Calculus
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Nikhil MauryaCalculus
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Nikhil Maurya(a) State the value of ƒ(–1).
EXERCISE # 1 (b) For what values of x is f(x) = 2
Subjective Type Questions (c) State the domain and range of ƒ.
1. Find the domain of definition of the given functions :
1 (d) On what interval is ƒ increasing ?
(i) y px (p 0) (ii) y= (e) Estimated value of ƒ(2) is :
x2 1
1 1 (A) 2.2 (B) 2.8 (C) 2.5 (D) 3
(iii) y= (iv) y=
x3 x x 2 4x (f) Estimated value of x such that ƒ(x) = 0, is :
x
(v) y= x 2 4x 3 (vi) y= (A) –2.5 (B) 0.8 (C) –2.9 (D) 0.3
x 2 3x 2
(vii) y = 1 | x | (vii) y = logx2. 3 x, x 1
4. Graph the function F(x) =
1 1 2x, x 1
(ix) y= x2 (x) y= x3 log10 (2 x 3)
log10 (1 x) x2
5. Find a formula for each function graphed
log10 x 3 x
3 1
(xi) y= (xii) y= 3 sin x
4 x2 sin x
(xiii) y = log10 x 4 6x (xiv) y = log10[1 – log10(x2 – 5x + 16)]
2. Find the range of the following functions : (a) (b)
x 1 2
(i) ƒ(x) = (ii) ƒ(x) =
x2 x
1 x2 x 1
(iii) ƒ(x) = 2 (iv) ƒ(x) = 2 The graphs of ƒ and g are given.
x x 1 x x 1 6.
(v) ƒ(x) = e (x 1)2
(vi) 3 2
ƒ(x) = x – x + x + 1 (a) State the value of ƒ(–4) and g(3)
(b) For what value of x is ƒ(x) = g(x)?
(vii) ƒ(x) = log(x8 + x4 + x2 + 1) (viii) ƒ(x) = sin2x – 2sinx + 4 (c) Estimate the solution of the equation ƒ(x) = –1.
2 (d) On what interval is f decreasing?
(ix) ƒ(x) = sin(log2x) (x) ƒ(x) = 2 x + 1
(e) State the domain and range of ƒ.
e2x e x 1 1 (f) State the domain and range of g.
(xi) ƒ(x) = (xii) ƒ(x) =
e2x e x 1 8 3sin x
3. The graph of a function ƒ is given.
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7. Solve the following inequalities using graph of f(x) : 13. The range of the function ƒ : N I; ƒ(x) = (– 1)x–1, is -
(a) 0 ƒ(x) 1 (b) –1 ƒ(x) 2 (c) 2 ƒ(x) 3 (d) ƒ(x) > –1 & ƒ(x) < 0 (A) [–1, 1] (B) {– 1, 1} (C) {0, 1} (D) {0, 1, – 1}
14. The range of the function ƒ(x) = e–xex, is -
(A) ƒ(x) 1 (B) ƒ(x) 1 (C) ƒ(x) 2 (D) ƒ(x) 2
4x
15. If ƒ(x) = , then ƒ(x) + ƒ(1 – x) is equal to-
4x 2
(A) 0 (B) –1 (C) 1 (D) 4
16. The range of the function ƒ(x) = 4 x 2 x 2 1 is
Straight Objective Type (A) 3, 7 (B) 3, 5 (C) 2, 3 (D) 3, 6
8. If [a] denotes the greatest integer less than or equal to a and –1 x < 0, 0 y < 1, 1 z < 2,
17. A function ƒ has domain [–1, 2] and range [0, 1]. The domain and range respectively of the
[x] 1 [y] [z]
function g defined by g(x) = 1 – ƒ(x + 1) is
then [x] [y] 1 [z] is equal to –
(A) [–1, 1] ; [–1, 0] (B) [–2, 1] ; [0, 1] (C) [0, 2] ; [–1, 0] (D) [1, 3] ; [–1, 0]
[x] [y] [z] 1
(A) [x] (B) [y] (C) [z] (D) none of these
ex 1
18. For the function ƒ(x) = , if n(d) denotes the number of integers which are not in its
ex 1
9. If [x] and {x} denotes the greatest integer function less than or equal to x and fractional part
domain and n(r) denotes the number of integers which are not in its range, then n(d) + n(r) is
function respectively, then the number of real x, satisfying the equation (x–2)[x] = {x} – 1, is-
equal to -
(A) 0 (B) 1 (C) 2 (D) infinite
(A) 2 (B) 3 (C) 4 (D) Infinite
sin 2 x 2sin x 4
10. The range of the function ƒ(x) = sgn 2 is (where sgn(.) denotes signum 19. If x4 ƒ(x) – 1 sin 2x =| ƒ(x) | 2ƒ(x), then ƒ(–2) equals
sin x 2sin x 3 1 1 1
function)- (A) (B) (C) (D) 0
17 11 19
(A) {–1,0,1} (B) {–1,0} (C) {1} (D) {0,1}
15 1 x 10
1 20. Let ƒ : R – R be defined by ƒ(x) = then ƒ(x) is -
11. If 2ƒ(x) – 3ƒ x 2 , x is not equal to zero, then ƒ(2) is equal to- 2 2 2x 15
x (A) one-one but not onto (B) many one but not-onto
7 5
(A) (B) (C) – 1 (D) none of these (C) one-one and onto (D) many one and onto
4 2
2x 2 5x 3
21. ƒ : R R ƒ(x) = , then ƒ is -
5 2x 8x 2 9x 11
12. The number of integers lying in the domain of the function ƒ(x) = log 0.5 is -
x (A) one-one onto (B) many-one onto
(A) 3 (B) 2 (C) 1 (D) 0 (C) one-one into (D) many one into
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sin([x]) 1 | sin x |
22. If ƒ : R R & ƒ(x) = 2x 1 x(x 1) (where [x] denotes integral part of x), 29. Which of the following is the graph of y = ?
x 2 2x 3 4 sin x
then ƒ(x) is -
(A) one-one but not onto (B) one-one & onto
(C) onto but not one-one (D) neither one-one nor onto
(A) (B)
23. Which of the following function is surjective but not injective
(A) ƒ : R R ƒ(x) = x4 + 2x3 – x2 + 1 (B) ƒ : R R ƒ(x) = x3 + x + 1
(C) ƒ : R R+ ƒ(x) = 1 x 2 (D) ƒ : R R ƒ(x) = x3 + 2x2 – x + 1
24. If ƒ(x) = x|x| then ƒ–1(x) equals- (C) (D)
(A) | x | (B) (sgn x). | x | (C) | x | (D) Does not exist
(where sgn(x) denotes signum function of x)
25. If ƒ : (–,3] [7,) ; ƒ(x) = x2 – 6x + 16, then which of the following is true - x x
30. Period of function ƒ(x) = min{sinx, |x|} + (where [.] denotes greatest integer function)
(A) ƒ–1(x) = 3 +x 7 (B) ƒ1(x) 3 – x 7
–1 1 is -
(C) ƒ (x) = 2 (D) ƒ is many-one
x 6x 16 (A) /2 (B) (C) 2 (D) 4
26.
2
ƒ : R R such that ƒ(x) = n x x 1 . Another function g(x) is defined such that 1
A
tan x
then let us define a function ƒ(x) = det. (ATA–1) then which of the
1
31.
goƒ(x) = x, xR. Then g(2) is - tan x
(A)
e2 e2
(B) e2 (C)
e2 e 2
(D) e–2
following can not be the value of ƒ ƒ ƒ ƒ........ƒ(x) is (n ≥ 2)
2 2 n times
(A) ƒn(x) (B) 1 (C) ƒn–1(x) (D) nƒ(x)
27. Let P(x) = kx3 + 2k2x2 + k3. The sum of all real numbers k for which (x – 2) is a factor of P(x),
is 32. The number of integral values of x satisfying the inequality [x – 5] [x – 3] + 2 < [x – 5] + 2[x – 3]
(A) 4 (B) 8 (C) –4 (D) –8 (where [.] represents greatest integer function) is -
(A) 0 (B) 1 (C) 2 (D) 3
28. Which of the following is the graph of y = |x – 1| + |x – 3|?
4
33. Range of function ƒ(x) = log2 is given by
x 2 2x
(A) (B)
1 1
(A) (0, ) (B) ,1 (C) [1, 2] (D) ,1
2 4
34. A lion moves in the region given by the graph y – |y| – x + |x| = 0. Then on which of the
(C) (D) following curve a person can move so that he does not encounter lion -
1
(A) y = e–|x| (B) y = (C) y = signum(x) (D) y = –|4 + |x||
x
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n
k Let ƒ : R R be defined by ƒ(x) = ln (x + x 2 1 ), then number of solutions of |ƒ–1(x)| = e–|x|
35. Suppose, ƒ(x, n) = log x , then the value of x satisfying the equation ƒ(x, 10) = ƒ(x, 11), 42.
k 1 x is :-
is (A) 1 (B) 2 (C) 3 (D) Infinite
(A) 9 (B) 10 (C) 11 (D) none
43. ƒ(x) = [x – 1] + {x}[x], x (1,3), then ƒ–1(x) is -
sec x tan x 1 (where [.] denotes greatest integer function and {.} denotes fractional part function)
36. Range of ƒ(x) = ; x 0, is -
tan x sec x 1 2 x 1 x (1, 2) x 1 x (1, 2)
(A) (B)
2 x 1 x [2,3) 2 x 1 x [2,3)
(A) (0,1) (B) (1,) (C) (–1, 0) (D) (–,–1)
x 1 x (0,1) x 1 x (0,1)
37. If ƒ(x, y) = max(x, y) + min(x, y) and g(x, y) = max(x, y) – min(x, y), then the value of (C) (D)
2 x 1 x [1, 2) 2 x 1 x [1, 2)
2 3
ƒ g , ,g( 3, 4) is greater than -
3 2
(A) 1 (B) 2 (C) 3 (D) 4 44. Let ƒ : R R and ƒ(x) = x3 + ax2 + bx – 8. If ƒ(x) = 0 has three real roots & ƒ(x) is a bijective
function, then (a + b) is equal to
(A) 0 (B) 6 (C) –6 (D) 12
x 3 , x rational
38. If functions ƒ(x) and g(x) are defined on R R such that ƒ(x) = ,
4x , x irrational 45. Which of the following functions is an odd function ?
x 5 , x irrational 1 1
g(x) = then (ƒ – g)(x) is - (A) |x – 2| + (x + 2) sgn(x + 2)
x e x 1 2x
(B)
x , x rational
(A) one-one & onto (B) neither one-one nor onto
(D) e4x e2x 1
4
(C) one-one but not onto (D) onto but not one-one (C) log(sin x + 1 sin 2 x )
(where sgn(x) denotes signum function of x)
Let ƒ : A B be an onto function such that ƒ(x) = x 2 2 x 3 x 2 2 x 3 then set 1 2
39. 46. Period of ƒ(x) = {x} + x x is equal to (where{.} denotes fractional part function)
'B' is - 3 3
(A) [–2,0] (B) [0,2] (C) [–3,0] (D) [–1,0] 2 1 1
(A) 1 (B) (C) (D)
3 2 3
x 1 x
40. Let ƒ(x) = and let be a real number. If x0 = , x1 = ƒ(x0), x2 = ƒ(x1), ....... & x2011 = – 47. Let ƒ(x) = 2x – and g(x) = cosx, where {.} denotes fractional part function, then period of
1 x 2012
then the value of is - goƒ(x) is -
2011 3
(A) (B) 1 (C) 2011 (D) –1 (A) (B) (C) (D)
2012 2 2 4
1
f 4 (x) sin x sin 5x
If ƒ1(x) = 2
f 2 (x)
, where ƒ2(x) 2012 f3 (x) , where ƒ3(x) = 48. The period of the function is -
41. where cos x cos5x
2013
ƒ4(x) = log2013logx2012, then the range of ƒ1(x) is - (A) (B) (C) (D) 2
3 2
(A) (2, ) (B) (2012, ) (C) (0, ) (D) (–,)
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49. Let ƒ : R R be a real valued function such that ƒ(10 + x) = ƒ(10 – x) x R and EXERCISE # 2
ƒ(20 + x) = –ƒ(20 – x) x R. Then which of the following statements is true - 1. Find the domains of definitions of the following functions :
(A) ƒ(x) is odd and periodic (B) ƒ(x) is odd and aperiodic (Read the symbols [*] and {*} as greatest integers and fractional part functions respectively.)
(C) ƒ(x) is even and periodic (D) ƒ(x) is even and aperiodic
(i) ƒ(x) = cos 2x 16 x 2 (ii) ƒ(x) = log7 log5 log3 log2 (2x3 + 5x2 – 14x)
Linked Comprehension Type
Paragraph for Question 50 & 51
(iii) ƒ(x) = ln x 2 5x 24 x 2 (iv) ƒ(x) =
1 5x
7 x 7
x2 ; x 1 2log10 x 1
x ; x0 (v) y = log10sin(x – 3) + 16 x 2 (vi) ƒ(x) = log100x
Let ƒ(x) = & g(x) = 2x 3 ; 1 x 1 x
1 x ; x 0 x ; x 1 1
(vii) ƒ(x) = x2 | x | (viii) ƒ(x) = (x 2 3x 10)ln 2 (x 3)
On the basis of above information, answer the following questions :
9 x2
1
7
(ix) ƒ(x) = (5x 6 x 2 ) { n{x}} (7 x 5 2 x 2 ) ln x
50. Range of ƒ(x) is - 2
(A) (–,1] (B) (–) (C) (–,0] (D) (–,2] (x) ƒ x log 1
x 2 x 6 16x C2x 1 203x P2x 5
x x
51. Range of g(ƒ(x)) is -
(A) (–,) (B) [1,3) (3,) (C) [1,) (D) [0,) 2. Find the domain & range of the following functions.
(i) y log 5 2(sinx cosx) 3 (ii) y=
2x
1 x2
x 2 3x 2 x
(iii) ƒ(x) = (iv) ƒ(x) =
x2 x 6 1 | x |
x 4 3
(v) y= 2 x 1 x (vi) ƒ(x) =
x 5
3. (a) Draw graphs of the following function, where [ ] denotes the greatest integer function.
(i) ƒ(x) = x + [x]
(ii) y = (x)[x] where x = [x] + (x) & x > 0 & x 3
(iii) y = sgn[x]
(iv) y = sgn(x – |x|)
(b) Identify the pair(s) of functions which are identical ?
(where [x] denotes greatest integer and {x} denotes fractional part function)
(i) ƒ(x) = sgn(x2 – 3x + 4) and g(x) = e[{x}]
1 cos 2x
(ii) ƒ(x) = and g(x) = tanx
1 cos 2x
(iii) ƒ(x) = ln(1 + x) + ln(1 – x) and g(x) = ln(1 – x2)
cos x 1 sin x
(iv) ƒ(x) = and g(x) =
1 sin x cos x
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4. Classify the following functions ƒ(x) defined in R R as injective, surjective, both or none. 11. Find whether the following functions are even or odd or none :
(a) ƒ(x) = 2
x 2 4x 30
x 8x 18
(b) 3 2
ƒ(x) = x – 6x + 11x – 6 (a)
ƒ(x) = log x 1 x
2
(b) ƒ(x) =
x(a x 1)
a x 1
(c) ƒ(x) = (x2 + x + 5)(x2 + x – 3) (c) ƒ(x) = sinx + cosx (d) ƒ(x) = xsin2x – x3
(e) ƒ(x) = sinx – cosx (f) ƒ(x) =
1 2 x 2
5. Solve the following problems from (a) to (d) on functional equation :
2x
(a) The function ƒ(x) defined on the real numbers has the property that ƒ(ƒ(x)).(1 + ƒ(x)) = –ƒ(x)
x x
for all x in the domain of ƒ. If the number 3 is in the domain and range of ƒ, compute (g) ƒ(x) = 1 (h) ƒ(x) = [(x + 1)2]1/3 + [(x – 1)2]1/3
ex 1 2
the value of ƒ(3).
(b) Suppose ƒ is a real function satisfying ƒ(x + ƒ(x)) = 4ƒ(x) and ƒ(1) = 4. Find the value
12. (i) Write explicitly, functions of y defined by the following equations and also find the
of ƒ(21).
domains of definition of the given implicit functions :
(c) Let ƒ be function defined from R+ R+. If [ƒ(xy)]2 = x(ƒ(y))2 for all positive numbers x
(a) 10x + 10y = 10 (b) x + |y| = 2y
and y and ƒ(2) = 6, find the value of ƒ(50).
(d) Let ƒ be a function such that ƒ(3) = 1 and ƒ(3x) = x + ƒ(3x – 3) for all x. Then find the (ii) The function ƒ(x) is defined on the interval [0, 1]. Find the domain of definition of the
value of ƒ(300). functions.
(a) ƒ(sinx) (b) ƒ(2x + 3)
6. Suppose ƒ(x) = sinx and g(x) = 1 – x . Then find the domain and range of the following (iii) Given that y = ƒ(x) is a function whose domain is [4,7] and range is [–1, 9]. Find the
functions. range and domain of
(A) ƒog (B) goƒ (C) ƒoƒ (D) gog 1
(a) g(x) = ƒ(x) (b) h(x) = ƒ(x – 7)
3
1 x
7. A function ƒ : R R is such that ƒ x for all x – 1. Prove that following.
1 x 13. Compute the inverse of the functions :
(a) ƒ(ƒ(x)) = x (b) ƒ(1/x) = –ƒ(x), x 0 x
ƒ(x) = n x x 1 ƒ(x) = 2 x 1
2
(a) (b)
(c) ƒ(–x – 2) = –ƒ(x) –2.
10x 10 x
x (c) y=
8. (a) Find the formula for the function ƒogoh, given ƒ(x) = ; g(x) = x10 and h(x) = x + 3. 10x 10 x
x 1
Find also the domain of this function. Also compute (ƒogoh)(–1).
+ 8 and hence solve the equation ƒ(x) = ƒ–1(x).
log10 x
14. Find the inverse of ƒ(x) = 2
(b) If ƒ(x) = max(x, 1/x) for x > 0 where max (a, b) denotes the greater of the two real
numbers a and b. Define the function g(x) = ƒ(x) ƒ(1/x) and plot its graph. 1
15. (a) Function ƒ & g are defined by ƒ(x) = sinx, x R; g(x) = tanx, x R – K
2
9. Let ƒ be a one-one function with domain {x, y, z} and range {1, 2, 3}. It is given that exactly
where K I.
one of the following statements is true and the remaining two are false.
Find (i) periods of ƒog & goƒ
ƒ(x) = 1; ƒ(y) 1; ƒ(z) 2. Determine ƒ –1(1)
(ii) range of the function ƒog & goƒ
1 x if x0 x if x 1 (b) Suppose that ƒ is an even, periodic function with period 2, and that ƒ(x) = x for all x in
10. ƒ(x) = 2 and g(x) = find (ƒog)(x) and (goƒ)(x).
x if x0 1 x if x 1 interval [0, 1]. Find the value of ƒ(3.14).
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16. The graph of the function y = ƒ(x) is as follows : 18. The sum of integral values of the elements in the domain of ƒ(x) log 1 | 3 x | is -
2
19. Number of integers in range of ƒ(x) = x(x + 2) (x + 4) (x + 6) + 7, x [–4, 2] is
tan 2 x 8tan x 15
20. The number of even integral value(s) in the range of the function ƒ(x) = is
1 tan 2 x
Match the function mentioned in Column-I with the respective graph given in Column-II.
Column-I Column-II 1 x 1
+ bx + c sinx + 5 and ƒ(log32) = 4, then ƒ log3 is equal to
3
21. If ƒ(x) = a log
1 x 2
(A) y = |ƒ(x)| (P) 22. Let P(x) = x4 + ax3 + bx2 + cx + d be a polynomial such that P(1) = 1, P(2) = 8, P(3) = 27, P(4) = 64
then find P(10).
0 x 1
23. If ƒ(x) = ; then the number of solutions of the equation ƒ(ƒ(ƒ(x))) = x is
(B) y = ƒ(|x|) (Q) 2x 2 x 1
24. Let 'ƒ' be an even periodic function with period '4' such that ƒ(x) = 2x – 1, 0 x 2.
The number of solutions of the equation ƒ(x) = 1 in [–10, 20] are
(C) y = ƒ(–|x|) (R)
2x 1 ax b
25. Let ƒ(x) = . If ƒ–1= , then a + b + c is
x 3 cx
1 26. Let ƒ(x) be a periodic function with period 'p' satisfying ƒ(x) + ƒ(x + 3) + ƒ(x + 6) +.....+ ƒ(x + 42) =
(D) y= (|ƒ(x)| – ƒ(x)) (S) constant x R, then sum of digits of 'p' is
2
17. Column-I Column-II
ƒ(x) Range
cos 2 x cos x 2 7
(A) (P) 0,
cos 2 x cos x 1 3
(B)
cos x sin x cos x sin x (Q)
4 7
,
3 cos x sin x 3 3
7 1
3 x 6 2x 4 3x 2 1 0, 3
(C) (R)
(D) log8(x2 + 2x + 2) (S) [0, )
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11. Range of ƒ(x) = cot–1(loge(1 – x2)) is -
EXERCISE # 1
(A) (0, ) (B) 0, (C) , (D) 0,
Straight Objective Type 2 2 2
1 x
1. The domain of the function sin log 2 is - 2 3
3 12. The value of sin 1 cot sin 1 cos 1
12
sec1 2 is -
4 4
1
(A) ,3
1
(B) ,3
3
(C) ,6
1
(D) , 2
2 2 2 2 (A) 5 (B) 6 (C) 0 (D) 10
2. Domain of the function ƒ(x) = logecos–1 { x} is, where {.} represents fractional part function - 1
13. Number of solution(s) of the equation cos 1 x sin 1 x 1 cos 1 1 x sin 1 is -
(A) x R (B) x [0, ) (C) x (0, ) (D) x R – {x | x I} x 2
(A) 0 (B) 1 (C) 2 (D) 4
1 1
3. tan 1 1 x 2 2 sin 1 x 2 2 1 (where x 0) is equal to
x x
r((r 1)!)
3 14. tan 1
2 is equal to -
(A)
2
(B)
4
(C)
4
(D) r 0 (r 1) ((r 1)!)
4. The value of tan2(sec–13) + cot2(cosec–14) is - (A) (B) (C) cot–13 (D) tan–12
2 4
(A) 9 (B) 16 (C) 25 (D) 23 if | x | 3
x2 4
15. Let ƒ(x) =
5.
12 16
If x > 0, cos–1 cos 1 then x is - 5sgn | x 3 | if | x | 3
x 2 x
(A) 12 (B) 16 (C) 20 (D) 320 and g(x) = 2tan–1(ex) – for all x R, then which of the following is wrong ?
2
(where sgn(x) denotes signum function of x)
6. If 2 a < 3, then the value of cos1 cos[a] cosec1 cosec[a] cot1 cot[a], (where [.] denotes
(A) ƒ(x) is an even function (B) goƒ(x) is an even function
greatest integer less than equal to x) is equal to
(C) g(x) is an odd function (D) ƒoƒ(x) is an odd function
(A) 2 – (B) 2 + (C) (D) 6
7. If cos–1(2x2 – 1) = 2 – 2cos–1x, then - Linked Comprehension Type
1 1 1
(A) x [–1, 0] (B) x [0, 1] (C) x 0, (D) x ,
2
Paragraph for Question 16 to 18
2 2 Consider a continuous function such that each image has atmost three pre image & atleast one
8. Number of integral ordered pairs (a, b) for which image has exactly three pre images. This type of function is to be called as three-one function.
a 2 a3 On the basis of above information, answer the following questions :
sin–1(1 + b + b2 +.... ) +cos–1 a .... = is -
3 9 2 16. Which of the following function is a three-one function ?
(A) 0 (B) 4 (C) 9 (D) Infinitely many (A) |n|x|| (B) e|x| (C) x3 + 3x2 – 7x + 6 (D) cos(cos–1x)
n
2r 1 If ƒ(x) = sin–1(sinx) is a three-one function, then possible interval of x is -
9. lim tan 1 is equal to - 17.
n
r 1 r 4 2r 3 r 2 1 3 3
(A) [–] (B) , (C) (–2, 0] (D) ,
(A)
(B)
3
(C)
(D)
2 2 2 2
4 4 2 8
10. The range of the function ƒ(x) = sin–1(log2(–x2 + 2x + 3)) is - 18. If ƒ(x) is a three-one function such that ƒ(a) = ƒ(b) (where a b), then number of maximum
possible values of b is -
(A) , (B) ,0 (C) 0, (D) [–1, 1]
2 2 2 2 (A) 1 (B) 2 (C) 3 (D) 4
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1 1 2 2
EXERCISE # 2 3. Find the domain and range of the following functions.
1. (a) Find the following : (Read the symbols [ * ] and { * } as greatest integers and fractional part function respectively)
(i) ƒ(x) = cot–1(2x – x2) (ii) ƒ(x) =sec–1 (log3 tan x + logtan x3)
1 1
(i) tan cos 1 tan 1
2 3 2x 2 1
(iii) ƒ(x) cos –1 2 (iv) ƒ(x) tan –1 log 4 (5x 2 – 8x 4)
7 x 1 5
(ii) cos 1 cos
6
1 3
(iii) cos tan 4. Identify the pair(s) of functions which are identical. Also plot the graphs in each case.
4
1 3 3 1 x2 1
(iv) tan sin cot 1 (a) y = tan(cos–1 x); y = (b) y = tan(cot–1x); y =
5 2 x x
(b) Find the following :
3 5. Let y = sin–1(sin8) – tan–1(tan10) + cos–1(cos12) – sec–1(sec9) + cot–1(cot6) – cosec–1(cosec7).
(i) sin sin 1 If y simplifies to a b , then find (a – b).
2 2
3
(ii) cos cos 1 33 1 46 1 13 1 19 13
2 6 6. Show that : sin–1 sin cos cos tan tan cot cot
7 7 8 8 7
3
(iii) tan–1 tan
4
1 2 6 1
63 7. Prove that : (a) 2cos–1
3 16 1
cot 1 cos 1
7
(b) arc cos arccos
(iv) sin arc sin 13 63 2 25 3 2 3 6
4 8
2. Find the domain of definition the following functions.
(Read the symbols [ * ] and { * } as greatest integers and fractional part functions respectively)) 8. If and are the roots of the equation x2 + 5x – 49 = 0, then find the value of cot(cot–1 + cot–1).
2x
(i) ƒ(x) = arc cos
1 x 1 ab 1 1 bc 1 1 ca
9. If a > b > c > 0, then find the value of : cot 1 cot cot .
1 x2 a b bc ca
1
(ii) ƒ(x) = cos(sinx) sin
2x
10. Find all values of k for which there is a triangle whose angles have measure
x 3
(iii) ƒ(x) = sin–1 – log10(4 – x) 1 1 1
2 tan 1 , tan 1 k and tan 1 2k .
2 2 2
(iv) ƒ(x) = sin–1(2x + x2)
1 sin x 11. Find the simplest value of
(v) ƒ(x) = cos 1 (1 {x}) , where {x} is the fractional part of x.
log5 (1 4 x 2 ) x 1 1
(a) ƒ(x) = arc cosx + arc cos 3 3x 2 , x ,1
2 2 2
3 2x –1
(vi) ƒ(x) = 3 x cos 1 + log6(2|x| – 3) + sin (log2x) 1 x2 1
5 (b) ƒ(x) = tan–1 , x R – {0}
x
x
x
sin 1
(vii) ƒ(x) = e 2
tan 1 1 n x [x]
2
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3 3 4 4
12. Prove that the identities : LIMIT
–1 –1 –1
–1 EXERCISE # 1
(a) sin cos(sin x) + cos sin(cos x) = , |x| 1
2 [SINGLE CORRECT CHOICE TYPE]
(b) tan(tan–1x + tan–1y + tan–1z) = cot(cot–1x + cot–1y + cot–1z) 1 3
1. lim is equal to
x 1 1 x 1 x3
13. Solve the inequality : (arc secx)2 – 6(arc secx) + 8 > 0 (A) –1 (B) 0 (C)1 (D) D.N.E.
14. Solve the following : 1 x 1 x
2. lim is equal to
x 0
–1 –1 2x
(a) sin x sin 2x = 1 1
3 (A) 0 (B) 1 (C) (D)
–1 1 1 1 2 2 4
(b) tan tan tan 1 2
1 2x 1 4x x
(c) tan–1(x 1) tan–1(x) tan1(x 1) tan–1(3x) 1 2 x 3
3. lim is equal to
x2
x 2
(d) 3cos1 x sin1 1 x 2 (4 x 2 1) 1 1 1
(A) (B) 3 (C) (D)
2 3 4 3 8 3
(e) sin–1x + sin–1y = & cos1 x cos–1y
3 3 n
x 1
4. lim (m and n integers) is equal to
x 1 m x 1
15. Find the sum of the series :
m n
(a) cot–17 + cot–113 + cot–121 + cot–131 + ..... to n terms. (A) 0 (B) 1 (C) (D)
n m
1 2 2n 1
(b) tan–1 + tan–1 + …. +tan–1 + ….
3 9 1 22n 1 2x x 2 3a 2
5. If lim = 2 (where a R+), then a is equal to –
(c) tan 1 2
1
tan 1 2
1
tan 1 2
1
tan 1 2
1
to n terms
x a x a 2a
x x 1 x 3x 3 x 5x 7 x 7x 13 1 1 1 1
1 (A) (B) (C) (D)
1 1 1
(d) sin–1 + sin–1 + sin–1 + …. + sin–1 + …. terms. 3 2 2 3 2 9
5 65 325 4n 4 1
n(sin 3x)
6. lim is equal to
x 0 n(sin x)
(A) 0 (B) 1 (C) 2 (D) Non existent
1 x 2 4 1 2x
3
7. lim is equal to
x 0 x x2
1 1
(A) (B) (C)1 (D) D.N.E.
4 2
3
7 x3 3 x 2
8. lim is equal to
x 1 x 1
1 1 1 1
(A) (B) (C) – (D) –
4 6 4 6
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5 5
(n 1) 4 (n 1) 4 sin{x}
9. lim is equal to 18. Let ƒ(x) = . If ƒ(5+) and ƒ(3+) exists finitely and are not zero, then the value of
n (n 1) (n 1)
4 4
x 2 ax b
(A) –1 (B) 0 (C) 1 (D) D.N.E. (a + b) is (where {.} represents fractional part function) -
(A) 7 (B) 10 (C) 11 (D) 20
(x 1)10 (x 2)10 ..... (x 100)10
10. lim is equal to | cos(sin(3x)) | 1
x x10 1010 19. lim
x 0 x2
equals
(A) 1 (B) 100 (C) 200 (D) 10
9 3 3 9
(A) (B) (C) (D)
11. lim
x
x 2x 1 x 7x 3 is equal to :
2 2 2 2 2 2
1 cos n
(A) –
5
2
(B)
5
2
(C)0 (D) D.N.E. 20. Let a = min {x2 + 2x + 3, x R} and b = lim
0 2
. then value of a .b
r 0
r n r
is :
2n 1 1 2n 1 1 4n 1 1
1 (A) (B) (C) (D) none of these
12. If lim 2n n 2n n =
2 2
(where is a real number), then - 3.2n 3.2n 3.2n
n 2
(A) = 1 (B) = – 1 (C) = ±1 (D) (–, 1) 21. Let BC is diameter of a circle centred at O. Point A is a variable point, moving on the
BM
circumference of circle. if BC = 1 unit, then lim is equal to –
n! n A B (Area of sector OAB) 2
13. Let Un = where n N. If Sn = U n then lim Sn equals
(n 2)! n 1
n
(A) 2 (B) 1 (C) 1/2 (D) Non existent
2k 1 . Then lim
n n
14. For n N, let an = 2k and bn= a n bn is equal to -
n
k 1 k 1
1 (A) 1 (B) 2 (C) 4 (D) 16
(A) 1 (B) (C) 0 (D) 2
2
x
x 2 2x 1
22. lim is equal to
n
1 a
x 4x 2
2
15. Let Pn = 1 k 1 . If lim Pn can be expressed as lowest rational in the form , then value x
n b
k 2 C2 1
of (a + b) is: (A) 1 (B) e (C) (D) e2
e2
(A) 4 (B) 8 (C) 10 (D) 12
cos x
23. lim (1 + sin x) is equal to
x 0
cos 2 cos 2x
16. lim is equal to 1
x 1 x2 | x | (A) 0 (B) e (C) 1 (D)
e
(A) 0 (B) cos2 (C) 2sin2 (D) sin1 24. 1/x
lim (cos x + asinbx) is equal to :
x 0
sin x 2sin 2x 10sin10x (A) ea (B) eab (C) eb (D) ea/b
17. Let ƒ(x) =
x x
+ + ......+
x (where [y] is the largest integer y). The
1/ x
value of lim ƒ(x) equals :
25. lim tan x is equal to
x 0 x 0
4
(A) 55 (B) 164 (C) 165 (D) 375 1
(A) e–2 (B) (C) e (D) e2
e
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26. lim(4n 5n )1/n is equal to
n
34. lim tan2x
x
2sin 2 x 3sin x 4 sin 2 x 6sin x 2 is equal to
2
(A) 5 (B) 4 (C) 0 (D) D.N.E. 3 1 1 5
(A) (B) (C) (D)
nx
4 6 12 12
11/x 21/x 31/x ..... n1/x x 1 x
27. lim is equal to 35. lim x arctan arctan is equal to
x
n x
x2 x2
1 1 1
(A) n! (B) 1 (C) (D) 0 (A) (B) (C) 1 (D) D.N.E.
n! 2 2
tan(a 2h) 2 tan(a h) tana
28. If lim (1 + ax + bx2)2/x = e3 , then 36. lim is equal to
x 0
h 0 h2
2
3 3 (A) tan a (B) tan a (C) sec a (D) 2(sec2a)(tan a)
(A) a = and b R (B) a = and b R+
2 2 1/ x
(C) a = 0 and b = 1 (D) a = 1 and b = 0 x 1 1
37. lim 2 equals
x 0
2
1/ x
f (x) x 2 1
29. If ƒ(x) is a polynomial of least degree, such that lim 1 2
= e , then ƒ(2) is - (A) 2 (B) n2 (C) n2 (D) 2
x 0
x2 2
(A) 2 (B) 8 (C) 10 (D) 12 1
38.
x 0
If lim cos x a 3 sin(b6 x) x = e512, then the value of ab2 is equal to
n
e
30. lim n2
equals - (A) – 512 (B) 512 (C) 8 (D) 8 8
n
1
1
n sin x n(1 3x) is equal to
3
tan x e 1
1 39. The value of lim 2 3
(A) 1 (B) (C) e (D) e x 0 1 5 x
2
2ƒ(tanx) 2ƒ(sinx) 1 3 2 4
31. If ƒ(x) is odd linear polynomial with ƒ(1) = 1, then lim is : (A) (B) (C) (D)
x 0 x 2 ƒ(sin x) 5 5 5 5
1 40. The figure shows an isosceles triangle ABC with B = C. The bisector of angle B intersects
(A) 1 (B) n2 (C) n2 (D) cos2
2 the side AC at the point P. Suppose that BC remains fixed but the altitude AM approaches 0, so
that A M (mid-point of BC). Limiting value of BP, is
x(1 acosx) bsinx
32. lim = 1 then
x 0 x3
(A) a = –5/2 (B) a = –3/2, b = –1/2
(C) a = –3/2, b = –5/2 (D) a = –5/2, b = –3/2
sin(a 3h) 3sin(a 2 h) 3sin(a h) sina
33. lim is equal to
h 0 h3 a a 2a 3a
(A) cos a (B) – cos a (C) sin a (D) sin a cos a (A) (B) (C) (D)
3 2 3 4
where a is fixed side BC.
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secx tan x 1 EXERCISE # 2
41. The value of lim is equal to-
x 2 x2
x x. n x n x 1
2
(A) sec2. n sec + tan2.n tan (B) sec2.n tan + tan2. n sec 1. lim
x 1 x 1
(C) sec2. n tan – tan2.n sec (D) sec2.n sec – tan2.n tan
100 k
1 x, 0 x 1 x 100
42. Consider the function ƒ(x) = x 2, 1 x 2 . Let lim ƒ(ƒ(x)) = and lim ƒ(ƒ(x)) = m then 2. lim K 1
x 1 x 2
x 1 x 1
4 x, 2 x 4
which one of the following hold good ? 1 tan x
3. lim
(A) exists but m does not. (B) m exists but does not. x
4
1 2 sin x
(C) Both and m exist (D) Neither nor m exist.
8 x2 x2 x2 x2
4. lim
x 0 x8 1 cos 2 cos 4 cos 2 cos 4
cos cos 2 x
43. Let ƒ(x) be a quadratic function such that ƒ(0) = ƒ(1) = 0 and ƒ(2) = 1, then lim 2
x 0 ƒ 2 (x) 2 cos sin
5. lim
is equal to- (4 ) 2
4
(A) (B) (C) 2 (D) 4
2
sin 4h 4sin 3h 6sin 2h 4sin h sin
6. 3 3 3 3 3
lim
h 0 h4
x2 x 3
7. lim x 2 3
x
x x
3x 4
2x 2 sin
1
x
| x |3 5
8. lim
x | x |3 | x |2 | x | 1
n
9. If = lim (r 1)sin r sin then find {}.
n
r 2 r 1 r
(where { } denotes the fractional part function)
x2 1
10. Find a and b if : (i) lim ax b = 0
x
x 1
(ii) lim x 2 x 1 ax b = 0
x
2 2
11. lim [n (1 + sin x). cot (n (1 + x))]
x 0
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27 x 9x 3x 1 CONTINUITY
12. lim
x 0 2 1 cos x EXERCISE # 1
[SINGLE CORRECT CHOICE TYPE]
–1 a
13. (a) lim tan , where a R ;
x 0 x2 ax 1 if x 1
2x 1 x 1. Let ƒ(x) = 3 if x 1 . If f(x) is continuous at x = 1 then (a – b) is equal to-
(b) Plot the graph of the function ƒ(x) = lim tan 2 bx 2 1 if x 1
t 0
t
(A) 0 (B) 1 (C) 2 (D) 4
14. Let {an}, {bn}, {cn}be sequences such that
(i) an + bn + cn = 2n + 1 ; 1
(ii) an bn + bncn + cnan = 2n – 1 ; 2. For the function ƒ(x) = 1
, x 2 which of the following holds ?
(iii) anbncn = – 1 ; x 2 x 2
(iv) an < bn< cn (A) ƒ(2) = 1/2 and f is continuous at x = 2 (B) ƒ(2) 0, 1/2 and ƒ is continuous at x = 2
Then find the value of L im (nan). (C) ƒ can not be continuous at x = 2 (D) ƒ(2) = 0 and f is continuous at x = 2.
n
15. Let ƒ(x) = ax3 + bx2 + cx + d and g(x) = x2 + x – 2. 4 x2
3. The function ƒ(x) = , is-
ƒ(x) ƒ(x) c2 d 2 4x x 3
If lim = 1 and lim = 4, then find the value of 2 2 . (A) Discontinuous at only one point in its domain.
x 1 g(x) x 2 g(x) a b (B) Discontinuous at two points in its domain.
(C) Discontinuous at three points in its domain.
8x 2 3 (D) Continuous everywhere in its domain.
2x 2 3
16. lim
2x 5
x
2
x 2 bx 25
4. If ƒ(x) = for x 5 and f is continuous at x = 5, then ƒ(5) has the value equal to-
x 2 7x 10
xc
x
17. (A) 0 (B) 5 (C) 10 (D) 25
lim
x x c
= 4 then find c
x e x cos 2x
5. If ƒ(x) = , x 0 is continuous at x = 0, then –
x x2
x
tan
18. 2
lim tan 5
x 1
4 (A) f(0) = (B) [f(0)] = –2 (C) {f(0)} = –0.5 (D) [f(0)]. {f(0)} = –1.5
2
1 where [.] and {.} denotes greatest integer and fractional part function
19. x 1 cos x x
lim
x 0
x 6. y = ƒ(x) is a continuous function such that its graph passes through (a,0). Then
log e (1 3ƒ(x))
lim is-
20. If n N and an = 22 + 42 + 62 + ....... +(2n)2 and bn = 12 + 32 + 52 + ...... + (2n – 1)2. x a 2ƒ(x)
a n bn 3 2
Find the value lim . (A) 1 (B) 0 (C) (D)
n n 2 3
7. In [1,3], the function [x2 + 1], [.] denoting the greatest integer function, is continuous -
(A) For all x
(B) For all x except at nine points
(C) For all x except at seven points
(D) For all x except at eight points
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8. Number of points of discontinuity of ƒ(x) = [2x3 – 5] in [1,2), is equal to- x
(Where [x] denotes greatest integer less than or equal to x) [x] if 1 x 2
(A) 14 (B) 13
15. Consider the function f(x) = 1 if x2
(C) 10 (D) 8
6 x if 2 x3
| x 1| if x 2
2x 3 if 2 x 0 Where [x] denotes step up function then at x = 2 function -
9. Given ƒ(x) = 2 . Then number of point(s) of discontinuity of ƒ(x) is- (A) has missing point removable discontinuity
x 3 if 0 x 3 (B) has isolated point removable discontinuity
x 15 if
3
x3 (C) has non removable discontinuity finite type
(A) 0 (B) 1 (C) 2 (D) 3 (D) is continuous
9 2 1 cos 3x x[x]2 log (1 x) 2 , for 1 x 0
10. If ƒ(x) is continuous and f = , then the value of lim ƒ is-
2
2 9
9
x 0
x2 16.
Consider ƒ(x) = n e x 2 {x}
2
, for0 x 1
(A) (B) (C) 0 (D) data insufficient tan x
9 2
where [*] & {*} are the greatest integer function & fractional part function respectively, then :-
(A) ƒ(0) = n2 f is continuous at x = 0
11. ƒ is a continuous function on the real line. Given that x2 + (ƒ(x) – 2)x – 3 ·ƒ(x) + 2 3 – 3 = 0.
(B) ƒ(0) = 2 f is continuous at x = 0
Then the value of ƒ( 3 ) (C) ƒ(0) = e2 f is continuous at x = 0
(A) cannot be determined (B) is 2 (1 – 3) (D) ƒ has an irremovable discontinuity at x = 0
2( 3 2) 2x 1
(C) is zero (D) is
3 17. The function ƒ(x) = [x]. cos , where [·] denotes the greatest integer function, is discontinuous
2
at :-
12. The function ƒ(x) = [x]2 – [x2] (where [y] is the greatest integer less than or equal to y), is
(A) all x (B) all integer points (C) no x (D) x which is not an integer
discontinuous at :
(A) all integers (B) all integers except 0 & 1 sin x x cos x
(C) all integers except 0 (D) all integers except 1 18. Consider the function defined on [0, 1] R, ƒ(x) = if x 0 and ƒ(0) = 0, then
x2
the function ƒ(x):-
13. Let ƒ : R R be a continuous function x R and f(x) = 5 x irrational. Then the value
of f (3) is - (A) has a removable discontinuity at x = 0
(A) 1 (B) 2 (C) 5 (D) cannot determine (B) has a non removable finite discontinuity at x = 0
(C) has a non removable infinite discontinuity at x = 0
1 1 (D) is continuous at x = 0
14. If ƒ(x) = and g(x) = 2 , then points of discontinuity of ƒ{g(x)} are -
(x 1)(x 2) x
x 2 px 1
1 1 1 19. Let ƒ(x) = . If ƒ(x) is discontinuous at exactly 2 values of x then number of
(A) 1, 0,1, (B) , 1, 0,1, x2 p
2 2 2
integers in the range of p is
1
(C) {0,1} (D) 0,1, (A) 1 (B) 2 (C) 3 (D) 4
2
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1
EXERCISE # 2
(1 tan x) e ,
x
20. Let ƒ(x) = if x 0 3x 2 ax a 3
x 1. If the function ƒ(x) = is continuous at x = – 2. Find ƒ(–2).
x2 x 2
k, if x 0
If ƒ(x) is continuous at x = 0, then the value of k is
2. Find all possible values of a and b so that ƒ(x) is continuous for all x R if
e e
(A) e (B) (C) (D) None
2 4 | ax 3 | if x 1
| 3 x a | if 1 x 0
px 2 px q, x 1 ƒ(x) = b sin 2x
x 2b if 0x
21. Let ƒ(x) = x 1, 1 x 3
lx 2 mx 2, x 3 cos x 3
2
if x
ql m
If ƒ(x) is continuous x R, then the value of is equal to
l tan 6x
6 tan 5x
(A) 1 (B) 2 (C) 3 (D) 4 if 0x
5 2
sin x 2
x2 , x 0 3. The function ƒ(x) = b2 if x
2
22. Let ƒ(x) = . a|tanx|
1 | cosx | b if x
3 1 2
,x0
4 4
If ƒ(x) is continuous at x = 0, then can be Determine the values of 'a' & 'b', if ƒ is continuous at x = /2.
(A) 1 (B) 2 (C) 3 (D) 4
23. Let ƒ : [0, 1] R be a continuous function and assumes only rational values. If ƒ(0) = 2 then ƒ(x)
, x3
1 3 1 4. Suppose that ƒ(x) = x3 – 3x2 – 4x + 12 and h(x) = x 3 then
the value of tan–1 ƒ + tan–1 ƒ is
2 2 2 K , x3
3 (a) find all zeroes of ƒ(x).
(A) (B) (C) (D)
4 6 4
(b) find the value of K that makes h continuous at x = 3.
x 1; x 0 (c) using the value of K found in (b) determine whether h is an even function.
24. ƒ(x) = at x = 0, ƒ(x) is
cos x; x 0
(A) continuous (B) having removable discontinuity 1 sin x 1
(C) discontinuous (D) none , x
1 cos 2x 2
1
5. Let ƒ(x) = p, x . Determine the value of p, if possible, so that the
2
2x 1 1
, x
4 2x 1 2 2
function is continuous at x = 1/2.
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6. Given the function g(x) = 6 2x and h(x) = 2x2 – 3x + a. Then
DEFFERENTIABILITY
g(x), x 1 EXERCISE # 1
(a) evaluate h(g(2)) (b) If ƒ(x) = , find 'a' so that ƒ is continuous.
h(x), x 1 [SINGLE CORRECT CHOICE TYPE]
1. Let ƒ(x) = [tan2x], (where [.] denotes greatest integer function). Then -
(A) lim ƒ(x) does not exist (B) ƒ(x) is continuous at x = 0.
1 x, 0 x 2 x 0
7. Let ƒ(x) = . Determine the form of g(x) = ƒ[ƒ(x)] & hence find the point of
3 x, 2 x 3 (C) ƒ(x) is not differentiable at x = 0 (D) ƒ'(0) = 1
discontinuity of g, if any. 2. The number of points where ƒ(x) = [sin x + cos x] (where [·] denotes the greatest integer function),
x (0,2) is not continuous is -
n cos x (A) 3 (B) 4 (C) 5 (D) 6
4 if x 0
1 x2 1
8. Let ƒ(x) = 3. If 6,8 and 12 are th, mth and nth terms of an A.P. and ƒ(x) = nx2 + 2x – 2m, then the equation
e sin 4x
1
if x 0 ƒ(x) = 0 has -
n(1 tan 2 x) (A) a root between 0 and 1 (B) both roots imaginary.
Is it possible to define f(0) to make the function continuous at x = 0. If yes what is the value of (C) both roots negative. (D) both roots greater than 1.
ƒ(0), if not then indicate the nature of discontinuity.
ƒ(h) ƒ(2h)
4. Let ƒ be differentiable at x = 0 and ƒ'(0) = 1. Then lim =
1 sin x 3
h 0 h
3cos 2 x if x (A) 3 (B) 2 (C) 1 (D) –1
2
9. Determine a & b so that f is continuous at x = where f(x) = a if x 3x 2 4 x 1 for x 1
2 2 5. Let g(x) =
ax b for x 1
b(1 sinx) if x
If g(x) is continuous and differentiable for all numbers in its domain then -
( 2 x)
2
2
(A) a = b = 4 (B) a = b = – 4 (C) a = 4 and b = – 4 (D) a = – 4 and b = 4
sin(a 1) x sinx 6. If ƒ(x) ƒ(y) + 2 = ƒ(x) + ƒ(y) + ƒ(xy) and ƒ(1) = 2, ƒ'(1) = 2 then sgn ƒ(x) is equal to
for x0 (where sgn denotes signum function) -
x
(A) 0 (B) 1 (C) –1 (D) 4
10. Determine the values of a, b & c for which the function ƒ(x) = c for x 0
(x bx 2 )1/2 x1/2
for x 0 x b, x 0
bx 3/2 7. The function g(x) = can be made differentiable at x = 0-
cos x x 0
is continuous at x = 0. (A) if b is equal to zero (B) if b is not equal to zero
(C) if b takes any real value (D) for no value of b
8. Which one of the following functions is continuous everywhere in its domain but has atleast
one point where it is not differentiable ?
|x|
(A) ƒ(x) = x1/3 (B) ƒ(x) = (C) ƒ(x) = e–x (D) ƒ(x) = tanx
x
9. If the right hand derivative of ƒ(x) = [x] tan x at x = 7 is k, then k is equal to
([y] denotes greatest integer y)
(A) 6 (B) 7 (C) –7 (D) 49
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10. Let ƒ : R R be a continuous onto function satisfying ƒ(x) + ƒ(–x) = 0, x R. If ƒ(–3) = 2 18. Let ƒ(x) = [n + p sin x], x (0, ), n I and p is a prime number. The number of points where
and ƒ(5) = 4 in [–5,5], then the equation ƒ(x) = 0 has- ƒ(x) is not differentiable is :-
(A) exactly three real roots (B) exactly two real roots (A) p – 1 (B) p + 1 (C) 2p + 1 (D) 2p – 1
(C) atleast five real roots (D) atleast three real roots Here [x] denotes greatest integer function.
x
n
19. The function ƒ(x) = (x2 – 1) | x2 – 3x + 2 | + cos (|x|) is NOT differentiable at :
ax(x 1) cot (px 2 2)
4 (A) –1 (B) 0 (C) 1 (D) 2
lim , x (0,1) (1, 2)
x
n
11. Let ƒ(x) = n
cot 1 2x tan 1 x a, x 0
4 20. Let g(x) = 3 .
x x bx, 0x
2
0 , x 1
If g(x) is differentiable for all x (–, ) then (a2 + b2) is equal to
If ƒ(x) is differentiable for all x (0,2) then (a2 + p2) equals -
(A) 20 (B) 13 (C) 9 (D) 4
(A) 18 (B) 20 (C) 22 (D) 24
ax b, x 2
12. If 2x + 3|y| = 4y, then y as a function of x i.e. y = ƒ(x), is -
21. If the function ƒ(x) = x 2 5x 6, 2x3
(A) discontinuous at one point px 2 qx 1, 3 x
(B) non differentiable at one point
(C) discontinuous& non differentiable at same point is differentiable in (–, ), then
(D) continuous& differentiable everywhere 4 5 5
(A) a = –1, p = (B) b = 2, q = (C) a = 1, b = 2 (D) a = –1, q =
9 3 3
13. If ƒ(x) = (x5 + 1) |x2 – 4x – 5| + sin|x| + cos(|x – 1|), then ƒ(x) is not differentiable at - x , x [0, 1)
(A) 2 points (B) 3 points (C) 4 points (D) zero points x 1, x [1, 2)
22. Let ƒ(x) = [x] and g(x) = .
x 3 2x 2 x Q x 2, x [2, 3)
14. Let ƒ(x) = 3 , then the integral value of 'a' so that ƒ(x) is differentiable at
x 2x ax x Q
2
0, x3
x = 1, is Then ƒ(x) + g(x) is
(A) 2 (B) 6 (C) 7 (D) not possible (A) discontinuous at x = 1 and x = 2.
(B) continuous in [0, 3] but non derivable in [0,3].
15. Let R be the set of real numbers and ƒ: R R, be a differentiable function such that (C) not twice differentiable in [0, 3].
| ƒ(x) – ƒ(y)| |x – y|3x,y R. If ƒ(10) = 100, then the value of ƒ(20) is equal to - (D) twice differentiable in [0, 3]
(A) 0 (B) 10 (C) 20 (D) 100 [Note :[k] denotes the greatest integer function less than or equal to k.]
x 2, x0
16. For what triplets of real numbers (a, b, c) with a 0 the function
x x 1 23. Let ƒ(x) = (2 x ),
2
0 x 1
ƒ(x) = 2 is differentiable for all real x ? x, x 1
ax bx c otherwise
(A) {(a, 1 – 2a, a) | a R, a 0} Then the number of points where |ƒ(x)| is non-derivable is
(B) {(a, 1 – 2a, c) | a, c R, a 0} (A) 3 (B) 2 (C) 1 (D) 0
(C) {(a, b, c) | a, b, c R, a + b + c = 1}
(D) {(a, 1 – 2a, 0) | a R, a 0} g(1 x) g(1)
24. Let g(x) = min.(x, x2) where x R, then lim equals
x 0 x
17. Number of points of non-differentiability of the function (A) 0 (B) 1 (C) 2 (D) does not exist
g(x) = [x2]{cos24x} + {x2}[cos2 4x] + x2 sin2 4x + [x2][cos2 4x] + {x2}{cos2 4x} in (–50, 50)
where [x] and {x} denotes the greatest integer function and fractional part function of x
respectively, is equal to:-
(A) 98 (B) 99 (C) 100 (D) 0
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IIT MATHEMATICS
ENTHUSE
EXERCISE # 2
1. Discuss the continuity & differentiability of the function ƒ(x) = sin x + sin | x |, x R. Draw a Diffe ntiate the n s with t to x
rough sketch of the graph of ƒ(x). x 1 3
y y tan x tan x x y x sec 2 x tan x
sin x cos x 3
2. Examine the continuity and differentiability of ƒ(x) = | x | + | x – 1 | + | x – 2 | x R. Also draw x x 1
y a cos y tan y 1 2 tan x
the graph of ƒ(x). 3 2
x
x2 y cos 3 4x y tan y sin 1 x 2
2
2 for x 0
3. If the function ƒ(x) defined as ƒ(x) = is continuous but not derivable at 1
x n sin 1 y cot 3 1 x 2 y (1 sin 2 x) 4 y 1 tan x
for x 0 x
x
1 x
x = 0 then find the range of n. y cos 2 y sin 2 (cos 3x) y x arcsin x
1 x
arcsin x 1
y y (arcsin x)2 y
1 for x 0 arccos x arcsin x
x
y x sin x arctan x y (arccos x arcsin x) n y arctan x
4. A function ƒ is defined as follows: ƒ(x) = 1 | sin x | for 0x 1 x2
2 14
y arctan x 2 y arcsin x 2 2x
2
2
2 x for x
2 2 b a cos x
y arccos ; (a, b > 0, sinx > 0) y x 2 log3 x
a b cos x
Discuss the continuity & differentiability at x = 0 & x =/2. nx
y nx y x sin x nx y
xn
5. Examine the origin for continuity & derivability in the case of the function ƒ defined by 1 nx
ƒ(x) = x tan–1 (1/x), x 0 and f(0) = 0. y y n(x 2 4x) y n tan x
1 nx
3 x 3
6. Let ƒ(0) = 0 and ƒ' (0) = 1. For a positive integer k, show that y log 2 [log 3 (log 5 x)] y n arctan 1 x 2 y n sin
4
1 x x 1 1 1
lim ƒ(x) ƒ ....... ƒ = 1 + + + ......+ x 3 2x cos x
x 0 x
2 k 2 3 k y x.10 x y y
ex ex
1 1 x
Let ƒ(x) = xe |x| x
; x 0, ƒ(0) = 0, test the continuity & differentiability at x = 0 1 10 x
7. y 2 nx y y 3sin x
1 10x
2 2
8. If ƒ(x) = | x – 1 | . ([x] – [–x]), then find ƒ' (1+) & f ' (1–) where [x] denotes greatest integer y ae b x y
2
Ae k x sin( x ) y xx
x
function. y (sin x)cos x y ( nx)x y x nx
(x 1)3 4 x 2 1
ax 2 b if | x | 1 y y x sin x 1 e x y xx
5
(x 3)2
9. If ƒ(x) = 1 is derivable at x = 1. Find the values of a & b.
if | x | 1 3 x(x 2 1)
| x | y 2x x
y (x 2 1)sin x y
(x 2 1) 2
a x 2, 0 x 2 1
10. Let g(x) = . If g(x) is derivable on (0, 5), then find (2a + b). Prove that the function y = n satisfies the relationship xy' + 1 = ey (where dash denotes derivative)
bx 2, 2 x 5 1 x
arcsin x
Prove that the function y satisfies the relationship (1 – x2)y' – xy = 1(where dash denotes
1 x2
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IIT MATHEMATICS
ENTHUSE
sin x cos x x(sin x cos x) sin x
tan4x 2x
1 sin 2x cos 3 x
1
a x 1 EXERCISE # 1
sin x 1
3 3 2 cos 2 1 2 tan x.cos 2 x
2 1 1 1 dy
1 1. If y = + + , then is equal to-
x cos 1 x 2 1 x x 1 x x 1 x x dx
–12cos24x sin4x x x (A) 0 (B) 1
4 tan .cos 2 1 x2
2 2 (C) ( + + )x + + –1 (D)
2x x2 1
4(1 + sin2x)3sin2x
3sin 2 3
1 x . (1 x ) 2 3 2 2 1 1
2x 2 cos2 x 1 tan x 3
x x 2. If f(x) = |cosx|, then f ' is equal to -
1 x 4
sin 2 x
1 x –3sin3x sin(2cos3x) arcsin x 1 1
x (1 x) 2 1 x2 (A) – (B) (C) 1 (D) –1
2 2
2 arcsin x 1
2(arccos x) 2 1 x 2 1 x2 (arcsin x)2 1 x 2 d x
x sin x 3. (e sin 3 x) equals-
sin x.arctan x x cos x.arctan x 0 dx
1 x2
2x 2 2x x 1 (A) ex sin( 3 x + /3) (B) 2ex sin( 3 x + /3)
(1 x 2 ) 2 1 x4 8 4 (arcsin x 2 2x)3 (1 2x x 2 )(x 2 2x)
1
(C) ex sin( 3 x + /3)
1
(D) ex sin( 3 x – /3)
2 2
x 1 2 2
a b
2x log 3 x
a b cos x n3 2x nx
1 n nx 2 d
4. (n sin x ) is equal to-
sin x nx x cos x nx sin x xn 1 x(1 nx)2 dx
2x 4 2 1 tan x cot x cot x cot x
x log 5 x log 3 (log 5 x) n2 n3 n5 (A) (B) (C) (D)
x 2 4x sin 2x 2 x 2 x 2x 2 x
x 3
cot
x 4
3 x 3 10x (1 + x n10) 1 x dy
arctan 1 x 2 (2 x 2 ) 1 x 2 12 n 2 sin 5. If y = , then equals –
4 1 x dx
x
2 x ( n2 1) 3x 2 x 3 sin x cos x ( nx 1) n2 nx y y y y
– 2 (A) (B) 2 (C) (D)
ex ex 2
n x 1 x2 x 1 1 x2 y2 1
2.10x n10
b2x 2
(1 10 x ) 2 3sin x cos x. n3 2ab 2 xe
x (a 2 x 2 ) dy
x 1 If y = n
Ae k2 x
[ cos( x 2
) k sin( x )] x x .x x n2x nx 6. , then the value of is-
x a
dx
cos x cos 2 x 1
(sin x) sin x n sin x ( nx) x n nx a2 x2
1
(B) a a x a2 x2
2 2
sin x nx (A) (C) (D) x
a2 x2
57x 2 302x 361 (x 1) 2 4 x 2
2x nx 1
nx .
20(x 2)(x 3) 5
(x 3)2 dx
7. If x = y n(xy), then equals-
1 1 1 ex 1 dy
x sin x 1 e x cot x . 2
2 x 2 1 ex x x
(1 nx) y(x y) x(x y) y(x y) x(x y)
2x sin x (A) (B) (C) (D)
x
1
(x 2
1) sin x
cos x n x 2
1 x(x y) y(x y) x(x y) y(x y)
x 2
(2 nx) x2 1
x 6x 1 3 x(x 2 1)
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3x(1 x 4 ) (x 2 1) 2
dy d 1 x x
If (cosx)y = (siny)x, then
equals- (x 0)
8. equals- tan 3/2
dx 16.
dx 1 x
log sin y y tan x log sin y y tan x
(A) (B) 1 1 1 1
log cos x x cot y log cos x x cot y (A) (B)
2 x (1 x) 1 x 2 x (1 x) 1 x
2 2
log sin y y tan x log sin y y tan x
(C) (D) 1 1 1 1
log cos x x cot y log cos y y cot x (C) (D)
1 x 1 x2 1 x 1 x2
dy 1
9. If 2x + 2y = 2x+y, then is equal to- 17. If g is the inverse of f and f '(x) = then g'(x) is equal to-
dx 1 x3
2x 2y 2x 2y 2y 1 2x y 2x 1 1 1
(A) (B) (C) 2x–y 1 2 x (D) (A) 1 + [g(x)]3 (B) (C) (D)
2x 2y 1 2x y 2y 2(1 x 2 ) 2(1 x 2 ) 1 g(x)
3
dy 18. If x2 + y2 = 1, then-
10. If x = a(t – sint), y = a(1 + cost), then equals-
dx (A) yy" – 2(y')2 + 1 = 0 (B) yy" + (y')2 + 1 = 0
t t t t (C) yy" + (y')2 – 1 = 0 (D) yy" + 2(y')2 + 1 = 0
(A) –tan (B) cot (C) –cot (D) tan
2 2 2 2
19. Let f be a function defined for all x R. If f is differentiable and f(x3) = x5 for all x R
1 (x 0), then the value of f '(27) is-
11. The differential coefficient of sec–1 2 w.r.t. 1 x 2 is- (A) 15 (B) 45 (C) 0 (D) 35
2x 1
(A) 1/x2 (B) 2/x3 (C) x/2 (D) 2/x
x b b
x b
dy 20. If 1 = a x b and 2 = are given, then –
12. If x3 – y3 + 3xy2 – 3x2y + 1 = 0, then at (0, 1) equals- a x
dx a a x
(A) 1 (B) –1 (C) 2 (D) 0 d
(A) 1 = 3(2)2 (B) 1 = 32
dx
d 1 1 cos d
13. tan sin equals, if –<< - (C) 1= 3(2)2 (D) 1 = 3(2)3/2
d dx
(A) 1/2 (B) 1 (C) sec (D) cosec
dy dx
21. Let y = x3 – 8x + 7 and x = f(t). If = 2 and x = 3 at t = 0, then at t = 0 is given by :
d 1 x dt dt
14. cot–1 is equal to, if x > –1
dx 1 x (A) 1 (B)
19
(C)
2
(D) None of these
1 1 1 1 2 19
(A) (B) (C) (D)
1 x2 1 x2 1 x2 1 x2
dy 5
22. If y = cos–1(cos x), then at x = is equal to :
dy dx 4
15. If y = tan–1(cot x) + cot–1(tanx), then is equal to- 1 1
dx (A) 1 (B) –1 (C) (D)
(A) 1 (B) 0 (C) –1 (D) –2 2 2
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x2 1 x2 1 dy 32. Suppose the function f (x) – f (2x) has the derivative 5 at x = 1 and derivative 7 at x = 2. The
23. If y = sin–1 + sec–1 2 , |x| > 1, then is equal to : derivative of the function f (x) – f (4x) at x = 1, has the value equal to
x 1
2
x 1 dx
2
(A) 19 (B) 9 (C) 17 (D) 14
x x
(A) 4 (B) 4 (C) 0 (D) 1
x 1 x 1 x tan 1 x sec 1 (1/ x), x ( 1,1) {0}
33. If f(x) = , then f '(0) is -
24. If y = x – x2, then the derivative of y2 w.r.t. x2is : / 2, if x 0
(A) 2x2 + 3x – 1 (B) 2x2 – 3x + 1 (C) 2x2 + 3x + 1 (D) none of these (A) equal to –1 (B) equal to 0 (C) equal to 1 (D) non existent
cos x x 1 34. Given: f(x) = 4x3 – 6x2cos 2a + 3x sin 2a .sin 6a + n 2a a 2 then
f '(x)
25. Let f(x) = 2sin x x 2 2x . Then lim = (A) f(x) is not defined at x = 1/2 (B) f ' (1/2) < 0
x 0 x
tan x x 1 (C) f '(x) is not defined at x = 1/2 (D) f ' (1/2) > 0
(A) 2 (B) –2 (C) –1 (D) 1
d2 y
35. If x = t3 + t + 5 & y = sint, then =
26. Let ef(x) = lnx. If g(x) is the inverse function of f(x) then g'(x) equal to : dx 2
(C) e x e (3t 2 1)sint 6 tcost (3t 2 1)sint 6 tcost
x
(A) ex (B) ex + x (D) ex + ln x
(A) (B)
(3t 2 1)3 (3t 2 1)2
27. Consider f(x) be a polynomial function of second degree. If f(1) = f(–1) and a, b, c are in A.P.,
(3t 2 1)sint 6 tcost cos t
then f '(a), f '(b) and f '(c) are in : (C) (D)
(A) G.P. (B) H.P. (C) A.G.P. (D) A.P. (3t 2 1)2 3t 2 1
1 dy a a2 x2 x
28. If 8f(x) + 6f = x + 5 and y = x2 f(x), then at x = –1 is equal to : 36. If f(x) = where a > 0 and x < a, then f '(0) has the value equal to –
x dx a2 x2 a x
1 1 1 1
(A) 0 (B) (C) (D) none of these (A) a (B) a (C) (D)
14 14 a a
29. Let f(x) be a polynomial in x. Then the second derivative of f(ex) w.r.t. x is : 37. Suppose that f (0) = 0 and f '(0) = 2, and let g (x) = f (– x + f (f (x))). The value of g ' (0) is
(A) f "(ex)ex + f '(ex) (B) f "(ex)e2x + f '(ex)e2x equal to
(C) f "(ex)e2x (D) f "(ex)e2x + f '(ex)ex (A) 0 (B) 1 (C) 6 (D) 8
x
x 1 1 f (4) f(x 2 )
30. Let y = x 1 then y ' (1) equals 38. If f is differentiable in (0, 6) & f '(4) = 5, then L imit =
x x 2 2x
(A) (ln 2) + 1 (B) (2 ln 2) + 1 (C) (ln 2) – 1 (D) (2 ln 2) – 1 (A) 5 (B) 5/4 (C) 10 (D) 20
'
u(x) u '(x) u(x) d2x
31. Let u(x) and v(x) are differentiable functions such that = 7. If = p and = q, 39. If y = x + ex then is :
v(x) v '(x) v(x) dy 2
pq ex ex 1
then has the value equal to - (A) ex (B) (C) – (D)
pq (1 e x )3 (1 e x ) 2 (1 e x )3
(A) 1 (B) 0 (C) 7 (D) –7
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40. If f is twice differentiable such that EXERCISE # 2
f "(x) = –f(x), f '(x) = g(x)
h'(x) = [f(x)]2 + [g(x)]2 and 1. Let f, g and h are differentiable functions. If f(0) = 1 ; g(0) = 2 ; h(0) = 3 and the derivatives of
h (0) = 2, h(1) = 4 their pair wise products at x = 0 are (f g)' (0) = 6 ; (g h)' (0) = 4 and (h f)' (0) = 5 then compute
then the equation y = h(x) represents :
the value of (fgh)'(0).
(A) a curve of degree 2
(B) a curve passing through the origin dy ex xe ex dy
2. (a) If y = (cos x)nx + (nx)x find (b) If y = e x + e x + x e Find .
(C) a straight line with slope 2 dx dx
(D) a straight line with y intercept equal to – 2.
x2 1
3. If y = x x 2 1 + n x x 2 1 prove that 2y = xy'+ny'. Where y' denotes the
2 2 2
d y dy
41. If y = (A + Bx) emx + (m - 1)–2 ex then – 2m + m2y is equal to - derivative of y.
dx 2 dx
x mx –mx (1 – m) x
(A) e (B) e (C) e (D) e x
.a y
yx dy
4. If y = ln x e find .
3 2 dx
d 2 x dy d y
42. If + dx 2 = K then the value of K is equal to x1 x 2 .x x 3 .x 2
dy 2 dx 5. If y = 1 + + + + .....upto (n + 1) terms then
(A) 1 (B) –1 (C) 2 (D) 0 x x1 (x x1 )(x x 2 ) (x x1 )(x x 2 )(x x 3 )
(x h) f(x) 2 hf(h) dy y x x2 x3 xn
43. Let f(x) be differentiable at x = h, then L im is equal to - prove that = 1 ........
x h xh dx x x1 x x 2 x x 3 x xn x
(A) f(h) + 2hf '(h) (B) 2f(h) + hf '(h) 2
dy
(C) hf(h) + 2f '(h) (D) hf(h) – 2f '(h) 6. If x = cosec – sin ; y = cosecn – sinn, then show that (x2 + 4) – n2(y2 + 4) = 0.
dx
d3 y 7. If a curve is represented parametrically by the equations
44. If y = at2 + 2bt + c and t = ax2 + 2bx + c, then equals
dx 3
7 3
(A) 24 a2 (at + b) (B) 24 a (ax + b)2 x = sin t + sin t + sin t ,
(C) 24 a (at + b)2 (D) 24 a2 (ax + b)
12 12 12
7 3
1 y = cos t + cos t + cos t
45. Let f (x) = x + sin x. Suppose g denotes the inverse function of f. The value of g' 12 12 12
4 2
d x y
at t = .
has value equal to then find the value of
dt y x 8
2 1
(A) 2 –1 (B) (C) 2 – 2 (D) 2 1
2
1 nt 3 2nt dy dy
8. If x = and y = . Show that y = 2x + 1.
t2 t dx dx
1 x2 1 x2
9. Differentiate w.r.t. 1 x 4 .
1 x2 1 x2
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g(x), x0 1 1 1 1
17. If y = tan–1 + tan–1 2 + tan–1 2 + tan–1 2 +...... to n
10. Let g(x) be a polynomial, of degree one & f(x) be defined by f(x) = 1 x 1/x x2 x 1 x 3x 3 x 5x 7 x 7x 13
, x0
2 x terms. Find dy/dx, expressing your answer in 2 terms.
Find the continuous function f(x) satisfying f ' (1) = f (–1).
u 1 1 1 dy
18. If y = tan–1 and x = sec–1 , u 0, ,1 prove that 2 1 = 0.
1 u 2 2u 2 1 2 2 dx
dy x2 1 y6
11. If 1 x6 + 1 y6 = a3.(x3 – y3), prove that = 2 .
dx y 1 x6
1 sin x 1 sin x dy
19. If y = cot–1 , find if x 0, , .
1 sin x 1 sin x dx 2 2
1 dy 1
12. If y = x + , prove that =
1 dx x
x 2
x
1
x
1 (x a) 4 (x a)3 1 (x a) 4 (x a) 2 1
x ......... x
1
20. If f(x) = (x b) 4 (x b) 1 , then f ' (x) = . (x b) 4
3
(x b) 2 1 . Find the value of .
x .........
(x c) 4 (x c)3 1 (x c) 4 (x c) 2 1
1 1 1
13. Let f(x) = x + ........ Compute the value of f(100). f ' (100).
2x 2x 2x
21. (a) If y = y(x) and it follows the relation exy + y cos x = 2, then find (i) y'(0) and (ii) y"(0).
(b) A twice differentiable function f(x) is defined for all real numbers and satisfies the
14. Find the derivative with respect to x of the function :
following conditions
2x
(logcosxsinx) (logsinxcosx)–1 + arcsin at x = . f(0) = 2; f '(0) = –5 and f "(0) = 3.
1 x2 4
The function g(x) is defined by g(x) = eax + f (x) x R, where 'a' is any constant.
If g'(0) + g"(0) = 0. Find the value(s) of 'a'.
15. Suppose f (x) = tan(sin–1 (2x))
(a) Find the domain and range of f.
22. If x = 2cost – cos2t & y = 2sint – sin2t, find the value of (d2y/dx2) when t = (/2).
(b) Express f(x) as an algebraic function of x.
(c) Find f ' (1/4)
d2 y
23. Find the value of the expression y3 on the ellipse 3x2 + 4y2 = 12.
dx 2
16. (a) Let f(x) = x2 – 4x – 3, x > 2 and let g be the inverse of f. Find the value of g' where f(x) = 2.
cos(x x 2 ) sin(x x 2 ) cos(x x 2 )
(b) Let f : RR be defined as f(x) = x3 + 3x2 + 6x – 5 + 4e2x and g(x) = f–1(x), then find g'(–1). 24. If f(x) = sin(x x 2 ) cos(x x 2 ) sin(x x 2 ) then find f ' (x).
1 sin 2x 0 sin 2x 2
(c) Suppose f –1 is the inverse function of a differentiable function f and let G(x) = .
f 1 (x)
1 25. Let P(x) be a polynomial of degree 4 such that P(1) = P(3) = P(5) = P'(7) = 0. If the real number
If f(3) = 2 and f ' (3) = , find G ' (2).
9 x 1, 3, 5 is such that P(x) = 0 can be expressed as x = p/q where 'p' and 'q' are relatively
prime, then (p + q) equals.
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EXERCISE # 3
Evaluate the following limits using L'Hopital's Rule or otherwise (Q. No. 1 to 5) :
1. Lim
1 1 x2
2 2. L im
x n x2 1 x ENTHUSE
x 0 1
x sin x x x 0 x3
1 1 1 sin x cos x n(1 x)
3. L im 2 2 4. L im
x 0
x sin x x 0 x.tan 2 x
x 6000 (sinx)6000
5. L im
x 0 x 2 .(sinx)6000
1 cos x.cos 2x.cos3x....cos nx
6. If L im has the value equal to 253, find the value of n
x 0 x2
(where n N)
7. Given a real valued function f (x) as follows :
f(x) =
x 2 2cos x 2
x 4 for x < 0 ; f(0) =
1
12
and f(x) =
sin x n(e x cosx)
6x 2
for x > 0. Test the IIT MATHEMATICSN
continuity and differentiability of f (x) at x = 0.
8. Let a1> a2 > a3.............. an > 1 ; p1 > p2 > p3 ........ > pn > 0 ; such that p1 + p2 + p3 + .....+ pn = 1. INDEFINITE & DEFINITE INTEGRATION
Also F(x) = p a p a ....... pn a
x
1 1
x
2 2 n
x 1/ x
. Compute (SPECIAL DPP)
(a) Lim F(x) (b) Lim F(x) (c) Lim F(x)
x 0 x x
9. If x1, x1, x2, x3, ............xn – 1 be n zero's of the polynomial P(x) = xn + x + ,
where xixj i & j = 1, 2, 3, ........... (n – 1).
Prove that the value of Q(x) = (x1– x2)(x1– x3)(x1– x4)...(x1– xn – 1), is equal to nC2x1n – 2.
10. Column-I contains function defined on R and Column-II contains their properties. Match them:-
Column - I Column – II
n
1 tan
(A) L im 2n equal (P) e
n
1 sin
3n
1
(B) L im 1 equals (Q) e2
x 0
(1 cosecx) n(sinx)
1/ x
21
(C) L im cos x equals (R) e–2/
x 0
(S) e/6
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RESULTS OF BEST MENTORSHIP BY THE NUCLEUS TEAM
INDEX
S.No. Contents Page No.
AIR-1 AIR-3 AIR-6 AIR-8 INDEFINITE INTEGRATION
CHITRAANG MURDIA GOVIND LAHOTI NISHIT AGARWAL AMEY GUPTA
Gen. Category - 2014 Gen. Category - 2014 Gen. Category - 2012 Gen. Category - 2014
1. DPP-1 01
JEE MAIN RESULTS 2020 (January Attempt) OF NUCLEUS EDUCATION 2. DPP-2 02-03
3. DPP-3 04
4. DPP-4 05
5. DPP-5 06
6. Answer key 07-12
100 Percentile 100 Percentile 100 Percentile 100 Percentile 100 Percentile 100 Percentile
(Physics) (Maths & Physics) (Physics) (Maths) (Maths) (Maths)
DAKSH KHANDELWAL VAIBHAV SAHA ANISH MOHAN ARCHIT PATNAIK SWAPNIL YASASVI PARSHANT ARORA
2020 2020 2020 2020 2020 2020
DEFINITE INTEGRATION
JEE ADVANCED RESULTS OF NUCLEUS EDUCATION 1. DPP & Answer key 13-14
AIR-10 AIR-12 AIR-23 AIR-24 AIR-37 AIR-42 AIR-66 AIR-98
LAKSHAY SHARMA YATEESH AGRAWAL ABHEY GOYAL TUSHAR GAUTAM PIYUSH TIBAREWAL SATVIK MAYANK DUBEY HRITHIK
Gen. Category - 2017 Gen. Category - 2017 Gen. Category - 2017 Gen. Category - 2017 Gen. Category - 2017 Gen. Category - 2017 Gen. Category - 2017 Gen. Category - 2017
AIR-20 AIR-27 AIR-32 AIR-61 AIR-67 AIR-78 AIR-61 AIR-91
SHASHANK AGRAWAL RAAGHAV RAAJ SHREYA PATHAK SIDDHANT CHOUDAHRY ANISWAR S KRISHNAN AAYUSH KADAM SARTHAK BEHERA ANDREWS G. VARGHESE
Gen. Category - 2018 Gen. Category - 2018 Gen. Category - 2018 Gen. Category - 2018 Gen. Category - 2018 Gen. Category - 2018 Gen. Category - 2018 Gen. Category - 2018
DLP *SDCCP *SDCCP
AIR-2 AIR-19 AIR-33 AIR-48 AIR-51 AIR-53 AIR-86
HIMANSHU GAURAV SINGH VIBHAV AGGARWAL S. PRAJEETH SOHAM MISTRI SAYANTAN DHAR GAURAV KRISHAN GUPTA SATVIK JAIN
Gen. Category - 2019 Gen. Category - 2019 Gen. Category - 2019 Gen. Category - 2019 Gen. Category - 2019 Gen. Category - 2019 Gen. Category - 2019
*SDCCP *SDCCP DLP DLP DLP
MATHEMATICS MATHEMATICS
ENTHUSE ENTHUSE
DPP # 01 (SPECIAL DPP ON INDEFINITE INTEGRATION)
DPP # 02 (SPECIAL DPP ON INDEFINITE INTEGRATION)
ELEMENTARY DPP
Find the antiderivative/primitive/integrals of the following by simple manipulation/simplifying and con- 3
d 1 x 2
verting them into standard integrals. 1. sin xd(sin x) 2. tan xd tan x 3. 1 x2
1 cos 2 x 1 tan 2 x
x x
dx
1. 2 .e dx 2. 1 cos 2x dx 3. 1 tan 2 x dx 15
2x 3
dz
c 1
4. (x 1) dx 5. 5 6. a bz c
1 tan 2 x e5nx e4nx anx m
4. 1 cot dx 5. e dx 6. (e e xna )dx (a 0) dx
2
x 3nx
e 2nx 7. 5
(8 3x)6 dx 8. 8 2xdx 9. 3
(a bx)2
cos 2x 1 2x 2 x 21 x 2 1 dx 12. x2 5 x3 2 dx
7. cos dx 8. x 2
dx 9. 4 cos 2 .cos x.sin xdx 10. 2x 11. x 1 x 2 dx
2
x sin 2 x (1 x 2 ) 2
xdx x4 dx x 3 dx
cos x sin x 2 3
13. x2 1
14. 4 x5
15. 3
x4 1
10. (2 2 sin 2x)dx 11. (3sin x cos x sin x)dx 12. cos xdx
cos x sin x
(6x 5)dx 3 sin x dx
(1 x)2 x sec 2x 1 16. 2 3x 2 5x 6
17. sin x cos xdx 18. cos 2 x
13. dx 14. dx 15. dx
x(1 x 2 ) 2x 1 sec 2x 1 cos x dx 3 nx
19. 3
sin2 x
20. cos x sin 2xdx 21. x
dx
2x 1 e 2x 1
16. dx 17. x dx
x2 e (arctan x) 2 dx dx dx
22. 1 x2
23. (arcsin x) 3
1 x2
24. cos 2
x 1 tan x
sin x cos x cos 2x cos 2
18. dx (cos x sin x 0) 19. dx 2
1 sin 2x cos x cos d 1 nx
25 cos 2 26. cos cos2x dx 27. cos 2x dx
1 nx 4
x 6 1 sin 3 x cos3 x x4 x 2 1
20. x dx 21. dx 22. dx d(1 x 2 ) d arcsin x
2
1 sin 2 x cos2 x 2(1 x 2 ) 28. e sin e dx
x x
29. 30.
1 x2 arcsin x
sin 6 x cos6 x 2 9 x 2 7 x (2x 3)dx xdx x 2 dx
23. dx 24. sin sin dx 31. x2 3x 8 32. 33. x3 1
sin 2 x.cos 2 x 8 4 8 4 x2 1
cos 4x 1 sin 2x sin 5x sin 3x e xdx e 2 x dx
34. e 35. 36. tan3x dx
25. cot x tan x dx 26. cos x 1 2 sin 2 2x
dx x
1 e 2x a 2
sin 2x dx
37. cot (2x +1) dx 38. 1 cos dx 39.
cot 2 2x 1 cos4 x sin 4 x 2
x xnx
27. 2 cot 2x cos8x cot 4x dx 28. dx (cos 2x 0)
1 cos 4x ( nx)m
40. dx 41. esinx d (sinx) 42. e sin x cos xdx
x
3
2x 3x 4x 5 2 2 2
(x sin x) sec x 2 dx
29. 2x 1
dx 30. 1 x2
dx 31. 9 16x 2
43. e–3x+1 dx 44. e x2
x dx 45. e x3
x 2 dx
dx 2x 3 dx x
d dx
32. 25 4x 2 33. 3x 2 dx 34. 1 sin x 46.
3
47. 2 48. 1 9x
dx
2
x
2 1 25x
2 1
2 3x 3
cos 8x cos 7x sin 2x sin 2k
35. 1 2 cos 5x
dx 36. x 1 x dx
2 2 37. sin x sin k cos x cos k dx dx dx dx
49. 4 x2
50. 2x 2
9
51. 4 9x 2
x n(ex)dx
x
38. sin x cos x cos 2x cos 4x dx 39.
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MATHEMATICS
ENTHUSE MATHEMATICS
xdx xdx x2 dx ENTHUSE
52. x4 1 53. a 2 x4
54. 6
x 4 DPP # 03 (SPECIAL DPP ON INDEFINITE INTEGRATION)
x
3
x dx x
e dx 2 dx dx dx dx
55. 56. 57. 1. x 2. x 3. 4x
1 x8 e 2x
4 1 4 x
2
7x 10 2
3x 10 2
9
dx dx dx
cos d e 2x 1 4. 2 3x 5. (x 1) 2 6. x
58. a 2 59. dx 60. e x 3
1 dx
2
4 2
2x 3
sin 2 ex
dx dx dx
1 x
dx
3x 1 1 x 7. xx 2
2.5
8. 4x 2
4x 5
9. 1 (2x 3) 2
61. 1 x2
62. x2 9 dx 63. 1 x
dx
dx dx dx
1 x x 2
dx
10. 4x 3 x 2
11. 8 6x 9x 2 12. 2 6x 9x 2
x 1 x 2 dx
64. dx 65. 1 x 2 3 66. (x
1 x4 2
x 1) 2
2
dx
13. cos xdx 14. sin 2
x dx 15. 1 cos x
2x arcsin x x (arccos 3x) 2 (1 x)2
67. dx 68. dx 69. 2 dx dx 1 cos x 1 sin x
1 9x 2 x 1 1 sin x dx
1 x2 16. 1 sin x 17. 1 cos x dx 18.
x4 cos 2x dx
70. 1 x dx 19. (tan
2
x tan 4 x)dx 20. 1+sinx cosx 21. cos x sin 3xdx
dx 1 sin x
22. cos x cos 2x cos 3x dx 23. cos x 24. dx
cos x
sin 3 x cos 3 xdx sin 3
25. dx 26. 27. d
cos x sin 4 x cos
dx 4 5
28. cos 4
x
29. tan xdx 30. sin xdx
4
dx
31. sin xdx 32. tan 3
x dx 33. sin 6
x
x
e
34. (x 1) x 2 2x dx 35. dx 36. 1 e x e x dx
x
2x 5 3x 2 xdx 2x 3
37. 1 3x dx 38. 3 39. dx
3
x6 1 x 2 1 x2
dx x2 x 1 (arctan x) n
x dx dx
40.
3 n x 2 41.
(x 1)2 3 42. 1 x2
d x 3 dx xdx
43. sin 2 cos2 44.
x 1
45. (x 1) 3
(x 2)dx (x 3)dx (3x 1)dx
46. x 47. 48.
2
2x 2 3 2x x 2 x 2 2x 2
cot x e 1x
ex x
49. n(sin x) dx 50. e x
1
dx 51. e dx
2x 2 nx sin 2x dx d
52. e dx 53. 4 cos 2
2x
54.
3 cos sin
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MATHEMATICS MATHEMATICS
ENTHUSE ENTHUSE
DPP # 05 (SPECIAL RACE ON INDEFINITE INTEGRATION)
DPP # 04 (SPECIAL DPP ON INDEFINITE INTEGRATION)
PART-I dx x dx (2x 2 5)dx
1. 6x 3
7x 2 3x
2. x 4
3x 2 2
3. x 4
5x 2 6
1. xe–x dx 2. x n nx dx (n -1)
dx 3x 2 1 xdx
x arctan x 4. x 5. (x 2
dx 6. x
dx
4
x2 1)3 3
1
3. arctan xdx 4. 1 x2
x 2 dx dx x3 1
arcsin x x dx2
7. 1 x 4 8. (1 x ) 2 4 9. 4x 3
x
dx
5. dx 6. 2
1 x 1 x2 x 2 dx
2
x 2dx dx
10. x 1 11. (x 2) (x 4) 2 2 12. 4
(x 1)3 (x 2) 5
3 x
x dx n 3 x
7. 1 x 2
8. x2
dx
dx sin 2 x 1 x4
1 x dx
2
13. 1 sin 4 x 14. 3
cos14 x
dx 15. 4 3/ 2
9. arctan x xdx 10. eax cos nx dx
dx dx sin xdx
x 2 dx 16. cos x.sin 3
x
17. cos 3
x.sin 3 x
18. (1 cos x) 2
11. sin(nx)dx 12. 1 x 2
cos xdx dx x 4 dx
2
19. (1 cos x) 2 20. 4 3
sin x cos x 5 21. x 15
1
3x 1
arctan xdx
13. 2x x
14. n(x 1 x 2 )dx
(x 2 1)dx
sin 2xdx (e 3x e x )dx
PART-II 22. cos 4
x sin 4 x
23. x x 4 3x 2 1
24. e 4x
e 2x 1
dx 4x 3
15. 1 16. (x 2) dx (x sin x)dx x 2 1 dx
x 1
3
25. 26. x 2
.
1 cos x 1 1 x4
dx xdx x cos3 x sin x x5 x4 8
17. 1 3
x 1
18. x3x
27. e
sin x
cos2 x
dx 28. x 3 4x
dx
15
1 x2 dx x 1 x2 sin 3 x.dx
19. x4
dx 20. 2
(a x ) 2 3
29. 1 x 2
.dx 30. cos 4
x 3cos x 1 tan 1 sec x cos x
2
dx dx dx dx
21. x 2 22. x 31. 5 4 sin x 3cos x 32. 4 3cos 2
x 5sin 2 x
x2 9 1 x2
2x 2 41x 91 dx (cos 2x 3)dx
dx (x 1)dx 33. (x 1)(x 3)(x 4) dx 34. (sin x 2sec x) 2 35. cos
23.
x x2
24. x(1 xe x ) 4
x 4 cot 2 x
dx dx
x 4 dx dx e x 1dx cos
25. 26. x
36.
x x
37. 38. 3
x sin 2x
(1 x 2 )3 4
x2 4 sin cos3
2 2
dx
27. x 4
x2 3
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MATHEMATICS MATHEMATICS
ENTHUSE ENTHUSE
DPP # 02
ANSWER KEY
sin 2 x tan 4 x
DPP # 01 1. C 2. C 3. 2 1 x 2 C
2 4
2 x. e x 1 1 (x 1)16 1 (a bz)1 c C
1. C 2. (tan x x) C 3. sin 2x C 4. tan x – x + C 4. C 5. C 6.
1 n2 2 2 16 8(2x 3) 4 b(1 c)
x3 x a 1 a x 5 (8 2x) 3 3m 3
C C 7. C (8 3x) 11/ 5 8. C 9. a bx C
5. 6. 7. – (cot x + tan x ) + C 33 3 b
3 a 1 na
2 1 5
1 1 1 1 1 10. (x2 1)3 C 11. C (1 x 2 )3 12. 5
(x3 2)6 C
8. tan 1 x C 9. cos 9x cos10x cos11x cos12x C 3 3 18
x 9 10 11 12
2 33 4
13. x2 1 C 14. .4 x 5 C 15. (x 1)2 C
cos 3x 180 5 8
10. sin 2x + C 11. C 12. sin x C 13. nx + 2 tan x + C
–1
3 1
16. 3x 2 5x 6 C 17. sin 4 x C 18. secx + C
1 n(2x 1) 4
14. x C 15. tan x – x + C 16. 2x + 3n (x – 2) + C 2 2
2 2 19. 20. C cos 5 x 21. ( nx)3 C
3 3 sin x C 5 3
x –x
17. e +e +C 18. x+C 19. 2(sinx + x cos ) + C
(arctan x) 3 1
x 5 x3 1 x3 22. C 23. C 2(arcsin x) 2 24. 2 1 tan x C
x 2 tan 1 x C tan 1 x C 3
20.
5 3
21. secx – cosec x + C 22. 2 3 1
25 tan(1 + nx) +C 26. x cos sin2x C
x cos 4x 2
23. tan x – cot x – 3x + C 24. 2 cos C 25. C
2 8 1 1
27. tan 2x C or (tan 4x sec 4x) C 28. C – cos(ex)
2 4 2
cos 8x x
26. –2cos x + C 27. C 28. C
8 2
29. n(1 x 2 ) C 30. n|arcsin x| C 31. n(x2 3x 8) C
1 1 3
x 3 x 2 3x 7 1 1 4 32. n(x2 1) C 33. n|x 1| C 34. n(e x 1) C
29. n(2x 1) 30. tan x – tan–1 x + C 31. sin xC 2 3
3 2 2 4 4 3
1 1 1
1 2x 2 5 35. n(e2 x a 2 ) C 36. C n|cos 3x| 37. n|sin(2x 1)| C
tan 1 C x n(3x 2) C 2 3 2
32. 33. 34. tan x – sec x + C
10 5 3 9 38. C n(1 cos 2 x) 39. n| nx|
C
sin 3x sin 2x 2 m 1
n x
35. C 36. tan 1 x C 37. (sinx – cosx) + (sink + cosk)x + C 40. C if m –1 and n| nx|
C if m = –1 41. e sin x C
3 2 x m 1
1 e 1 3x
38. cos 8x C 39. xx + C 42. e sin x C 43. C 44. 0.5e x C 2
64 3
1 x3 x 1
45. C e 46. arcsin C 47. arcsin 5x C
3 3 5
1 x 1 2
48. arctan 3x C 49. arcsin C 50. arctan x C
3 2 3 2 3
1 3x 1 1 x2
51. arcsin C 52. arctan x 2 C 53. arcsin C
3 2 2 2 a
1 x3 1 1 ex
54. arctan C 55. arcsin x 4 C 56. arctan C
6 2 4 2 2
arcsin 2 x 1 sin
57. C 58. arctan C 59. e x e x C
n2 a a
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MATHEMATICS MATHEMATICS
ENTHUSE ENTHUSE
3 3
1 3x 3 2x
2 1 2
e e 3e x x C 36. c (1 e x ) 2 37. c n | 1 3x 3 x 6 | 38. n(1 x 2 ) c
60. 61. arcsin x 1 x 2 C 3 3 3
3 2
3 1 x
62. n(x2 9) arctan C 63. arcsin x 1 x 2 C nx
2 3 3 39. 2 1 x 2 3n(x 1 x 2 ) c 40. arcsin c
3
1 1 1
64. arctan x 2 n(x 4 1) C 65. arcsin x C
2 4 1 x2 1
n(x x 2 1) c 42. (arctan x) n 1
2 3 2 41. c if n 1 and n|arctanx| if n = –1
66. [x (x2 1)3 ] x C 67. C 2 1 x
2
(arcsin x) 3 x2 1 n 1
3 3
1 x3 x 2
68. C [ 1 9x 2 (arccos 3x) 3] 69. x n(x2 1) C x n | x 1 | C
9 43. c 2 cot 2 44.
3 2
70. 1 4 1 3 1 2
C x x x x n|1 x|
4 3 2 –1 1 1
45. + +C 46. n(x 2 2x 2) arctan(x 1) c
x +1 2(x+ 1)2 2
DPP # 03
x 1
1 x 5 1 x 2 1 2x 3 47. c 3 2x x 2 4 arcsin
1. n C 2. n C 3. 12 n 2x 3 C 2
3 x 2 7 x 5
1 2 x 3 1 x 1 1 x 1 48. 3 x 2 2x 2 4n(x 1 x 2 2x 2 c 49. n | n(sin x) | C
4. n C 5. arctan C 6. arctan C
2 6 2 x 3 2 2 2 2
x 1 2x2
2 1 2x 1 2x 1 1 50. 2n(e x / 2 e x / 2 ) C 51. ee C 52. e c
7. arctan C 8. arctan C 9. arcsin(2x 3) C 4
3 3 4 2 2
1 3x 1 1 3x 1
10. arcsin(x 2) C 11. arcsin C 12. 3 arcsin C 1 2 cos 2x 1
3 3 3 53. C n 54. n tan c
8 2 cos 2x 2 2 6
x sin 2x x sin 2x x
13. C 14. C 15. C cot DPP # 04
2 4 2 4 2
x x
PART-I
tan C x
16. 17. 2 tan x C 18. 2 tan x C
2 4 2 2 4 n 1
x 1
1. C e x (x 1) 2. nx C 3. x arctan x – x + arctan x + C
19.
1
tan 3 x C 20. n(2 sin2x) C n 1 n 1
3
C
1 cos 4x
cos 2x
1 1 1
2x sin 2x sin 4x sin 6x C
4. 1 x 2 arctan x n(x 1 x 2 ) C 5. 2 x 1 x arcsin x C
21. 4 2
22. 8 2 3
2
x 1 x2 1 x2 (1 x 2 ) 3 C
6. C arctan x 7.
cos 2 x 2(1 x 2 ) 2 3
n tan x C
23. 24. n(1 sin x) C 25. n|cos x| C
4 2 2
1 x2 1 1
cos 2 8. C (n 3 x 3n 2 x 6nx 6) 9. (arctan x)2 x arctan x n(1 x 2 ) C
1 1 2 cos
1 C 1 x 2 2
26. C 27. 28. tan x tan 3 x C
sin x 3sin 3 x 5 3
e ax x
1 2 1 10. (n sin nx a cos n x) C 11. (sin nx cos nx) C
29. tan 3 x tan x x C 30. C cos x cos 3 x cos 5x a n2
2
2
3 3 5
3 1 1 1 x 1 (x 2 1) arctan x
31. x sin 2x sin 4x C 32. tan 2 x n|cos x| C 12. C 1 x 2 arcsin x 13. 2 x C
8 4 32 2 2 2 x
2 1 1 xn(x 1 x 2 ) 1 x 2 C
33. C cot x cot 3 x cot 5x 34. (x 2 2x)3 c 35. 2e x
c 14.
3 5 3
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MATHEMATICS MATHEMATICS
ENTHUSE ENTHUSE
PART-II
11 4 1 1 ( z 1)2 2z 1
15. 2 x 1 n(1 x 1) C 16. C 20. 4 4 tan x c 21. n 2 3 arctan c, where z x
5
2(x 2)2 x 2 15 2 z z 1 3
3 x 2 1 x 4 3x 2 1
17. (x 1) 2 / 3 3(x 1)1/ 3 3n 1 3 x 1 C arctan(tan 2 x) c n C
2 22. 23.
x
6 6 x5 3 3 x2 x
18. x 2 x 3 3 x 6 6 x 6n 6
x 1 C 24. arctan(e x e x ) C 25. x tan C
5 2 2
(1 x 2 )3 x x2 9 1 x 2
19. c 20. c 21. c 26. arccos 2 C 27. esin x (x sec x) c
3x 3 a 2 2
x a 2 9x 2 x 1
|x| xe x x3 x 2 x 2 (x 2)5
22. n c 23. 2 arcsin x C 24. n C 4x n c ( 1 x 2 x)15
2 1 xe x 28. 29. c
1 x 1 3 2 (x 2)3 15
x(x 2 3)
3 4 x 2 (x 2 2) x 2 3(2x 2 3) 1
25. C arcsin x 26. C 27. c 30. log(tan–1(cosx + secx)) + c 31. c
2 1 x 2 2 24x 3 27x3 x
2 tan
DPP # 05 2
3 2 1 x2 2 (x 1) 4 (x 4)5
1. n 3x 1 n 2x 3 n x c 2. n C 1
11 33 3 x2 1 32. arctan(3 tan x) C 33. n c
3 (x 3)7
1 x 2 1 x 3 1 1 x 1
3. n n c 4. n c cos 2x 15 4 4 sin 2x 1
2 2 x 2 2 3 x 3 x 2 x 1 34. arcsin C
15(4 sin 2x) 15 15 4 sin 2x
x 1 | x 1 | 1 2x 1 or
5. c 6. n arctan c
(x 2 1)2 3 x2 x 1 3 3 z 1 z 1 15
3 tan 1 c, where z tan x & a 2
1 1 x 1
n arctan x c
15x5 40x 3 33x 15
arctan x C
8a 2 z 2 a 2
8a a 4 16
7. 8.
4 1 x 2 48(1 x 2 )3 48
1 7 9 1
1 cos x
9. x n | x | n | 2x 1| n | 2x 1| c C tan x(2 tan 2 x) 4 cot 2 x
4 x 2 C
4 16 16 35. 36. 2arctan cos n
3 2 x
cos x 1 cos
9 x4 5x 12 2 2
10. 4n | x | 3n | x 1| C 11. 2n c
x 1 x 2 x 2 6x 8
2
1 1 37. 2 ex 1 2 arctan ex 1 C 38. (tan 2 x 5) tan x C
4 4 x 1 tan x arctan( 2 tan x) C 5
12. C 13.
3 x2 2 2 2
3 3 1
14. tan 5 x (5 tan 2 x 11) c 15. +c
55 1 2
x
x2
1 1
16. n | tan x | c 17. (tan 2 x cot 2 x) 2n | tan x | c
2 sin 2 x 2
1 1 x 1 x
18. c 19. cot cot3 C
cos x 1 2 2 6 2
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MATHEMATICS
MATHEMATICS
ENTHUSE
ENTHUSE
Special DPP On Definite Integration
/4 /2 /2
sin cos 1 2 cos x x sin x
31. d 32. sin2 cos d 33. dx 34. dx
0 9 16 sin 2 0 0 ( 2 cos x) 2 0 1 cos x
4/3
Evaluate the following definite integrals. 2 x 2 x 1 dx
35. Let A = 3 2 then find the value of eA.
n 2 e 3/ 4 x x x 1
1
sin 1 x 1 ln x
1. dx 2. x ex dx 3.
x
dx 1
2 x2
1
d 1 e
0 x (1 x) 0 x ln x dx
1
36.
0 (1 x) 1 x 2
dx 37. dx 1 e
1
1/ x dx
38. ln( x x e x )
3 2 1
cos x 3
4. Given f ' (x) = , f = a, f = b. Find the value of the definite integral f ( x ) dx .
2 3 x
x 2 2 11 x
2 39. cos cos 2 dx 40. If f() = 2 & (f(x)+ f (x)) sin x dx = 5, then find
1 e /2 0 8 4 8 4 0
x dx 1 1 /4
sin 2 x cos x dx
5. 6. 2 dx
n x n x
7. dx 8. (1 sin x) (2 sin x)
f(0)
1 5 4x 2 0 sin 4 x cos4 x 0
b n 3
|x|
1 x 41. dx 42. f (x)dx, where f(x) = ex + 2e2x + 3e3x + ..
/4 3 sin 3 x n 2
sin 2 x . cos2 x 1 x 2 dx dx a
9. 2 dx 10. 11. /2 1
0 sin 3
x cos3 x 13
x 2 ( x 1) (5 x )
43.
sec x tan x cos ec x
dx 44. x f ' ' ( x ) dx , where f (x) =cos(tan–1x)
0 sec x tan x 1 2 cos ecx
2 12 0
x 1 /4 /2
dx
3
dx 45. (a) If g (x) is the inverse of f (x) and f (x) has domain x [1, 5], where f (1) = 2 and f (5) = 10 then
3 x dx (x cos x · cos 3x)dx
12. 13. 14. 15.
32 0 0 5 4 sin x 2 ( x 1) x 2 2 x 5 10
ln 3 find the value of f ( x ) dx g ( y) dy .
1 2
/2
dx
2
ex 1 /4 3
x
1
1
16. (0, ) 17. dx 18. cos 2x 1 sin 2 x dx 19. dx (b) Suppose f is continuous, f (0) = 0, f (1) = 1, f ' (x) > 0 and f ( x ) dx = . Find the value of the
0 1 cos . cos x
0
e2 x 1 0 0 3 x
1 0
3
definite integral f ( y) dy .
1
1/ 2 2
dx dx /2
20.
0 1 2x 2
1 x2
21.
1
x x4 1
22. sin cos a 2
sin 2 b 2 cos2 d a b (a > 0,b > 0) 0
ANSWER KEY
0
2
1 e 1 2
3 4 1 1. 2. n 3. 2 e 4. 2 ( a 3b) 5. 6. e 7.
2 6 n 2 4
(1 x ) sin x (1 x) cos x dx
sin x 4 2 2
23. (a) (b) x (1 x cos x · ln x sin x) dx 24. x (tan1 x)2 dx
0
0 2 4 1 ln 3 3 3
8. ln 9. 10. 11. 12. –1+ 13.
25. Suppose that f, f ' and f '' are continuous on [0, ln 2] and that f (0) = 0, f ' (0) = 3, f (ln 2) = 6, f ' (ln 2) = 4 3 6 2 6 2 6 16
ln 2 ln 2
2 1 1 3
and e
2 x
· f ( x ) dx = 3. Find the value of e
2 x
· f ' ' (x ) dx . 14. tan1 15. 16. sin 17. ln 3 ln 2 18. 1 19.
3 3 3 26 3 2
0 0
1
dx
b
dx e e 1 e2 e 2 1 1 32 1 a 3 b3
2
26. where << 27. where a = & b= 20. n 2 3 21. ln 22. 23. (a) 2 2 1 ; (b) 4
0 x 2 2 x cos 1 a 1 x 2 2 2 2 4 17 3 a 2 b2
1 1 1
1 x2 1
1 x 2 24. 1 n 2 25. 13 26. if 0 ; if 0 27. 1
28. 1 x dx 29. 5
dx 4 4 2
0
2
x4 x 1 x2
2 sin 2
0 1 3 8 1 4 1
30. Suppose that the function f, g, f ' and g ' are continuous over [0, 1], g (x) 0 for x [0, 1], f (0) = 0, 28. l n 3 29. 30. 2009 31. ln 3 32. 33.
2 24 20 9 2
2009 16 2
g(0) = , f (1) = and g (1) = 1. Find the value of the definite integral, 34. 35. 36. 37. 38. ln 2 39. 2 40. 3
2 2 9 2 1 e
1
f ( x ) · g' ( x ) g 2 ( x) 1 f ' ( x ) · g( x) g 2 ( x ) 1
dx .
1 3
g 2 ( x) 41. | b | – | a | 42. 43. /3 44. 1 45. (a) 48 ; (b) 2/3
0 2 2 2
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2019
100 Percentile 99.99 Percentile 99.98 Percentile 99.98 Percentile 99.97 Percentile 99.97 Percentile 99.96 Percentile 99.96 Percentile
HIMANSHU GAURAV SINGH GAURAV KRISHAN GUPTA SARTHAK ROUT VIBHAV AGGARWAL RITVIK GUPTA BHAVYA JAIN AYUSH PATTNAIK SAYANTAN DHAR
2019 (*SDCCP) 2020 (DLP) 2020 (CCP) 2019 (CCP) 2020 (DLP) 2020 (CCP) 2019 (CCP) 2020 (DLP)
IIT MATHEMATICS
INDEFINITE & DEFINITE INTEGRATION
Corporate Office: NAIVEDHYAM, Plot No. SP-11, Old INOX, Indra Vihar, Kota (Raj.) 324005
Call: 0744-2799900 Mob. 97831-97831, 70732-22177, 0744-2423333
RESULTS OF BEST MENTORSHIP BY THE NUCLEUS TEAM
INDEFINITE INTEGRATION
EXERCISE # 1
1 x7
1. x 1 x 7 dx equals -
ln 1 x 7 C ln 1 x 7 C
2 2
(A) l n x (B) l n x
AIR-1 AIR-3 AIR-6 AIR-8 7 7
(C) l n x ln 1 x 7 C (D) l n x ln 1 x 7 C
CHITRAANG MURDIA GOVIND LAHOTI NISHIT AGARWAL AMEY GUPTA 2 2
Gen. Category - 2014 Gen. Category - 2014 Gen. Category - 2012 Gen. Category - 2014
7 7
JEE MAIN RESULTS 2020 (January Attempt) OF NUCLEUS EDUCATION 2. Primitive of 3x 4 1
w.r.t.x is
x 4 x 1
2
x x x 1 x 1
(A) C (B) C (C) C (D) C
x4 x 1 x4 x 1 x4 x 1 x4 x 1
3. If
cos x sin x 1 x
e x sin x x
dx ln f x g x C where C is the constant of integration and
100 Percentile 100 Percentile 100 Percentile 100 Percentile 100 Percentile 100 Percentile f(x) is positive, then f(x) + g (x) has the value equal to
(Physics) (Maths & Physics) (Physics) (Maths) (Maths) (Maths)
DAKSH KHANDELWAL VAIBHAV SAHA ANISH MOHAN ARCHIT PATNAIK SWAPNIL YASASVI PARSHANT ARORA (A) ex + sin x + 2x (B) ex + sin x (C) ex– sin x (D) ex + sin x +x
2020 2020 2020 2020 2020 2020
4. Integral of 1 2cot x cot x cos ecx w.r.t.x is
JEE ADVANCED RESULTS OF NUCLEUS EDUCATION x x
(A) 2 n cosC (B) 2 n sin C
2 2
1 x
(C) n cos C (D) n sinx - n(cosecx – cot x ) + C
2 2
ln x 1 x 2 dx equals
AIR-10 AIR-12 AIR-23 AIR-24 AIR-37 AIR-42 AIR-66 AIR-98
5. x. 1 x2
LAKSHAY SHARMA YATEESH AGRAWAL ABHEY GOYAL TUSHAR GAUTAM PIYUSH TIBAREWAL SATVIK MAYANK DUBEY HRITHIK
Gen. Category - 2017 Gen. Category - 2017 Gen. Category - 2017 Gen. Category - 2017 Gen. Category - 2017 Gen. Category - 2017 Gen. Category - 2017 Gen. Category - 2017
x 2 x
(A) 1 x 2 ln x 1 x 2 x C (B) . ln x 1 x 2 C
2 1 x2
(C)
x 2
2
. ln x 1 x 2
x
1 x2
C
(D) 1 x 2 ln x 1 x 2 x C
AIR-20
SHASHANK AGRAWAL
Gen. Category - 2018
AIR-27
RAAGHAV RAAJ
Gen. Category - 2018
AIR-32
SHREYA PATHAK
Gen. Category - 2018
AIR-61
SIDDHANT CHOUDAHRY
Gen. Category - 2018
AIR-67
ANISWAR S KRISHNAN
Gen. Category - 2018
DLP
AIR-78
AAYUSH KADAM
Gen. Category - 2018
AIR-61
SARTHAK BEHERA
Gen. Category - 2018
*SDCCP
AIR-91
ANDREWS G. VARGHESE
Gen. Category - 2018
*SDCCP
6.
Let g(x) be an antiderivative for f x .Then n 1 g x is an antiderivative for
2
2 f xgx 2 f xgx 2 f x
(A) (B) (C) (D) none
1 f x 1 g x 1 f x
2 2 2
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AIR-2 AIR-19 AIR-33 AIR-48 AIR-51 AIR-53 AIR-86
HIMANSHU GAURAV SINGH VIBHAV AGGARWAL S. PRAJEETH SOHAM MISTRI SAYANTAN DHAR GAURAV KRISHAN GUPTA SATVIK JAIN
Gen. Category - 2019 Gen. Category - 2019 Gen. Category - 2019 Gen. Category - 2019 Gen. Category - 2019 Gen. Category - 2019 Gen. Category - 2019
*SDCCP *SDCCP DLP DLP DLP
A function y = f(x) satisfies f " x
1 1
2 sin x ;f ' 2 and f 1 0. The value of x 2 1 n x
n4x x 4 dx equals
7.
x2 2 12.
1
1 x 1
n n x– x C
f is
2 (A) n 2 2
2 n x 4
(A)n2 (B) 1
1 n x x 1 –1 n x
(B) n tan C
(C) n 2 (D) 1-n2 4 n x x 2 x
2
1 n x x 1 –1 n x
(C) n tan C
x 2 4 n x x 2 x
8. Consider f x ; g(t) f t dt. If g 1 0 then g x equals
1 x3 1 n x x –1 n x
(D) n tan C
1 1 x3 4 n x x x
(A) n 1 x 3
1
(B) n
3 3 2
2x 3 1
1 1 x3 1 1 x3 x x 1 x 2 x 3 1 C f x , where ƒ(x) is of the form of ax
2
13. + bx + c then (a + b + c)
(C) n (D) n
2 3 3 3
equals
(A) 4 (B) 5 (C) 6 (D) none
x
e
9. x
(x x )dx
x2 3
e x 1 dx is equal to
x
14.
(A) 2e x
x x 1 C (B) e x
x 2 x 1 C 2
x 3 x 3
(C) e (x x) C (D) e (x x 1) C C C
x
x x
(A) e x (B) e
x 1 x 1
x 1 1
2
C
x
dx (C) e (D) e x C
10. x 1 x 1
x 5/2 x 1
3 7/2
(where ‘C” is integration constant)
1/6
x 1 x 1
1/6
(A) C (B) 6 C
x x x3
15. dx is equal to
2x 1
2 3
5/6 5/6
x x
(C) C (D) C
x 1 x 1 1 1
2
1 1
2
(A) 2 2 C (B) 2 2 C
4 x 4 x
2sin 2 x 1 cos x 2sin x 1
2 2
1 1 1 1
11. Let f x then ex f x f ' x dx (where C is the constant (C) 2 2 C (D) 2 2 C
cos x 1 sin x 2 x 4 x
of integration) (where ‘C” is integration constant)
(A) ex tan x + C (B) ex cot x + C
x 2
(C) e cosec x + C (D) ex sec2x + C
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EXERCISE # 2 e x
e x
cos e x
e x
e
4
x
e x
cos e x
e x
4
17. dx
x 1 x2 x dx x
1. x x x x x2 x
ecos x x sin 3 x cos x
18. dx 19. dx
e sin 2 x
2
x
x 1
2. A function g defined for all positive real numbers, satisfies g'(x2) = x3 for all x > 0 and g(1) =1.
Compute g(4).
5x 4 4x 5
(sin x)
11/ 3
20. dx 21. (cos x )1/ 3 dx
x5 x 1
2
x x2 3
sin sin x sin dx
2
3.
2
4. x x 6 2
1
dx
dx 4x 5 7x 4 8x 3 2x 2 4x 7
cos ecx cot x sec x
22. sin x sec x 23. x 2 x 2 1
2
dx
5. x x x
dx 6. .
cos ecx cot x 1 2sec x
dx
cot .cot .cot
2 3 6
f '(x)g x g ' x f x f x g x
24. Let dx m tan 1 C.
1 x f x g x f x g x g2 x
ng x
ln ln
1 x x x e x
7. 1 x 2 dx 8. e n xdx
x
Where m, n N and ‘C’ is constant of integration (g(x) > 0). Find the value of (m2 + n2).
1 cot x
2008
x 1 dx 1
9.
x 5 3x 4 x 3 8x 2 x 8
dx 10.
25. If the value tan x cot x 2009
dx l n | sin k x cos k x | C, then find k.
k
x2 1 x 3
x 1
dx
x xnx 26. x x x
sin
1
11. dx 12. .dx
ax x 1
2 3/2
1 7 cos 2 x g x
x 2 1 n x 2 1 2nx 27. Suppose dx C, where C is arbitrary constant of integration. Then find
dx tan 2
13. x4
14. cos6 sin 6
d sin 7 x cos 2 x sin 7 x
the value of g' (0) + g"
4
3x 2 1 ax 2
b dx
15. dx 16.
x 2 1 x c x ax 2 b
3 2
2 2
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/2
DEFINITE INTEGRATION 8. The value of the definite integral sinx sin2x sin3x dx
0
is equal to :
EXERCISE # 1
1 2 1 1
x
(A) (B)– (C) (D)
1. If g x cos t dt, then g (x + ) equals -
4
3 3 3 6
0
(A) g(x) g( ) (B) g(x) g( ) (C) g(x)g( ) (D) g(x) / g( )
sin 3x 4x cos 4x
1/2
9. Value of the definite integral 1 3 1 3
3x dx :
1/2
tan 1 x 7
1
2.
0
x
dx (A) 0 (B)–
2
(C)
2
(D)
2
/4 /2 /2 /4
sin x x 1 x 1 x
(A) x
dx (B) sin x dx (C)
2 sin x dx (D)
2 sin x dx 1/ x ln 1 t 2
0 0 0 0
10. Lim x 3 dt equals
x 1 et
1/ x
/2
ex (A) 1/3 (B) 2/3 (C) 1 (D) 0
3. The value of definite integral sin x cos x .
0
sin x
dx equals .
/2 /2 /2 1 /4 C0 C1 C2
(A) 2 e (B) e (C) 2 e .cos1 (D) e 11. If 0 , where C0, C1, C2 are all real, the equation C2x2+ C1x + C0 = 0 has :
2 1 2 3
(A) atleast one root in (0, 1)
y d2 y (B)one root in (1, 2) & other in (3, 4)
dt
4. Variable x and y are related by equation x . The value of is equal to (C) one root in (–1,1) & the other in (–5, –2)
0 1 t2 dx 2
(D) both roots imaginary
y 2y
(A) (B)y (C) (D) 4y
1 y2 1 y2 e
x 1 e . nx dx is
x
12. The value of the definite integral
1
x 1 1
5. If f t dt x t 2 .f t dt
0 x
4
1, then the value of the integral f x dx is equal to
1
(A) e (B) ee+1 (C) ee(e –1) (D) ee(e – 1) + e
(A) 0 (B)/4 (C)/2 (D)
1 1 1
13. Lim ...... is equal to
/2 /4 n n n 1 n n2 n 4n
6. If I n sin x dx then n sin x cos x dx
0 /4 (A) 2 (B) 4 (C) 2 2 1 (D) 2 2 1
I I I
(A) (B) (C) (D) I
2 4 2
x
37
3 sin 2x dx where {x} denotes the fractional part
2
14. The value of the definite integral
7. If f (x) = x sinx ; g(x) = x cosx for x [–1, 2]
2 2
19
2 2 function.
A f x dx ; B g(x) dx, then (A) 0 (B) 6 (C) 9 (D) can’t be determined
1 1
(A) A > 0 ; B < 0 (B) A < 0 ; B > 0 (C) A > 0 ; B > 0 (D) A < 0 ; B < 0
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x /2 /2 /2
sin t dt
2
21. Let I1 e
x2
sin x dx; I 2 e
x2
dx ; I 3 e
x2
(1 x) dx
15. lim 0
equals 0 0 0
x 0 x 1 cos x and consider the statements
1 1 2 I I1 < I2 II I2 < I3 III I1 = I3
(A) (B) 2 (C) (D)
3 2 3 Which of the following is (are) true ?
(A) I only
16. If g (x) is the inverse of f(x) and f(x) has domain x [1, 5], where f (1) = 2 and f (5) = 10 then (B) II only
5 10
(C) Neither I nor II nor III
the values of f (x)dx g(y) dy equals
1 2 (D) Both I and II
(A) 48 (B) 64
(C) 71 (D) 52 1
l n x 1 /2
22. Let u dx and v l n sin 2x dx then
4 x 1
2
x 3 x 4 x 6 x 10 x sin x dx equals
0 0
17. The value of the definite integral
2 (A) u = 4v (B) 4u + v = 0
(A) cos 2 + cos 4 (B) cos 2 – cos 4 (C) u + 4v = 0 (D) 2u +v = 0
(C) sin 2 + sin 4 (D) sin 2 – sin 4
1
3
1
2
1
dx 23. 2 x 3 1 x 4 dx equals
18. The value of
1 x
is 1
2
3 9
1 (A) (B)
(A) (B) 2 (C) 4 (D) undefined 2 8
2
1 3
(C) (D)
4 2
1
x x 1
3
19. x
1
2
2 x 1
dx a l n 2 b then Where {.} denotes the fraction part function.
(A) a= 2 ; b =1 (B) a = 2 ; b = 0
24. Suppose that F(x) is an antiderivative of
(C) a = 3 ; b = – 2 (D) a =4 ; b = –1
3
sin x sin 2 x
f x , x 0 then dx can beexpressed as
x 1
x
x
20. The true solution set of the inequality, 5x 6 x 2 dz x sin 2 xdx is: 1
20 0 (A) F(6) – F(2) (B) (F(6) – F(2))
2
(A) R (B) (1, 6)
1
(C) (– 6, 1) (D) (2, 3) (C) (F(3) – F(1)) (D) 2(F(6) – F(2))
2
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k x
25. Let f be a positive function. Let I1 x f x 1 x dx ;
1 k
31. The angle between the tangent lines to the graph of the function ƒ(x) = (2 t 5) dt at the points
2
k where the graph cuts the x-axis is
I2
I2 f x 1 x dx ; where 2k 1 0. Then is
1 k
I1 (A) (B) (C) (D)
6 4 3 2
1
(A) k (B) (C) 1 (D) 2
1 1
x
2 2 x
32. Number of critical points of the function, ƒ(x) = x3 – + cos 2 t t dt which lie
3 2 1
2 2
x h x in the interval [ – 2, 2] is :
n t dt n 2 t dt
2
(A) 2 (B) 4 (C) 6 (D) 8
26. Li m a a
h 0 h
x2
2 n x
(A) 0 (B) n x 2
(C) (D) does not exist 33. If ƒ(x) = (t 1) dt, 1 x 2, then global maximum value of ƒ(x) is
x x
(A) 1 (B) 2 (C) 4 (D) 5
1 n x
27. f x x . dx x
sin t
0
x 34. For the function ƒ(x) = 0
t
dt, where x > 0,
(A) is equal to zero (B) is equal to one
(A) maximum occurs at x = n, n is even (B) minimum occurs at x = n, n is odd
1
(C) is equal to (D) cannot be evaluated (C) maximum occurs at x = n, n is odd (D) None of these
2
/2
cos sin
28. 1 cos 1 sin d equals
0
1 1
(A) cos (B) cos–1(0) (C) cos–1(1) (D) cos-1(–1)
2
29. Let f(x) be a continuous function on [0, 4] satisfying f(x) f(4–x) =1.
4
1
The value of the definite integral dx equals
0
1 f x
(A) 0 (B) 1 (C) 2 (D) 4
x x3
If g(x) = e t dt then the value of e
2
t2
30. dt equals
1 3
(A) g(x3) –g(3) (B) g(x3) + g(3) (C) g(x3) –3 (D) g(x3) –3g(x)
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/4 /4
sin x cos x
2 2
EXERCISE # 2 cos x v
12. Let u = sin x cos x dx and v cos x
dx. Find the value of u .
x 0 0
1 3 sin 1
1 x dx
2
e
1
1. Evaluate: (i) n tan x
. sin 1 (cosx).dx (ii) /4
xdx
0 1/3
x 13. Evaluate : cos x(cosx sin x)
0
sin 1 x
1
2. Evaluate : x(1 x)
dx 1
sin 1 x
0 14. Evaluate : x
0
2
x 1
dx
/2
1 2cos x
3. Evaluate : (2 cosx) 2
dx 1 5
0 2
x2 1 1
/2
15. Evaluate : n 1 x dx
x x2 1
4
x
2 x
1
2 x
e cos(sinx)cos sin(sinx)sin dx
x
4. Evaluate :
0 2 2 1/n
16. lim n 2
n
1/n
(2010sinx 2012cosx) | x | dx
e
5. Evaluate : {(1 x)e (1 x)e } n x dx
x x
1
17. If (cosx cos 2 x cos3x) 2 (sinx sin 2 x sin 3x) 2 dx has the value equal to w
2 0 k
x xdx dx
6. If P dx; Q and R , the prove that: where k and w are positive integers, find the value of (k2 + w2).
0
1 x4 0
1 x4 0
1 x4
(a) Q , (b) P = R, (c) P 2Q R /2
a sin x b cos x
4 2 2 18. Evaluate :
dx
0 sin x
4
(x 2 1)dx
2
u (1000) u
7. If where u and v are in their lowest form. Find the value of . 19. A continuous real function f satisfies f(2x) = 3f(x) x R
1 x . 2x 4 2x 2 1
3 v v
1 2
/2
1 sin 2x
If f (x) dx 1, then compute the value of definite integral f (x)dx
0 1
8. Evaluate : dx
0
1 sin 2x
(ax b)sec x tan x
9. If a1, a2 and a3 are the three values of a which satisfy the equation 20. Evaluate : dx (a, b > 0)
0
4 tan 2 x
/2 /2
4a
0 (sinx a cos x) dx 2 0 x cos xdx 2 then find the value of 1000(a1 a 2 a3 ).
3 2 2 2
(2 x 3)sinx
21. Evaluate : dx
(1 cos 2 x)
2x 3x 10x 7x 12x x 1
2 7 6 5 3 2 0
10. Evaluate : x2 2
dx 3
x
2 22. Evaluate : 3 x
dx
x x
2 2 0
11. Evaluate : dx n
x2 4 Let In ({x 1}·{x 2} {x 2 2}{x 3 4})dx,
2
2 23.
n
where {·} denotes the fractional part of x. Find I1.
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a
n(1 ax) 35. Let f(x) be a function defined on R such that f'(x) = f'(3 – x) x [0, 3] with f(0) = – 32 and
24. Evaluate : dx,a N
1 x2 3
0 f(3) = 46. Then find the value of f (x) dx.
0
n3
2
e 1
x 36. Let f and g be function that are differentiable for all real numbers x and that have the following
25. Evaluate : e
0
2x
1
dx properties.
(i) f'(x) = f(x) – g(x); (ii) g'(x) = g(x) – f(x) ;
1
2 3x 4x 2 (iii) f(0)= 5 ; (iv) g(0) = 1
26. Let I dx. Find the value of I2. (a) Prove that f(x) + g(x) = 6 for all x. (b) Find f(x) and g(x)
0 2 1 x x2
(2x 332 x 998 4 x1668 . sin x 691 ) 1 1 n r
. Let n f and
1
27. Evaluate : dx 37. Consider a function f(n) =
1 n 2
n r 1 n
1
1 x 666
1 n 1 r
p q
n f for n = 1,2,3….. Also lim
n r 0 n n
n & lim n . Then prove
n
28. Show that | cosx | dx 2q sinp where q N & p .
0
2 2
(a) n < n (b) = (c) n n
2 4
x sin 2x.sin .cos x
29. Evaluate : 2 dx
2x 2
100U10 1
Let U10 x sin xdx, then find the value of
0 10
38. .
0 U8
x
30. If (x) = cosx – (x t)(t) dt. Then find the value of "(x) + (x).
0 39. Prove the inequalities :
2
1 2
dx
(b) 2e1/4 e x x dx 2e2 .
6 0 4 x 2 x 3
2
x
f '(x) (a)
31. (a) Let g(x) = xc. e2x& let f(x) = e2t .(3t 2 1)1/2 dt. For a certain value of 'c', the limit of 8 0
0
g '(x)
as x is finite and non-zero. Determine the value of 'c' and the limit. 9 1
I k n
x
t 2dt 40. Let e
x
(1 {x} {x}2 )dx I and e
x
(1 {x} {x}2 )dx J, if e , then k is equal to
0 a t 0 0
J n 0
(b) Find the constants 'a' (a > 0) and 'b' such that lim 1.
x 0 bx sin x (where {·} denotes fractional part function)
d
3 x
3t 4 1 41. A cubic ƒ(x) vanishes at x = –2 & has relative minimum/maximum at x = –1 and x = 1/3.
32. Evaluate : lim
x dx 1 (t 3) (t 3)
2
dt 1
14
2sin If ƒ (x) dx = , find the cubic ƒ (x).
x
1
3
33. Evaluate x
Investigate for maxima & minima for the function, ƒ(x) = [2(t 1) (t – 2)3 + 3(t – 1)2(t – 2)2]dt
1/n
1 22 32 n 2 1 1 2 3n 42.
(a) lim 1 2 1 2 1 2 ......1 2 (b) lim .....
n n n 1
1
n
n n n n n 2 4n
34. Find a positive real valued continuously differentiable functions f on the real line such that for all x
x
f 2 (x) ((f(t))2 (f '(t))2 ) dt e2
0
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7. Water is drained from a vertical cylindrical tank by opening a valve at the base of the tank. It is
DIFFERENTIAL EQUATION
known that the rate at which the water level drops is proportional to the square root of water
EXERCISE # 1 depth y, where the constant of proportionality k > 0 depends on the acceleration due to gravity
1. Number of value of m N for which y = emx is a solution of the differential equation 1
and the geometry of the hole. If t is measured in minutes and k = then the time to drain the
D3y – 3D2y – 4Dy + 12y = 0, is 15
(A) 0 (B) 1 (C) 2 (D) more than 2 tank if the water is 4 meter deep to start with is
2 (A) 30 min (B) 45 min (C) 60 min (D) 80 min
dy dy
2. Number of straight lines which satisfy the differential equation x y 0 is :
dx dx dy 1 x
8. The general solution of the differential equation is a family of curves which looks
(A) 1 (B) 2 (C) 3 (D) 4 dx y
most like which of the following?
3. The value of the constant 'm' and 'c' for which y = mx + c is a solution of the differential
equation D2y – 3Dy –4y = –4x.
(A) is m = –1; c = 3/4 (B) is m = 1; c= – 3/4 (C) no such real m, c (D) is m = 1; c = 3/4 (A) (B)
4. Consider the two statement
Statement-1 : y = sin kt satisfies the differential equation y" + 9y = 0.
Statement-2 : y = ekt satisfy the differential equation y" + y' – 6y = 0
The value of k for which both the statement are correct is (C) (D)
(A) –3 (B) 0 (C) 2 (D) 3
x
5. If y (where c is an arbitrary constant) is the general solution of the differential
ln | cx | 9. Spherical rain drop evaporates at a rate proportional to its surface area. The differential
dy y x x equation corresponding to the rate of change of the radius of the rain drop if the constant of
equation then the function is :
dx x y y proportionality is K > 0, is
x2 x2 y2 y2 dr dr dr
(A) 2 (B) 2 (C) 2 (D) 2 (A) K 0 (B) K 0 (C) Kr (D) none
y y x x dt dt dt
6. The differential equation corresponding to the family of curve y = ex(ax + b) is 10. The x-intercept of the tangent to a curve is equal to the ordinate of the point of contact. The
2
d y dy equation of the curve through the point (1, 1) is
(A) 2 y0
dx 2 dx x x y y
2 (A) ye y e (B) xe y e (C) xe x e (D) ye x e
d y dy
(B) 2 y0
dx 2 dx 1
11. A function f(x) satisfying f (tx)dt nf (x), , where x > 0, is
d2 y dy
(C) 2 y0 0
dx 2 dx
1 n n 1
d2 y dy (A) f x c x n
(B) f x c x n 1 (C) f x c x n (D) f(x) = c x(1 - n)
(D) 2 y0
dx 2 dx
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12. Which one of the following curves represents the solution of the initial value problem
Dy = 100 – y, where y (0) = 50 EXERCISE # 2
E-1
[FORMATION & VARIABLES SEPARABLE]
(A) (B) 1. State the order and degree of the following differential equations:
3/2
d 2 y dy
3
d 2 x dx
4 2
(i) 2 xt 0 (ii) 1
dt dt dx
2
dx
2. (a) Form a differential equation for the family of curves represented by ax2 + by2 = 1,
(C) (D)
where a & b are arbitrary constants.
(b) Obtain the differential equation of the family of circles x2 + y2 + 2gx + 2fy + c = 0 ;
where g, f & c are arbitrary constants.
13. The real value of m for which the substitution, y = um will transform the differential equation,
(c) Obtain the differential equation associated with the primitive,
dy
2x4y + y4 = 4x6 into a homogeneous equation is : y = c1 e3x + c2 e2x + c3 ex, where c1, c2, c3 are arbitrary constants.
dx
(A) m = 0 (B) m = 1 (C) m = 3/2 (D) no value of m Solve the following differential equation for Q.3 to Q.9.
n(sec x tan x) n(sec y tan y)
3. dx dy 4. (1 – x2) (1 – y) dx = xy (1 + y) dy
14. A curve C passes through origin and has the property that at each point (x, y) on it the normal cos x cos y
line at that point passes through (1, 0). The equation of a common tangent to the curve C and
the parabola y2 = 4x is
5.
dy
x 2
1 y 2 1
0 6. yx
dy dy
a y2
(A) x = 0 (B) y = 0 (C) y = x + 1 (D) x + y + 1 = 0
dx xy dx dx
2
ex dy dy x(2nx 1)
15. A function y = f (x) satisfies (x + 1). f '(x) – 2 (x2 + x) f (x) = x > – 1 7. = sin(x + y) + cos(x + y) 8.
(x 1) , dx dx sin y y cos y
If f (0) = 5 then f(x) is
dy xy xy dy
3x 5 x2 6x 5 x 2 6x 5 x 2 5 6x x 2 9. (a) sin sin (b) sinx y . lny if y = e, when x =
(A) .e (B) .e (C) 2
.e (D) .e dx 2 2 dx 2
x 1 x 1 (x 1) x 1
10. The population P of a town decreases at rate proportional to the number by which the
2
population exceeds 1000, proportionality constant being k > 0. Find
16. The equation to the orthogonal trajectories of the system of parabolas y = ax is
(a) Population at any time t, given initial population of the town being 2500.
x2 y2 x2 y2
(A) y2 c (B) x 2 c (C) y2 c (D) x 2 c (b) If 10 years later the population has fallen to 1900, find the time when the population
2 2 2 2 will be 1500.
(c) Predict about the population of the town in the long run.
x
t y(t)dt x y(x) then y as a function of x is
2
17. If
a 11. It is known that the decay rate of radium is directly proportional to its quantity at each given
x 2 a 2 x 2 a 2
instant. Find the law of variation of a mass of radium as a function of time if at t = 0, the mass
(A) y = 2 – (2 + a2) e 2
(B) y = 1 – (2 + a2) e 2
x 2 a 2
of the radium was m0 and during time t0, % of the original mass of radium decay.
(C) y = 2 – (1 + a2) e 2
(D) none
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12. A normal is drawn at a point P(x, y) of a curve. It meets the x-axis at Q. If PQ is of constant (E-3)
dy
length k, then show that the differential equation describing such curves is, y k 2 y2 . [LINEAR]
dx
dy
Find the equation of such a curve passing through (0,k). 1. If y = y(x) is solution of differential equation y 1 e x and y(0) = y0 has a finite value,
dx
13. A curve is such that the length of the polar radius of any point on the curve is equal to the when x , then find y0.
length of the tangent drawn at this point. Form the differential equation and solve it to find the
equation of the curve. dy x 1
2. y
dx 1 x 2 2x(1 x 2 )
dy
14. Given y(0) = 2000 and = 32000 – 20y2, then find the value of Lim y(x) .
dx x
dy
3. (1 x 2 ) 2xy x(1 x 2 )1/2
dx
15. Let f(x) is a continuous function which takes positive values for x 0 and satisfy
x
1
0 f (t) dt x f (x) with f(1) = 2 , Find the value if f 2 1 . 4. (a) Find the curve such that the area of the trapezium formed by the co-ordinate axes,
ordinate of an arbitrary point & the tangent at this point equals half the square of its
abscissa.
1
dy (b) A curve in the first quadrant is such that the area of the triangle formed in the first
16. y y dx given y = 1, where x = 0
dx 0
quadrant by the x-axis, a tangent to the curve at any of its point P and radius vector of
E-2 the point P is 2sq. units. If the curve passes through (2, 1), find the equation of the
curve.
[HOMOGENEOUS]
dy x 2 xy
1. 5. x(x – 1)
dy
–(x – 2) y = x3 (2x – 1)
dx x 2 y 2 dx
2. Find the equation of a curve such that the projection of its ordinate upon the normal is equal to
dy
its abscissa. 6. sin x +3y = cosx
dx
3. The perpendicular from the origin to the tangent at any point on a curve is equal to the abscissa
of the point of contact. Find the equation of the curve satisfying the above condition and which dy
7. x(x2 + 1) = y(1 – x2) + x3. nx
passes through (1, 1). dx
4. Find the curve for which any tangent intersects the y-axis at the point equidistant from the point
dy
of tangency and the origin. 8. x y 2x 2 cos ec2x
dx
dy x 2y 3
5. (x – y)dy = (x + y + 1)dx 6. 9. (1 + y2)dx = (tan–1y – x)dy
dx 2x y 3
dy y x 1 dy x y 1
7. 8.
dx y x 5 dx 2x 2y 3 10. Let the function nf(x) is defined where f(x) exists for x 2 & k is fixed positive real number,
d
dy 2(y 2) 2 prove that if (x . f (x)) – k f (x) then f(x) A x – 1– k where A is independent of x.
9. dx
dx (x y 1) 2
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x x
11. Find the differential function which satisfies the equation f(x)= – f (t) tan t d t tan(t x)dt AREA UNDER THE CURVE
0 0
where x (–/2, /2)
EXERCISE # 1
1. The area bounded in the first quadrant by the normal at (1, 2) on the curve y2 = 4x, x-axis & the
2 curve is given by :
12. y – x Dy = b(1 + x Dy)
10 7 4 9
(A) (B) (C) (D)
3 3 3 2
2. Suppose y = f(x) and y = g(x) are two functions whose graphs intersect at the three points
13. Find all function f(x) defined on , with real values and has a primitive F(x) such that (0, 4), (2, 2) and (4, 0) with f(x) > g(x) for 0 < x < 2 and f(x) < g(x) for 2 < x < 4. If
2 2 4 4
f(x) + cosx . F(x) =
sin 2x
. [f (x) g(x)]dx 10 and [g(x) f (x)]dx 5 , the area between two curves for 0 < x < 2, is
(1 sin x) 2 0 2
(A) 5 (B) 10 (C) 15 (D) 20
dy 3. Let ‘a’ be a positive constant number. Consider two curves C1: y = ex, C2 : y = ea – x. Let S be
14. 2 ysec x y3 tan x
dx S
the area of the part surrounding by C1, C2 and the y-axis, then Lim 2 equals
a 0 a
(A) 4 (B) 1/2 (C) 0 (D) 1/4
dy
15. x2y – x3 = y4 cos x
dx 4. The area of the region(s) enclosed by the curve y = x2 and y = | x | is
(A) 1/3 (B) 2/3 (C) 1/6 (D) 1
16. y(2xy + ex) dx – ex dy = 0
5. Area enclosed by the graph of the function y = ln2x – 1 lying in the 4th quadrant is
2 4 1 1
17. A tank contains 100 liters of fresh water. A solution containing 1 gm/liter of soluble lawn (A) (B) (C) 2 e (D) 4 e
e e e e
fertilizer runs into the tank at the rate of 1 lit/min and the mixture in pumped out of the tank at
the rate of 3 liters/min. Find the time when the amounts of fertilizer in the tank is maximum. 6. The area bounded by the curve y = f(x) (where f(x) 0), the co-ordinate axes & the line x = x1
is given by x1.e x1 . Therefore f(x) equals :
(A) ex (B) xex (C) xex – ex (D) xex + ex
E-4
(GENERAL – CHANGE OF VARIABLE BY A SUITABLE SUBSTITUTION) 7. The slope of the tangent to curve y = f(x) at (x, f(x)) is 2x + 1. If the curve passes through the
point (1, 2) then the area of the region bounded by the curve, the x-axis and the line x = 1 is
1. (x – y2) dx + 2 xy dy = 0 2. (x3 + y2 + 2) dx + 2y dy = 0
5 6 1
dy dy tan y (A) (B) (C) (D) 1
3. x y ln y xye x 4. (1 x)e x sec y 6 5 6
dx dx 1 x
2
8. The area bounded by the curves y = x(x – 3)2 and y = x is (in sq. units) :
dy e y 1 dy dy (A) 28 (B) 32 (C) 4 (D) 8
5. 6. (x y) xy 0
dx x 2 x dx dx
9. Area of the region enclosed between the curves x = y2 – 1 and x = |y| 1 y2 is
dy y2 x
7. 8. 2 2 2
(1 – xy + x y ) dx = x dy (A) 1 (B) 4/3 (C) 2/3 (D) 2
dx 2y(x 1)
10. The curve y = ax2 + bx + c passes through the point (1, 2) and its tangent at origin is the line y = x.
dy dy The area bounded by the curve, the ordinate of the curve at minima and the tangent line is
e x y (ex e y )
2
9. 10. (x + y) = 6x
dx dx 1 1 1 1
(A) (B) (C) (D)
24 12 8 6
Corporate Office: NAIVEDHYAM, Plot No. SP-11, Old INOX, Indra Vihar, Kota (Raj.) 324005 7 Corporate Office: NAIVEDHYAM, Plot No. SP-11, Old INOX, Indra Vihar, Kota (Raj.) 324005 31
Call: 0744-2799900 Online Partner UNACADEMY Call: 0744-2799900 Online Partner UNACADEMY
EXERCISE # 2
1. Find the area bounded on the right by the line x + y = 2, on the left by the parabola y = x 2 and
below by the x-axis.
2. Find the area of the region {(x, y) : 0 y x2 + 1 , 0 y x + 1 , 0 x 2}.
3. Find the area of the region bounded by curves f(x) = (x – 4)2, g(x) = 16 – x2 and the x-axis.
x
4. A figure is bounded by the curves y = 2 sin , y = 0, x = 2 & x = 4. At what angles to the
4
positive x-axis straight lines must be drawn through (4, 0) so that these lines partition the figure
into three parts of the same area.
5. Find the area bounded by the curves y = 1 x 2 and y = x3– x. Also find the ratio in which the
y-axis divided this area.
6. If the area enclosed by the parabolas y = a – x2 and y = x2 is 18 2 sq. units. Find the value of
'a'.
7. The line 3x + 2y = 13 divides the area enclosed by the curve,9x 2 + 4y2– 18x – 16y – 11 = 0 into
two parts. Find the ratio of the larger area to the smaller area.
1
8. Consider two curves C1 : y = and C2 : y = ln x on the xy plane. Let D1 denotes the region
x
surrounded by C1, C2 and the line x = 1 and D2 denotes the region surrounded by C1, C2 and the
line x = a. IfD1 = D2. Find the value of 'a'.
9. Find the area enclosed between the curves : y = loge (x + e) , x = loge(1/y) & the x-axis.
10. For what value of 'a' is the area bounded by the curve y = a2x2 + ax + 1 and the straight line
y = 0,x = 0 & x = 1 the least ?
11. Find the positive value of 'a' for which the parabola y = x 2 + 1 bisects the area of the rectangle
with vertices(0, 0), (a, 0), (0, a2 + 1) and (a, a2 + 1).
12. Compute the area of the curvilinear triangle bounded by the y-axis & the curve, y = tan x &
y=(2/3)cosx.
13. Find the value of 'c' for which the area of the figure bounded by the curve, y = 8x2– x5, the
straight lines x = 1 & x = c & the abscissa axis is equal to 16/3.
Find the area bounded by the curve y = xe x , the x-axis, and the line x = c where y(c) is
2
14.
maximum.
15. A polynomial function f (x) satisfies the condition f (x + 1) = f (x) + 2x + 1. Find f (x) if
f (0) = 1. Find also the equations of the pair of tangents from the origin on the curve y = f (x)
and compute the area enclosed by the curve and the pair of tangents.
16. The figure shows two regions in the first quadrant.
A(t) is the area under the curve y = sin x2 from 0 to t and B(t) is the area of the triangle with
A( t )
vertices O, P and M(t, 0). Find Lim
t0 B( t )
Corporate Office: NAIVEDHYAM, Plot No. SP-11, Old INOX, Indra Vihar, Kota (Raj.) 324005 32
Call: 0744-2799900 Online Partner UNACADEMY
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