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IIT-JEE – 2014 Date : 22-02-13

Subject : Mathematics

M.M.: 90

Time : 1:30 Hours

Batch : XI(AG)

STRAIGHT OBJECTIVE TYPE (+3, –1) 1.

In ABC, If A – B = 120° and R = 8r, then the value of

1 cos C equals 1 cos C

(All symbols used have their usual meaning in a triangle) (A) 12 (B) 15 (C) 21 2.

3.

(D) 31

The coefficient of t8 in the expansion of (1 + 2t2 – t3)9, is (A) 1680 (B) 2140 (C) 2520

(D) 2730

Inside the unit circle S = {(x, y) | x2 + y2 = 1} there are three smaller circles of equal radius a , tangent to each other and to S. The value of a equals (A)

2

(B)

2 1

3 2 3

2 2 3

(C)

(D)

3 3 1

4.

A committee of 6 members is to be chosen from among 5 democrates and 3 republicians so that atleast two members of each party serve on the committee. Number of possible ways it can be done, is (A) 15 (B) 20 (C) 25 (D) 28

5.

The value of

1

ln 1 n 2

equals

n 2

(A) – ln 3

(B) 0

(C) – ln 2

(D) – ln 5

6.

Let the lines (y – 2) = m1(x – 5) and (y + 4) = m2(x – 3) intersect at right angles at P (where m1 and m2 are parameters). If locus of P is x2 + y2 + gx + fy + 7 = 0, then (f – g) equals (A) 1 (B) 2 (C) 8 (D) 10

7.

The slope of the line, which belongs to family of lines (1 +)x + ( – 1)y + 2(1–) = 0 and makes shortest intercept on

x2 4 y 4 is: (A) 3 8.

(B) –1

(C) 0

(D) 2

Circles with centres O, O' and P each tangent of the line L and also mutually tangent. If the radii of circle O and circle O' are equal and the radius of the circle P is 6, then the radius of the larger circle is: (A) 22 (B) 23 (C) 24 (D) 25

O

O' P

Paragraph for question nos. 9 to 11 Consider the family of circles x2 + y2 – 2x – 2y – 8 = 0 passing through two fixed points A and B. Also S = 0 is a circle of this family, the tangent to which at A and B intersect on the line x + 2y + 5 = 0. 9.

The distance between the points A and B, is (A) 4

10.

(B) 4 2

(C) 6

(D) 8

The area of an equilateral triangle inscribed in S, is (A)

27 3 4

FIITJEE

(B)

9 3 2

(C)

27 3 2

(D) 9 3

1

11.

If the circle x2 + y2 – 10x + 2y + c = 0 is orthogonal to S = 0, then the value of c equals (A) 8 (B) 9 (C) 10 (D) 12 Paragraph for question nos. 12 to 14

12.

Consider an isosceles triangle T with base 10 and height 12. Define a sequence C1, C2 , C3 ............... of circles such that C1 is the incircle of T and Ci + 1 is tangent to Ci and both legs of the isosceles triangle, for i 1. The radius of C1 , is (A)

13.

(B)

7 3

(C)

10 3

(D)

11 3

(C)

5 9

(D)

6 9

(C)

117 13

(D)

180 13

The ratio of the radius of Ci + 1 to the radius of Ci , is (A)

14.

5 3

2 9

(B)

4 9

The total area contained in all the circles, is (A)

164 13

(B)

169 13

MULTIPLE OPTION TYPE QUESTIONS (+4, 0) 15.

Let one of the vertices of the equilateral triangle circumscribing the circle

|z – 2

3 i| = 1 is

z1 = 1 + 3 3 i. If the other two vertices are represented by z2 and z3, then which of the following statement(s) is/are CORRECT? (A) The area of triangle is

3 3 . 4

(B) Re(z 2 z 3 ) I m (z 2 z 3 ) 8 . (C) The radius of circle circumscribing the triangle is 2. (D) The perimeter of the escribed circle drawn opposite to vertex z1 of triangle is

3 . 2

Paragraph for question nos. 16 to 18 : y2 = 4x. Let PQR

Consider a conic C be an equilateral triangle with side length k where P be any point on C, Q be the foot of perpendicular from P upon the directrix of C and R be the focus of C. A circle C2 is inscribed in another conic C1 : y2 = k(x + 1) which touches C1 at the points where C1 cuts the y-axis. C3 is an ellipse whose auxiliary circle is C2 and major axis coincides with the axis of symmetry of C1 and whose length of minor axis is 4. 16.

Identify the correct statement(s) for C1. (A) Minimum length of focal chord is 4. (B) Locus of point of intersection of perpendicular tangents is x + 2 = 0. (C) Distance between focus and tangent at vertex is 1. (D) Foot of the directrix is (–2, 0).

17.

Identify the correct statement(s) for C3. (A) Eccentricity is 1/2

(B) Focal length is 4.

(C) Length of latus-rectum is 2 2

(D) Director circle is x2 + y2 – 4x – 8 = 0

18.

The triangle formed by common tangents to curves C1 and C2 and latus rectum of C1, is (A) equilateral. (B) isosceles. (C) of area 4 square units.

FIITJEE

(D) of area 4 2 square units.

2

MATCH THE COLUMN (2 × 3) 19.

In a ABC, BC = 2, CA = 1 3 and C = 60°. Feet of the perpendicular from A, B and C on the opposite sides BC, CA and AB are D, E and F respectively and are concurrent at P. Now match the entries of column-I with respective entries of column-II. Column-I

Column-II

(A) Radius of the circle circumscribing the DEF, is

(P)

(B) Area of the DEF, is

(Q)

(C) Radius of the circle inscribed in the DEF, is

(R)

(S) 20. (A)

Column-I Number of ways in which 4 indistinguishable balls can be kept in 7 different boxes if each box can have atmost one ball, is

(B)

6 2 4 1 2 3 4 6 2 4 (P)

Column-II 30

1 2 3 5 , is The sum of the rational terms in the expansion of

(Q)

32

Number of ways in which 5 balls can be selected from a bag containing 5 identical and 5 different balls, is

(R) (S)

35 41

10

(C)

INTEGER TYPE (SINGLE DIGIT ANSWER) (+4, 0) 21.

In ABC, circumradius is 3 and inradius is 1.5 units. If the value of a cot2A + b2cot3B + c3cot4C is m n where m and n are prime numbers, then find the value of (m + n – 10).

22.

Let a, b, c, d be four distinct real numbers in A.P. If the smallest positive value of k satisfying 2(a – b) + k(b – c)2 + (c – a)3 = 2(a – d) + (b – d)2 + (c – d)3 is m then find

m.

23.

Let P(x) = 5/3 – 6x – 9x2 and Q(y) = – 4y2 + 4y + 13/2. If there exist unique pair of real numbers (x, y) such that P(x) Q(y) = 20, then find the value of (6x + 10y).

24.

India and South Africa play one day international series untill one team wins 4 matches. No match ends in a draw. If the number of ways that the series can be won is m then sum of digits of m is :

25.

Let S denote sum of the series

3 4 5 6 ........................ 23 2 4 ·3 2 6 ·3 27 ·5 Compute the value of S–1.

*****

FIITJEE

3

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Subject : Mathematics

M.M.: 90

Time : 1:30 Hours

Batch : XI(AG)

STRAIGHT OBJECTIVE TYPE (+3, –1) 1.

In ABC, If A – B = 120° and R = 8r, then the value of

1 cos C equals 1 cos C

(All symbols used have their usual meaning in a triangle) (A) 12 (B) 15 (C) 21 2.

3.

(D) 31

The coefficient of t8 in the expansion of (1 + 2t2 – t3)9, is (A) 1680 (B) 2140 (C) 2520

(D) 2730

Inside the unit circle S = {(x, y) | x2 + y2 = 1} there are three smaller circles of equal radius a , tangent to each other and to S. The value of a equals (A)

2

(B)

2 1

3 2 3

2 2 3

(C)

(D)

3 3 1

4.

A committee of 6 members is to be chosen from among 5 democrates and 3 republicians so that atleast two members of each party serve on the committee. Number of possible ways it can be done, is (A) 15 (B) 20 (C) 25 (D) 28

5.

The value of

1

ln 1 n 2

equals

n 2

(A) – ln 3

(B) 0

(C) – ln 2

(D) – ln 5

6.

Let the lines (y – 2) = m1(x – 5) and (y + 4) = m2(x – 3) intersect at right angles at P (where m1 and m2 are parameters). If locus of P is x2 + y2 + gx + fy + 7 = 0, then (f – g) equals (A) 1 (B) 2 (C) 8 (D) 10

7.

The slope of the line, which belongs to family of lines (1 +)x + ( – 1)y + 2(1–) = 0 and makes shortest intercept on

x2 4 y 4 is: (A) 3 8.

(B) –1

(C) 0

(D) 2

Circles with centres O, O' and P each tangent of the line L and also mutually tangent. If the radii of circle O and circle O' are equal and the radius of the circle P is 6, then the radius of the larger circle is: (A) 22 (B) 23 (C) 24 (D) 25

O

O' P

Paragraph for question nos. 9 to 11 Consider the family of circles x2 + y2 – 2x – 2y – 8 = 0 passing through two fixed points A and B. Also S = 0 is a circle of this family, the tangent to which at A and B intersect on the line x + 2y + 5 = 0. 9.

The distance between the points A and B, is (A) 4

10.

(B) 4 2

(C) 6

(D) 8

The area of an equilateral triangle inscribed in S, is (A)

27 3 4

FIITJEE

(B)

9 3 2

(C)

27 3 2

(D) 9 3

1

11.

If the circle x2 + y2 – 10x + 2y + c = 0 is orthogonal to S = 0, then the value of c equals (A) 8 (B) 9 (C) 10 (D) 12 Paragraph for question nos. 12 to 14

12.

Consider an isosceles triangle T with base 10 and height 12. Define a sequence C1, C2 , C3 ............... of circles such that C1 is the incircle of T and Ci + 1 is tangent to Ci and both legs of the isosceles triangle, for i 1. The radius of C1 , is (A)

13.

(B)

7 3

(C)

10 3

(D)

11 3

(C)

5 9

(D)

6 9

(C)

117 13

(D)

180 13

The ratio of the radius of Ci + 1 to the radius of Ci , is (A)

14.

5 3

2 9

(B)

4 9

The total area contained in all the circles, is (A)

164 13

(B)

169 13

MULTIPLE OPTION TYPE QUESTIONS (+4, 0) 15.

Let one of the vertices of the equilateral triangle circumscribing the circle

|z – 2

3 i| = 1 is

z1 = 1 + 3 3 i. If the other two vertices are represented by z2 and z3, then which of the following statement(s) is/are CORRECT? (A) The area of triangle is

3 3 . 4

(B) Re(z 2 z 3 ) I m (z 2 z 3 ) 8 . (C) The radius of circle circumscribing the triangle is 2. (D) The perimeter of the escribed circle drawn opposite to vertex z1 of triangle is

3 . 2

Paragraph for question nos. 16 to 18 : y2 = 4x. Let PQR

Consider a conic C be an equilateral triangle with side length k where P be any point on C, Q be the foot of perpendicular from P upon the directrix of C and R be the focus of C. A circle C2 is inscribed in another conic C1 : y2 = k(x + 1) which touches C1 at the points where C1 cuts the y-axis. C3 is an ellipse whose auxiliary circle is C2 and major axis coincides with the axis of symmetry of C1 and whose length of minor axis is 4. 16.

Identify the correct statement(s) for C1. (A) Minimum length of focal chord is 4. (B) Locus of point of intersection of perpendicular tangents is x + 2 = 0. (C) Distance between focus and tangent at vertex is 1. (D) Foot of the directrix is (–2, 0).

17.

Identify the correct statement(s) for C3. (A) Eccentricity is 1/2

(B) Focal length is 4.

(C) Length of latus-rectum is 2 2

(D) Director circle is x2 + y2 – 4x – 8 = 0

18.

The triangle formed by common tangents to curves C1 and C2 and latus rectum of C1, is (A) equilateral. (B) isosceles. (C) of area 4 square units.

FIITJEE

(D) of area 4 2 square units.

2

MATCH THE COLUMN (2 × 3) 19.

In a ABC, BC = 2, CA = 1 3 and C = 60°. Feet of the perpendicular from A, B and C on the opposite sides BC, CA and AB are D, E and F respectively and are concurrent at P. Now match the entries of column-I with respective entries of column-II. Column-I

Column-II

(A) Radius of the circle circumscribing the DEF, is

(P)

(B) Area of the DEF, is

(Q)

(C) Radius of the circle inscribed in the DEF, is

(R)

(S) 20. (A)

Column-I Number of ways in which 4 indistinguishable balls can be kept in 7 different boxes if each box can have atmost one ball, is

(B)

6 2 4 1 2 3 4 6 2 4 (P)

Column-II 30

1 2 3 5 , is The sum of the rational terms in the expansion of

(Q)

32

Number of ways in which 5 balls can be selected from a bag containing 5 identical and 5 different balls, is

(R) (S)

35 41

10

(C)

INTEGER TYPE (SINGLE DIGIT ANSWER) (+4, 0) 21.

In ABC, circumradius is 3 and inradius is 1.5 units. If the value of a cot2A + b2cot3B + c3cot4C is m n where m and n are prime numbers, then find the value of (m + n – 10).

22.

Let a, b, c, d be four distinct real numbers in A.P. If the smallest positive value of k satisfying 2(a – b) + k(b – c)2 + (c – a)3 = 2(a – d) + (b – d)2 + (c – d)3 is m then find

m.

23.

Let P(x) = 5/3 – 6x – 9x2 and Q(y) = – 4y2 + 4y + 13/2. If there exist unique pair of real numbers (x, y) such that P(x) Q(y) = 20, then find the value of (6x + 10y).

24.

India and South Africa play one day international series untill one team wins 4 matches. No match ends in a draw. If the number of ways that the series can be won is m then sum of digits of m is :

25.

Let S denote sum of the series

3 4 5 6 ........................ 23 2 4 ·3 2 6 ·3 27 ·5 Compute the value of S–1.

*****

FIITJEE

3

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