Application of Derivatives
* Column 1 contains information about zeros of ƒ(x), ƒ'(x) and ƒ"(x). * Column 2 contains information about the limiting behavior of ƒ(x), ƒ'(x) and ƒ"(x) at infinity. * Column 3 contains information about increasing/decreasing nature of ƒ(x) and ƒ'(x). 

Column1  Column2  Column3 
(I) ƒ(x) = 0 for some x ∈ (1,e^{2})  (i) lim_{x→∞} ƒ(x) = 0  (P) ƒ is increasing in (0,1) 
(II) ƒ'(x) = 0 for some x ∈ (1,e)  (ii) lim_{x→∞} ƒ(x) = –∞  (Q) ƒ is decreasing in (e,e^{2}) 
(III) ƒ'(x) = 0 for some x ∈ (0,1)  (iii) lim_{x→∞} ƒ'(x) = –∞  (R) ƒ' is increasing in (0,1) 
(IV) ƒ"(x) = 0 for some x ∈ (1,e)  (iv) lim_{x→∞} ƒ"(x) = 0  (S) ƒ' is decreasing in (e,e^{2}) 
Which of the following options is the only CORRECT combination ?
* Column 1 contains information about zeros of ƒ(x), ƒ'(x) and ƒ"(x). * Column 2 contains information about the limiting behavior of ƒ(x), ƒ'(x) and ƒ"(x) at infinity. * Column 3 contains information about increasing/decreasing nature of ƒ(x) and ƒ'(x). 

Column1  Column2  Column3 
(I) ƒ(x) = 0 for some x ∈ (1,e^{2})  (i) lim_{x→∞} ƒ(x) = 0  (P) ƒ is increasing in (0,1) 
(II) ƒ'(x) = 0 for some x ∈ (1,e)  (ii) lim_{x→∞} ƒ(x) = –∞  (Q) ƒ is decreasing in (e,e^{2}) 
(III) ƒ'(x) = 0 for some x ∈ (0,1)  (iii) lim_{x→∞} ƒ'(x) = –∞  (R) ƒ' is increasing in (0,1) 
(IV) ƒ"(x) = 0 for some x ∈ (1,e)  (iv) lim_{x→∞} ƒ"(x) = 0  (S) ƒ' is decreasing in (e,e^{2}) 
Which of the following options is the only CORRECT combination ?
* Column 1 contains information about zeros of ƒ(x), ƒ'(x) and ƒ"(x). * Column 2 contains information about the limiting behavior of ƒ(x), ƒ'(x) and ƒ"(x) at infinity. * Column 3 contains information about increasing/decreasing nature of ƒ(x) and ƒ'(x). 

Column1  Column2  Column3 
(I) ƒ(x) = 0 for some x ∈ (1,e^{2})  (i) lim_{x→∞} ƒ(x) = 0  (P) ƒ is increasing in (0,1) 
(II) ƒ'(x) = 0 for some x ∈ (1,e)  (ii) lim_{x→∞} ƒ(x) = –∞  (Q) ƒ is decreasing in (e,e^{2}) 
(III) ƒ'(x) = 0 for some x ∈ (0,1)  (iii) lim_{x→∞} ƒ'(x) = –∞  (R) ƒ' is increasing in (0,1) 
(IV) ƒ"(x) = 0 for some x ∈ (1,e)  (iv) lim_{x→∞} ƒ"(x) = 0  (S) ƒ' is decreasing in (e,e^{2}) 
Which of the following options is the only INCORRECT combination ?
A line L : y mx + 3 meets y − axis at E(0, 3) and the arc of the parabola y^{2} = 16x, 0 ≤ y ≤ 6 at the point F(x_{0}, y_{0}). Then tangent to the parabola at F(x_{0}, y_{0}) intersects the yaxis at G(0, y_{1}). The slope m of the line L is chosen such that the area of the triangle EFG has a local maximum.
Match List I with List II and select the correct answer using the code given below the lists:

List I 

List II 
P. 
m = 
1. 

Q. 
Maximum area of ∆EFG is 
2. 
4 
R. 
y_{0} = 
3. 
2 
S. 
y_{1} = 
4. 
1 
Match the statements given in Column I with the intervals/ union of intervals given in Column II
Column I  Column II  
(A)  The set  (p)  
(B)  The domain of the function  (q)  
(C)  If , then the set is  (r)  
(D)  If , then f(x) is increasing in  (s)  
(t) 
Match the conditions / expressions in Column I with statements in Column II and mark your answers in 4 × 4 matrix given in the ORS.
Column I  Column II  
(A)  If −1 < x < 1, then f(x) satisfies  (p)  0 < f(x) < 1 
(B)  If 1 < x < 2, then f(x) satisfies  (q)  f(x) < 0 
(C)  If 3 < x < 5, then f(x) satisfies  (r)  f(x) > 0 
(D)  If x > 5, then f(x) satisfies  (s)  f(x) < 1 