(c)^1
1
1(^21)
2
e
ex
ex
c
−
−
- log sin +
cos
, (e is greater than 1)
(d)^2
1
1
(^21)
2
e
exe x
ex
c
−
++−
log cos sin +
cos
,
(e is greater than 1)
- Let f be a differentiable function such that
f′(x) =+fx()∫fxdxf() , () 0 =^4 −e ,
3
202
then f(x) is(a) ex− e −
⎛
⎝⎜⎞
⎠⎟(^21)
3
(b) ex−()e −
(^22)
3
(c) ex+ez−−[arg |^1 |]
3
(d) None of these
Multiple Correct Answer Type
- IfI x
x
dx I x
x==∫∫sin(sin ) dx
sin, sin
02
2
02ππandI x
x
3 dx
02
=∫sin(tan )
tanπ, then which of the followingis true?
(a) I 1 > I 3 (b) I 2 > I 3
(c) I 1 > I 2 (d) I 1 < I 2
- If f(x) is monotonic and differentiable function,
then 2 xb f^1 x dx
fa
fb
(())
()()
∫ − − =(a) (() ())fx fadx
a
b(^22) −
∫ (b) (() ())fx fbdx
a
b
(^22) −
∫
(c) fxdx abfb
a
b
(^22) () +(− ) ()
∫
(d) fxdx abfa
a
b
(^22) () +(− ) ()
∫
- dx
()xxx()()1
2 1
11
+^20101=
+−
+⎡
⎣⎢
⎢⎤
⎦⎥
∫ αβαβ⎥
+ cwhere α, β > 0 then
(a) |α – β| = 1
(b) (β + 2)(α + 1) = (2010)^2
(c) β, α, 2010 are in A.P.
(d) α + 1 = β + 2 = 2010
- x
x
x
x+ dx
−⎛
⎝⎜⎞
⎠⎟ +−
+⎛
⎝⎜⎞
⎠⎟ −⎧
⎨⎪
⎩⎪⎫
⎬⎪
− ⎭⎪∫
1
11
12221212
is
//(a) 4 4
3ln⎛
⎝⎜⎞
⎠⎟(b) 4 3
4ln⎛⎝⎜ ⎞⎠⎟(c) − ⎛
⎝⎜⎞
⎠⎟ln^81
256(d) ln^256
81⎛
⎝⎜⎞
⎠⎟- Letfx x then
t
dtx
() sin
/ cos=
2 ∫ 41 +^22π
(a) f′⎛⎝⎜π⎟⎞⎠=π
2(b) f′⎛⎝⎜−π⎟⎞⎠=π
2(c) f′⎛⎝⎜^3 ⎟⎞⎠=−
2π 3 π (d) ′ =
∫ +
f dx
x()
cos/
π
ππ1 2422- The value of tdt
t
dt
e ttxex1 11 +^212+
∫∫/ +tan/cot()is(a)^1
2 +tan^2 x(b) 1(c) π
4(d)^2
11 21
πdt
−∫ +t
Comprehension Type
Paragraph for Q. No. 41 to 43
Let f(x) defined in [a, b] has discontinuities C 1 , C 2 , C 3 ,
...., Cn such that a < C 1 < C 2 < .... < Cn < b thenfxdx fxdx fxdx
abCCaC
∫∫( ) =+ +∫ ( ) ( ) ....
11 2++∫ ∫
−fxdx fxdx
CbCCn nn
() ()
1- [] 23
1
1
xdx− =
−∫ (where [.] is greatest integer
function)
(a) –7 (b) –9 (c) 5 (d) 11/2- ∫[tan−^1 ] =
0
50
xdxπ
(where [.] is greatest integerfunction)
(a) tan1 + 50π (b) –tan1
(c) 50π – tan1 (d) 20π – 2tan1