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86. Let f be a real valued function on [a, 0] such that fis differentiable on (a, 0) and both f(a*) and 10൦)

87.

88.

89.

90.

91.

92.

93.

94.

exist. Then there exists ௦ € (8, 0) such that
A) f(b) - f(a) = f(c).(b — a)
©) f(b) - f(a*) = f(c).(b — a)

⋅ (Hy
lim | ——| =
x0 x41
A) 1

1
©) >

೫
ചി =

4

^) ठ 8) log,2

1
Let f: [0, 1] ‏ج‎ ೫ be defined by f(x) = on if

(x) dx = °
ர்‌ (x) dx =
0 ० %
x2 7
If f(x) = ர்‌ cos Vt dt , then f(x) =
A) 2xcosx 8) 006%

| >)
೫೮ ಹ
amt’ on

8) f(b-) - f(a) = f(c).(b — a)
0) 102) - f(a*) = f(c).(b - a)

1 1
Cc) = ०) 7

for some
n=0,4,2...)- Then

otherwise

௦) % 21

D) cosx

0) (ടഫി

Which of the following is not a subspace of M,(R) = The space of all n x ೧ matrices over R ۶

A) {Ac M,(B); A= ^
©) {Ae M,(®); Ais upper triangular}

Consider the vectors
X =(1,0, 1, 0, 1)
۷ =(0, 1, 0, 1, 0)
2 = (1, 1, 1, 1, 1)
വ = {1, -1, 1, -1, 1)

8) {Ae M,(R); det(A) = 0}
0) (8 € M,(B); trace (A) = 0}

The dimension of the subspace spanned by these 4 vectors in Ris

A) 5 B) 4

LetT : ೫3 ‏ب‎ R? be defined by T(x, y, 2) = (x + y, 2). Then which of the following forms a basis

for the Kernel of 2

03 0) 2

8) {(1, 0, 1)
0) {(1, 0, 1), (1, -1, 0)}

What is the dimension of the vector space of n xn real symmetric matrices with trace 0, over R ?

A) {(1, -1, 0)}
©) (60, -1, 1)}
‏رم‎ 12411 B) won

2 ‏ے2‎
‎0) ‏لكا‎ D) १01

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