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  • Teko Classes IIT JEE/AIEEE MATHS by SUHAAG SIR Bhopal, Ph. (0755)32 00 000www.tekoclasses.com Question. & Solution. Int. Cal. Page: 1 of 26

    THE BOND || Phy. by Chitranjan|| ||Chem. by Pavan Gubrele|| ||Maths by Suhaag Kariya||

    TOPIC = INTEGRAL CALCULUS

    Single Correct Type

    Que. 1. Let 3 3 32 4

    f (x) sin x sin x sin x3 3

    = + + + +

    then the primitive of f(x) w.r.t. x is

    (a) 3sin 3x

    C4

    + (b) 3cos3x

    C4

    + (c) sin 3x

    C4

    + (d) cos3x

    C4

    +

    where C is an arbitrary constant. (code-V2T3PAQ5)

    Que. 2. If the dependent variable y is changed to z by the substitution y = tan z then the differential equa-

    tion

    22

    2 2

    d y 2(1 y) dy1

    dx 1 y dx

    + = +

    + is changed to

    2

    2

    d z

    dx =

    2

    2 dzcos z k ,dx

    +

    then the value of k equals

    (a) 1 (b) 0 (c)1 (d) 2 (code-V2T3PAQ10)

    Que. 3. ( )xx n ex dx is equal to (code-V2T5PAQ1)

    (a) xx C+ (b) x. n x (c) ( )

    xc n x C+ (d) None

    Que. 4. The value of the definite integral 2008

    x

    2008

    f '(x) f '( x)dx

    (2008) 1

    +

    + equals (code-V2T5PAQ3)

    (a) f (2008) f ( 2008)+ (b) f (2008) f ( 2008)

    (c) 0 (d) f ( 2008) f (2008)

    Que. 5.e

    1

    1 n xdx

    xx n x

    +

    equals (code-V2T5PAQ4)

    (a) e (b) 2e (c) 2 e (d) 2e

    Que. 6. If x

    4

    0

    g(x) cos t dt,= then g(x )+ equals (code-V2T5PAQ5)

    (a) g(x) g( )+ (b) g(x) g( ) (c) g(x)g( ) (d) [ ]g(x) g( )

    Que. 7. Let f be a positive function. Let ( ) ( )k k

    1 2

    1 k 1 k

    I x f x(1 x) dx; I f x(1 x) dx,

    = = where 2k 1 0. >

    Then 2

    1

    I

    I is (code-V2T5PAQ7)

    (a) k (b) 1/2 (c) 1 (d) 2

    Que. 8. 2 2

    dx

    x 16 x has the value equal to (code-V2T5PAQ8)

    (a) 1 x

    C arc sec4 4

    (b)

    1 xarc sec C

    4 4

    +

    (c) 2

    16 xC

    16x

    (d)

    216 x

    C16x

    +

  • Teko Classes IIT JEE/AIEEE MATHS by SUHAAG SIR Bhopal, Ph. (0755)32 00 000www.tekoclasses.com Question. & Solution. Int. Cal. Page: 2 of 26

    THE BOND || Phy. by Chitranjan|| ||Chem. by Pavan Gubrele|| ||Maths by Suhaag Kariya||

    Que. 9. If ( )2 2x 2x 2x .e dx e ax bx c d, = + + + then (code-V2T5PAQ10)

    (a) 1 1 1

    a ,b ,c2 2 4

    = = = (b) 1 1 1

    a ,b ,c2 2 4

    = = =

    (c) 1 1

    a ,b 1,c2 2

    = = = (d) 1

    a 1,b 1,c2

    = = =

    Que. 10. If

    1

    -1

    0

    1tan x dx= - ln 2

    4 2 then the value of the definite integral ( )1

    1 2

    0

    tan 1 x x dx

    + equals

    (a) n 2 (b) n 24

    + (c) n 2

    4

    (d) 2 n 2 (code-V2T5PAQ11)

    Que. 11.

    2

    n

    nn0

    nlim x dx

    2 equals (code-V2T5PAQ12)

    (a) 1 (b) 2 (c) 1

    2(d) 4

    Que. 12. If f is continuous function and x 2

    0 t

    F(x) (2t 3). f (u)du dt

    = + then F"(2) is equal to

    (a) 7f (2) (b) 7f '(2) (c) 3f '(2) (d) 7f (2) (code-V2T5PAQ15)

    Que. 13. ( )( )sin x/ 2

    x 1 x cos x n x sin x dx

    + + is equal to (code-V2T5PAQ16)

    (a)

    2

    2

    (b)

    2

    (c)

    24

    4

    (d) 1

    2

    Que. 14. Let f :[0, ) R be a continuous strictly increasing function, such that (code-V2T5PAQ17)

    x

    3 2

    0

    f (x) t.f (t)dt= for every x 0. The value of f (6) is

    (a) 1 (b) 6 (c) 12 (d) 36

    Que. 15. If the value of definite integral

    / 4

    x

    / 6

    1 cot xdx,

    e sin x

    + is equal to / 6 / 4ae be + then (a + b) equals

    (a) 2 2 (b) 2 2+ (c) 2 2 2 (d) 2 3 2 (code-V2T5PAQ18)

    Que. 16. Let 30

    n xJ dx

    1 x

    =+

    and 20

    x n xK dx

    1 x

    =+

    then (code-V2T5PAQ20)

    (a) J K 0+ = (b) J K 0 = (c) J K 0+ < (d) none

  • Teko Classes IIT JEE/AIEEE MATHS by SUHAAG SIR Bhopal, Ph. (0755)32 00 000www.tekoclasses.com Question. & Solution. Int. Cal. Page: 3 of 26

    THE BOND || Phy. by Chitranjan|| ||Chem. by Pavan Gubrele|| ||Maths by Suhaag Kariya||

    Que. 17. The value of x > 1 satisfying the equation x

    1

    1t n t dt ,

    4= is (code-V2T5PAQ22)

    (a) e (b) e (c) 2e (d) e 1

    Que. 18. If

    x

    1

    F(x) f (t)dt= where 2

    t 4

    1

    1 uf(t) du

    u

    += then the value of F"(2) equals (code-V2T5PAQ23)

    (a) 7

    4 17(b)

    15

    17(c) 257 (d)

    15 17

    68

    Que. 19. Let f be a continuous function on [a, b]. If ( )( )x b

    a x

    F(x) f (t)dt f (t)dt 2x a b

    = + then there exist

    some ( )c a,b such that (code-V2T8PAQ6)

    (a)

    c b

    a c

    f (t)dt f (t)dt= (b) ( )c b

    a c

    f (t)dt f (t)dt f (c) a b 2c = +

    (c) ( )( )c b

    a c

    f (t)dt f (t)dt f (c) 2c a b = + (d) ( )( )c b

    a c

    f (t)dt f (t)dt f (c) 2c a b+ = +

    Que. 20. The value of the definite integral ( ) ( )/ 2

    x 2 2

    0

    x xI e cos sin x cos sin sin x sin dx,

    2 2

    = +

    is

    (a) ( )/ 21 e cos1 sin1 1

    2

    + (b) ( )/ 2e

    cos1 sin12

    + (codeV2T10PAQ2)

    (c) ( )/ 21

    e cos1 12

    (d) [ ]

    / 2ecos1 sin1 1

    2

    +

    (a) ( )/ 21 e cos1 sin1 1

    2

    + (b) ( )/ 2e

    cos1 sin12

    +

    (c) ( )/ 21

    e cos1 12

    (d) [ ]

    / 2ecos1 sin1 1

    2

    +

    Que. 21. A tank with a capacity of 1000 liters originally contains 100 gms of salt dissolved in 400 liters of

    water. Beginning at time t 0= and ending at time t 100= minutes, water containing 1 gm of salt per

    liters enters the tank at the of 4 liters per minute, and the well mixed solution is drained from the tank

    at a rate of 2 liter/minute. The differential equation for the amount of salt y in the tank at time t is

    (a) dy y

    4dt 400 2t

    = +

    (b) dy y

    4dt 200 t

    = +

    (code-V2T11PAQ2)

    (c) ( )

    dy y4

    dt 200 t 2=

    + (d) dy y

    4dt 500 t

    = +

  • Teko Classes IIT JEE/AIEEE MATHS by SUHAAG SIR Bhopal, Ph. (0755)32 00 000www.tekoclasses.com Question. & Solution. Int. Cal. Page: 4 of 26

    THE BOND || Phy. by Chitranjan|| ||Chem. by Pavan Gubrele|| ||Maths by Suhaag Kariya||

    Que. 22. Let y y(t)= be solution to the differential equation 2y ' 2ty t ,+ = then t

    ylim

    t is (code-V2T11PAQ4)

    (a) zero (b) 1

    2(c) 1 (d) Non existent.

    Que. 23. The area of the region bounded below by 1y sin x,= above by 1y cos x= and on the left by y-axis,

    is

    (a) 2 1 (b) 2 2 (c) 2 1+ (d) 2 (code-V2T11PAQ7)

    Que. 24. 2

    0

    xdx

    4 x is equal to (code-V2T13PAQ20)

    (a) 2

    (b) 1

    2

    (c) 1 (d) 2

    Que. 25. If 2

    4 2

    2

    x . 4 x dx

    has the value equal to k then the value of k equals (code-V2T14PAQ20)

    (a) 0 (b) 2 (c) 8 (d) 4

    Que. 26. Let q

    2008 r

    dx 1 xn C

    x x p 1 x

    = +

    + + where p,q.r N and need not be distinct, then the value of

    ( )p q r+ + equals (code-V2T14PAQ25)

    (a) 6024 (b) 6022 (c) 6021 (d) 6020

    Que. 27. ( ) ( )/ 2

    x

    0

    sin x n(sin x) x cot x dx

    + is (code-V2T17PAQ7)

    (a) 1 (b) 1 (c) 0 (d) Indeterminant

    Que. 28. Let ( )2

    y n 1 cos x= + then the value of 2

    2 y / 2

    d y 2

    dx e+ equals (code-V2T17PAQ8)

    (a) 0 (b) 2

    1 cos x+(c)

    4

    (1 cos x)+(d) 2

    4

    (1 cos x)

    +

    Que. 29. The value of definite integral 1

    x 2

    1

    dx

    (1 e )(1 x )

    + + is (code-V2T18PAQ1)

    (a) / 2 (b) / 4 (c) / 8 (d) /16

    Que. 30. The expression 2

    3

    2

    d yy

    dx on the ellipse 2 23x 4y 12+ = is equal to (code-V2T19PAQ5)

    (a) 9

    4(b)

    9

    4 (c)

    4

    9(d)

    4

    9

    Que. 31. The value of the definite integral ( )n

    0

    x sin xdx n N

    1 cos x

    +

    is equal (code-V2T19PAQ6)

    (a) 2n n 2 (b) 2n n 2 (c) n n 2 (d) 2

    n n 2

    2

  • Teko Classes IIT JEE/AIEEE MATHS by SUHAAG SIR Bhopal, Ph. (0755)32 00 000www.tekoclasses.com Question. & Solution. Int. Cal. Page: 5 of 26

    THE BOND || Phy. by Chitranjan|| ||Chem. by Pavan Gubrele|| ||Maths by Suhaag Kariya||

    Comprehesion Type# 1 Paragraph for Q. 1 to Q. 3

    A curve in represented parametrically by the equations t tx e cos t and y e sin t= = is a parameter. Then

    1. The relation between the parameter t and the angle between the tangent to the given curve and the

    x-axis is given by, t equals (code-V2T4PAQ1,2,3)

    (a) 2

    (b)

    4

    + (c)

    4

    (d)

    4

    2. The value of 2

    2

    d y

    dx at the point where t = 0 is

    (a) 1 (b) 2 (c) 2 (d) 3

    3. If ( )F(t) x y dt= + the point the value of F F(0)2

    is

    (a) 1 (b) 1 (c) / 2e (d) 0

    # 2 Paragraph for Q. 4 to Q. 6

    Let f (x) is a derivable function satisfying ( )x

    2

    0

    xf (x) f (t) dt x n x 1 x = + + with f (0) n 2.= Let

    g(x) xf '(x)= then (code-V2T4PAQ4,5,6,)

    4. Range of g (x) is

    (a) [ )0, (b) [ )0,1 (c) [ )1, (d) [ ),

    5. For the function f which one of the following is correct ?

    (a) f is neither odd nor even (b) f is transcendental

    (c) f is injective (d) f is symmetric w.r.t. origin.

    6.

    1

    0

    f (x) dx equals

    (a)