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Solutions: Mathematics Paper 1 IITJEE 2008

1.
 
Ans.
 
2.  Consider the two curves

                        C1 : y2 = 4x

                        C2 : x2 + y2 - 6x + 1 = 0

Then,

(A)    C1 and C2 touch each other only at one point

(B)    C1 and C2 touch each other exactly at two points

(C)    C1 and C2 intersect (but do not touch) at exactly two points

(D)    C1 and C2 neither intersect not touch each other

 

Ans.  Replace y2 = 4x in C2 to get

x2 + 4x - 6x + 1 = 0

x2 -2x + 1 = 0  => (x - 1)2 =0

=> x = 1 As x = 1 is double root,

Parabola & circle touch each other.

C1 & C2 touch each other at two parts.

 
3.  The edges of a parallelepiped are of unit length and are parallel to non-coplanar unit vectors  such that

Then, the volume of the parallelepiped is
 
Ans. 
Volume of parallelepiped defined by as adjacent edges
 
4.  Let a and b non-zero real numbers. Then, the equation

                 

Represents

(A)                four straight lines, when c = 0 and a, b are of the same sign

(B)                two straight lines and a circle, when a = b, and c is of sign opposite to that of a

(C)                two straight lines and a hyperbola, when a and b are of the same sign and c is of sign opposite to that of a

(D)                a circle and an ellipse, when a and b are of the same sign and c is of sign opposite to that of a

 

Ans. 

(ax2 + by2 + c) (x2 - 5xy + 6y2) = 0

(ax2 + by2 + c) (x - 2y) (x - 3y) = 0

x - 2y = 0 & x = 3y are straight lines passing through origin.

ax2 + by2 + c = 0 represents Circle if a =b. As sign of C is opposite to that of a & b, it is a real Circle with positive radius.

ax2 + by2 - k =  0  (Assuming k > 0)

=> x2 + y2 =k/a      & a > 0

As k/a > 0,  it is real circle whose centre is (0, 0),

 

5.  The total number of local maxima and local minima of the functionis

 

(A)        0          (B)        1         

 

(C)        2          (D)        3

 

Ans.  Draw graph of f(x) as it is a easy catch.

From graph, there is 1 local max. & 1 local min.

Total local max./local min. = 2

 

6.  Let m and n are integers,and let p be the left hand derivative of  at x = 1. Ifthen

 

            (A)        n = 1, m = 1                  (B)        n = 1 , m = -1

 

            (C )       n = 2, m = 2                  (D)        n > 2 , m = n

 

Ans. 

m = -1

m = 1

-1

LHD of |x - 1| = -1

                        => p = -1

lim g(x) = lim (g1 + h) {RHL of g(x) at x = 1}

 

Apply LH Rute

           

As p = -1, n can not be more than 2 If n >  2, then lim g(x) = 0 which is wrong

=> n = 2

Also

 

7.  Let P (x1 , y1)     and Q( x2, y2) , y1 < 0, y2 < 0 , be the end points of the latus rectum of the ellipse x2 + 4y2 = 4. The equations of parabolas with latus rectum PQ are

 

Ans.
Replace P & Q in all choices & see parabolas given in (B) & (C) pass through P & Q
 
8.  A straight line through the vertex P of a triangle PQR intersects the side QR at the points S and circumcircle of the triangle PQR at the point T. If S is not the centre of circumcircle, then
 
Ans. 
 
 
9.  Let f(x) be a non-constant twice differentiable function defined on  such that f(x) = f(1-x) and  Then
 
Ans. 

f(x) = f(1- x)       => f’(x) = -f’(1-x)

put x =1/2

f’(1/2)  -f’(1/2)     => f’(1/2) = 0 => B

 

f(x) = f(1-x)

f’(x) = -f’(1-x)

f’(1/4) = -f(1-1/4) = -f’(3/4)

f’(1/4) = 0           => f’(3/4) = 0

Apply Rolle’s Them in {1/4, ½) & between [1/2, 1] on f’(x)

Hence f”(x) = 0 at two points       => A is correct

f(x) = f(1-x)
 
 
10. 
  
 
Ans.   
 
 
11.  Consider the system of equations

                                                x - 2y + 3z = - 1

                                                -x + y - 2z = k

                                                x - 3y + 4z = 1.

 

STATEMENT-1 : The system of equations has no solution for

And

STATEMENT-2 : The determinant 

           

(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 True

 

Ans. 

=> Statement -1 is correct.

 

Statement -2 is also correct as shown above.

=> Statement -2 is correct. 

It is obvious that statement -2 is correct explanation of statement -1. 

 

12.  Consider the system of equations

ax  by = 0, cx + dy = 0, where a, b, c, d {0, 1}.

STATEMENT-1 : The probability that the system of  equations has a unique solution is

and

STATEMENT-2 : The probability that the system of equations has a solution is 1.

 

(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 True

 

Ans.  x = 0, y = 0 will definitely satisfy ax + by = 0 & cx + dy = 0

=> system of equations is consistent.

=> Prob system has a solution = 1.

=> Statement -2 is correct.

Unique solution would exist for the following cases:

a,b,c,d can take values in 24 ways (Each can be either 0 or 1)

P(Unique Solution)

=> Statement -1 is correct

As prob (solution exists) Cannot be used to find P(Unique Solution), Statement -2 is not correct explanation of statement -1 => B

 

13.  Let f and g be real valued functions defined on interval (-1, 1) such that g” (x) is continuous, g (0)  0, g” (0)  0 , and f(x) = g(x) sin x.

STATEMENT-1 :

and

SATEMENT-2 : f’ (0) = g (0).

 

(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 True

 

Ans.  f(x) = g(x)sinx

f|(x) = g|(x) s9nx + g(x) cosx

f’(0) = 0 + g(0) => f| (0) = g(0)

                        => Statement -2 is correct

f||(x) = g||(x) sinx + g|(x) cosx + g|(x) cosx + g(x) (-sinx)

f ||(0) = 0 + g|(0) + g|(0) + 0 2g|(0)

As g|(0) = ), f||(0) = 2g|(0) = 0.

Statement-1

=> statement –1 is correct

Ans is A as statement –2 is used to prove statement-1
 
14.  Consider three planes

            P1 : x - y  + z = 1

            P2 : x + y - z = -1

            P3 : x - 3y + 3z = 2.

 

Let L1, L2, L3 be the lines of intersection of the planes P2 and P3, P3 and P1. and P1 and P2, respectively.

STATEMENT-1 : At least two of the lines L1, L2 and L3 are non-parallel.

and

STATEMENT-2 : The three planes do not have a common-point.

 

(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 True

 

Ans.  let n1, n2 & n3 be dRs of planes.

Combining (1), (2) & (3) L1, L2 & L3 are parallel.

=> P1, P2 & P3 do not have a common point.
 
15.  Let A, B, C be three sets of complex numbers as defined below

            A = {z : Im z 1}

            B = {z : |z -2 - i| = 3}

            C = {z : Re ((1 - i) z) = }.

 

The number of elements in the set is

 

(A)        0          (B)        1          (C)        2          (D)
 
Ans. 
 
16.  Let A, B, C be three sets of complex numbers as defined below

            A = {z : Im z 1}

            B = {z : |z -2 - i| = 3}

            C = {z : Re ((1 - i) z) = }.

 

Let z be any point in Then, |z + 1 - i|2 + | z - 5 - i|2 lies between

 

(A)        25 and 29          (B)        30 and 34          (C)        35 and 39     (D) 40 and 44
 
Ans.
 
|2 +1-i|2 + |2-5-i|2 = PQ2 + PR2 = QR2 = diameter2 = 36
 
17.  Let z be any point in and let be any point satisfying |w - 2 - i | < 3. Then, |z| - |w| + 3 lies between

 

(A)        -6 and 3    (B)   -3 and 6            (C)        -6 and 6            (D)        -3 and 9
 
Ans. 
 

Lows of w is interior of the circle

O < |z| < 3 (z is p see figure)

O < |w| < 6  -6 < |z| - |w| < 3

On combining => -3 < |z| - |w| + 3 < 6
 

Paragraph for Question Nos. 18 to 20

A circle C of radius 1 is inscribed in an equilateral triangle PQR. The points of contact of C with the sides PQ, QR, RP are D, E, F, respectively. The line PQ is given by the equation  and the point D is . Further, it is given that the origin and the centre of C are on the same side of the line PQ.

 
18.  The equation of circle C is
 
Ans. 
 
19. Points E and F are given by
 
Ans. 
 

As is equilateral,  Replace in circle to get yF = 0

Possible choices can be A or B

Ans. would be A as yE > yC

 
20.   Equations of the sides QR, RP are
 
Ans. 

 
 
equation of PR is of type y = k & slope of QR is
 
 

Paragraph for Question Nos. 21 to 23

Consider the functions defined implicitly by the equation y3 – 3y + x =0 on various intervals in the real line.

If , the equation implicitly defines a unique real valued differentiable function y = f(x).

If , the equation implicitly defines a unique real valued differentiable function y = g(x) satisfying g (0) =0.
 
 
21. 
 
Ans.  
 
 

22.  The area of the region bounded by the curve y=f(x), the axis, and the lines x = a and x = b, where  is

 

 
Ans. 
 
 
23. 
 
(A)       2g(-1)              (B)       0         
 
(C )      -2g(1)              (D)       2g(1)
 
 
Ans.

y3 - 3y + x = 0 Draw graph

{first draw y = 3x - x3 & then reflect in y = x line}