Aits Coprehensions

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    SECTION III

    [Linked Comprehension Type]

    Read the following passages and answer the questionsPASSAGE - 1

    If x 2y + 4 = 0 and2x + y 5 = 0 are sides AB, AC of an isosceles triangle ABC having area is 10 sq. units. Then,

    14. Orthocentre of the triangle is

    (A)6

    ,05

    (B)13

    0,5

    (C)6 13

    ,5 5

    (D) None of these

    14. C

    15. Perpendicular from vertex A to the side BC is

    (A)10

    (B)2 10

    (C)3 10 (D) None of these

    15. A

    16. Lengths of the side AB is

    (A) 5 (B) 2 5

    (C)3 5 (D) None of these

    16. B

    PASSAGE 2

    A quadratic expression f(x) whose leading co-efficient is positive may take both positive and

    negative value. Indeed if the corresponding equation has real roots c1and c2then the expressionwill be negative for all values of x between c1 and c2. From which we can conclude that the

    expression is negative for all x lying in the interval 1 2 1 1 2 2 1 2d ,d if c d d c i.e.d ,d both liebetween the roots or equivalently 1 2f(d ) 0,f (d ) 0. If f(x) contains a parameter k then in general, itis not very easy to find the value of k for which the expression f(x) is positive for values of n in(d1, d2).

    17. The expression x2mx + 1 is negative for values of x in (1, 2) if

    (A) m > 2 (B) 5

    m2

    (C) 5

    m2

    (D) m = 0

    17. C

    18. Both roots of the equation x2ax + 2 = 0 lie in (0, 3) if a lies in

    (A)

    112 2,

    3 (B)

    111,

    3

    (C)

    112 2,

    3 (D) none of these

    18. C

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    19. For the expression f(x) = x2kx + 1 to be positive for all value of x in (1, 2):(A) it is necessary that f(x) > 0 for all x(B) it is sufficient that f(x) > 0 for all x(C) it is necessary and sufficient that f(x) > 0 for all x(D) none of these

    19. B

    If x 2y + 4 = 0 and 2x + y 5 = 0 are sides AB, AC of an isosceles triangle ABC having

    area is 10 sq. units. Then,

    14. Orthocentre of the triangle is

    (A)6

    ,05

    (B)13

    0,5

    (C)6 13

    ,5 5

    (D) None of these

    14. C

    15. Perpendicular from vertex A to the side BC is

    (A) 10 (B) 2 10

    (C)3 10 (D) None of these15. A

    16. Lengths of the side AB is

    (A) 5 (B) 2 5

    (C)3 5 (D) None of these

    16. B

    PASSAGE 2

    A quadratic expression f(x) whose leading co-efficient is positive may take both positive andnegative value. Indeed if the corresponding equation has real roots c1and c2then the expression

    will be negative for all values of x between c1 and c2. From which we can conclude that theexpression is negative for all x lying in the interval 1 2 1 1 2 2 1 2d ,d if c d d c i.e.d ,d both lie

    between the roots or equivalently 1 2f(d ) 0,f (d ) 0. If f(x) contains a parameter k then in general, itis not very easy to find the value of k for which the expression f(x) is positive for values of n in(d1, d2).

    17. The expression x2mx + 1 is negative for values of x in (1, 2) if

    (A) m > 2 (B) 5

    m2

    (C) 5

    m2

    (D) m = 0

    17. C

    18. Both roots of the equation x2ax + 2 = 0 lie in (0, 3) if a lies in

    (A)

    112 2,

    3 (B)

    111,

    3

    (C)

    112 2,

    3 (D) none of these

    18. C

    19. For the expression f(x) = x2kx + 1 to be positive for all value of x in (1, 2):

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    (A) it is necessary that f(x) > 0 for all x(B) it is sufficient that f(x) > 0 for all x(C) it is necessary and sufficient that f(x) > 0 for all x(D) none of these

    19. B

    Read the following passages and answer the questionsPASSAGE - 1

    If 1 1x ,y be a fixed point on a straight line which makes an angle with positive direction of x-axis.Then equation of the line in parametric form can be written as

    1x x r cos

    1y y r sin

    Where the pt. (x, y) is at a distance r from 1 1x ,y , measured along the straight line.The parametric equation of a straight line is given as

    rx 2

    5

    3ry 3

    5 ,Then

    14. The Cartesian equation of the above given line is(A) x -3y + 11 = 0 (B) x + y + 9 = 0(C) 3x y + 7 = 0 (D) 3x y - 9 = 0

    14. C

    15. Area of triangle formed by the given straight line and the co-ordinate axes

    (A)27

    2sq. units (B) 9sq. units

    (C)121

    6sq. units (D) 3 sq. units

    15. D16. Which of the following points lie on the given line

    (A) (1, 6) (B) (1, - 6)(C) (0,0) (D) None of these

    16. D

    PASSAGE - 2

    One side of an equilateral triangle is 24 cm. The mid points of its sides are joined to form anothertriangle. Again another triangle is formed by joining the mid points of this triangle and so on.

    17. The sides of the triangles so formed are in(A) AP (B) GP(C) HP (D) None of these

    17. B

    18. The areas of triangles so formed are in(A) an AP with common difference 2 (B) a GP with common ratio 2(C) a GP with common ratio 1/2 (D) None of these

    18. D

    19. If the triangles are formed endlessly the sum of perimeters of all triangles will be(A) 144 cm (B) 180 cm(C) 160 cm (D) None of these

    19. A

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    Read the following passages and answer the questionsPASSAGE 1

    A function has a derivative at a point x0if the slopes of the secant lines through P(x0, f(x0))and a nearby point Q on the graph approach a limit as Q approaches P Whenever thesecants fails to take up a limiting position or becomes vertical as Q approaches P, thederivative does not exist. A function whose graph is otherwise smooth, will fail to have aderivative at a point where the graph has

    (1) a corner, where the one sided derivatives differ(2) a cusp , where the slope of PQ approaches from one side and -from other side(3) A vertical tangent, where the slope of PQ approaches or - from both sides(4) a discontinuity

    14. The total number of points of non differentiability of function f(x) = |x + 1| + |x 1| is /are(A) 1 (B) 2(C) 2 (D) 43

    14. B15. The function f(x) = |ln|x|| is non differentiable at x =

    (A) 1 , -1 (B) 1 , -1, 0(C) 1, 0 (D) -1, 0

    15. B

    16. The slope of tangent to the function f(x) = sin|x| + sinx at the point x = -2/5 is(A) 1 (B) 3(C) 0 (D) 1/ 3

    16. C

    PASSAGE - 2

    If we want to compare f(x) and g(x) consider a function Q(x) = f(x) g(x) or g(x) f(x) and checkwhether Q(x) is increasing or decreasing and Q(0) in the given domain of f(x) and g(x)e.g. sin x > x, x (0, )

    17. For x (, 0], the order relation between x and1

    tan x is(A) 1x tan x (B) 1x tan x

    (C) 1x tan x (D) 1x tan x 017. C

    18. If 1 22x tan x n 1 x , then x (A) (, 0] (B) [0, )(C) (1, 1) (D) (, )

    18. D19. For all x (0, 1), which option is correct

    (A) xe 1 x (B) n 1 x x

    (C) sin x > x (D) n x > x

    19. B

    If 1 1x ,y be a fixed point on a straight line which makes an angle with positive direction of x-axis.Then equation of the line in parametric form can be written as

    1x x r cos

    1y y r sin

    Where the pt. (x, y) is at a distance r from 1 1x ,y , measured along the straight line. The parametricequation of a straight line is given as

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    rx 2

    10

    3ry 1

    10

    14. The Cartesian equation of the above given line is(A) x + y + 2 = 0 (B) x + y + r = 0

    (C) 3x y + 7 = 0 (D) None of these14. C

    15. Area of triangle formed by the given straight line and the co-ordinate axes

    (A) 3 sq. units (B) 2 7 sq. units

    (C)4

    3sq. units (D)

    49

    6sq. units

    15. D

    16. Which of the following points lie on the given line(A) (1, 1) (B) (0, 0)(C) (8, 2) (D) None of these

    16. D

    PASSAGE 2

    Consider two circles S1and S2, S1having radius 3 and lies entirely in the first quadrant such that ittouches both the coordinate axes while S2has centre (8, 15).

    17. Equation of S2, such that S1touches S2internally is

    (A) 2 2x y 16x 30y 33 0 (B) 2 2x y 16x 30y 189 0

    (C) 2 2x y 16x 30y 33 0 (D) 2 2x y 16x 30y 189 0 17. A

    18. If S2touches S1externally then the point of contact is(A)

    54 75,

    13 13

    (B)6 15

    ,7 7

    (C)54 75

    ,13 13

    (D)6 15

    ,7 7

    18. C

    19. If radius of S2is 5 then point of intersection of direct common tangent is

    (A)39

    ,302

    (B)9

    , 152

    (C)39

    ,30

    2

    (D)9

    ,15

    2

    19. CRead the following passages and answer the questionsPASSAGE - 1

    If x 2y + 4 = 0 and2x + y 5 = 0 are sides AB, AC of an isosceles triangle ABC having area is 10 sq. units. Then,

    14. Orthocentre of the triangle is

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    (A)6

    ,05

    (B)13

    0,5

    (C)6 13

    ,5 5

    (D) None of these

    14. C

    15. Perpendicular from vertex A to the side BC is

    (A) 10 (B) 2 10

    (C) 3 10 (D) None of these15. A

    16. Lengths of the side AB is

    (A) 5 (B) 2 5

    (C) 3 5 (D) None of these16. B

    PASSAGE 2

    A quadratic expression f(x) whose leading co-efficient is positive may take both positive andnegative value. Indeed if the corresponding equation has real roots c1and c2then the expressionwill be negative for all values of x between c1 and c2. From which we can conclude that the

    expression is negative for all x lying in the interval 1 2 1 1 2 2 1 2d ,d if c d d c i.e.d ,d both liebetween the roots or equivalently 1 2f(d ) 0, f(d ) 0.If f(x) contains a parameter k then in general, itis not very easy to find the value of k for which the expression f(x) is positive for values of n in(d1, d2).

    17. The expression x2mx + 1 is negative for values of x in (1, 2) if

    (A) m > 2 (B) 5

    m2

    (C) 5

    m2

    (D) m = 0

    17. C

    18. Both roots of the equation x2ax + 2 = 0 lie in (0, 3) if a lies in

    (A)

    112 2,

    3 (B)

    111,

    3

    (C)

    112 2,

    3 (D) none of these

    18. C

    19. For the expression f(x) = x2kx + 1 to be positive for all value of x in (1, 2):(A) it is necessary that f(x) > 0 for all x(B) it is sufficient that f(x) > 0 for all x(C) it is necessary and sufficient that f(x) > 0 for all x(D) none of these

    19. B

    Mapping of function:A function is called one-one if each element of domain has a distinct image in co-domain orfor any two or more than two elements of domain, function doesnt have same value.Otherwise function will be many-one. Function is called onto if co-domain = Range otherwise

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    into. Function which is both one-one and onto, is called bijective. Inverse is defined only forbijective function.

    14. Which of the following function is one-one for xR.(A) f(x) = x2+ x (B) f(x) = x |x|

    (C)x

    f(x) sin2

    (D) f(x) [2x]

    14. B

    15. Let f: R Y. f(x) =2

    2

    x,

    x 2then set Y for which f(x) is onto

    (A) [0, 1) (B)1 1,

    3 2

    (C)1,1

    3

    (D)

    1,1

    2

    15. C

    16. Let f: X Y if f(x) =2x 12 is bijective then possible set of X and Y are

    (A) 1

    X 0, , Y ,2

    (B) X 0, , Y 0,

    (C) 1

    X ,0 Y ,2

    (D) X ,0 Y 0,

    16. A

    PASSAGE - 2

    A tangent to the curve y = f(x) at P(x, y) cuts the x-axis and y-axis at A and B, respectively such thatBP : AP = 3 : 1, if f(1) = 1

    17. The differential equation for the curve y = f(x) is

    (A)dx

    y 3x 0dy (B)dy

    x 3y 0dx

    (C)dy

    y 3x 0dx

    (D)dx

    x 3y 0dy

    17. B

    18. Equation of curve y = f(x) is

    (A) 2x y 1 (B) 2y x 1

    (C) 3x y 1 (D) 3xy 1 18. C

    19. The equation of normal to curve at the point (1, 1) is(A) x + y 2 = 0 (B) 2x + y 3 = 0(C) x + 3y 4 = 0 (D) x 3y + 2 = 0

    19. D

    Read the following passages and answer the questionsPASSAGE - 1

    If 1 1x ,y be a fixed point on a straight line which makes an angle with positive direction of x-axis.Then equation of the line in parametric form can be written as

    1x x r cos

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    1y y r sin

    Where the pt. (x, y) is at a distance r from 1 1x ,y , measured along the straight line.The parametric equation of a straight line is given as

    rx 2

    5

    3ry 3

    5 ,Then

    14. The Cartesian equation of the above given line is(A) x -3y + 11 = 0 (B) x + y + 9 = 0(C) 3x y + 7 = 0 (D) 3x y - 9 = 0

    14. C

    15. Area of triangle formed by the given straight line and the co-ordinate axes

    (A)27

    2sq. units (B) 9sq. units

    (C)121

    6sq. units (D) 3 sq. units

    15. D16. Which of the following points lie on the given line

    (A) (1, 6) (B) (1, - 6)(C) (0,0) (D) None of these

    16. D

    PASSAGE - 2

    One side of an equilateral triangle is 24 cm. The mid points of its sides are joined to form anothertriangle. Again another triangle is formed by joining the mid points of this triangle and so on.

    17. The sides of the triangles so formed are in(A) AP (B) GP(C) HP (D) None of these

    17. B

    18. The areas of triangles so formed are in(A) an AP with common difference 2 (B) a GP with common ratio 2(C) a GP with common ratio 1/2 (D) None of these

    18. D

    19. If the triangles are formed endlessly the sum of perimeters of all triangles will be(A) 144 cm (B) 180 cm(C) 160 cm (D) None of these

    19. AA function has a derivative at a point x0if the slopes of the secant lines through P(x0, f(x0)) and a

    nearby point Q on the graph approach a limit as Q approaches P Whenever the secants failsto take up a limiting position or becomes vertical as Q approaches P, the derivative does notexist. A function whose graph is otherwise smooth, will fail to have a derivative at a pointwhere the graph has(1) a corner, where the one sided derivatives differ(2) a cusp , where the slope of PQ approaches from one side and -from other side(3) A vertical tangent, where the slope of PQ approaches or - from both sides(4) a discontinuity

    14. The total number of points of non differentiability of function f(x) = |x + 1| + |x 1| is /are(A) 1 (B) 2(C) 2 (D) 43

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    15. D

    16. Which of the following points lie on the given line(A) (1, 1) (B) (0, 0)(C) (8, 2) (D) None of these

    16. D

    PASSAGE 2

    Consider two circles S1and S2, S1having radius 3 and lies entirely in the first quadrant such that ittouches both the coordinate axes while S2has centre (8, 15).

    17. Equation of S2, such that S1touches S2internally is

    (A) 2 2x y 16x 30y 33 0 (B) 2 2x y 16x 30y 189 0

    (C) 2 2x y 16x 30y 33 0 (D) 2 2x y 16x 30y 189 0 17. A

    18. If S2touches S1externally then the point of contact is

    (A)54 75

    ,13 13

    (B)6 15

    ,7 7

    (C)54 75

    ,13 13

    (D)6 15

    ,7 7

    18. C

    19. If radius of S2is 5 then point of intersection of direct common tangent is

    (A)39

    ,302

    (B)9

    , 152

    (C)39

    ,302

    (D)9

    ,152

    19. CRead the following passages and answer the questions

    PASSAGE - 1

    Consider the points A (1, 2)

    B 5 cos , 5 sin

    C 5 sin , 5 cos . Then14. Circumcentre of ABC is

    (A) (1, 0) (B) (0, 1)(C) (1, 1) (D) (0, 0)

    14. D

    15. Radius of the circumcircle is

    (A) 1 (B) 5(C) 5 (D) None of these

    15. C

    16. Locus of the orthocenter is

    (A) 2 2

    x 1 y 2 5 (B) 2 2

    x 1 y 1 10

    (C) 2 2

    x 1 y 2 10 (D) None of these16. B

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    PASSAGE - 2

    The roots of f(x) = 2x 2 b 3 x 9 lie in (6, 1) where b is a +ve integer .Consider the geometricprogression 2, a1, a2, . a20, b then

    17. The value of b is(A) 6 (B) 7(C) 8 (D) 9

    17. A

    18. Nature of the roots of the equation f(x) = 0(A) real and distinct (B) equal positive roots(C) equal negative roots (D) None of these

    18. C

    19. The value of a3a18is equal to(A) 18 (B) 16(C) 14 (D) 12

    19. DRead the following passages and answer the questions

    PASSAGE - 1

    Consider the points A (1, 2)

    B 5 cos , 5 sin

    C 3 sin , 5 cos . Then14. Circumcentre of ABC is

    (A) (1, 0) (B) (0, 1)(C) (1, 1) (D) (0, 0)

    14. D

    15. Radius of the circumcircle is(A) 1 (B) 5

    (C) 5 (D) None of these15. C

    16. Locus of the orthocenter is

    (A) 2 2

    x 1 y 2 5 (B) 2 2

    x 1 y 1 10

    (C) 2 2

    x 1 y 2 10 (D) None of these

    16. B

    PASSAGE 2

    A quadratic expression f(x) whose leading co-efficient is positive may take both positive andnegative value. Indeed if the corresponding equation has real roots c1and c2then the expressionwill be negative for all values of x between c1 and c2. From which we can conclude that the

    expression is negative for all x lying in the interval 1 2 1 1 2 2 1 2d ,d if c d d c i.e.d ,d both liebetween the roots or equivalently 1 2f(d ) 0,f (d ) 0. If f(x) contains a parameter k then in general, itis not very easy to find the value of k for which the expression f(x) is positive for values of n in(d1, d2).

    17. The expression x2mx + 1 is negative for values of x in (1, 2) if

    (A) m > 2 (B) 5

    m2

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    (C) 5

    m2

    (D) m = 0

    17. C

    18. Both roots of the equation x2ax + 2 = 0 lie in (0, 3) if a lies in

    (A)

    112 2,

    3 (B)

    111,

    3

    (C)

    112 2,3

    (D) none of these

    18. C

    19. For the expression f(x) = x2kx + 1 to be positive for all value of x in (1, 2):(A) it is necessary that f(x) > 0 for all x(B) it is sufficient that f(x) > 0 for all x(C) it is necessary and sufficient that f(x) > 0 for all x(D) none of these

    19. BRead the following passages and answer the questionsPASSAGE - 1

    If x 2y + 4 = 0 and2x + y 5 = 0 are sides AB, AC of an issociless triangle ABC having area is 10 sq. units. Then,

    14. Orthocentre of the triangle is

    (A)6

    ,05

    (B)13

    0,5

    (C)6 13

    ,5 5

    (D) None of these

    14. C

    15. Perpendicular from vertex A to the side BC is

    (A) 10 (B) 2 10 (C) 3 10 (D) None of these

    15. A

    16. Lengths of the side AB is

    (A) 5 (B) 2 5

    (C) 3 5 (D) None of these16. B

    PASSAGE - 2

    If 1 1x ,y be a fixed point on a straight line which makes an angle with positive direction of x-axis.Then equation of the line in parametric form can be written as

    1x x r cos

    1y y r sin

    Where the pt. (x, y) is at a distance r from 1 1x ,y , measured along the straight line. The parametricequation of a straight line is given as

    rx 2

    10

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    3ry 1

    10

    17. The Cartesian equation of the above given line is(A) x + y + 2 = 0 (B) x + y + r = 0(C) 3x y + 7 = 0 (D) None of these

    17. C

    18. Area of triangle formed by the given straight line and the co-ordinate axes

    (A) 3 sq. units (B) 2 7 sq. units

    (C)4

    3sq. units (D)

    49

    6sq. units

    18. D

    19. Which of the following points lie on the given line(A) (1, 1) (B) (0, 0)(C) (8, 2) (D) None of thes

    19. DRead the following passages and answer the questionsPASSAGE - 1

    Consider the points A (1, 2)

    B 5 cos , 5 sin

    C 5 sin , 5 cos . Then14. Circumcentre of ABC is

    (A) (1, 0) (B) (0, 1)(C) (1, 1) (D) (0, 0)

    14. D

    15. Radius of the circumcircle is(A) 1 (B) 5

    (C) 5 (D) None of these15. C

    16. Locus of the orthocenter is

    (A) 2 2

    x 1 y 2 5 (B) 2 2

    x 1 y 1 10

    (C) 2 2

    x 1 y 2 10 (D) None of these16. B

    PASSAGE 2

    We know that

    n(n 1)1 2 3 ... f(n);2

    2 2 2 2

    n(n 1)(2n 1)1 2 3 ... n g(n)

    6

    2

    3 3 3 3 n(n 1)1 2 3 ... n h(n)2

    and so on. We note that f(n), g(n) and h(n) make sense if n is a positive integer but the functionsf(n), g(n) and h(n) are defined for fractional and negative values also. For example,

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    1 3f( 1) g( 1)h( 1) 0; f

    2 8etc.

    Let r r r r f(n) 1 2 3 ... n

    17. f(n) f(n 1) must be equal to(A) nr (B) (n 1)r(C) nn (D) (n + 1)r

    17. A

    18. f(1) must be equal to(A) 0 (B) 1(C) (1)r (D) None of these

    18. A

    19. f(n) must be equal to

    (A) r

    1 f n 1 (B) r

    1 f n 1

    (C)

    r 1

    1 f n 1 (D) None of these

    19. CRead the following passages and answer the questionsPASSAGE - 1

    Suppose you are given a graph of function y = f(x).Observing the shape of the graph you can conclude following points about the nature of thefunction.(i) If graph is symmetrical about y-axis, then function is even(ii) If graph is symmetrical about origin, then function is odd.(iii) If graph rises when moving along positive x-axis the function is increasing, i.e. f(x) > 0 and

    f(x) < 0 if graph falls.(iv) If graph touches x-axis at x = , then it has as repeated root, i.e. f() = f() = 0.(v) If the graph is convex upward, then slope of tangent is a decreasing function, i.e. f(x) < 0

    and f(x) > 0 if convex upward.(vi) If the convex and concave parts of a graph meet at x = , then it is point of inflexion, i.e. f

    () = 0 if exist f() changes sign.Now analyse the following graphs of derivative of a function f(x) i.e. y = g(x), where g(x) = f(x) andanswer the following questions a x b . Given f(c) = 0

    a c b

    Y

    X

    y = g(x) = f(x)

    14. The graph of y = f(x) will intersect x-axis(A) never (B) once

    (C) twice (D) touches once14. B15. The equation f(x) = 0; a x b has

    (A) no real roots (B) two distinct real roots(C) two repeated roots (D) atleast three repeated roots

    15. D16. The graph of y = f(x), a x b has

    (A) no point of inflexion (B) one point of inflection(C) two points of inflexion (D) none of these

    16. B

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    PASSAGE 2

    Let f(x) is a cube polynomial which has local maximum at x = 1, if f(2) = 18, f(1) = 1 and f(x) haslocal minima at x = 0, then

    17. The cube polynomial f(x) is

    (A) 31

    x 45x 548 (B) 3

    x x 1

    (C) 3 2x x 9x 12 (D) 31

    19x 57x 344

    17. D18. f(x) is increasing for

    (A)

    1x ,

    3 (B) x 1,2 5

    (C) x R (D) x 1,18. D19. f(x) has local minimum at

    (A) x = 0 (B) x = 1

    (C) x = 2 (D) x 2

    19. BRead the following passages and answer the questionsPASSAGE - 1

    If x 2y + 4 = 0 and2x + y 5 = 0 are sides AB, AC of an issociless triangle ABC having area is 10 sq. units. Then,

    14. Orthocentre of the triangle is

    (A)6

    ,05

    (B)13

    0,5

    (C) 6 13,5 5

    (D) None of these

    14. C

    15. Perpendicular from vertex A to the side BC is

    (A) 10 (B) 2 10

    (C) 3 10 (D) None of these15. A

    16. Lengths of the side AB is

    (A) 5 (B) 2 5

    (C)3 5 (D) None of these16. B

    PASSAGE 2

    Consider the unequal positive real numbers a, b, c and f(x) = ax2+ 2bx + c. Then

    17. If 2b = a + c then roots of the equation f(x) = 0 are(A) real and equal (B) real and distinct(C) imagionary (D) None of these

    17. B

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    18. If b(a + c) = 2ac then roots of the equation f(x) = 0 are

    (A) real and equal (B) real and distinct(C) imagionary (D) None of these

    18. C

    19. If b is the GM(geometric mean) of a and c then roots of the equation f(x) = 0 are(A) real and equal (B) real and distinct

    (C) imagionary (D) None of these19. ARead the following passages and answer the questionsPASSAGE - 1

    Suppose you are given a graph of function y = f(x).Observing the shape of the graph you can conclude following points about the nature of thefunction.(i) If graph is symmetrical about y-axis, then function is even(ii) If graph is symmetrical about origin, then function is odd.(iii) If graph rises when moving along positive x-axis the function is increasing, i.e. f(x) > 0 and

    f(x) < 0 if graph falls.(iv) If graph touches x-axis at x = , then it has as repeated root, i.e. f() = f() = 0.(v) If the graph is convex upward, then slope of tangent is a decreasing function, i.e. f(x) < 0

    and f(x) > 0 if convex upward.(vi) If the convex and concave parts of a graph meet at x = , then it is point of inflexion, i.e. f

    () = 0 if exist f() changes sign.Now analyse the following graphs of derivative of a function f(x) i.e. y = g(x), where g(x) = f(x) andanswer the following questions a x b . Given f(c) = 0

    a c b

    Y

    X

    y = g(x) = f(x)

    14. The graph of y = f(x) will intersect x-axis(A) never (B) once(C) twice (D) touches once

    14. B15. The equation f(x) = 0; a x b has

    (A) no real roots (B) two distinct real roots(C) two repeated roots (D) atleast three repeated roots

    15. D16. The graph of y = f(x), a x b has

    (A) no point of inflexion (B) one point of inflection(C) two points of inflexion (D) none of these

    16. B

    PASSAGE 2

    A functional equation is an equation, which relates the values assumed by a function at two or morepoints, which are themselves related in a particular manner. For example, we define an odd functionby the relation f(x) = f(x) for all x. This definition can be paraphrased to say that it is a functionf(x), which satisfies the functional relation f(x) + f(y) = 0, whenever x + y = 0. Of course, this doesnot identify the function uniquely, sometimes with some additional information, a function, a functionsatisfying a given functional equation can be identified uniquely.

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    Suppose a functional equation has a relation between f(x) and1

    fx

    , then due to the reason that

    reciprocal of a reciprocal gives back the original number, we can substitute1

    xfor x. This will result

    into another equation and solving these two, we can find f(x) uniquely. Similarly, we can solve anequation, which contains f(x) and f(x). Such equations are of repetitive nature.

    17. If f(x) + f(y) = 2 2f x 1 y y 1 x then

    (A) f(4x2+ 3x) + 3f(x) = 0 (B) f(3x 4x3) + 3f(x) = 0(C) f(4x3+ 3x) 3f(x) = 0 (D) f(4x33x) + 3f(x) = 0

    17. D

    18. If f(x) + f(y)

    x y,

    1 xyfor x, y, > 0, xy 1 then

    1

    1f(x)dx is

    (A) 2f(1) (B) 2f(1) (C) f (1)f(1)

    (D) 0

    18. D

    19. If f(x) be a polynomial in x such that1 1

    f(x) f f(x).f ,x 0.

    x x

    If f(2) = 63 then f(4) will

    be equal to

    (A) Cant be determined (B) 64 1

    (C) 61 4 (D) None of these19. CRead the following passages and answer the questionsPASSAGE - 1

    Let P and Q be the two points on the ellipse x 2+ 4y2= 16 whose eccentric angles are

    4and

    3

    4

    respectively. The tangent at P and the normal at Q cut each other at R and the normal at Q cuts the

    ellipse again at M.

    14. The coordinates of M is

    (A)

    10 2 11 2,

    3 3 (B)

    14 2 23 2,

    17 17

    (C)

    5 2 3 2,

    6 6 (D) 2 2 , 2

    14. B15. The coordinates of Q is

    (A)

    10 2 11 2,

    3 3 (B)

    14 2 23 2,

    17 17

    (C)

    5 2 3 2,

    6 6 (D) 2 2 , 2

    15. D16. The coordinates of R is

    (A)

    10 2 11 2,

    3 3 (B)

    14 2 23 2,

    17 17

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    (C)

    5 2 3 2,

    6 6 (D) 2 2 , 2

    16. A

    PASSAGE 2

    Regular hexagon A1A2A3A4A5A6whose side length is 2k, Q isthe centre of hexagon; as shown in the figure. Q1as centre circleis drawn, Q2as centre circle is drawn Q3as centre circle is drawn.The three circles touch each other pairwise and each touches twosides of hexagon.

    A1 A2

    A3

    A4A5

    A6 Q2

    Q3

    Q1

    B2

    B1

    B3

    B4

    B5

    B6

    2k

    17. Radius of each circle is

    (A)3k

    2 (B)

    3k

    4

    (C)k

    4

    (D)k

    2

    17. A18. Area of the quadrilateral A1B1Q1B2is

    (A) 23 3

    k4

    (B)2k

    2

    (C) 23

    k4

    (D) 23 3

    k16

    18. C

    19. Radius of the circumcircle of Q1 Q2 Q3is

    (A)k

    2 (B)

    k

    3

    (C)2k

    3 (D) k

    19. DRead the following passages and answer the questionsPASSAGE 1

    A function is called one-one if each element of domain has a distinct image in co-domain orfor any two or more than two elements of domain, function doesnt have same value.Otherwise function will be many-one. Function is called onto if co-domain = Range otherwiseinto. Function which is both one-one and onto, is called bijective. Inverse is defined only forbijective function.

    14. Let f: R Y. f(x) =

    2

    2

    x,

    x 2then set Y for which f(x) is onto

    (A) [0, 1) (B)

    1 1,

    3 2

    (C)

    1,1

    3 (D)

    1,1

    2

    14. C

    15. Let f: X Y if f(x) = 2x 12 is bijective then possible set of X and Y are

    (A)

    1X 0, , Y ,

    2 (B) X 0, , Y 0,

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    (C)

    1X ,0 Y ,

    2 (D) X ,0 Y 0,

    15 A

    16. If f: [0, ) [0, ) and f(x) =

    x,

    1 xthen f(x) is

    (A) one-one & onto (B) one-one & into(C) many one & onto (D) many one & into

    16. B

    PASSAGE 2

    Let f(x) is a cube polynomial which has local maximum at x = 1, if f(2) = 18, f(1) = 1 and f(x) haslocal minima at x = 0, then

    17. The cube polynomial f(x) is

    (A) 31

    x 45x 548

    (B) 3x x 1

    (C) 3 2x x 9x 12 (D) 31

    19x 57x 344

    17. D18. f(x) is increasing for

    (A)

    1x ,

    3 (B) x 1,2 5

    (C) x R (D) x 1, 18. D19. f(x) has local minimum at

    (A) x = 0 (B) x = 1

    (C) x = 2 (D) x 2 19. BRead the following passages and answer the questionsPASSAGE - 1

    A function f(x) is said to be increasing if 1 2 1 2x x f(x ) f(x ) { 1 2x and x are in domain of f(x)} and

    f(x) is said to be decreasing if 1 2 1 2x x f(x ) f(x ) . Concept of increasing and decreasingbehavior of a function is used to prove certain inequalities and solving the problems e.q.

    x sin x x 0, can be proved by taking

    f(x) = x sin xSo that f(x) = 1 cos x 0f(x) is increasingx 0f(x) f(0) f(x) 0 x 0

    14. For x (, 0], the order relation between x and1

    tan x is(A) 1x tan x (B) 1x tan x

    (C) 1x tan x (D) 1x tan x 014. C

    15. If 1 22x tan x n 1 x , then x (A) (, 0] (B) [0, )(C) (1, 1) (D) (, )

    15. D

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    16. n (n x) = a n x, will have two solutions if a belongs to

    (A)

    1,e

    (B)

    10,

    e

    (C)

    1,1

    e (D) (1, )

    16. B

    PASSAGE - 2

    Let f(x) is a cube polynomial which has local maximum at x = 1, if f(2) = 18, f(1) = 1 and f(x) haslocal minima at x = 0, then

    17. The cube polynomial f(x) is

    (A) 31

    x 45x 548

    (B) 3x x 1

    (C) 3 2x x 9x 12 (D) 31

    19x 57x 344

    17. D

    18. f(x) is increasing for

    (A)

    1x ,

    3 (B) x 1,2 5

    (C) x R (D) x 1, 18. D

    19. f(x) has local minimum at(A) x = 0 (B) x = 1

    (C) x = 2 (D) x 219. BRead the following passages and answer the questions

    PASSAGE 1

    Highest power of a prime p in the expression n!

    2 3 s

    n n n n.... ,

    p p p pwhere [x] denotes

    the greatest integer x and s s 1p n p e.g. the highest power of 5 in 50!

    2

    50 5010 2 12

    5 5

    The highest power of 6 in 50 ! = common of highest powers of 2 and 3 in 50!. Highest power of 2 in50!

    2 3 4 5

    50 50 50 50 50

    2 2 2 2 2

    = 25 + 12 + 6 + 3 + 1 = 47Highest power of 3 in 50!

    =

    2 350 50 50

    16 5 1 223 3 3

    Highest power of 6 in 50! = 22

    14. The number of zeros at the end of 200! is(A) 20 (B) 22(C) 49 (D) 26

    14. C

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    15. The exponent(power) of 7 in 100 50C is

    (A) 8 (B) 16(C) 0 (D) 2

    15. C

    16. The largest integer n for which 45! is divisible by 3nis(A) 16 (B) 20(C) 21 (D) 28

    16. C

    PASSAGE 2

    Consider (i) n

    2 2 2n0 1 2 2n1 2x 2x a a x a x ... a x

    (ii) n2 2 2n

    0 1 2 2n1 x x b b x b x .... b x

    Now answer the following

    17. The value of

    0 2n 1 2n 1 2n 0

    a a a a .... a a will be if n = 11

    (A) 0 (B) 1(C) 1 (D) None of these

    17. A

    18. The value of 0 2n 1 2n 1 2n 0a a a a ....a a will if n = 10(A) 0 (B) 1

    (C) 10 10 52 C (D)10 10

    22 C

    18. C

    19. The value of 0 1 1 2 2 3 3 4 2n 1 nb b b b b b b b ...b b will be if n = 11

    (A) 1 (B) 1(C) 2 (D) 0

    19. D

    Read the following passages and answer the questionsPASSAGE - 1

    The roots of f(x) = 2x 2 b 3 x 9 lie in (6, 1) where b is a +ve integer .Consider the geometricprogression 2, a1, a2, . a20, b then

    14. The value of b is(A) 6 (B) 7(C) 8 (D) 9

    14. A

    15. Nature of the roots of the equation f(x) = 0(A) real and distinct (B) equal positive roots(C) equal negative roots (D) None of these

    15. C16. The value of a3a18is equal to

    (A) 18 (B) 16(C) 14 (D) 12

    16. D

    PASSAGE 2

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    The geometrical meaning of 1 2| z z | is the distance between points 1 2z & z in the argand plane.One of the many applications of this is in solving least value problems. The fact that sum of twosides of a triangle can never be less than the third side is also widely used.

    17. The least value of | 5z 13 | | 3z 11| equal to

    (A)8

    3

    (B)16

    5

    (C)12

    5 (D)

    7

    3

    17. B18. The least value of | z 5 | + | z + 11 | is

    (A) 6 (B) 16(C) 8 (D) 3

    18. B19. The least value of | z 1 i | | z 3 5i | equal to

    (A) 13 (B) 11

    (C) 2 13 (D) 2 11 19. C

    Read the following passages and answer the questionsPASSAGE - 1

    There are ten points in a plane. Of these ten points four points are collinear and except these noother three points are collinear

    14. The number of straight lines formed by these points are(A) 116 (B) 115(C) 39 (D) 40

    14. D

    15. The number of triangle formed by these points are(A) 115 (B) 116

    (C) 40 (D) 18515. B

    16. The number of quadrilateral formed by these points are(A) 116 (B) 205(C) 185 (D) 182

    16. C

    PASSAGE 2

    The geometrical meaning of 1 2| z z | is the distance between points 1 2z & z in the argand plane.One of the many applications of this is in solving least value problems. The fact that sum of two

    sides of a triangle can never be less than the third side is also widely used.

    17. The least value of |5z 13 | |3z 11| equal to

    (A)8

    3 (B)

    16

    5

    (C)12

    5 (D)

    7

    3

    17. B

    18. The least value of | z 5 | + | z + 11 | is

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    (A) 6 (B) 16(C) 8 (D) 3

    18. B

    19. The least value of | z 1 i | | z 3 5i | equal to

    (A) 13 (B) 11

    (C) 2 13 (D) 2 11

    19. C

    Read the following passages and answer the questionsPASSAGE - 1An urn contains 4 white and 9 black balls. r balls drawn with replacement. Let P(r) be the probabilitythat no two white balls appear in succession. Answer the following questions

    14. The value of P(4)must be

    (A)

    2

    81

    169 (B)

    2

    81 273

    169

    (C)4

    41

    13

    (D) None of these

    14. B

    15. The recursion relation for P(r) must be

    (A)9 4 9

    P(r) P(r 2) P(r 1)13 13 13

    (B)2

    9 4P(r) P(r 2) P(r 1)

    13 13

    (C)2

    4 9P(r) P(r 2) P(r 1)

    13 13

    (D) None of these

    15. A

    16. P(r) must be equal to

    (A)

    r r

    r

    16 12 ( 3)

    12 13

    (B)

    r r

    r

    16 12 ( 3)

    9 13

    (C)r r

    r

    16 12 ( 3)

    15 13

    (D) None of these

    16. C

    PASSAGE 2

    Let z be a complex number of magnitude unity and z be a complex number given by 2z z z. Answer the following questions17. The arg z = then | z| must be equal to

    (A) 2 sin2

    (B) 2 cos

    2

    (C) 2 sin2

    (D) 2 cos2

    17. A

    18. If 4 K< < (4K + 2) (K is an integer) then arg zmust be equal to

    (A)3

    2

    (B)

    3

    2

    (C)3

    2

    (D) None of these

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    18. C

    19. If (4K + 2) < < (4K + 4) (K is an integer) then arg z

    (A) 32

    (B)

    3 3

    2 2

    (C)3

    32

    (D) None of these

    19. BRead the following passages and answer the questionsPASSAGE - 1

    Highest power of a prime p in the expression n!

    2 3 s

    n n n n.... ,

    p p p pwhere [x] denotes

    the greatest integer x and s s 1p n p

    14. The number of zeros at the end of 200! is(A) 20 (B) 22(C) 49 (D) 26

    14. C

    15. The exponent(power) of 7 in 100 50C is

    (A) 8 (B) 16(C) 0 (D) 2

    15. D

    16. The largest integer n for which 45! is divisible by 3nis(A) 16 (B) 20(C) 21 (D) 28

    16. A

    PASSAGE 2

    The geometrical meaning of 1 2| z z | is the distance between points 1 2z & z in the argand plane.One of the many applications of this is in solving least value problems. The fact that sum of twosides of a triangle can never be less than the third side is also widely used.

    17. The least value of |5z 13 | |3z 11| equal to

    (A)8

    3 (B)

    16

    5

    (C)12

    5 (D)

    7

    3

    17. B

    18. The least value of | z 5 | + | z + 11 | is(A) 6 (B) 16(C) 8 (D) 3

    18. B

    19. The least value of | z 1 i | | z 3 5i | equal to

    (A) 13 (B) 11

    (C) 2 13 (D) 2 11

    19. C

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    Read the following passages and answer the questionsPASSAGE - 1

    The roots of f(x) = 2x 2 b 3 x 9 lie in (6, 1) where b is a +ve integer .Consider the geometricprogression 2, a1, a2, . a20, b then

    14. The value of b is(A) 6 (B) 7(C) 8 (D) 9

    14. A15. Nature of the roots of the equation f(x) = 0

    (A) real and distinct (B) equal positive roots(C) equal negative roots (D) None of these

    15. C16. The value of a3a18is equal to

    (A) 18 (B) 16(C) 14 (D) 12

    16. D

    PASSAGE 2

    The geometrical meaning of 1 2| z z | is the distance between points 1 2z & z in the argand plane.One of the many applications of this is in solving least value problems. The fact that sum of twosides of a triangle can never be less than the third side is also widely used.

    17. The least value of |5z 13 | |3z 11| equal to

    (A)8

    3 (B)

    16

    5

    (C)12

    5 (D)

    7

    3

    17. B

    18. The least value of | z 5 | + | z + 11 | is(A) 6 (B) 16(C) 8 (D) 3

    18. B19. The least value of | z 1 i | | z 3 5i | equal to

    (A) 13 (B) 11

    (C) 2 13 (D) 2 11 19. CPASSAGE 1

    A function is called one-one if each element of domain has a distinct image in co-domain orfor any two or more than two elements of domain, function doesnt have same value.Otherwise function will be many-one. Function is called onto if co-domain = Range otherwise

    into. Function which is both one-one and onto, is called bijective. Inverse is defined only forbijective function.

    14. Let f: R Y. f(x) =2

    2

    x,

    x 2then set Y for which f(x) is onto

    (A) [0, 1) (B)1 1

    ,3 2

    (C)1

    ,13

    (D)1

    ,12

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    14. C

    15. Let f: X Y if f(x) =2x 12 is bijective then possible set of X and Y are

    (A) 1

    X 0, , Y ,2

    (B) X 0, , Y 0,

    (C) 1

    X ,0 Y ,2

    (D) X ,0 Y 0,

    15 A

    16. If f: [0, ) [0, ) and f(x) =x

    ,1 x

    then f(x) is

    (A) one-one & onto (B) one-one & into(C) many one & onto (D) many one & into

    16. B

    PASSAGE - 2

    Let f(x) is a cube polynomial which has local maximum at x = 1, if f(2) = 18, f(1) = 1 and f(x) haslocal minima at x = 0, then

    17. The cube polynomial f(x) is(A) 3

    1x 45x 54

    8 (B) 3x x 1

    (C) 3 2x x 9x 12 (D) 31

    19x 57x 344

    17. D

    18. f(x) is increasing for

    (A)1

    x ,3

    (B) x 1,2 5

    (C) x R (D) x 1,

    18. D

    19. f(x) has local minimum at(A) x = 0 (B) x = 1

    (C) x = 2 (D) x 2 19. B

    Read the following passages and answer the questionsPASSAGE - 1

    There are ten points in a plane. Of these ten points four points are collinear and except these noother three points are collinear

    14. The number of straight lines formed by these points are(A) 116 (B) 115(C) 39 (D) 40

    14. D

    15. The number of triangle formed by these points are(A) 115 (B) 116(C) 40 (D) 185

    15. B

    16. The number of quadrilateral formed by these points are

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    (A) 116 (B) 205(C) 185 (D) 182

    16. C

    PASSAGE 2

    The geometrical meaning of 1 2| z z | is the distance between points 1 2z & z in the argand plane.One of the many applications of this is in solving least value problems. The fact that sum of two

    sides of a triangle can never be less than the third side is also widely used.

    17. The least value of | 5z 13 | | 3z 11| equal to

    (A)8

    3 (B)

    16

    5

    (C)12

    5 (D)

    7

    3

    17. B

    18. The least value of | z 5 | + | z + 11 | is(A) 6 (B) 16

    (C) 8 (D) 318. B

    19. The least value of | z 1 i | | z 3 5i | equal to

    (A) 13 (B) 11

    (C) 2 13 (D) 2 11

    19. CRead the following passages and answer the questionsPASSAGE 1

    Let f(x) is a cube polynomial which has local maximum at x = 1, if f(2) = 18, f(1) = 1 and f(x) haslocal minima at x = 0, then

    14. The cube polynomial f(x) is

    (A) 31

    x 45x 548

    (B) 3x x 1

    (C) 3 2x x 9x 12 (D) 31

    19x 57x 344

    14. D15. f(x) is increasing for

    (A)

    1x ,

    3 (B) x 1,2 5

    (C) x R (D) x 1,

    15. B16. f(x) has local minimum at

    (A) x = 0 (B) x = 1

    (C) x = 2 (D) x 216. B

    PASSAGE - 2

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    If we want to compare f(x) and g(x) consider a function Q(x) = f(x) g(x) or g(x) f(x) and checkwhether Q(x) is increasing or decreasing and Q(0) in the given domain of f(x) and g(x)e.g. sin x > x, x (0, )

    17. For x (, 0], the order relation between x and 1tan x is

    (A) 1x tan x (B) 1x tan x

    (C) 1x tan x (D) 1x tan x 0

    17. C18. If 1 22x tan x n 1 x , then x

    (A) (, 0] (B) [0, )(C) (1, 1) (D) (, )

    18. D19. For all x (0, 1), which option is correct

    (A) xe 1 x (B) n 1 x x

    (C) sin x > x (D) n x > x

    19. BRead the following passages and answer the questionsPASSAGE 1

    If roots of 22z 2z 0 R and origin form an equilateral triangle ABC then14. The value of is

    (A)1

    3 (B)

    2

    3

    (C) 1 (D) None of these14. B

    15. Centroid of ABC is

    (A)1

    3 (B)

    i

    3

    (C)1

    3 (D) i

    3

    15. C

    16. Area of ABC is

    (A)1

    3 (B)

    1

    2 3

    (C)1

    3 3 (D)

    1

    4 3

    16. D

    PASSAGE 2

    Sixteen players 1 2 16S ,S ...S play in a tournament they are divided into groups at random. Each

    group consists of two players.17. The number of ways in which the grouping can be done is

    (A)

    8

    16!

    2! (B)

    8

    16!

    2! 8!

    (C)

    16

    16!

    2! 8! (D)

    8

    16!

    4! 8!

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    17. B

    18. The number of ways in which grouping can be done so that 1S and 2S are grouped together

    is

    (A)

    7

    7!

    2! 7! (B)

    8

    16!

    2!

    7

    14!

    2!

    (C) 714!

    2! 7! (D) 716!

    2! 7!

    18. C

    19. The number of ways in which grouping can be done so that 1S and 2S are not together is

    (A)8

    224 14!

    2 8!

    (B)

    8

    102 14!

    2 7!

    (C)8

    15 14!

    (2) 8!

    (D)

    7

    224 14!

    (2) 8!

    19. A

    Read the following passages and answer the questions

    PASSAGE 1

    In the 2007 World Cup in Westindies, the tournament will be arranged as per the following rules:In the beginning 16 teams are taken and divided into 2 groups of 8 teams each. Teams of eachgroup will play a match against each other in the same group. From each group 4 top teams willqualify for the next round. In the next round the team at first position played with team at secondposition and team at third position played with fourth one in each group and the losing team goesout of the tournament. Then four winning teams play for semi-final round and finally there is onefinal. The rules of the tournament are such that every match can result only in a win or a loss andnot in a tie. In case, number of wins by two teams are equal, qualifier will be decided by run rate andassume no two teams have the same run rate.14. The total number of matches played in the tournament is

    (A) 51 (B) 64

    (C) 63 (D) 5214. C15. The maximum number of matches that a team going out of the tournament in the first round

    can win is(A) 1 (B) 2(C) 5 (D) 4

    15. C16. The number of ways of conducting all the matches except the final in 6 different places so

    that three are atleast 2 matches each in any three places and atleast 3 matches each inother places is

    (A) 626

    C (B) 6 523 5

    C C

    (C) 6 453 5

    C C (D) None of these

    16. B

    PASSAGE 2

    The geometrical meaning of 1 2| z z | is the distance between points 1 2z & z in the argand plane.One of the many applications of this is in solving least value problems. The fact that sum of twosides of a triangle can never be less than the third side is also widely used.17. The least value of | 3z 13 | | 3z 11 | equal to

    (A) 0 (B) 2

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    (C)12

    5 (D)

    7

    3

    17. B18. The least value of | z 5 | + | z + 11 | is

    (A) 6 (B) 16(C) 8 (D) 3

    18. B19. The least value of | z 1 i | | z 3 5i | equal to

    (A) 13 (B) 11

    (C) 2 13 (D) 2 11 19. CPASSAGE 1

    Let the equation of a family of circle is 2 2x y 2x 2ay 8 0, where a is a variable.

    14. The equation represents a family of circles passing through two fixed points. The coordinateof one of these point is(A) (2, 0) (B) (2, 0)(C) (5, 0) (D) (4, 0)

    14. A

    15. Let P and Q be two fixed points. Equation of a circle C of this family, tangents to which atthese fixed points intersects on the lime x + 2y + 5 = 0 is

    (A) 2 2x y 2x 8y 8 0 (B) 2 2x y 2x 6y 8 0

    (C) 2 2x y 2x 8y 8 0 (D) 2 2x y 2x 6y 8 0 15. D

    16. If the chord PQ subtends an angle at the centre of the circle C, then =

    (A)6

    (B)4

    (C)

    3 (D)

    2 16. D

    PASSAGE 2

    Consider two circles S1and S2, S1having radius 3 and lies entirely in the first quadrant such that ittouches both the coordinate axes while S2has centre (8, 15).

    17. Equation of S2, such that S1touches S2internally is

    (A) 2 2x y 16x 30y 33 0 (B) 2 2x y 16x 30y 189 0

    (C) 2 2x y 16x 30y 33 0 (D) 2 2x y 16x 30y 189 0 17. C

    18. If S2touches S1externally then the point of contact is

    (A)54 75

    ,13 13

    (B)6 15

    ,7 7

    (C)54 75

    ,13 13

    (D)6 15

    ,7 7

    18. C

    19. If radius of S2is 5 then point of intersection of direct common tangent is

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    (A)39

    ,302

    (B)9

    , 152

    (C)39

    ,302

    (D)9

    ,152

    19. B

    Read the following passages and answer the questions

    PASSAGE 1

    The geometrical meaning of 1 2| z z | is the distance between points 1 2z & z in the argand plane.One of the many applications of this is in solving least value problems. The fact that sum of twosides of a triangle can never be less than the third side is also widely used.14. The least value of | 3z 13 | | 3z 11 | equal to

    (A) 0 (B) 2

    (C)12

    5 (D)

    7

    3

    14. B

    15. The least value of | z 5 | + | z + 11 | is

    (A) 6 (B) 16(C) 8 (D) 3

    15. B

    16. The least value of | z 1 i | | z 3 5i | equal to

    (A) 13 (B) 11

    (C) 2 13 (D) 2 11 16. C

    PASSAGE 2

    Consider two circles S1and S2, S1having radius 3 and lies entirely in the first quadrant such that it

    touches both the coordinate axes while S2has centre (8, 15).

    17. Equation of S2, such that S1touches S2internally is

    (A) 2 2x y 16x 30y 33 0 (B) 2 2x y 16x 30y 189 0

    (C) 2 2x y 16x 30y 33 0 (D) 2 2x y 16x 30y 189 0 17. C

    18. If S2touches S1externally then the point of contact is

    (A)54 75

    ,13 13

    (B)6 15

    ,7 7

    (C)54 75

    ,

    13 13

    (D)6 15

    ,

    7 7

    18. C

    19. If radius of S2is 5 then point of intersection of direct common tangent is

    (A)39

    ,302

    (B)9

    , 152

    (C)39

    ,302

    (D)9

    ,152

    19. BRead the following passages and answer the questions

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    PASSAGE 1

    Let the equation of a family of circle is 2 2x y 2x 2ay 8 0, where a is a variable.

    14. The equation represents a family of circles passing through two fixed points. The coordinateof one of these point is(A) (2, 0) (B) (2, 0)(C) (5, 0) (D) (4, 0)

    14. A15. Let P and Q be two fixed points. Equation of a circle C of this family, tangents to which at

    these fixed points intersects on the lime x + 2y + 5 = 0 is

    (A) 2 2x y 2x 8y 8 0 (B) 2 2x y 2x 6y 8 0

    (C) 2 2x y 2x 8y 8 0 (D) 2 2x y 2x 6y 8 0 15. D16. If the chord PQ subtends an angle at the centre of the circle C, then =

    (A)

    6 (B)

    4

    (C)3

    (D)2

    16. D

    PASSAGE 2

    The geometrical meaning of 1 2| z z | is the distance between points 1 2z & z in the argand plane.One of the many applications of this is in solving least value problems. The fact that sum of twosides of a triangle can never be less than the third side is also widely used.17. The least value of | 3z 13 | | 3z 11 | equal to

    (A) 0 (B) 2

    (C)12

    5 (D)

    7

    3

    17. B

    18. The least value of | z 5 | + | z + 11 | is(A) 6 (B) 16(C) 8 (D) 3

    18. B19. The least value of | z 1 i | | z 3 5i | equal to

    (A) 13 (B) 11

    (C) 2 13 (D) 2 11 19. C

    Read the following passages and answer the questionsPASSAGE 1

    The geometrical meaning of 1 2| z z | is the distance between points 1 2z & z in the argand plane.One of the many applications of this is in solving least value problems. The fact that sum of twosides of a triangle can never be less than the third side is also widely used.14. The least value of | 3z 13 | | 3z 11| equal to

    (A) 0 (B) 2

    (C)12

    5 (D)

    7

    3

    14. B

    15. The least value of | z 5 | + | z + 11 | is

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    (A) 1 (B) 1(C) 2 (D) 0

    16. D

    PASSAGE 2

    Consider two circles S1and S2, S1having radius 3 and lies entirely in the first quadrant such that ittouches both the coordinate axes while S2has centre (8, 15).

    17. Equation of S2, such that S1touches S2internally is

    (A) 2 2x y 16x 30y 33 0 (B) 2 2x y 16x 30y 189 0

    (C) 2 2x y 16x 30y 33 0 (D) 2 2x y 16x 30y 189 0 17. C

    18. If S2touches S1externally then the point of contact is

    (A)54 75

    ,13 13

    (B)6 15

    ,7 7

    (C)54 75

    ,13 13

    (D)6 15

    ,7 7

    18. C

    19. If radius of S2is 5 then point of intersection of direct common tangent is

    (A)39

    ,302

    (B)9

    , 152

    (C)39

    ,302

    (D)9

    ,152

    19. BComprehension IIn the given figure. ABC is a triangle inscribed in a circle and AL,BM and CNare diameters of the circle, then mark the correct answers.

    15. Area of BLC is(A) 22R sinA sinBsinC

    (B) 22R sinA cosBcosC

    (C) 22R cosA cosBcosC

    (D) 2R cosA cosBcosC [R is the radius of the circumcircle of ABC]

    B

    L

    C

    M

    A

    N

    O

    15 B16 Area of CMA is

    (A) 22R sinAcosBcosC (B) 22R cos AcosBcosC

    (C) 22R sinBcosCcosA16 C

    (D) 22R sinA sinBcosC

    17. Area of ANB is(A) 22R sinA sinBsinC (B) 22R cosA cosBcosC

    (C) 22R sinA cosBcosC (D) 22R sinCcosA cosB17 DComprehension IIIf the equation of a rectangular hyperbola is xy=8 then

    18. The focus of the hyperbola is(A) (2,2) (B) (4,4)

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    (C)( 3,3) (D) 2 2,2 2 18. B19. The directrix of the hyperbola is

    (A) x + y = 3 (B)x +y = -4

    (C)x+y= 2 2 (D) None of these19. B20. The center of the hyperbola (x-3)(y-2)=8 is

    (A) (0,0) (B) (3,2)(C)( 3,3) (D) 2 2,2 2

    20. B

    Comprehension IIIConsider an equation sin x + sin y = 2 (i)We know that sin x 1 and sin y 1 for all x, ySo, sin x + sin y 2 for all x and yTherefore, sin x + sin y = 2 if and only if sin x = 1 and sin y = 1

    x 2n

    2and

    y 2m

    2

    Which is the required solution of given equation. To solve the equation (1), we have used theboundedness of sin x rather than using conventional methods of solving equationIn general we employ one or more of the following extreme value conditions.

    1. 21 sinx 1 | sin x | 1 and sin x 1

    2. 21 cos x 1 | cos x | 1 andcos x 1

    3. 2 2 2 2 2 2a b asinx bcosx a b | asinx bcosx | a b

    21. The minimum value of cos 2x. sin 2x27 81 is

    (A) 1 (B)1

    9

    (C)

    1

    81 (D)

    1

    243 21. D

    22. Number of roots of the equation 7 4cos x sin x 1in the interval [0, 2] is(A) 0 (B) 1(C) 2 (D) 4

    22. D

    23. The value of a for which the eqution 2 2a 2a sec a x 0 has solution is(A) 1 (B) 2(C) 0 or 1 (D) 1 or 2

    23. AComprehension I

    (i) General solution for sin =0 is =n, n I and for cos = 0 is = 2n 1 2

    ,n I and for tan

    =0 is = n+, n I .

    (ii) sin = sin n

    n 1 ,n I

    cos = cos 2n ,n I tan tan n ,n I

    14. The sum of all the solution of the equation

    cosx.cos 1

    x cos x ,x 0,63 3 4

    is

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    (A) 20 (B)30(C) 50 (D) none of those

    14. B

    15. The number of solutions of sin5cos3=sin 9cos 7in 0,2

    is

    (A) 6 (B) 7(C) 8 (D) 9

    15. D16. The value of satisfying

    3 2cos 2 3 sin -3 2sin 0 are

    (A)2

    n ,n3 6

    (B) n ,n

    3 6

    (C) 2n ,n3

    (D) none of these

    16. B

    Comprehension II

    (i) 1 1sin x cos x , 1 x 12

    (ii) 1 1sec x cosec x2

    , x 1or x 1

    (iii) 1 1tan x cot x , x2

    17. If 1 1 1 12

    sin x sin y , thencos x cos y3

    (A)2

    3

    (B)

    3

    (C)6

    (D) .

    17. B

    18. If 1 12

    tan x 2cot x ,then x3

    (A) 3 (B) 3

    (C) 2 (D)3 1

    3 1

    18. B

    19. If x= 1 1sin K,y cos K, 1 K 1, then the correct relationship is(A) x + y= 2 (B) x - y= 2

    (C) x + y =2

    (D) x y =

    2

    .

    19. CRead the following passages and answer the questionsPASSAGE - 1

    Let f(x) is a cube polynomial which has local maximum at x = 1, if f(2) = 18, f(1) = 1 and f(x) haslocal minima at x = 0, then

    14. The cube polynomial f(x) is

    (A) 31

    x 45x 548

    (B) 3x x 1

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    (C) 3 2x x 9x 12 (D) 31

    19x 57x 344

    14. D15. f(x) is increasing for

    (A)

    1x ,

    3 (B) x 1,2 5

    (C) x R (D) x 1, 15. B

    16. f(x) has local minimum at(A) x = 0 (B) x = 1

    (C) x = 2 (D) x 216. B

    PASSAGE 2

    If we want to compare f(x) and g(x) consider a function Q(x) = f(x) g(x) or g(x) f(x) and checkwhether Q(x) is increasing or decreasing and Q(0) in the given domain of f(x) and g(x)e.g. sin x > x, x (0, )

    17. For x (, 0], the order relation between x and 1tan x is(A) 1x tan x (B) 1x tan x

    (C) 1x tan x (D) 1x tan x 0 17. C

    18. If 1 22x tan x n 1 x , then x (A) (, 0] (B) [0, )(C) (1, 1) (D) (, )

    18. D19. For all x (0, 1), which option is correct

    (A) xe 1 x (B) n 1 x x

    (C) sin x > x (D) n x > x

    19. BRead the following passages and answer the questionsPASSAGE - 1

    Let C and D are representing two parabola i.e. 2 2 1 2C : y x 3,D : y kx , andL and L are

    representing two straight lines i.e. 1 2L : x a,L : x 1. a 014. If the parabola C and D intersect at a point A on the line L1. then equation of the tangent line

    L at A to the parabola D is

    (A) 3 32 a 3 x ay a 3a 0 (B) 3 32 a 3 x ay a 3a 0

    (C) 3 3

    a 3 x 2ay 2a 6a 0 (D) None of these14. B

    15. If the line L meets the parabola C at a point B on the line L2, other A then a can be equal to(A) 3 (B) 4(C) 2 (D) 3

    15. D

    16. If a > 0, the angle subtended by the chord AB at the vertex of the parabola C is

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    (A)

    1 5tan7

    (B)

    1 1tan2

    (C) 1tan 2 (D)

    1 1tan8

    16. B

    Passage 2

    2 22 2 x yC : x y 9, E : 1

    9 4

    L : y = 2xC, E, L represents circle, ellipse & line respectively.

    17. If P is a point on the circle C in the first quadrant, the perpendicular PN to the major axis of

    the ellipse E meets the ellipse at M, thenMN

    PNis equal to

    (A)1

    3 (B)

    2

    3

    (C) 12

    (D) None of these

    17. B

    18. If L represent line joining the point P on circle C to its centre O, then equation of the tangentat M to the ellipse is

    (A) x 3y 3 5 (B) x 3y 3 5 0

    (C)3x y 3 5 0 (D) None of these18. A

    19. If R is the point of intersection of line L with the line x = 1, then R lies(A) insides both C and E (B) outside both C and E(C) on both C and E (D) inside C but outside E

    19. D

    Read the following passages and answer the questionsPASSAGE - 1

    Let P and Q be the two points on the ellipse x 2+ 4y2= 16 whose eccentric angles are4

    and34

    respectively. The tangent at P and the normal at Q cut each other at R and the normal at Q cuts theellipse again at M.

    14. The coordinates of M is

    (A)

    10 2 11 2,3 3

    (B)

    14 2 23 2,17 17

    (C)

    5 2 3 2,

    6 6 (D) 2 2 , 2

    14. B

    15. The coordinates of Q is

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    (A)

    10 2 11 2,

    3 3 (B)

    14 2 23 2,

    17 17

    (C)

    5 2 3 2,

    6 6 (D) 2 2 , 2

    15. D

    16. The coordinates of R is

    (A)

    10 2 11 2,

    3 3 (B)

    14 2 23 2,

    17 17

    (C)

    5 2 3 2,

    6 6 (D) 2 2 , 2

    16. A

    Passage 2

    Let the equation of a family of circles is 2 2x y 2x 2ay 8 0, where a is a variable.

    17. The equation represents a family of circles passing through two fixed points. Thecoordinates of one of these points is(A) (2, 0) (B) (2, 0)(C) (5, 0) (D) (4, 0)

    17. A

    18. Let P and Q be two fixed points. Equation of a circle C of this family, tangents to which atthese fixed points intersect on the line x + 2y + 5 = 0 is

    (A) 2 2x y 2x 8y 8 0 (B) 2 2x y 2x 6y 8 0

    (C) 2 2x y 2x 8y 8 0 (D) 2 2x y 2x 6y 8 018. D

    19. If the chord PQ subtends an angle at the centre of the circle C, then =

    (A)

    6 (B)

    4

    (C)

    3 (D)

    2

    19. DRead the following passages and answer the questionsPASSAGE - 1

    Let P and Q be the two points on the ellipse x 2+ 4y2= 16 whose eccentric angles are

    4and

    34

    respectively. The tangent at P and the normal at Q cut each other at R and the normal at Q cuts the

    ellipse again at M.

    14. The coordinates of M is

    (A)

    10 2 11 2,

    3 3 (B)

    14 2 23 2,

    17 17

    (C)

    5 2 3 2,

    6 6 (D) 2 2 , 2

    14. B15. The coordinates of Q is

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    (A)

    10 2 11 2,

    3 3 (B)

    14 2 23 2,

    17 17

    (C)

    5 2 3 2,

    6 6 (D) 2 2 , 2

    15. D16. The coordinates of R is

    (A)

    10 2 11 2,3 3

    (B)

    14 2 23 2,17 17

    (C)

    5 2 3 2,

    6 6 (D) 2 2 , 2

    16. A

    PASSAGE 2Let a hyperbola whose centre is at origin. A line x + y = 2 touches this hyperbola at P(1, 1) and

    intersects the asymptotes at A and B such that AB = 6 2 units. (you can use the concept thatincase of hyperbola portion of tangent intercepted between asymptotes is bisected at the point ofcontact).

    17. Equation of asymptotes are(A) 2 25xy 2x 2y 0 (B) 2 23x 4y 6xy 0

    (C) 2 22x 2y 5xy 0 (D) None of these

    17. A18. Angle subtended by AB at centre of the hyperbola is

    (A) 14

    sin5

    (B) 12

    sin5

    (C) 13

    sin5

    (D) none of these

    18. C

    19. Equation of the tangent to the hyperbola at7

    1,2

    is

    (A) 5x + 2y = 2 (B) 3x + 2y = 4(C) 3x + 4y = 11 (D) none of these

    19. B

    PASSAGE - 1

    Let f(x) is a cube polynomial which has local maximum at x = 1, if f(2) = 18, f(1) = 1 and f(x) haslocal minima at x = 0, then

    14. The cube polynomial f(x) is

    (A) 31

    x 45x 548

    (B) 3x x 1

    (C) 3 2x x 9x 12 (D) 31 19x 57x 344

    14. D

    15. f(x) is increasing for

    (A)

    1x ,

    3 (B) x 1,2 5

    (C) x R (D) x 1, 15. D

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    16. f(x) has local minimum at

    (A) x = 0 (B) x = 1

    (C) x = 2 (D) x 2 16. B

    PASSAGE 2Let a hyperbola whose centre is at origin. A line x + y = 2 touches this hyperbola at P(1, 1) and

    intersects the asymptotes at A and B such that AB = 6 2 units. (you can use the concept thatincase of hyperbola portion of tangent intercepted between asymptotes is bisected at the point ofcontact).

    17. Equation of asymptotes are

    (A) 2 25xy 2x 2y 0 (B) 2 23x 4y 6xy 0

    (C) 2 22x 2y 5xy 0 (D) None of these

    17. A

    18. Angle subtended by AB at centre of the hyperbola is

    (A) 14

    sin5

    (B) 12

    sin5

    (C) 13

    sin5

    (D) none of these

    18. C

    19. Equation of the tangent to the hyperbola at7

    1,2

    is

    (A) 5x + 2y = 2 (B) 3x + 2y = 4(C) 3x + 4y = 11 (D) none of these

    19. BPASSAGE - 1An urn contains 4 white and 9 black balls. r balls drawn with replacement. Let P(r) be the probabilitythat no two white balls appear in succession. Answer the following questions

    14. The value of P(4) must be

    (A)

    2

    81

    169 (B)

    2

    81 273

    169

    (C)4

    41

    13

    (D) None of these

    14. B

    15. The recursion relation for P(r) must be

    (A)9 4 9

    P(r) P(r 2) P(r 1)13 13 13

    (B)2

    9 4P(r) P(r 2) P(r 1)

    13 13

    (C)2

    4 9P(r) P(r 2) P(r 1)

    13 13

    (D) None of these

    15. A

    16. P(r) must be equal to

    (A)r r

    r

    16 12 ( 3)

    12 13

    (B)

    r r

    r

    16 12 ( 3)

    9 13

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    (C)r r

    r

    16 12 ( 3)

    15 13

    (D) None of these

    16. C

    Passage 2

    Consider an equation sin x + sin y = 2 (i)

    We know that sin x 1 and sin y 1 for all x, ySo, sin x + sin y 2 for all x and yTherefore, sin x + sin y = 2 if and only if sin x = 1 and sin y = 1

    x 2n

    2and

    y 2m

    2

    Which is the required solution of given equation. To solve the equation (1), we have used theboundedness of sin x rather than using conventional methods of solving equationIn general we employ one or more of the following extreme value conditions.

    1. 21 sinx 1 | sin x | 1 and sin x 1

    2. 21 cos x 1 | cos x | 1 andcos x 1

    3. 2 2 2 2 2 2a b asinx bcosx a b | asinx bcosx | a b

    17. The minimum value of cos 2x. sin 2x27 81 is

    (A) 1 (B)1

    9

    (C)1

    81 (D)

    1

    243

    17. D

    18. Number of roots of the equation 7 4cos x sin x 1in the interval [0, 2] is(A) 0 (B) 1(C) 2 (D) 4

    18. D

    19. The value of a for which the eqution 2 2a 2a sec a x 0 has solution is(A) 1 (B) 2(C) 0 or 1 (D) 1 or 2

    19. A

    PASSAGE - 1

    The roots of f(x) = 2x 2 b 3 x 9 lie in (6, 1) where b is a +ve integer. Consider the geometricprogression 2, a1, a2, . a20, b then

    14. The value of b is(A) 6 (B) 7

    (C) 8 (D) 914. A

    15. Nature of the roots of the equation f(x) = 0(A) real and distinct (B) equal positive roots(C) equal negative roots (D) None of these

    15. C

    16. The value of a3a18is equal to(A) 18 (B) 16(C) 14 (D) 12

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    16. D

    PASSAGE 2

    Consider (i) n2 2 2n

    0 1 2 2n1 2x 2x a a x a x ... a x

    (ii) n

    2 2 2n0 1 2 2n1 x x b b x b x .... b x

    Now answer the following

    17. The value of 0 2n 1 2n 1 2n 0a a a a .... a a will be if n = 11(A) 0 (B) 1(C) 1 (D) None of these

    17. A

    18. The value of 0 2n 1 2n 1 2n 0a a a a ....a a will if n = 10

    (A) 0 (B) 1

    (C) 10 10 52 C (D)10 10

    22 C

    18. C

    19. The value of 0 1 1 2 2 3 3 4 2n 1 nb b b b b b b b ...b b will be if n = 11(A) 1 (B) 1(C) 2 (D) 0

    19. D

    PASSAGE - 1

    Let P and Q be the two points on the ellipse x 2+ 4y2= 16 whose eccentric angles are

    4and

    34

    respectively. The tangent at P and the normal at Q cut each other at R and the normal at Q cuts theellipse again at M.

    14. The coordinates of M is

    (A)

    10 2 11 2,

    3 3 (B)

    14 2 23 2,

    17 17

    (C)

    5 2 3 2,

    6 6 (D) 2 2 , 2

    14. B15. The coordinates of Q is

    (A)

    10 2 11 2,

    3 3 (B)

    14 2 23 2,

    17 17

    (C)

    5 2 3 2,

    6 6

    (D) 2 2 , 2

    15. D16. The coordinates of R is

    (A)

    10 2 11 2,

    3 3 (B)

    14 2 23 2,

    17 17

    (C)

    5 2 3 2,

    6 6 (D) 2 2 , 2

    16. A

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    PASSAGE 2

    Regular hexagon A1 A2 A3 A4A5 A6whose side length is 2k, Q is thecentre of hexagon; as shown in the figure. Q1as centre circle is drawn,Q2 as centre circle is drawn Q3 as centre circle is drawn. The threecircles touch each other pairwise and each touches two sides ofhexagon.

    17. Radius of each circle is

    (A)3k

    2 (B)

    3k

    4

    (C)k

    4 (D)

    k

    2

    A1 A2

    A3

    A4A5

    A6 Q2

    Q3

    Q1

    B2

    B1

    B3

    B4

    B5

    B6

    2k

    17. A18. Area of the quadrilateral A1B1Q1B2is

    (A) 23 3

    k4

    (B)2k

    2

    (C) 23

    k

    4

    (D) 23 3

    k

    16

    18. C19. Radius of the circumcircle of Q1 Q2 Q3is

    (A)k

    2 (B)

    k

    3

    (C)2k

    3 (D) k

    19. DPASSAGE - 1

    The roots of f(x) = 2x 2 b 3 x 9 lie in (6, 1) where b is a +ve integer .Consider the geometric

    progression 2, a1, a2, . a20, b then

    14. The value of b is(A) 6 (B) 7(C) 8 (D) 9

    14. A15. Nature of the roots of the equation f(x) = 0

    (A) real and distinct (B) equal positive roots(C) equal negative roots (D) None of these

    15. C16. The value of a3a18is equal to

    (A) 18 (B) 16(C) 14 (D) 12

    16. D

    PASSAGE 2

    The geometrical meaning of 1 2| z z | is the distance between points 1 2z & z in the argand plane.One of the many applications of this is in solving least value problems. The fact that sum of twosides of a triangle can never be less than the third side is also widely used.

    17. The least value of |5z 13 | |3z 11| equal to

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    (A)8

    3 (B)

    16

    5

    (C)12

    5 (D)

    7

    3

    17. B18. The least value of | z 5 | + | z + 11 | is

    (A) 6 (B) 16

    (C) 8 (D) 318. B19. The least value of | z 1 i | | z 3 5i | equal to

    (A) 13 (B) 11

    (C) 2 13 (D) 2 11 19. C

    Passage I

    Regular hexagon A1A2A3A4A5A6whose side length is 2k, Q is the centre ofhexagon; as shown in the figure. Q1as centre circle is drawn, Q2as centrecircle is drawn Q3 as centre circle is drawn. The three circles touch eachother pairwise and each touches two sides of hexagon.

    14. Radius of each circle is

    (A)3k

    2 (B)

    3k

    4

    (C)k

    4 (D)

    k

    2

    A1 A2

    A3

    A4A5

    A6 Q2

    Q3

    Q1

    B2

    B1

    B3

    B4

    B5

    B6

    2k

    14. A15. Area of the quadrilateral A1B1Q1B2is

    (A) 23 3

    k4

    (B)2k

    2

    (C) 23

    k4

    (D) 23 3

    k16

    15. C16. Radius of the circumcircle of Q1 Q2 Q3is

    (A)k

    2 (B)

    k

    3

    (C)2k

    3 (D) k

    16. DPassage 2

    2 22 2 x yC : x y 9, E : 1

    9 4

    L: y = 2xC, E, L represents circle, ellipse & line respectively.

    17. If P is a point on the circle C in the first quadrant, the perpendicular PN to the major axis of

    the ellipse E meets the ellipse at M, thenMN

    PNis equal to

    (A)1

    3 (B)

    2

    3

    (C)1

    2 (D) None of these

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    17. B18. If L represent line joining the point P on circle C to its centre O, then equation of the tangent

    at M to the ellipse is

    (A) x 3y 3 5 (B) x 3y 3 5 0

    (C)3x y 3 5 0 (D) None of these18. A19. If R is the point of intersection of line L with the line x = 1, then R lies

    (A) insides both C and E (B) outside both C and E(C) on both C and E (D) inside C but outside E19. DPassage 1

    r a bt

    be a line and r.n q

    be a plane. Answer the following questions

    14. The equation of a line passing through any point of the given line and normal to the givenplane must be of the type

    (A) r a tb t n

    for some t and t (B) r tb t n

    for some t and t(C) r a t n

    for some t and t (D) None of these

    14. C

    15. The correct tdescribed in Q. 17 of this passage must be(A) q a bt .n

    (B) q a bt .n

    (C)

    2

    q (a bt).n

    | n |

    (D) none of these

    15. D

    16. If r c d

    be the projection of the line r a tb

    on the plane r.n q

    then c

    must be

    (A) q a.n

    (B) q a.n

    (C)2

    q a.na .n

    | n |

    (D) 2q a.n

    a .n| n |

    16. B

    Comprehension - IIConsider an equation sin x + sin y = 2 (i)We know that sin x 1 and sin y 1 for all x, ySo, sin x + sin y 2 for all x and yTherefore, sin x + sin y = 2 if and only if sin x = 1 and sin y = 1

    x 2n

    2and

    y 2m

    2

    Which is the required solution of given equation. To solve the equation (1), we have used theboundedness of sin x rather than using conventional methods of solving equationIn general we employ one or more of the following extreme value conditions.

    1. 21 sinx 1 | sinx | 1 and sin x 1

    2. 21 cos x 1 | cos x | 1 andcos x 1

    3. 2 2 2 2 2 2a b asin x bcos x a b | a sin x bcos x | a b

    17. The minimum value of cos 2x. sin 2x27 81 is

    (A) 1 (B)1

    9

    (C)1

    81 (D)

    1

    243

    17. D

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    18. Number of roots of the equation 7 4cos x sin x 1in the interval [0, 2] is(A) 0 (B) 1(C) 2 (D) 4

    18. D

    19. The value of a for which the eqution 2 2a 2a sec a x 0 has solution is(A) 1 (B) 2(C) 0 or 1 (D) 1 or 2

    19. ARead the following passages and answer the questionsPASSAGE - 1

    Let 2 2

    1 2C y x 1 ,C y x 1 and 31

    C y4

    then

    14. The area bounded by the curves 1 2C ,C and 3C is

    (A) 4 (B)4

    3

    (C)1

    3 (D)

    1

    6

    14. C

    15. The area bounded by curves 2C and 3C is

    (A)1

    6 (B)

    1

    3

    (C)16

    3 (D)

    1

    4

    15. A

    16. Let f(x) = min 1 2 3C , C , C , then area bounded by y = f(x) and x-axis is

    (A)1

    6 (B)

    1

    8

    (C) 23

    (D) 13

    16. D

    PASSAGE - 2

    A tangent to the curve y = f(x) at P(x, y) cuts the x-axis and y-axis at A and B, respectively such thatBP : AP = 3 : 1, if f(1) = 1

    17. The differential equation for the curve y = f(x) is

    (A)dx

    y 3x 0dy

    (B)dy

    x 3y 0dx

    (C) dyy 3x 0dx

    (D) dxx 3y 0dy

    17. B

    18. Equation of curve y = f(x) is

    (A) 2x y 1 (B) 2y x 1

    (C) 3x y 1 (D) 3xy 1 18. C

    19. The equation of normal to curve at the point (1, 1) is

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    (A) x + y 2 = 0 (B) 2x + y 3 = 0(C) x + 3y y = 0 (D) x 3y + 2 = 0

    19. DPASSAGE - 1

    A function f(x) is said to be increasing if 1 2 1 2x x f(x ) f(x ) { 1 2x and x are in domain of f(x)} and

    f(x) is said to be decreasing if 1 2 1 2x x f(x ) f(x ) . Concept of increasing and decreasing

    behavior of a function is used to prove certain inequalities and solving the problems e.q. x sin x x 0, can be proved by taking

    f(x) = x sin xSo that f(x) = 1 cos x 0f(x) is increasingx 0f(x) f(0) f(x) 0 x 0

    14. For x (, 0], the order relation between x and 1tan x is

    (A) 1x tan x (B) 1x tan x

    (C) 1x tan x (D) 1x tan x 0 14. C

    15. If 1 22x tan x n 1 x , then x (A) (, 0] (B) [0, )(C) (1, 1) (D) (, )

    15. D

    16. n (n x) = a n x, will have two solutions if a belongs to

    (A)

    1,e

    (B)

    10,

    e

    (C)

    1,1

    e (D) (1, )

    16. B

    PASSAGE - 2

    Let f(x) is a cube polynomial which has local maximum at x = 1, if f(2) = 18, f(1) = 1 and f(x) haslocal minima at x = 0, then

    17. The cube polynomial f(x) is

    (A) 31

    x 45x 548

    (B) 3x x 1

    (C) 3 2x x 9x 12 (D) 31

    19x 57x 344

    17. D

    18. f(x) is increasing for

    (A)

    1x ,

    3 (B) x 1,2 5

    (C) x R (D) x 1, 18. D

    19. f(x) has local minimum at(A) x = 0 (B) x = 1

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    (C) x = 2 (D) x 2 19. B

    Passage 1

    r a bt

    be a line and r.n q

    be a plane. Answer the following questions

    17. The equation of a line passing through any point of the given line and normal to the givenplane must be of the type

    (A) r a tb t n

    for some t and t (B) r tb t n

    for some t and t(C) r a t n

    for some t and t (D) None of these

    17. D

    18. The correct tdescribed in Q. 17 of this passage must be

    (A) q a bt .n

    (B) q a bt .n

    (C)

    2

    q (a bt).n

    | n |

    (D) none of these

    18. D

    19. If r c d

    be the projection of the line r a tb

    on the plane r.n q

    then c

    must be

    (A) q a.n

    (B) q a.n

    (C)2

    q a.na .n

    | n |

    (D) 2q a.n

    a .n| n |

    19. A

    Passage 2

    Consider an equation sin x + sin y = 2 (i)We know that sin x 1 and sin y 1 for all x, ySo, sin x + sin y 2 for all x and yTherefore, sin x + sin y = 2 if and only if sin x = 1 and sin y = 1

    x 2n

    2and

    y 2m

    2

    Which is the required solution of given equation. To solve the equation (1), we have used theboundedness of sin x rather than using conventional methods of solving equationIn general we employ one or more of the following extreme value conditions.

    1. 21 sin x 1 | sin x | 1 and sin x 1

    2. 21 cosx 1 | cosx | 1 andcos x 1

    3. 2 2 2 2 2 2a b asin x bcos x a b | a sin x bcos x | a b

    17. The minimum value of

    cos 2x. sin 2x

    27 81 is(A) 1 (B)

    1

    9

    (C)1

    81 (D)

    1

    243

    18. Number of roots of the equation 7 4cos x sin x 1in the interval [0, 2] is(A) 0 (B) 1(C) 2 (D) 4

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    18. The value of a for which the eqution 2 2a 2a sec a x 0 has solution is(A) 1 (B) 2(C) 0 or 1 (D) 1 or 2

    PASSAGE - 1

    An urn contains 4 white and 9 black balls. r balls drawn with replacement. Let P(r) be the probabilitythat no two white balls appear in succession. Answer the following questions

    14. The value of P(4) must be

    (A)

    2

    81

    169 (B)

    2

    81 273

    169

    (C)4

    41

    13

    (D) None of these

    14. B

    15. The recursion relation for P(r) must be

    (A)9 4 9

    P(r) P(r 2) P(r 1)

    13 13 13

    (B)2

    9 4P(r) P(r 2) P(r 1)

    13 13

    (C)2

    4 9P(r) P(r 2) P(r 1)

    13 13

    (D) None of these

    15. A

    16. P(r) must be equal to

    (A)r r

    r

    16 12 ( 3)

    12 13

    (B)

    r r

    r

    16 12 ( 3)

    9 13

    (C)r r

    r

    16 12 ( 3)

    15 13

    (D) None of these

    16. C

    Passage 2

    r a bt

    be a line and r.n q

    be a plane. Answer the following questions

    17. The equation of a line passing through any point of the given line and normal to the givenplane must be of the type

    (A) r a tb t n

    for some t and t (B) r tb t n

    for some t and t(C) r a t n

    for some t and t (D) None of these

    17. A

    18. The correct tdescribed in Q. 17 of this passage must be

    (A) q a bt .n

    (B) q a bt .n

    (C)

    2

    q (a bt).n

    | n |

    (D) none of these

    18. C

    19. If r c d

    be the projection of the line r a tb

    on the plane r.n q

    then c

    must be

    (A) q a.n

    (B) q a.n

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    (C)2

    q a.na .n

    | n |

    (D) 2q a.n

    a .n| n |

    19. D