Q1.
Evaluate 
(x + 2)3 - 8/
x
Solution.
y = x + 2 , x → 0
x = y - 2, y → 2
=
y3 - 23/y - 2
= 3(2)
2 = 12
Solution
Q2.
Find 
log x - log 5/
x - 5
=
=
log(1 +
x - 5/5
)
/(x - 5/
5
) . 5 =
1
/5
Q3.
Evaluate 
sin x - sin a
/x - a
Solution.
Using D' hospital law derive the function
cosx
/1
= cos a
Solution
Q4.
Evaluate
sin x . cos x
/4x
Solution.
1/4
(
sin x
/x
)(cos x) =
1/4
(
sin x
/x
) .
(cos x)
=
1/4
Solution
Q5.
Evaluate
9x cosx - 4sinx
/5x + 3tanx
Solution.
=
( 9x cosx - 4sinx
/x
)
/ (5x + 3tanx
/x
)
=
9cosx - 4sinx
/x
/5 + 3tanx
/x
=
9 - 4/5 + 3
=
5/8
Solution
Q6.
Evaluate 
xx
/(1 + x)x
Solution.
=
1/e
(∴
(1 + x)
1/x = e)
Solution
Q7.
Evaluate
1
/x2
(1 + 2 + 3 + --------+ x) .
Solution.
=
1
/x2
∑ x =
x(x + 1)
/2x2
=
x + 1
/2x
(
x
/2x
+
1
/2x
) =
1/2
+
1
/2x
=
1/2
Solution
Q8.
Find 
x + x2 + x3 + ----------+ xn - n
/x - 1
Solution.
=
(x - 1) + (x2 - 1) + (x3-1) + ----------+ (xn - 1)
/x - 1
{
x - 1
/x - 1
+
x2 - 1/x - 1
+
x3-1/x - 1
+
------ +
xn - 1/x - 1
}
1 + 2 + 3 + -------- + n
=
n(n + 1)
/2
(∴
xn - an
/x - a
= na
n - 1)
Solution
Q9.
Evaluate
5x - 1
/√(1 + x) - 1
Solution.
=
5x - 1
/√(1 + x) - 1
x
√(1 + x) + 1/√(1 + x) + 1
=
5x - 1
/x
(√(1 + x) + 1)
= (log 5) . 2 = 2 log 5
Solution
Q10.
Evaluate
(ex + e-x - 4)(x2 - 3x + 2)
/x - 1
Solution.
=
(ex + e-x - 4)(x - 1)(x - 2)
/x - 1
=
(e
x + e
-x - 4)(x - 2)
= (e +
1/e
- 4)(-1)
=
- (
e2 + 1/e
- 4)
Solution
