1.Algebra Booster

1.24 Algebra Booster
22. No questions asked in 1989.
x Ê x 7ˆ
23. If log32, log 3(2 -5), log3Á2
- ˜ are in AP, determine
the value of x. [IIT-JEE, 1990]
Ë 2¯ 24. Let p be the first of the n arithmetic means between
two numbers and q the first of the n harmonic means
between the same numbers. Show that q does not lie
2
Ên
+ 1ˆ
between p and Á p
Ën
- 1
˜ . [IIT-JEE, 1991]
¯
25. If S 1
, S 2
, S 3
, …, S n
are the sums of infinite geometric
series whose first terms are {1, 2, 3, …, n} and whose
Ï1 1 1 1 ¸
common ratios are Ì , , ,…, ˝
Ó2 3 4 n + 1˛ , respectively,
2 2 2
find the value of S1 + S2 +º+ S2n
- 1 .
[IIT-JEE, 1991]
26. Let the harmonic mean and geometric mean of two
positive numbers be in the ratio 4 : 5. Then the two
numbers in the ratio… [IIT-JEE, 1992]
p
2
27. For 0 < j < , if cos n
2
x = Â j , y = Â sin n j and
2
n=
0
2n
= Â cos
2n
sin , then
n=
0
z j j
n=
0
(a) xyz = xz + y (b) xyz = xy + z
(c) xyz = x + y + z (d) xyz = yz + x
[IIT-JEE, 1993]
28. If ln (a + c), ln (a – c) and ln (a – 2b + c) are in AP, then
(a) a, b, c are in AP (b) a 2 , b 2 , c 2 are in AP
(c) a, b, c are in GP (d) a, b, c are in HP
[IIT-JEE, 1994]
29. No questions asked in 1995.
30. For any odd integer n ≥ 1,
n 3 – (n – 1) 3 + … + (–1) n – 1 ◊ 1 3 = …
[IIT-JEE, 1996]
31. The real number x 1
, x 2
and x 3
satisfying the equation
x 3 – x 2 + bx + g = 0 are in AP. Find the intervals in which
b and g lie. [IIT-JEE, 1996]
32. Let p and q be the roots of the equation x 2 – 2x + A = 0
and let r and s be the roots of x 2 – 18x + B = 0. If
p < q < r < s are in arithmetic progression, then A = …
and B = … [IIT-JEE, 1997]
33. Let x be the arithmetic mean and y and z be the two
geometric means between any two positive numbers .
Then
y
3 3
+ z
xyz
= …. [IIT-JEE, 1997]
34. Let T r
be the rth term of an AP for r = 1, 2, 3. If for
some positive integers m and n, we have T m
= 1/n and
T n
= 1/m, then T mn
equals
1
1 1
(a)
(b) +
mn
m n
(c) 1 (d) 0 [IIT-JEE, 1998]
35. If x > 1, y > 1 and z > 1 are in GP, then
1 1 1
, ,
1+ ln x 1+ ln y 1+
ln z
are in
(a) AP (b) GP (c) HP (d) None
[IIT-JEE, 1998]
36. Let a 1
, a 2
, …, a 10
be in AP and h 1
, h 2
, …, h 10
be in HP. If
a 1
= 2 = h 1
and a 10
= 3 = h 10
, the value of a 4
h 7
is
[IIT-JEE, 1999]
(a) 2 (b) 3 (c) 5 (d) 6
37. The harmonic mean of the roots of the equation
(5 +
2
2) x -(4 + 5) x+ (8 + 2 5) = 0 is
(a) 2 (b) 4 (c) 6 (d) 8
[IIT-JEE, 1999]
38. Consider an infinite series with first term a and common
ratio r. If its sum is 4 and the second term is 3/4,
then
(a) a = 4/7, r = 3/7 (b) a = 2, r = 3/8
(c) a = 3/2, r = 1/2 (d) a = 3, r = 1/4
[IIT-JEE, 2000]
39. The fourth power of the common difference of an AP
with integer entries in added to the product of any four
consecutive terms of it, prove that the resulting sum is
square of an integer. [IIT-JEE, 2000]
40. Let T n
denotes the number of triangles which can be
formed using the vertices of a regular polygon of n-
sides. If T n+1
– T n
= 21, then n equals
(a) 5 (b) 7 (c) 6 (d) 4
[IIT-JEE, 2001]
41. Let the positive numbers a, b, c, d be in AP, then abc,
abd, acd, bcd are
(a) not in AP/GP/HP (b) in AP
(c) in GP
(d) in HP
[IIT-JEE, 2001]
42. If the sum of the first 2n terms of the AP = {2, 5, 8, …}
is equal to the sum of the first n terms of the AP =
{57, 59, 61, …}, then n equals
(a) 100 (b) 12 (c) 11 (d) 13
[IIT-JEE, 2001]
43. Let a 1
, a 2
, a 3
, …, a n
be positive real number in GP for
each a, let A n
, G n
, H n
, be the arithmetic mean, geometric
mean, harmonic mean respectively. Find an expression
for the geometric mean G 1
, G 2
, G 3
, …, G n
in terms of
A 1
, A 2
, A 3
, …, A n
and H 1
, H 2
, …, H n
[IIT-JEE, 2001]
44. If a 1
, a 2
, a 3
, …, a n
are positive real numbers whose
product is a fixed number c, the minimum value of
a 1
+ a 2
+ … + a n–1
+ 2a n
is [IIT-JEE, 2002]
(a) n(2c) 1/n (b) (n + 1)c 1/n
(c) 2n.c 1/n (d) (n + 1)(2c) 1/n
45. Suppose a, b and c are in AP and a 2 , b 2 and c 2 are in GP.
3
If a < b < c and a + b+ c= , the value of a is
2
1 1
(a) (b) (c) 1 + 1 (d) 1 -
1
2 2 2 3 2 2 2 2
[IIT-JEE, 2002]