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Solutions: Mathematics paper 2 IITJEE 2008

1.  Let the function g : be given by g(u) =2 tan-1 (eu) Then, g is

 

(A) even and is strictly increasing in

(B) odd and is strictly decreasing in

(C) odd and is strictly increasing in

(D) neither even nor odd, but is strictly increasing in
 
Ans. 
 
2.  A particle P starts from the point z0 = 1 + 2i, where It moves first horizontally away from origin by 5 units and then vertically away from origin by 3 units to reach a point z1. From z1 the particle moves units in the direction of the vector and then it moves through an angle in anticlockwise direction on a circle with centre at origin, to reach a point z2. The point z2 is given by

(A) 6+7i         (B)  -7+6i         

 

(C) 7+6i         (D) –6+7i

 

Ans. 

 

3.  Consider a branch of the hyperbola With vertex at the point A. Let B be one of the end points of its latus rectum. If C is the focus of the hyperbola nearest to the point A, then the area of the triangle ABC is
 
 
Ans. 
 
4.  
 
Ans. 
 
5.  Let two non-collinear unit vectors  form an acute angle. A point P moves so that at any time t the position vector (where O is the origin) is given by
 When P is farthest from origin O, let M be the length of  and be the unit vector along . Then,
Ans. 
 
6.  Let g(x) = log f(x) where f(x) is a twice differentiable positive function on  such that f(x + 1) = x f(x). Then for N = 1, 2, 3,…….
 
Ans. 

g(x) = log f(x)

f(x+1) = xf(x)

f(x) = (x-1)!f(1)

g(x) = log f(x) = log (x -1)! + log f(1)

g(x) = log(x-1)+log(x-2)+------+logf(1)
 
7.  The area of the region between the curves  and  bounded by the lines x = 0 and  is
 
Ans. 
 
8.   An experiment has 10 equally likely outcomes. Let A and B be two non-empty events of the experiment. If A consists of 4 outcomes, the number of outcomes that B must have so that A and B are independent, is

 

(A)  2, 4 or 8       (B) 3, 6 or 9        (C) 4 or 8        (D)  5 or 10
 
Ans. 
 
9.  Consider three points and  Then
 

(A) P lies on the line segment RQ            (B) Q lies on the line segment PR

(C) R lies on the line segment QP            (D) P, Q, R are non-collinear

 

Ans. 

 

10.  Consider

L1: 2x+3y+p-3=0

L2: 2x+3y+p+3=0

Where p is a real number and C: x2+y2+6x-10y+30=0.

STATEMENT-1: If line L1 is a chord of circle C, then line L2 is not  always a diameter of circle C.

And

STATEMENT-2: If line L1 is a diameter of circle C, then line L2 is not a chord of circle C.

 

(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2  is a correct      explanation for STATEMENT-1

(B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2  is not a correct explanation for STATEMENT-1

(C)STATEMENT-1 is True, STATEMENT-2 is False

(D) STATEMENT-1 is False, STATEMENT-2 is False

 

Ans. 

L1: 2x+3y+p-3=0

L2: 2x+3y+p+3=0

C: x2+y2+6x-10y+30=0, => C(-3, 5),

If L1 is the chord of C then

for L2 to be diameter

2(-3)+3(5)+p+3=0 => p = -12

hence statement 1 is true

as p = -12 lies in the range [L2 is dia only if p = -12]

statement-2:

if L1 is dia then

2(-3)+3(5)+p-3=0 => p = -6

if L2 is the chord then

p = -6 does not imply that L2 cannot be the chord of circle

hence statement 2 is false

 

11.  Let a solution y = y(x) of the differential equation

satisfy

STATEMENT-1

:

And

STATEMENT-2: y(x) is given by

(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2  is a correct      explanation for STATEMENT-1

(B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2  is not a correct explanation for STATEMENT-1

(C)STATEMENT-1 is True, STATEMENT-2 is False

(D) STATEMENT-1 is False, STATEMENT-2 is False

 

Ans.    12.  Suppose four distinct positive number a1, a2, a3, a4 are in G.P. Let b1 = a1, b2 = b1+a2,

b3 = b2+a3 and b4 = b3+a4 .

STATEMENT-1: The number b1, b2, b3, b4 are neither in A.P. nor in G.P.

And

STATEMENT-2: The number b1, b2, b3, b4 are in H.P.

(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2  is a correct      explanation for STATEMENT-1

(B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2  is not a correct explanation for STATEMENT-1

(C)STATEMENT-1 is True, STATEMENT-2 is False

(D) STATEMENT-1 is False, STATEMENT-2 is False

 

Ans. 

a1, a2, a3, a4 in G.P,  b1 = a1, b2 = b1+a2, b3 = b2 + a3, b4 = b3 + a4

a1 = a, a2 = ar, a3 = ar2, a4 = ar3

13.  Let a, b, c, p, q be real numbers. Suppose are the roots of the equation x2+2px+q = 0 and are the roots of the equation , where
.

STATEMENT-1:

And

STATEMENT-2: 

(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2  is a correct      explanation for STATEMENT-1

(B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2  is not a correct explanation for STATEMENT-1

(C)STATEMENT-1 is True, STATEMENT-2 is False

(D) STATEMENT-1 is False, STATEMENT-2 is False

 

Ans.   let a, b, c, p, q

x2+2px+q = 0 => D = 4p2-4q

obviously statement 1 is correct as D of both are real

 

                    Paragraph for question Nos. 14 to 16

Consider the function  defined by

14.  Which of the following is true?
 
Ans. 
 
15.  Which of the following is true?
 

(A) f(x) is decreasing on (-1, 1) and has a local minimum at x = 1

(B) f(x) is increasing on (-1, 1) and has a local maximum at x = 1

(C) f(x) is increasing on (-1, 1) but has neither a local maximum nor a local minimum at x = 1

(D) f(x) is decreasing on (-1, 1) but has neither a local maximum nor a local minimum at x = 1
 
Ans. 
 
16.  Let

which of the following is true?

 

(A) g’(x) is positive onand negative on

(B) g’(x) is negative on  and positive on

(C) g’(x) change sign on both  and

(D) g’(x) does not change on
 
Ans. 
 

                 Paragraph for question Nos.17 to 19

Consider the lines

17.  The unit vector perpendicular to both L1 and  L2  is

Ans.   

18.  The shortest distance between L1 and L2  is   Ans.  shortest  distance between L1 & L2

    

Ans. shortest  distance between L1 & L2

 

19.  The distance of the point ( 1,1,1 ) from the plane passing through the point ( -1, -2,-1) and whose normal is perpendicular to both the lines  L1 and L2  is

 
Ans.

                                                                                                   

                                      SECTION IV  

This section contains 3 questions. Each question contains statements given in two columns, which have to be matched. Statements in Column I are labeled as A, B, C and D whereas statements in Column II are labeled as p, q, r and s. The answers to these questions have to be appropriately bubbled as illustrated in the following example.

If the correct matches are A-q, A-r, B-p, B-s, C-r, C-s and D-q, then the correctly bubbled matrix will look like the following:  

 

20. Consider the lines given by

L1: x+3y-5=0

L2: 3x-ky-1=0

L3: 5x+2y-12=0

Match the Statements / Expressions in Column I with the Statements / Expressions in Column II  and indicate your answer by darkening the appropriate bubbles in the 4 X 4 matrix given in the ORS.

  

Ans.  L1: x+3y-5=0

L2: 3x-ky-1=0

L3: 5x+2y-12=0

1(12k+2)-3(-36+5)-5(6+5k) = 0

=> k = 5 => A = S

L1, L2, L3 from a  if any two of them are not parallel hence  & lines are not

concurrent if k = 5

L1, L2, L3 do not form a  if any two of them become parallel or all the lines are concurrent 

 

21.  Consider all possible permutations of the letters the word ENDEANOEL.Match the Statements/Expressions in Column I with the Statements / Expressions in Column  II  and indicate your answer by darkening the appropriate bubbles in the 4 X 4 matrix given in the ORS.

 

Ans.  consider it as a packet

  

22.   Match the Statements / Expressions in Column I with the Statements / Expressions    in    Column II and indicate your answer by darkening the appropriate bubbles in the  4 X 4 matrix given in the ORS.

  

Ans.