IIT-JEE-Mathematics-Screening–2001

SCREENING 

Time : Three hours                                                              Max. Marks : 100 
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Notations : 
R : set of real numbers. 
[x] : the greatest integer £ x. 

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1. Let f∶R→R be a function defined by (x)=max { x,x3 }. The set of all points where is NOT differentiable is: 
(A) {-1, 1 }
(B) {-1, 0 } 
(C) {0, 1 } 
(D) {-1, 0, 1 } 

2. Let f∶(0,∞)→R and F(x)= ∫0x f(t)dt. If F(x2 )= x2  (1+x), then f(4)equals:
(A) 5/4
(B) 7
(C) 4
(D) 2

3. The left hand derivative of f(x)=[x] sin(πx) at x=k,k an integer, is:
(A) (-1)k (k-1)π
(B) (-1)(k-1) (k-1)π
(C) (-1)k kπ
(D) (-1)(k-1) kπ

4. If f(x)=x e(x(1-x)), then f(x) is:

(A) Increasing on [-1/2 ,1]
(B) Decreasing on R
(C) Increasing on R
(D) Decreasing on [-1/2 ,1]

5. limx→0 sin(π cos2 x) / x2

(A) –π
(B) π
(C) π/2
(D) 1


6. The triangle formed by the tangent to the curve f(x)= x2+ bx-b at the point

(1, 1)and the coordinate axes, lies in the first quadrant. If its area is 2, then the value of b is: 
(A) -1
(B) 3
(C) -3
(D) 1

7. Let g(x)=1+x-[x] and 

 

Then for all x,f[g(x)] is equal to: 
(A) x 
(B) 1 
(C) f(x) 
(D) g(x) 

8. If f:[1,∞) is given by f(x)=x+1/x then f-1 (x) equals :

(A) (x+√(x2-4))/2 
(B) x/1+x2 
(C) (x-√(x2-4))/2 
(D) 1+√(x2-4) 

9. The domain of definition of f(x)= (log2 (x+3))/(x2+3x+2) is:

(A) R\{-1, -2} 
(B) (-2, ∞) 
(C) R/{-1,-2,-3} 
(D) (-3, ∞)\{-1, -2} 

10. The equation of the common tangent touching the circle (x-3)2+ y2= 9 and the parabola y2=4x above the x-axis is :

(A) √3 y=3x+1 
(B) √3 y=-(x+3) 
(C) √3 y=x+3 
(D) √3 y=-(3x+1) 

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