NARAYANA IIT ACADEMY
(G\ NARAYANA
or excuience T - JEE ACADEMY - INDIA
APEX_PT-4_ MATHS ASSIGNMENT
1. The integral /- is equal to:
tan (4) ee be {an'(e+3)) +0
A) . B) ‘
af iw RY),
ome ele (eel) =
J axdx
2. The integral ave equal
5) log) +V2+"] +0 py eel 2= +6
eS ~loglt-V2=" +0 A, «log|t— bees
3. TE s(x)= (= a a xdvand f (0)=0,then f (1) equal
Le = z
A)tan 4 B) tanl+1 G4 D)1-4
4 Saamesaa 7
“aot +C Pram
Omens te Daciag te
5. Let I(x)=f
A) 6v3 B) V3 oO D) 3V3
(sin"?+cos*?0)d0
=
sin’ 8.cos’ @.sin(0 +c)
2 : 2
vleosar.tand +sina ~-—— cosa + cot8.cosa +e
a sina
dx. 1f KO) = 3, then (Bs equal to
6. The value of J Eee NARAYANA IIT ACADEMY
2 : 2
B) Veosq.tand + sina -—~—eosa + cot@.sina +¢
cose sina
2 2
Cc) Vsina.tan@ + cosa -—— cosa + cot @.cosa +¢
cosa sina
a -
Vcosa.tan@ + cosa -——Vcosa@ + cot@.cosa +e
cosa sina
1 1 1 7 A
7. I fdr = — tan" (3 tan x) + constant, then the maximum value of a sin
@sin'x+B cos'x 12
x +b cos.x, is
D)
A) V2 B) Var ©) V9 D) V40
x'(x'+x°—1)dx | zs
8 El Geaareay Oro f(x)
(where ¢ is integration constant)
whe) B) F(x? +x
©) -2 2x 43x 42°)" (= EN By
6 25°20 10
a <7 is equal to (Where C is a constant of integration)
9. The integral eae
x4) (x
\ \ 3
(3 ay +e ») (23) "+e o(B)re mas) +C
x+4 x+4 2la+4 13x44
smfart)
10. if f
R be a differentiable function such that
O41.
eri
I x(cos x—sinx) A g(xi(e +1—xe") i
e+ (e+
for all ¥>0, where c is an arbitrary constant. Then Eee NARAYANA IIT ACADEMY.
B) 8" is increasing
|
C) & is increasing in D) &* & is increasing in |
12. The integral
dv is equal to
J sin’
(sin’ x-+cos? xsin? x +-sin’ xcos? x-+cos* x)"
tse — TG mg |
A) root's p) 30+) Cy MF eM py Treot's
+€
13. Let / bea differentiable function such that {0 =? and /= S09 forall ER If
Wx)= (£0) then *'C) is equal to
A) 4e B) 4° ©) 2 D) 2e°
3a ae
14. The integral J dx,x> 0, is equal to: (where c is a constant of
é
Fm some ae
integration)
A) log, |x? +5x—7| +e log, |x? +5x—7|+¢
B)
dog, |x? +5 a+
oa er x 1+ D) be. we 45x—-T+e
(5 (cosx—sinx)
15, ‘The integral. f oe is equal to (where C is a constant of integration)
1+ sin2x
[epee
fn xi
1 (5 ml ‘|
Frog, +¢ —tiog +c
tan 42 2 tan 4%
A) a B) De Eee NARAYANA IIT ACADEMY
+c diog, c
2
D)
16. = Aftan{2—! coum +C, where C isa constant of
3) x —2x410
integration, then 64+ Bis equal to
20 i 5 28
A) 9 B) 6 C6 D) 9
1
S(m)=-5
17, {x)= f O522E= Asin’ xeos.e— Asindeos!2r— sind, g Oa ee
1—4sin” x-+2sin? 2x—2cos2x
of the following is true .
: “(| a
By 2 (@)/>1 co 4) 2 D) none
18. If f(x +242") Jeet 43x46)" ar" (p+ qr! +r)" 40
Where c is constant of integration a,b,p,q,t are integers then @+5—P +9
A)35 B)30 ©)25 D)20
19. f (Vian + Jeots) dr is
a V2sin-(sinx -cosx)+e py V2sin- "(sins Heosx) +e
¢) sie "sin —cosx) +e py sin" (sine +0981) +6
(where c is integrating constant)
20s
x+
20. iF ff =
(seex+cosecr + tan x-+cotx)
b
then Ja+b+¢
—— (Where d is constant of integration)
ef ete allt byl+x7 +C, then 34-0 = __
vite
(where C is integral constant)
al. Eee NARAYANA IIT ACADEMY
22. ForK(x)=" sin”
hao 2
“ mesh
23. The integral i
xsecx
+C
tanx—
A)
axsinx+cosx
xtanx
+
secx—
1)
Jet (2+ xsec’ x)dx =
xsinx+cosx
24.
A) 2e™" +C
“4C
C) Tanx + xe™
sec” x— 2022
a
jo
1a
}-0
A) ei(Z)-1(
3 6
cos’ x
For I(x)
° wu(2)-(2
AG
26. The integral
(xsinx + cosx)°
2)
sue GE
B)
SHE)
a
dx is equal to (where C is a constant of integration);
xtan.x
secx + +C
B)
tanx+
D)
xsinx+cosx
xseex
+C
xsinx+cosx
B) x'e™ +
D) x? +xe™+C
as (2 - 2!" then
un
no(a}-4g)
pens)
d= FO) +e such that SO =0, then
Ail
([Jis the greatest integer function, ¢ is integrating constant)
A)l B)O
fan Vicdeis equal to
7. 9
A) + Dian Vx Ve +C
©) Venstan V+
2
3 D)2
py stan Ve +C
py ¥¥-(+Dtan"' Vs +C
(where C is integrating constant)
Sets
28,
dx is equal to 29.
30.
31.
34,
NARAYANA IIT ACADEMY
SE <(log, “|
——_+¢ xtan( 22) 4.¢
A) 1+ cos(logx) B) 2
—xeot tees). C A3* 46
° _ py Hsinioa)
If fee ade=— Le /(a)+C where C is a constant of integration, then f°) is
equal to
(2
wf ‘seox(2-+ see) 4. f(x) +C.where c is the integration constant and the curve
(1+2secx)
y= F(®)passes through the origin, then 2 & + f(x) equals
1 x x
If [| log(logx) + ——, |dv =—log(logx) + B——+C. Then 4+B=
i log( mes) fr = “log (logx) cm en
(Ayo (B) 2 (2 ()1
ex
is equal to:
J sin2x-+e lyin 2x40 :
A)2 B) 2 D) -sin’ x+c
The integral equal (for some arbitrary constant K)
( j seex-+ tan oy box
ay (eo +tans
1
B) exon"
alee
ome See JEE MAIN-2026 MATHS ASSIGNMENT NARAYANA IIT ACADEMY
1 fil :
Fr pleer+ tans)
D) (secx+tanx)"* U1
Tx
sin
35. i 2a is equal to (where c is a constant of integration)
sin
2sin3x 2sin3x
+sin2x+2sinx—x+e
A) 3 B)
+sin2x—2sinx+x+e
2sin3x
D)
2x +2sinx+x+e
+sin2x+2sinx+x+0
36. [—*~ a=
J x +l)vx* +1
37.
a2
A) 3 B) C=1/3 © D=1 D) B+C+D=-1
38.
Ay Se
il
©) loxls
39. te fae ae Bin
ie —5e
8B-A
then 2
+e (Where c is integrating constant) REE NARAYANA IIT ACADEMY
3e0t3x—cotx,,
40. If fc = p+ qi
Ul cme=sanacee aaad
+e then 2
(where c is constant of integration)
41.
For real numbers °9:7 474 6 ig
cep stan fF etl
1s fan-
432 +e [F Bs |
x
allog, | tan) * =) + pian (= =} btm [2H \;'
x Fe ;
where C is an ame constant, then the value of !0(¢ + 818) is equal to
42 —
If Ie ae 4 +C, where C is the constant of integration and
(x—1)"(x +3)" Bx+3
4.8 are constants, then the value of © 9+20AB jg
43. 1 0) = fae (20) ang £0 =9 then the value of 4)
oe 2x
(A) 2 (B) 2 ©-1 (D)1
dx
¥2(a=3)x+(a-1)
(x+a-3)- f(a)|
(«+a-3)+ f(a)
44. Assertion (A): For2D. 15, + Lyng = a tan? x + bx
+, where ¢ is a constant of
integration, then the ordered pair (4-5) is equal to
A) (a4) B) (aa) ° las] D) bas
54. The value of integral /, - 8in9X (y, equals to (Where c is integrating constant)
cosx
cos8x _cos6x , cos4x
+cos2x + In|secx| +e
Ay 4 3
cos8x . cos6x 4
SOSBE 5 086% | CO8FN 5 cos 24 4 Infseca] +e
B) 4 3
A S0SBR | LOSER _ COSAT | gs 2x + Ineo} +c
Ce 3
_cos7x | cosSx _cos3x
+ +c082x + In]seex| +e
pb) 4 A [pees]
55. Let 1, = ftan'xdx, (n> 1). 1, +1, =atan’ x+ bx’ +C, where C is a constant of
integration, then the ordered pair (54) js equal to
a (-1,0) (B) (-11) © (1,0) (D) (1-1)
1+10x? - x
56. fet 1
we(E vem e( EA} oe a re a
I-x I-x Tex
de is equal to: (where c is an integral constant)
lex! REFER ee
then &+B° ig equal to :
58. Let /“be a twice differentiable function defined on R such that £0 =1/°0)=3 and
fe) #0 for all ER. If ra Pu 0, for all xR, then the value of f(1) lies in
the interval.
A) (9,12) B) (15,25) ©) (10,15) D) (25,30)
If ——_, d= Atan'x +Btan”’=+C then 24+ B=
59. (Gf +DQ +4) 2
(where C is integral constant)