Teko Classes IIT JEE/AIEEE MATHS by SUHAAG SIR ... ∫ = is (code-V2T5PAQ22) (a) e (b) e (c) e2...

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Transcript of Teko Classes IIT JEE/AIEEE MATHS by SUHAAG SIR ... ∫ = is (code-V2T5PAQ22) (a) e (b) e (c) e2...

  • Teko Classes IIT JEE/AIEEE MATHS by SUHAAG SIR Bhopal, Ph. (0755)32 00 000 www.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 3 2 4

    f (x) sin x sin x sin x 3 3

    π π    = + + + +   

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

    (a) 3sin 3x

    C 4

    − + (b) 3cos3x

    C 4

    − + (c) sin 3x

    C 4

    + (d) cos3x

    C 4

    +

    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) dy 1

    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) ( )

    x c 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 x dx

    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 )+ π equals (code-V2T5PAQ5)

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

    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 sec 4 4

      −  

      (b)

    1 x arc sec C

    4 4

      + 

     

    (c) 2

    16 x C

    16x

    − − (d)

    2 16 x

    C 16x

    − +

  • Teko Classes IIT JEE/AIEEE MATHS by SUHAAG SIR Bhopal, Ph. (0755)32 00 000 www.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 ,c 2 2 4

    = − = − = − (b) 1 1 1

    a ,b ,c 2 2 4

    = − = − = −

    (c) 1 1

    a ,b 1,c 2 2

    = − = − = − (d) 1

    a 1,b 1,c 2

    = = = −

    Que. 10. If

    1

    -1

    0

    π 1 tan 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 2 4

    π + � (c) n 2

    4

    π − � (d) 2 n 2� (code-V2T5PAQ11)

    Que. 11.

    2

    n

    nn 0

    n lim 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

    π

    π

    + +∫ � is equal to (code-V2T5PAQ16)

    (a)

    2

    2

    π (b)

    2

    π (c)

    24

    4

    π − π (d) 1

    2

    π −

    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 x dx,

    e sin x

    π

    π

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

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

    Que. 16. Let 3 0

    n x J dx

    1 x

    = +∫ �

    and 2 0

    x n x K 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 00 000 www.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

    1 t 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 u f(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 x I e cos sin x cos sin sin x sin dx,

    2 2

    π  

    = +   

    ∫ is

    (a) ( ) / 21 e cos1 sin1 1

    2

    π + −  (b) ( ) / 2e

    cos1 sin1 2

    π

    + (codeV2T10PAQ2)

    (c) ( )/ 2 1

    e cos1 1 2

    π − (d) [ ]

    / 2e cos1 sin1 1

    2

    π

    + −

    (a) ( ) / 21 e cos1 sin1 1

    2

    π + −  (b) ( ) / 2e

    cos1 sin1 2

    π

    +

    (c) ( )/ 2 1

    e cos1 1 2

    π − (d) [ ]

    / 2e cos1 sin1 1

    2

    π

    + −

    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

    4 dt 400 2t

    = − +

    (b) dy y

    4 dt 200 t

    = − +

    (code-V2T11PAQ2)

    (c) ( )

    dy y 4

    dt 200 t 2 = −

    + (d) dy y

    4 dt 500 t

    = − +

  • Teko Classes IIT JEE/AIEEE MATHS by SUHAAG SIR Bhopal, Ph. (0755)32 00 000 www.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

    y lim

    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

    x dx

    4 x−∫ is equal to (code-V2T13PAQ20)

    (a) 2

    π (b) 1

    2

    π − (c) 1π − (d) 2π −

    Que. 25. If 2

    4 2

    2

    x . 4 x dx −

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

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

    Que. 26. Let q

    2008 r

    dx 1 x n 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

    π

    +∫ � 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) / 2π (b) / 4π (c) / 8π (d) /16π

    Que. 30. The expression 2

    3

    2

    d y y

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