1.Algebra Booster

1.18 Algebra Booster
37. Observing that 1 3 = 1, 2 3 = 3 + 5, 3 3 = 7 + 9 + 1, 4 3 = 13
+ 15 + 17 + 19,
find a general formula for the cube of natural numbers.
[Roorkee, 1997]
38. Let a, b, c are the first three terms of a geometric series.
If the harmonic mean of a and b is 12 and that of b and
c is 36, find the first five terms of the series.
[Roorkee, 1998]
39. The sum of the infinite series is 162 and the sum of the
first n terms is 160. If the inverse of its common ratio is
an integer, find all possible values of the common ratio,
n and the first term of the series. [Roorkee, 1999]
Note No question asked in 2000.
40. The sum of three numbers in a GP is 42. If the first two
numbers are increased by 2 and third is decreased by 4,
the resulting numbers form an AP. Find the numbers of
the GP. [Roorkee, 2001]
41 If the sum of
1 1 1 1 1 1
1+ + + 1+ + + 1+ +
2 2 2 2 2 2
1 2 2 3 3 4
is equal to
Ê 1 ˆ
Á n - ˜ , n Œ N , find n.
Ë n¯
42. Find the value of
359
 k◊cos( k∞)
.
k = 0
43 Find the sum to n terms of
1 1
+º+ 1 + +
1999 2000
2 3
Ê 1ˆ Ê 1ˆ Ê 1ˆ
1+ 2Á1+ + 3 1+ + 4 1+ +º
Ë
˜
n¯ Á
Ë
˜ Á ˜
n¯ Ë n¯
44. Find the sum of
Ê2xˆ Ê x ˆ
f() x = Â sinÁ sin
n˜ Á n˜
.
Ë3 ¯ Ë3
¯
n=
1
2 2
45. If the roots of 10x 3 – cx 2 – 54x – 27 = 0 are in HP, find
the value of c.
LEVEL IV
(Tougher Problems For JEE-
Advanced)
1. In a triangle ABC, if cot A, cot B, cot C are in AP, prove
that a 2 , b 2 , c 2 are in AP.
2. The sum of the sequence of three distinct real numbers,
which are in GP is S 2 . If their sum is aS, show that
2 Ê1 ˆ
a Œ Á ,1 ˜ » (1,3) .
Ë3
¯
3. The sum of n terms of two arithmetic progressions are
in the ratio (7n + 1) : (4n + 17). Find the ratio of their
nth terms.
4. Solve the following equations for x and y:
log 10
x + log 10
x 1/2 + log 10
x 1/4 + ... = y
1+ 3+ 5 +º+ (2y
-1) 20
and
= .
4 + 7 + 10 +º+ (3y
+ 1) 7log10x
n
5. If a i
> 0 for every i in N such that ’ ai
= 1, prove that
i=
1
(1 + a 1
)(1 + a 2
)(1 + a 3
)…(1 + a n
) ≥ 2 n .
6. Obtain the sum of
n
1 2 4 2
+ + + … + .
2 4 2n
( x + 1) ( x + 1) ( x + 1) ( x + 1)
7. 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
common ratios 1 , 1 , 1 ,…,
1 , respectively, find
2 3 4 n + 1
2 2 2 2
the values of S1 + S2 + S3 + S2n
- 1.
8. If exp (sin 2 x + sin 4 x + sin 6 x + … ) In 2 satisfies
the equation x 2 – 9x + 8 = 0, find the value of
cos x
p
,0< x < .
cos x+
sin x 2
9. Find the sum of the following infinite series
Ê 2 -1ˆ Ê3- 2 2ˆ Ê5 2 - 7ˆ
1 + Á + Á ˜ +
Ë
˜
2 2 ¯ Ë 8 ¯ Á
Ë
˜
16 2 ¯
17 - 12 2
to
64
10. Find the natural number a for which
n
Â
k = 1
n
f( a + k) = 16(2 -1)
, where the function f satisfies
the relation f(x + y) = f(x) ◊ f(y) for all natural numbers
x and y and further f(1) = 2.
11. The real numbers 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.
12. Let x = 1 + 3a + 6a 2 + 10a 3 + …, |A| < 1, y = 1 + 4b +
10b 2 + 20b 3 + …, |b| < 1.
Find S = 1 + 3(ab) + 5(ab) 2 + … in terms of x and y.
13. Let
2 2
n
n+
4
(1 + x ) (1 + x) =Â a x . If a 1
, a 2
and a 3
are in
k
k = 0
arithmetic progression, find n.
14. Let cos (x – y), cos x and cos (x + y) are in HP, and
a, b and c are positive real numbers. If m is the value
of cos x secÁ
˜ and n is the minimum value of
Ê yˆ
Ë 2 ¯
Ê1 1 1ˆ
( a + b+ c)
Á + +
Ë
˜
a b c¯ , find the value of (n + m2 – 4).
15. Let x be the arithmetic mean, and y and z be the two
geometric means between any two positive numbers,
3 3
y + z
find .
xyz
k