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Let f(x) be a polynomial of degree 2 sat...

Let f(x) be a polynomial of degree 2 satisfying `f(0)=1, f(0) =-2 and f''(0)=6`, then `int_(-1)^(2) f(x)` is equal to

A

6

B

2

C

9

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
C
Let `f(x)=ax^(2)+bx+c`. Then,
`f(0)=1 rArr c=1`
Now,
`f'(x)=2ax +b and f''(x)=2a`
`f'(0)=-2 and f''(09) =6 rArr b=-2 and a=3`
`:. f(x)=3x^(2)-2x+1`
`rArr underset(-1)overset(2)int f(x)dxunderset(-1)overset(2)int(3x^(2)-2x+1)dx=[ x^(3)-x^(2)+x]_(-1)^(2)`
`rArr underset(-1)overset(2)int f(x)dx=(8-4+2)-(-1-1-1)=9`
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OBJECTIVE RD SHARMA-DEFINITE INTEGRALS-Section I - Solved Mcqs
  1. The value of the integral int(a)^(a+pi//2) (|sin x|+|cosx|)dx is

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  2. If rArrI(n)= int(a)^(a+pi//2)(cos^(2)nx)/(sinx) dx, "then" I(2)-I(1),I...

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  3. Let f(x) be a polynomial of degree 2 satisfying f(0)=1, f(0) =-2 and f...

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  4. The vlaue of int(-2)^(2) (sin^(2)x)/([(x)/(pi)]+(1)/(2))dx where [*]...

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  5. If int(0)^(x) f(t)dt=x+int(x)^(1) t f(t) dt, then the value of f(1), i...

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  6. If f(x)= int0^(sinx) cos^(-1)t dt +int(0)^(cosx) sin^(-1)t dt, 0 lt ...

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  7. Let f(x) be a continuous function such that int(n)^(n+1) f(x) dx=n^(3)...

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  8. Let f(x)=(e^(x)+1)/(e^(-x)-1) and int(0)^(1)x^(3)(e^(x)+1)/(e^(x)-1) d...

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  9. If int(0)^(1) x e^(x^(2) ) dx=alpha int(0)^(1) e^(x^(2)) dx, hten

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  10. If I=int(0)^(1) (1+e^(-x^2)) dx then, s

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  11. If I= int(0)^(1)(x)/(8+x^(3))dx then the smallest interval is which I ...

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  12. Let f: r in R be a contunuous function given be f(x+y)=f(x)+f(y)"for a...

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  13. Let the be an integrable function defined on [0,a] if I(1)=int(pi//2)^...

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  14. lim(x to 0) (2 int(0)^(cosx) cos^(-1) t dt)/( 2x -sin 2x) is equal to

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  15. If I(1)= int(1)^(sin theta) (x)/(1+x^(2)) dx and I(2) int(1)^("cosec" ...

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  16. If f(x)=int(1)^(x) (log t)/(1+t) dt"then" f(x)+f((1)/(x)) is equal to

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  17. Let F(x) =f(x) +f((1)/(x)),"where" f(x)=int(1)^(x) (log t)/(1+t) dt Th...

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  18. If int(0)^(x) (bt cos 4 t - a sin 4t)/( t^(2))dt=(a sin 4x)/(x) "for a...

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  19. Let f: R in R be given by f(x)={{:(|x-[x]|,"when[x]is odd"),(|x-[x]-...

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  20. If f(x) = sin x +cos x and g(x) = {:{((|x|)/(x),","x ne0),(2,","x=0):}...

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