Differential Calculus - DPP - 02 PDF
Differential Calculus - DPP - 02 PDF
Differential Calculus - DPP - 02 PDF
3 1 x3
01. Let f x 8x 3x and f x be the inverse function of f x , then Lt 1
x f 8x f 1 x
A) 0 B) 1 C) 2 D) 4
1
e x x 2 a x
02. If f x ; 1 x 0
x
3/ x
e1/ x e2/ x e
3/ x
; 0 x 1 b 0 is continuous at x 0, then f 0 is
a e2/ x b.e
A) e3/ 2 B) e3/ 2 C) 1 D) 0
2 1 sin x
x sin ; x 0 ; x 0
03. f x x and g(x) tan 1 x ; which of the following is CORRECT
0 ; x 0 1 ; x 0
A) y f x .g x is discontinuous
B) y g x is discontinuous at x = 1
C) y f x .g x is differentiable at x = 0
D) y f x .g x is differentiable x R
n
04. If f x ra k
r0
(k is constant , k R ) x R and a > 0, then period of f(x) is
A) na B) (n + 1) a C) n 2 n a D) n 2 a
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MULTIPLE CORRECT ANSWER TYPE
2
06. , are roots of 3x ax b 0,
Let f t Lim 1 tan 3x ax b
x t
2
xt
then which of the following is/are INCORRECT
A) f f B) f . f 1
C) f has only two elements in domain D) f has only two elements in Range
x 3 , x 1 2 x ,x2
07. If f x , g x If h x f x g x is
x 2 a , x 1 sgn x b , x 2
discontinuous at exactly one point, then which of the following can be correct?
(sgn(x) denote signum function)
A) a 3, b 0 B) a 3, b 1 C) a 2, b 1 D) a 0, b 1
1 x14
P e ,x 0
08. Let P x be a polynomial function of degree ‘n’ and f x x 3 , x 0,
,x 0
0
then
max . | x |, x 2 : 0 x 1
f x and g x | 3sin x | , then which of the following
x :1 x 2
statements is/are correct.
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10. Let f ( x ) be a polynomial of degree 5 with leading coefficient unity such that
f (1) 5, f (2) 4, f (3) 3, f (4) 2, f (5) 1 then
COMPREHENSION TYPE
Answer Q,11, Q,12 and Q,13 by appropriately matching the information given in the three columns of
the following table.
Column-I contains information about function f x
Column-II contains information about number of points of discontinuity of f x
Column-III contains information about number of points of non-differentiability of f x
Column-I Column-II Column-III
(I) x 2 2 x 1 x 1, 2 (i) Prime Number (P) 1
f x
sgn sin x x 2, 2
(II) 2 x 1 x 1 (ii) Not a Natural (Q) 2
f x 1 x2 x 1,1 Number
x 1 1 x 1
Let g(x) = x 3 g '' 1 x 2 3g ' 1 g '' 1 1 x 3g ' 1 ; f(x) = x g(x) – 12x + 1 and
2
f(x) = h x where h(0) = 1
15. Which of the following is/are true for the function y = g(x) ?
1 1
A) g(x) monotonically decreases in , 2 2 ,
3 3
1 1
B) g(x) monotonically increases in 2 , 2
3 3
C) There exists exactly one tangent to y = g(x) which is parallel to the chord joining
the points (1, g(1)) and (3, g(3))
D) There exists exactly two distinct Lagrange’s mean value in (0, 4) for the function
y = g(x)
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16. Which one of the following does not hold good for y = h(x) ?
C) Exactly one real zero in (0, 3) D) Exactly one tangent parallel to x-axis
INTEGER TYPE
x 3 x
tan
r 1 tan r 1
n
2 2
f x lim Sn Sn
17. Let where x , then the value of
1 tan 2 r 1
n r 0
2
x3
lim
x0 f x sin x equals
18. If f x be differentiable function and curve y f x passes through 1,1 and satisfies
f x 1
the relation 2 f x y f x y 3 y 2 3 f x 2 xy , then lim is equal to
x 1 x 1
19. Number possible value of x satisfying the equation x 3 x 2 4x 2sin x 0 such that
0 x 2
b 27ab 2
20. For positive real numbers a,b,c ax c , x 0 . The minimum value of
2
is
x c3
1
21. Let f be a cubic polynomial function having relative extremum at x 1 and x
3
1
14
Such that s f t dt 3
. If f 2 0 , f f f 0 1 ___
1
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MATCHING TYPE
Column – I Column – II
x3 1
P) Lt 1) 2
x1 ln x
x cos x cos 2 x
Q) Lt 2) 3
x0 2sin x sin 2 x
1 3
S) If f x cos x cos and 4)
x
4
g x
ln sec 2 x are continuous
x sin x
at x 0 then f 0 g 0
P Q R S P Q R S
A) 1 3 4 2 B) 2 3 4 1
C) 1 2 4 3 D) 1 3 2 4
23 Let f1 : R R , f 2 : , R , f 3 : 1, e 2 2 R and f4 : R R be functions
2 2
defined by
(i) f1 x sin 1 e x
2
sin x
if x 0
(ii) f 2 x tan 1 x , where the inverse trigonometric function tan 1 x assumes
1 if x 0
values in , ,
2 2
iii) f3 x sin loge x 2 , where , fort R, t denotes the greatest less than or equal to t,
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2 1
x sin if x 0
(iv) f 4 x x
0 if x 0
Column I Column II
(P) The function f1 is 1 NOT continuous at x0
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x
24. Let E1 x R : x 1 and 0 and
x 1
x
E 2 x E1 :sin 1 log e is a real number .
x 1
1
Here, theinverse trignomericfunction sin x assumes valuesin 2 , 2
x
Let f : E1 R bethefunctiondefinedby f (x) loge
x 1
x
and g : E2 R be thefunction defined by g ( x) sin 1 log e .
x 1
Match List-I with List-II and select the correct answer using the code given below
the list.
List – I List – II
1 e
P) The range of f is 1) , ,
1 e e 1
Q) The range of g contains 2) 0 ,1
R) The domain of f contains 3) ,
S) The domain of g is 4) ,0 0,
e
5) ,
e 1
P Q R S P Q R S
A) 4 2 1 1 B) 3 2 1 1
C) 4 2 1 5 D) 4 3 3 5
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