/(1+g(x)³) B) 1/(1+(f(x))³)
C) 1+ (g(x))³ D) 1+ (f(x))³
81) the function
f(x)= xⁿ sin(1/x), x≠ 0
0, x= 0
is continuous and differentiable at x= 0 if
A) n ∈ (0,1] B) n ∈ [1,∞)
C) n ∈ (1,∞) D) n ∈ (-∞,0)
82) If for a continuous function f, f(0)= f(1)= 0, f'(1)=2 and y(x)= f(eˣ) ₑf(x) , then y'(0) is equal to
A) 1 B) 2 C) 0 D) none
83) Let f(x) be the function such that f(x+y)= f(x)+ f(y) and f(x)= sin x g(x) for all x,y belongs to R. If g(x) is a continuous function such that g(0)= K, then f'(x) is equal to
A) K B) Kx C) Kg(x) D) none
84) let be twice differentiable function such that f"(x)= - f(x) and f(x) = g(x), h(x)= {f(x)}²+ {g(x)}². If h(5) = 11, then h(10) is
A) 22 B) 11 C) 0 D) none
85) suppose a function f(x) satisfies the following two condition for all x and y.
I) f(x+y)= f(x) f(y)
II) f(x)= 1+ xg(x) log a, where a> 1 and lim ₓ→₀ g(x) = 1. then f'(x) is equals to
A) log a B) log ₐf(x) C) log (f(x))ᵃ D) none of these
86) if the function
f(x)= Ax - B,. x≤1
3x, 1< x < 2
Bx² - A, x≥ 2 be the continuous at x=1 and discontinuous at x=2, then
A) A= 3+ B, B≠ 3
B) A= 3+ B, B= 3
C) A= 3+ B D) none
87) if f(x)= {|x| - |x - 1|}², then f'(x) equal
A) 0 for all x B) 2{|x|-| x-1|}²
C) 0 for x< 0 and for x> 1
4(2x-1) for 0 < x< 1
D) 0 for x< 0
4(2x-1) for x> 0
88) if the derivative of the function
f(x)= ax² + b, x<-1
bx²+ ax+4, x≥ -2 is everywhere continuous, then
A) a=2, b=3 B) a=3, b=2
C) a= -2, b=-3 D) a=-3 , b=-2
89) let f and g be differentiable functions satisfying g'(a)= 2, g(a)=b and fog= I( identify function). then f'(b) is equals to
A) 1/2 B) 2 C) 2/3 D) none
90) the set of all points, where the function f(x)= x/(1+|x|) is differentiable is
A) (-∞,∞) B) (0,∞)
C) (-∞,0)U(0,∞) D) none
91) let f(x)= (sin 4π[x])/(1+[x]²), where [x] is the greatest integer less than or equal to x, then
A) f(x) is not differentiable at some points
B) f'(x) exist but different from zero
C) f'(x)= 0 for all x
D) f'(x)= 0 but f is not constant function
92) if f(x)= ax²+ b, b≠0, x≤1
bx²+ax+c, x> 1 then f(x) is continuous and differentiable at x=1 if:
A) c=0, a= 2b
B) a=b, c belongs to R
C) a= b, c= 0
D) c≠ 0 a= b
93)
94) let f(x) be an even function, then f'(x)
A) is an even function
B) is an odd function
C) may be even or odd D) none
95) let f(x) be an odd function. then f'(x)
A) is an even function
B) is an odd function
C) may be even or odd D) none
96) if a function f(x) is defined as f(x) = x/√x², x ≠ 0
0, x= 0 then
A) f(x) is continuous at x= 0 but not differentiable at x= 0
B) f(x) is continuously as well as differentiable at x= 0
C) f(x) is differentiation at x= 0
D) none
97) The function f(x)=[x] cos{(2x-1)/2}π where [ ] denotes the greatest integer function, is discontinuous at
A) all x B) all integer points
C) no x D) x which is not an integer
98) let f(x) be defined for all x> 0 and be continuous. Let f(x) satisfy f(x/y)= f(x) - f(y) for all x, y and f(e)= 1. Then
A) f(x) is bounded
B) f(1/x) --> 0 as x-> 0
C) xf(x)--> 1 as x-> 0
D) f(x)= ln x
99) the function f(x)= Max {(1-x),(1+x),2}, x belongs to (-∞,∞), is
A) continuous at all points
B) differentiable at all points
C) differentiable at all point except at x= 1 and x=-1
D) continuous at all points except at x= 1 and x=-1, where it is discontinuous
100) g(x)= xf(x), where
f(x)= x sin(1/x), x≠ 0
0, x= 0 at x= 0
A) g is differentiable but g' is not continuous
B) g is differentiable while f is not differentiable
C) both f and g are differenciable
D) g is differentiable but g' is continuous
101) if f(x)=x/2 - 1, then on the interval [0,π].
A) tan[f(x)] and 1/f(x) are both continuous
B) tan[f(x)] and 1/f(x) are both discontinuous
C) tan[f(x)] and inverse of f(x) are both continuous
D) tan[f(x)] is continuous but 1/f(x) is not
102) If x + |y|= 2y, then y as a function of x is
A) defined for all real x
B) continuous at x= 0
C) differentiable for all x
D) such that dy/dx= 1/3 for x< 0
103) at the point x=1, the function
f(x)= x³ -1 ; 1< x <∞
x -1; - ∞< x ≤ 1
A) continuous and differentiable
B) continuous and not differentiable
C) discontinuous and differentiable
D) discontinuous and not differentiable
104) the value of k which makes
f(x)= sin(1/x), x≠ 0
k ,. x= 0 continuous at x= 0 is
A) 8 B) 1 C) -1 D) none
105) If f(x)= ˣ₋₁∫ | t| dt,.x≥ -1, then
A) f and f' are continuous for x+1> 0
B) f is continuous but f' is not so for x+1> 0
C) f and f' are continuous at x= 0
D) f is continuous at x= 0 but f' is not so
106) the value of the derivatives of |x-1|+|x-3| at x= 2 is
A) -2 B) 0 C) 2 D) not defined
107) if f(x)= 1 for x< 0
1+ sinx for 0≤x<π/2, then at x= 0, the derivative f'(x) is
A) 1 B) 0 C) infinite D) doesn't exist
108) Let a function f(x) be defined by f(x)={x - |x-1|}/x, then f(x) is
A) discontinuous at x= 0
B) discontinuous at x=1
C) not differentiable at x= 0
D) not differentiable at x= 1
109) if f(x)=[cos πx], x<. 1
|x-2|, 2>x≥ 1, then f(x)=
A) doscontinuous and non differentiable at x=-1 and x=1
B) continuous and differentiable at x =0
C) discontinuous at x=1/2
D) continuous but not differentiable at x= 2
110) If y= f(x)=1/(u²+u -1) where u= 1/(x-1), then the function is discontinuous at x=
A) 1 B) 1/2 C) 2 D) -2
111) if f(x)=✓[√2 sinx], where [x] represents the greatest integer function, then
A) f(x) is periodic
B) maximum value of f(x) is 1 in the interval [-2π,2π]
C) f(x) is discontinuous at x=nπ/2 +π/4, n belongs to Z
D) f(x) is differentiable at x= nπ, n belongs to Z
112) the number of points at which the function
f(x)=|x- 0.5|+|x- 1|+ tanx
does not have a derivatives in the interval (0,2) is
A) 1 B) 2 C) 3 D) 4
113) consider f(x)= x²/|x|, x≠0
0,. x=0
A) f(x) is discontinuous everywhere
B) f(x) is continuous everywhere
C) f'(x) exists in (-1,1)
D) f'(x) exists in (-2,2)
114) If the function f is defined by
f(x)= x/(1+| x|), then at what points is f differentiable ?
A) everywhere B) except x=±1
C) except at x=0
D) except at x= 0 or ±1
115) if f'(a)= 2 and f(a)= 4, then
lim ₓ→ₐ {xf(a)- a f(x)}/(x-a) equal
A) 2a-4 B) 4- 2a C) 2a+4 D) none
116) In [1,3] the function [x²+1], [x] denoting the greatest integer function, is continuous
A) for all x
B) for all x except Four Points
C) for all except seven points
D) for all except at eight points
⁻
117) the functions f(x) defined by
f(x)= log₄ₓ ₋₃(x²- 2x+5), 3/4< x <1 and x>1
4 , x= 1
A) is continuous at x=1
B) is discontinuous at x= 1 since f(1⁺) does not exist though f(1⁻) exists.
C) is continuous at x= 1 since f(1⁻) does not exist though f(1⁺) exists
D) is continuous at x= 1 since neither f(1⁺) nor f(1⁻) exists
118) If f(x)= min{tanx, cotx} then
A) f(x) is not differenciable at x= 0, π/4, 5π/4
B) f(x) is discontinuous at x= 0, π/2, 3π/2
C) ∫ f(x) dx at (π/2,0)= ln √2
D) f(x) is periodic with period π
119) if f(x)=sin ln{√(9-x²)/(2-x)}, then
A) domain of f(x) is x belongs (-3,2)
B) range of f(x) is y belongs (-1,1)
C) f(x) is discontinuous at x= 0
D) the right hand limit of y= (x-3) f(x) at x=-3 is zero
120) f(x)=|[x] x| in -1< x ≤ -2 is..
A) continuous at x =0
B) discontinuous at x=1
C) not differentiable at x=2
D) not differentiable at x= 0
121) if f(x)= (1-|x|)/(1+x), x≠ -1
1, x= -1 then f([2x])|is ( where [ ] represents the greatest integer function)
A) continuous at c=-1
B) continuous at x=0
C) discontinuous at x=1/2
D) discontinuous at x=1
122) Let f(x)=[x]+√(x-[x]), where [x] greatest integer function. Then
A) f(x) is continuous on R⁺
B) f(x) is continuous on R
C) f(x) is discontinuous on R - Z
D) none
123) Let f(x)=[2x³ -5], [ ] denotes the greatest integer function. Then number of points in (1,2) where the function is discontinuous, is
A) 0 B) 13 C) 10 D) 3
124) Given that f(x) is a differenciable function of x and that f(x). f(y) =f(x)+ f(y) + f(xy) - 2 and that f(2)= 5. Then f(3) is
A) 10 B) 24 C) 15 D) none
125) If f(x+y+z)=f(x).f(y). f(z) for all x,y,z and f(2)= 4, f'(0)= 3, then f'(2) is equal to
A) 12 B) 9 C) 16 D) 6
126) The function f(x)= a[x+1]+ b[x-1], where [ ] is the greatest integer function, is continuous at x= 1, if
A) a+b= 0 B)a= b C) 2a-b= 0 D) n
127) The function f(x)= (x²-1) |x²- 3x+2| + cos(|x|) is not differenciable at
A) -1 B) 0 C) 1 D) 2
FILL IN THE BLANKS
--------------------------
1) Under the condition___the domain of f/g is equal to domain f ∩ domain g.
2) The function f(x)={|2x-1|}/(2x-1) is defined and continuous throughout the entire number real line except at the point x=-----
3) The value of f(0) for which the function f(x)=(2- √(x+4))/(sin 2x) is continuous at x=0 is _____
4) The function f(x)=6. 5ˣ for x≤0
2a+x for x> 0 will be continuous at x= 0 if a=___
5) If the function
f(x)= (x³+x²-16x+20)/(x-2)², if x≠ 2
k,. If x= 2
is continuous for all x, then k=___
6) Let f(x)=(x³-3x+2)/(x-1)², if x≠ 1
k, if x= 1, if f(x) is continuous for all x, then the value of k is _____
7) A discontinuous function y= f(x) satisfying x²+ y²= 4 is given by f(x)=__
8) if f(x)= - 2 sinx for -π≤ x<-π/2
a sinx+ b for -π/2≤ x<π/2
Cosx for π/2≤ x<π the interval [-π,π], then a=____ and b=__
9) The function
f(x)=(1- x)/(logx), x> 0 and x≠ 1
k, x= 1 is continuous at x= 1, then the value of k is _____
10) The set of points of continuity of the function f(x)=√(1/2 - cos²x) is given by_____
11) If y= 1/(z²+z-2) and z=1/(x-1), then the points of discontinuous of y are____
12) if f(x)= |log x|, then the left hand and right hand derivatives of f(x) at x= 1 are ___ and___ respectively.
13) If f(x)= √(x²-4x+4) is defined on (0,3), than the left hand right hand derivatives of f(x) at x= 2 are___ and _____respectively.
14) the derivative of an even function may be an___ function.
15) let f(x)= (x-1)² sin{1/(x-1) -|x| if x≠ 1
-1 if x= 1 be a real valued function. then the set of points where f(x) is not differentiable is_____.
16) The function f(x)= ³√(x-2) has no finite derivatives at the point x= ___.
17) the set of points of discontinuity of the function f(x)= 1/{log|x|} is_____.
18) the set of the points of discontinuity of the function f(x)= 1/(x - [x]) is ____.
19) If f(x)=| x-2| and g(x)= fof(x), then for x> 20, g'(x)= ____.
20) If f(x)=| x-a| and h(x)= f(f(f(x))); then g'(x)= ____ for x > 4a.
21) If f(x)=log tan{(2x+1)/4}, then f'(0)= ______.
TRUE/FALSE
***********
1) every continuous function is differentiable.
2) every differentiable function is continuous
3) The derivatives of an odd function is always an even function.
4) if f(x)= x², x≤ 2
ax, x> 2, then the value of 'a' for which f(x) is continuous is 2.
5) the function
f(x)= |x - 3|,. x≥ 1
x²/2 - 3x/2 +13/4 , x< 1 is continuous and differentiable at x= 1.
6) the value of f(0) so that the function
f(x)={log(1+x² tanx)/(sinx³), x≠ 0 is continuous at x= 0, is 2.
7)lim ₓ→₀ {cot x - 1/x)= 1
8) the function f(x)=|sinx/√(2-x²)| is differentiable at x= 0.
9) the function f(x)= sin ⁻¹(cos x) is continuous at x= 0, but not derivable thereat.
10) If f(x+y)= f(x)+f(y) for all x,y and f(x) is continuous at x= 0, then f(x) is continuous all x
11) A function f defined as
f(x)= (x³-4x+3)/(x²-1), x≠ 1
2, x= 1 is continuous at x=1.
MORE QUESTIONS
----------------------------
1) f(x)={√(1+px)-√(1-px)}/x, -1≤ x< 0
(2x+1)/(x-2), 0≤ x≤ 1
is continuous in the interval [-,11], then p is equals to
A) -1 B) -1/2 C) 1/2 D) 1
2) the function
f(x)= x²/a, 0≤ x <1
a ,. 1≤ x <√2
(2b²-4b)/x², √2≤ x < ∞ is continuous for 0≤ x <∞, than the most suitable values of a and b are
A) a= 1, b1= -1 B) a=- 1, b=1+√2
C) a=-1, b= 1 D) none
3) if f(x)= (1-sinx)/(π-2x)², when x≠π/2 and f(π/2)= K, then f(x) will be discontinuous function at x=π/2, where K=
A) 1/8 B) 1/4 C) 1/2
4) the value of a for which the function
f(x)= (4ˣ -1)³/(sin(x/a) log (1+x²/3)), x≠0
12 (log 4)³, x= 0 may be continuous x= 0 is
A) 1 B) 2 C) 3 D) none
5) If f(x)= |x|³, then f'(x) at x= 0 is
A) -1 B) 0 C) 1/2 D) not defined
6) if f(x)=x |x|, then derivative of f(x) at x= 0 is
A) 0 B) -1 C) 1 D) not defined
7) let 3x+2y= |y|. than y as a function of x is
A) defined for all x
B) continuous everywhere
C) not differentiable at x= 0
D) f'(x)= -3 for x< 0
E) all of these
8) let f(x)= x - [x]. then for any integers n, lim ₓ→ₙ f(x)
A) 0 B) n C) - n D) does not exist
9) Let f(x)=[x] sin{π/[x+1]}, where [.] denotes the greatest integer function. The domain of f is____ and the points of discontinuity of f in the domain are____
10) let [ ] denotes the greatest integer function and f(x)=[tan²x]. then
A) lim ₓ→⁰ does not exist
B) f(x) is it continuous at x=0
C) f(x) is not differentiable at x= 0
D) f'(0) = 1
11) If f: R -> R is a function such that f(x)= x³ + x³f'(2)+ x f"(3)+ f"'(3) for all x belongs to R, then f(3)- f(1)
A) f(0) B) -f(0) C) f'(0) D) -f'(0)
12) the function
f(x)=lim ₓ→∞ (xⁿ - x⁻ⁿ)/(xⁿ + x⁻ⁿ), x> 0 is
A) everywhere differentiable
B) not differentiable at x= 1 only
C) everywhere continuous
D) such that f'(x)= 0 for all x -0 except at x= 1
13) the function f(x)= arc sin(2x/(1+x²)
A) is everywhere differentiable such that such that f'(x)= 2/(1+x²)
B) is such that
f'(x)= 2/(1+x²), if -1< x< 1
-2/(1+x²), if |x|> 1
C) is such that
f'(x)= 2/(1+x²) if -1≤ x < 1
-2/(1+x²), if |x|>1 D) n
14) let f(x)= (sin{π[x-π]}/(1+[x²]), where [ ]stands for the greatest integer function. then f(x) is
A) continuous at integers points
B) continuous everywhere
C) differentiable once but f"(x), f"'(x) do not exist
D) differentiable for all x
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