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Let f(x) = (1 - x)^2 sin^2 x + x^2 for ...

Let `f(x) = (1 - x)^2 sin^2 x + x^2 ` for all x ∈ R, and let `g(x) = ∫((2(t - 1))/(t + 1) - ln t)f(t)dt` for t ∈ [1, x] for all x ∈ (1, ∞).Which of the following is true ?

A

(a) 0

B

(b) 1

C

(c) 2

D

(d) 3

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The correct Answer is:
C
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VIKAS GUPTA (BLACK BOOK) ENGLISH-CONTINUITY, DIFFERENTIABILITY AND DIFFERENTIATION-EXERCISE (SUBJECTIVE TYPE PROBLEMS)
  1. Let f(x) = (1 - x)^2 sin^2 x + x^2 for all x ∈ R, and let g(x) = ∫((2...

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  2. Let f (x)= {{:(ax (x-1)+b,,, x lt 1),( x+2,,, 1 le x le 3),(px ^(2) +q...

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  3. If y= sin (8 sin ^(-1) x ) then (1-x ^(2)) (d^(2)y)/(dx ^(2))-x (dy)/...

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  4. If y ^(2) =4ax, then (d^(2) y)/(dx ^(2))=(ka ^(2))/( y ^(3)), where k ...

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  5. The number of values of x , x ∈ [-2,3] where f (x) =[x ^(2)] sin (pix)...

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  6. If f (x) is continous and differentiable in [-3,9] and f'(x) in [-2,8]...

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  7. In f (x)= [{:(cos x ^(2),, x lt 0), ( sin x ^(3) -|x ^(3)-1|,, x ge 0)...

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  8. Consider f(x) =x^(2)+ax+3 and g(x) =x+band F(x) = lim( n to oo) (f...

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  9. Let f (x)= {{:(2-x"," , -3 le x le 0),( x-2"," , 0 lt x lt 4):} Then f...

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  10. If f (x) +2 f (1-x) =x ^(2) +2 AA x in R and f (x) is a differentiable...

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  11. Let f (x)= signum (x) and g (x) =x (x ^(2) -10x+21), then the number o...

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  12. If (d^(2))/(d x ^(2))((sin ^(4)x+ sin ^(2)x+1)/(sin ^(2)x + si n x+1))...

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  13. f (x) =a cos (pix)+b, f'((1)/(2))=pi and int (1//2)^(3//2) f (x) dx =2...

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  14. Let alpha (x) = f(x) -f (2x) and beta (x) =f (x) -f (4x) and alpha '(1...

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  15. Let f (x) =-4.e ^((1-x)/(2))+ (x ^(3))/(3 ) + (x ^(2))/(2)+ x+1 and g ...

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  16. If y=e^(2 sin ^(-1)x) then |((x ^(2) -1) y ^('') +xy')/(y)| is equal t...

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  17. Let f be continuous function on [0,oo) such that lim (x to oo) (f(x)+ ...

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  18. Let f (x)=x+ (x ^(2))/(2 )+ (x ^(3))/(3 )+ (x ^(4))/(4 ) +(x ^(5))/(5)...

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  19. In f (x)= [{:(cos x ^(2),, x lt 0), ( sin x ^(3) -|x ^(3)-1|,, x ge 0)...

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  20. Let f :R to R be a differentiable function satisfying: f (xy) =(f(x)...

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  21. For the curve sinx+siny=1 lying in first quadrant. If underset(xrarr0...

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