1.Algebra Booster

1.16 Algebra Booster
51. In a series, if t
(a)
(c)
52. If 1 + l + l 2 + l 3 + … + l n
= (1 + l)(1 + l 2 )(1 + l 4 )(1 + l 8 )(1 + l 16 ),
the value of n is
(a) 15 (b) 31 (c) 32 (d) 16
53. If log 2
(a + b) + log 2
(c + d) ≥ 4, the minimum value of
(a + b + c + d) is
(a) 2 (b) 4 (c) 8 (d) 16
54. An infinite GP has first term x and sum ‘5’ then
(a) x < –10 (b) –10 < x < 0
(c) 0 < x < 10 (d) x > 10
55. If a and b are the roots of ax 2 + bx + c = 0 and a + b,
a 2 + b 2 , a 3 + b 3 are in GP and D = b 2 – 4ac, then
(a) cD = 0 (b) cb π 0 (c) bD = 0 (d) Dπ 0
56. If a > 0, b > 0, c > 0 and s = a + b + c, then
57. If
ÈÊ
1 1 1 9
ÍÁ
ÎËs -a s -b s -c s
(a) less than 0 (b) equal to 0
(c) greater than 0 (d) less than equal to 0
58. If U
(a) AP (b) GP (c) HP (d) AGP
(a) 2 –n + 2n – 1 (b) 2 n – 2n – 1
(c) 2 –n – 2n + 1 (d) 2 –n – 2n + 2
59. If a, b, c are in GP, x, y be the AMs between a and b; and
b and c, respectively, the value of
(a) 2 (b) 4 (c) 3 (d) None
60. The sum of n terms of the series
1+ 5 + 9 + 13 + …
(a) 4n 2 – 3n
(c) n 2 + 2
61. The sum of all the product of the first n positive integers,
taken two at a time, is
(a)
1
(b)
2
( 1)( – 2)
48 n n n
1 ( 1)( 2)( 5)
6 nn+ n+ n+
20
n
n = , the sum of  tn
is
(c)
( n + 1)!
n=
1
Ê 1 ˆ
Á1 -
Ë
˜
(b)
Ê 1 ˆ
(d)
1 -
20! ¯
Á
Ë
˜
21! ¯
Ê 1 ˆ
1
Á1
+
Ë
˜
(d)
Ê ˆ
1 +
21! ¯
Á
Ë
˜
20! ¯
1
(a) (3 n 1
- 1)
(b) (3
n + 1 n
- 3) +
4
4 2
1
(c) (3
n + 1 n 1
- 2) + (d) (3
n + 1 n
- 2) +
4 3 4 4
ˆ ˘
+ + ˜ - ˙ is
¯ 2 ˚
LEVEL III
Problems for JEE-Advanced
1. Prove that the value of the expression
p /2
Ê
2
sin nxˆ
In
= Ú Á dx
2 ˜ , then I 1
, I 2
, I 3
, … are in
Ê 1ˆÊ 1 ˆ Ê 1ˆÊ 1 ˆ
sin x
0
Ë ¯
Á1+ 1+ + 2+ 2+
Ë
˜Á 2 2
w¯Ë ˜
w ¯
Á
Ë
˜Á
w¯Ë ˜
w ¯
1 1 1 1
n
n
Ê 1
+ 3+ 3 + + … + n + n +
ˆ
2 2
n = Â Á
Ë n ˜ , then Un
n=
0 2 ¯
 is
w w
w w
n=
1
2
nn ( + 2)
=
3
2. Prove that the value of the expression
1 ◊ (2 – w)(2 – w 2 ) + 2 ◊ (3 – w)(3 – w 2 ) + …
Êa cˆÊb bˆ
Á + +
Ë x y
˜Á
¯Ë x y
˜
¯ is
+ (n – 1)(n – w)(n – w 2 )
2
Ênn
( + 1) ˆ
= Á - n
Ë
˜
2 ¯
3. Prove that the sum of the series
2 3
Ê 4n+ 1ˆ Ê 4n+ 1ˆ Ê 4n+
1ˆ
Á
Ë4 n –3
˜
¯
Á
Ë4 n –3
˜
¯
Á
Ë4 n –3
˜ is
n ◊ 1 + (n – 1) ◊ 2 + (n – 2) ◊ 3 + … + 1 ◊ n
¯
(b) 4n 2 + 3n
= 1 ( 1)( 2)
6 nn+ n+
(d) n 2 + 3n
1 ( 1)( 1)(3 2)
24 nn- n+ n+
d e f
that , , are in AP.
a b c -
1
nn ( –1)( n+ 2)
12
62. The nth term of the series 2 + 5 + 12 + 31 + 86 +... is
(a) n + 3 n–1 (b) (n – 1) + 3 n–2
(c) (n + 1) + 3 n (d) (n – 2) + 3 n
63. The sum to n terms of the series 2 + 5 + 14 + 41 + … is
64. If the numbers a, b, c, d and e are in AP, the value of
a – 4b + 6c – 4d + e is
(a) 1 (b) 2 (c) 0 (d) –2
65. If a 3 + b 3 + 6abc = 8c 2 and w is a cube root of unity, then
(a) a, c, b are in AP
(b) a, c, b are in GP
(c) a + bw – 2cw 2 = 0
(d) a + bw – 2cw = 0
Ê ˆÊ ˆ Ê ˆÊ ˆ
Á
Ë
˜Á
¯Ë
˜
¯
Á
Ë
˜Á
¯Ë
˜
¯
4. If a, b, c are three distinct real numbers in GP and
a + b + c = xb, prove that x £ –1 or x ≥ 3.
5. If a, b, c are in GP and the equations ax 2 + 2bx
+ c = 0 and dx 2 + 2ex + f = 0 have a common root, show
6. If a 1
, a 2
, a 3
, a 4
, …, a n
are in HP, prove that
a 1
a 2
+ a 2
a 3
+ a 3
a 4
+ … a n–1
= (n – 1)a 1
a n
.