VITEEE 2009 Sample Question Papers - Mathematics

VITEEE 2009 Sample Question Papers - Chemistry -  PART – 3 [MATHEMATICS]

81.  The system of equations -

        Ans:  

                   = 1(1 - 3) - 2(1 - 2) = - 2 + 2 = 0

                   Choice (B)

82.  , then -

        Ans:   = I

                   A ⁴ = I

A³ = I A^{- 1}

A² = I A^{- 2}

     =

A^{- 1} =

     =

A^{- 2} =  =

ab =

Þ a²b² = 1

Þ ab = 1

Choice (D)

83.  If D = diag (d₁, d₂, …….., d_n) where -

        Ans:   Choice (D)

84.  If x, y, z are different from zero and D = -

        Ans:   = 0

                    = 0

                =

           Choice (D)

85.  Probability of getting positive integral roots of the equation, -

        Ans:   x = ±

                   n = 1, 4, 9, 16, 25, 36

                   Probability =

                   Choice (C)

86.  The number of real roots of equation -

Ans:   = 22 - x⁴

           x ⁴ + 20 = (22 - x⁴)²

                = 484 + x ⁸ - 44x⁴

           x ⁸ - 45x⁴ + 464 = 0

           x ⁴ =

                = =

                = 29, 16

           x ⁴ = 29 is not admissible

           Þ x⁴ = 16

Choice (B)

87.  Let a , b be the roots of the equation -

        Ans:   a ²- aa + b = 0

                   A _n+1 - aA_n + bA_{n- 1}

                             = a ⁿ⁺¹ + b ⁿ⁺¹- a(a ⁿ + b ⁿ) + b(a ^{n- 1} + b ^{n- 1})

                   = a ^{n- 1}(a ²- aa + b) + b ^{n- 1}(b ²- ab + b)

                   = 0

                   Choice (C)

88.  If the sides of a right – angle triangle -

        Ans:   b, c, a ® AP

        a =

        sin B =

        Choice (A)

89.  The plane through the point -

        Ans:   x + 3y - z = 0

y + 2z = 0

Let the plane be

A(x + 1) + B(y + 1) + C(z + 1) = 0

Plane passes through the origin

A + B + C = 0

Choice (A)

90.  are one of the sides -

        Ans:       

                                    =

                   Area =

                            =

                   Choice (D)

91.  If be three unit vectors such that -

        Ans :       =

                        =

                        =

                        cos q ₂     =         Þ q ₂ =

                   cos q ₁ = 0           Þ q ₁ =

                   Choice (C)

92.  The equation  -

        Ans:   Equation is

                   x ² + y² + z² - 2xc₁ - 2yc₂ - 2zc₃ + h = 0

                   Choice (D)

93.  The simplified expression of -

        Ans:

                   Let tan^{- 1} x be a Þ tan a = x

                   Then from the figure sin a =

                   Þ sin (tan^{- 1} x)  

                   = sin

                   Choice (B)

94.  If -

        Ans:  

Þ z lies on the line perpendicular to the real axis and divides the line segment between 1 and 25 in the ratio 1 : 5 Þ   z = (5, 0) Þ |z| = 5

                     Choice (C)

95.  Argument of the complex number -

Ans:  

                                     = - (1 + i)

                   \ Arg

                   Choice (C)

96.  In a triangle ABC, the sides b and c are -

        Ans:   x ² - 61x + 820 = 0

                   x ² - 41x - 20x + 820 = 0

                   Þ x_1,2 = 41, 20

                   A = tan^{- 1}

                             Þ cosA =

                   \ By Cosine formula,

                   a ² = b² + c² - 2bc cos A

                   a ² = 41² + 20² - 2(41)(20)

                   = 2081 - 984 = 1097

                   Choice (C)

97.  The shortest distance between the straight lines through -

        Ans:           

                   \ Shortest distance =

                   =

                   =

                   Choice (D)

98.  The center and radius of the sphere -

        Ans:   Centre is at

                   Choice (C)

99.  Let A and B are two fixed points in a plane then locus of another -

        Ans:   Ellipse

                   Choice (B)

100.        The directrix of the parabola -

        Ans:   y ² = - 4x - 3

                        = - 4

                   Equation of the directrix is

                   x =

                   Choice (D)

101.        If g(x) is a polynomial satisfying g(x) -

        Ans:   g(x) . g(y) = g(x) + g(y) + g(xy) - 2 ¾ (1)

                   g(2) . g(y) = g(x) + g(y) + g(xy) - 2

5.g(y) = 5 + g(y) + g(xy) - 2

Þ 4g(y) = 3 + g(xy)

\ g(0) = 1

g(x) is given in a polynomial, and by the relation given g(x) cannot be linear.

Let g(x) = x² + k

Since g(0) = 1 Þ g(x) = x² + 1

Verifying (1) Þ

(x² + 1) (y² + 1)

             = x ² + 1 + y² + 1 + x²y² + 1 - 2

(1) is satisfied by g(x) = x² + 1

g(x) = g(3) (Q g(x) in a polynomial)

     = 10

                   Choice (B)

102.        The value of f(0) so that -

        Ans:  

                        =

                        = 2 ⁰?n2 - 1 = l n2 - 1

                        = f(0)

                   Choice (D)

103.           Let [   ]   denote the greatest integer -

        Ans :

                   f(x) is continuous at x = 0

                   Choice (B)

104.           A spherical balloon is expanding -

        Ans : Let r be the radius and V be the volume

                   \ = 2            r = 5

                   \ V = p r³

                   = 4p (5)² ´ (2)

                   = 200 p

                   Choice (C)

105.           The length of the parabola -

Ans :

                   Length =

                         =

                         =

                         =

                          =

                         =

                        =

        =

                   = 2

                   -

        =

                   =

                   =

                    = 

                   =

                   Choice (A)

106.           If I = -

        Ans :  I =

                   Put 1 + x³ = t Þ x² dx =

                   \ I =

                   =

                   =

                   =

                   Choice (D)

107.           Area enclosed by the curve -

        Ans :

                   Þ  = 1

                   \ Area of ellipse = p ab

                   = p ´

                   = 4

                   Choice (D)

108.The value of -

        Ans :

                   x = a sin² q

                   dx = 2a sin q cos q dq

                   x = 0 ® q = 0

                   x = a ® q =

                       =

                   = 2a ´

                   Choice (C)

109.           Let y be the number of people -

        Ans : 

                   l n y = - kt + c

                   y = ce^{- kt},  c > 0

                                    k ³ 0

                                Choice (B)

110.        The differential equation of -

         Ans:  x cos q + y sin q = a ¾ (1)

                differentiating cos q + y’ sin q = 0 ¾ (2)

                Eliminating sin q and cos q from (1) and (2)

                cos q =

                sin q =

                sin ² q + cos² q = 1

                Þ

                Þ a²y’ + a² = (xy’ - g)²

                Þ

                Choice (B)

111.           The differential equation
        admits -

        Ans:  

                   , |y| > 0 , 3 > 0

     Three positive quantities cannot add to give zero.

   \ No solution.

                   Choice (B)

112.           Solution of the differential equation xdy -

        Ans : ¾ (1)

                   which is homogeneous put y = v_x

                   \

                   \ (1) Þ

                   \ x

                   \

                   Integrating

                   log

                   log

                   \ y +

                   Choice (B)

113.           Let P, Q, R and S be statements and suppose-

        Ans:   p ® G ® R R ® p and ~ S ® R

                   Þ (C) and (D) are not true also ~ S ® R .

                   \ (A) is not true

                   Choice (B)

114. In how many number of ways -

        Ans  :   Required number of ways =

                                     = 2100

                     Choice (D)

115.        If R be a relation defined -

        Ans :       Relation is symmetric and transitive

                        Choice (D)

116.        Let S be a finite set containing n elements.
Then -

        Ans:        For commutative binary operations, there are  pairs available. For each of there pairs the result of the Binary operation should be among the n elements of S.