VIT 2008 PAPERS WITH SOLUTIONS

PART – III [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.

\ Total number of required operations

=

=

Choice (B)

117. A manufacturer of cotter pins knows that-

Ans: Probability of a cotter pin to be defective
=

Average number of defective cotter pins in a box of 100 is = 100 ´

= 5

We use Poisson distribution with parameter m = 5

Choice (B)

118. The probability that a certain kind -

Ans : p = , q = , n = 4

P(X = x) =

\ p(X = 2) =

=

Choice (D)

119. Mean and standard deviation -

Ans : For best performance & is less

Which true for = 75, s = 5

Choice (B)

120. A random variable X follows -

Ans : For Binomial distribution

0 <>

0 < b < a

Choice (B)