# AIEEE Mathematics 2008

### Mathematics 2008

Statement-*1*:

r is equivalent to either q or p .

Statement *2*:

r is equivalent to ~(p ~q)

(1) Statement -1 is true , Statement -2 is false

(2) Statement -1 is false, Statement -2 is true

(3) Statement -1 is true, Statement -2 is true:

Statement -2 is a correct explanation for

Statement -1

(4) Statement -1 is true, Statement -2 is true:

Statement -2 is ** not** a correct explanation for

_{ } Statement -1

Statement -1 :

The number of different ways the child can buy the six ice-creams is .

Statement -2 :

The number of different ways the child can buy the six ice-creams is equal of different ways of arranging 6 A’s and 4 B’s in a row

(1) Statement -1 is true, Statement -2 is false

(2) Statement -1 is false, Statement -2 is true

(3) Statement -1 is true , Statement -2 is true

Statement -2 is a correct explanation for Statement -1

(4) Statement -1 is true , Statement-2 is true :

Statement -2 is ** not **a correct explanation for Statement -1

_{1}, x

_{2}, x

_{3}, x

_{4}, x

_{5}be the number of icecreams selected from 5 types of ice-creams

x_{1}+ x_{2}+ x_{3}+ x_{4}+ x_{5} = 6

=> statement (1) is false but statement (2) is correct.

(1) Statement -1 is true, Statement -2 is false

(2) Statement -1 is false, Statement -2 is true

(3) Statement -1 is true, Statement -2 is true

Statement -2 is a correct explanation for Statement -1

(4) Statement -1 is true, Statement-2 is true :

Statement -2 is ** not **a correct explanation for Statement -1

Ans.

(1) Statement -1 is true, Statement -2 is false

(2) Statement -1 is false, Statement -2 is true

(3) Statement -1 is true, Statement -2 is true

Statement -2 is a correct explanation for Statement -1

(4 Statement -1 is true, Statement-2 is true :

Statement -2 is ** not **a correct explanation for Statement -1

Ans.

statement 1 is correct

^{2 }= 1

Statement -1 :

If A 1 and A- 1 , then A = - 1

Statement- 2:

If A 1 and A - 1, then tr (A) 0.

(1) Statement -1 is true, Statement -2 is false

(2) Statement -1 is false, Statement -2 is true

(3) Statement -1 is true, Statement -2 is true

Statement -2 is a correct explanation for Statement -1

(4) Statement -1 is true, Statement-2 is true :

Statement -2 is ** not **a correct explanation for Statement -1

^{o}. He moves away from the pole along the line BC to a point D such that CD = 7 m. From D the angle of elevation of the point A is 45

^{o}. Then the height of the pole is

^{7}+ 14x

^{5}+ 16x

^{3}+ 30x - 560 = 0 have ?

(1) 5 (2) 7 (3) 1 (4) 3

Ans.

= Oscillating between -1 & 1

=> RHD does not exists

Non diff. At x=1

(1) 4

(2) –4

(3) –12

(4) 12

br^{2} + br^{3} = 48 r<0

b(1+r) = 12 & br^{2} (1+r) = 48

Divide => r^{2} = 4 =>

As r<0, we take r = -2

Replace r in (1) to get

^{3}-px + q

^{2}- p

Arrange M|||PP| in

Between 7 letter there are 8 possibilities for 4

(1) (2, 0)

(2) (0, 2)

(3) (1, 0)

(4) (0, 1)

Ans.

distance of vertex from directrix = distance from focus

=> V=(1, 0)

22. The point diametrically opposite to the point P(1, 0) on the circle x^{2} + y^{2} + 2x + 4y – 3 = 0 is

(1) (3, 4)

(2) (3, -4)

(3) (-3, 4)

(4) (-3, -4)

satisfying the condition y(1) = 1 is

(1) y = x ln x + x

(2) y = ln x + x

(3) y = x ln x x + x^{2}

(4) y = x e^{(x-1)}

Ans.

^{2}+ b

^{2}+ c

^{2}+ 2abc is equal to

(1) 1

(2) 2

(3) –1

(4) 0

Ans. x - cy - bz = 0cx - y + az = 0

bx + ay - z = 0

It is given that system of equations is consistent.

i.e. posses a solution.

It is given that not all x, y, z are zero.

=> Non trivial solution exist

=> D = 0

Evalute 1(-1 - a^{2}) + c(-c - ab) - b(ac + b) = 0

=>-1 - a^{2 }- c^{2 }- abc - abc - b^{2} = 0

=> a^{2 }+ b^{2 }+ c^{2 }+ 2abc = -1

26. Let A be a square matrix all of whose entries are integers. Then which one of the following is true ?

(1) If det A = , then A^{-1} need not exist

(2) If det A = , then A^{-1} exists but all its entries are not necessarily integers

(3) If det A , then A^{-1} exists and all its entries are non-integers.

(4) If det A = , then A^{-1} exists and all its entries are integers

Ans. For example, let

det(A) = 1

A^{-1} exists & all entries are integers.

27. The quadratic equations x^{2} – 6x + a = 0 and x^{2} – cx + 6 = 0have one root in common. The other roots of the first and second equations are integers in the ration 4: 3. Then the common root is

(1) 2 (2) 1

(3) 4 (4) 3

Ans.

(1) a = 3, b = 4

(2) a = 0, b = 7

(3) a = 5, b = 2

(4) a = 1, b = 6

Ans.simplify (1) to get:

23 + a + b=30 => a + b = 7 —–(3)

simplify (2) to get

(a - 6)^{2 }+ (b - 6)^{2 }+ 4 + 1 + 16=34

=> (a - 6)^{2 }+ (7- a - 6)^{2} = 13

(a^{2}-12a + 36) + (a^{2 }+1 - 2a) = 13

2a^{2}-14a + 24 = 0

a^{2 }- 7a +12 = 0 => (a-4) (a-3) = 0 => b = 3, or 4

29. The vector lies in the plane of the vectors and and bisects the angle between Then which one of the following gives possible values of

(1) a = 8, b = 2

(2) a = 2, b = 8

(3) a = 4, b = 6

(4) a = 6, b = 4

Ans.

Equation of line is

let coordinates of intersection of l with yz plane be [2k+5, (1-b)k+1, (a-1)k+a]

it lies on yz plane,

2k+5 = 0 => k = -^{5}⁄_{2}

Also (1-b)k+1=^{17}⁄_{2} => 1-b = ^{15}⁄_{2}(-^{2}⁄_{5}) = -3 => b =4

Also (a-1)k+a =-^{13}⁄_{2}

A(1+k)-k = -^{13}⁄_{2} => a(1-^{5}⁄_{2}) = -^{13}⁄_{2}-^{5}⁄_{2} = -9

=> -^{3}⁄_{2} a = -9 => a = 6

32. If the straight lines

intersect at a point, then the integer k is equal to

(1) -2

(2) –5

(3) 5

(4) 2

Ans.

S = {(x, y) : y = x + 1 and 0 < x < 2}

T = {(x, y) : x – y is an integer}.

Which one of the following is true ?

(1) T is an equivalence relation on R but S is not

(2) Neither S nor T is an equivalence relations on R

(3) Both S and T are equivalence relation on R

(4) S is an equivalence relation on R but T is not

35. Let f : N Y be a function defined as

f(x) =4x + 3 where

Y = {y N : y = 4x + 3 for some x N}. Show that f is invertible and its inverse is

Ans. To find inverse of f, replace y by x and x by y.i.e.

x = 4f ^{-1}(x)+3 => f ^{–1}(x) = g(x) = x-^{3}⁄_{4}

Interims of y

g(y) = y-^{3}⁄_{4}