1.Algebra Booster

1.20 Algebra Booster
7. If a, b and c are real numbers such that a + b + c =
1 1 1 10
3 and + + = , find the value of
a + b b+ c c+
a 3
a b c
+ + .
b + c c + a a + b
Ê1 1 1 1 ˆ
log
5 Á + + +º+
Ë4 8 16 n -1˜
4.2 ¯
1
8. If
Ê ˆ
b n = Á ˜
, find ([ lim ( b n)] + 3) .
Ë5¯
n
9. If a, x, y, z and b are in AP, the value of (x + y + z) is 15
while 1 + 1 + 1 is 5 if a, x, y, z and b are in HP, find
x y z 3
the value (a – b – 2).
10. If (1 – p)(1 + 3x + 9x 2 + 27x 3 + 8ax 4 + 243x 5 ) = (1 – p 6 ),
Ê p ˆ
p π 1, find the value of Á + 2
Ë
˜
x ¯ .
n+
5
2
11. If  4( x - 3) = An + Bn+
C, find the value of (A +
x=
5
B – C – 4).
n
12. If  rr ( + 1)(2r+
3) = an 4 + bn 3 + cn 2 + dn + e,
r = 1
find the value of (a + c + 1).
13. If x and y be two positive real numbers such that x 2 + y 2
= 8, find the maximum value of (x + y).
n Ê k ˆ
2 4 3 2
14. If ÂÁÂ
m ˜ = an + bn + cn + dn + e, find the
k= 1Ëm=
1 ¯
value of 12(a + d).
15. Let a, b and c are positive real numbers such that ab +
bc + ca = 12, then find the greatest value of abc.
Passage I
Comprehensive Link Passage
(For JEE-Advanced Examination Only)
Let a, b and g be the roots of
where a, b > 0 and a > b > g .
(i) Then b is
x – a x - b a b
+ = +
b a x -b x - a
,
2 2
(a) a + b
(b)
a + b
a + b
(c) 0 (d)
a + b
2 2
a + b
(ii) If a = 2b, the maximum value of the area of a triangle
whose perimeter 3a, is
(a) a 2 (b)
3 2
4 a (c) 2
2
a 3 (d) 2 3a
(iii) If a – b – g = c, then
(a) a, b, c are in AP (b) a, c, b are in AP
(c) a, b, c are in HP (d) a, c, b are in HP
Passage II
Suppose A 1
, A 2
,…, A n
be AMs; G 1
, G 2
, …, G n
be GMs; H 1
, H 2
,
…, H n
be HMs between two positive real numbers a and b.
(i) A n
, G n
, H n
are in
(a) AP (b) GP (c) HP (d) AGP
(ii) H 1
is
(a)
a + (2 n – 1) b
a(2n+ 1) + b
(b)
2n
2n
(c)
a(2n+ 1) -b
2nab
(d)
2n
a + (2 n – 1) b
(iii)
H1
+ a H2n-1+
b
+
H1-a H2n-1-b
is
(a) 2(n – 1) (b) 4n (c) 4n – 2 (d) 4n + 2
Passage III
Suppose a, b and c are the sides of a triangle, which are in GP.
(i) If the common ratio, r of the series is less than unity,
then r is
(a)
Ê 1 ˆ
Ê
n,
5 - 1ˆ
Á
Ë 2 + 1
˜ (b) Á0,
¯
Ë
˜
2 ¯
(c)
Ê 5 + 1ˆ
Ê
Á0,
Ë
˜ (d)
5 - 1 ˆ ,1
4 ¯
Á
Ë
˜
2 ¯
(ii) If log a – log 2b, log 2b – log 3c and log 3c – log a are
in AP, the least side of the triangle is
(a) a (b) b (c) c (d) a = b = c
(iii) The greatest angle of the triangle is
(a) 135° (b) 90°
-1 Ê1ˆ
(c) 120°
(d) p - cos Á
Ë
˜
3¯
Passage IV
Let a, b and g are the real roots of ax 3 + 3bx 2 + 3cx + d = 0
such that a π b π g.
(i) If roots are in AP, the value of 2a 3 is
(a) 3abc + 2a 2 c (b) 3abc – ac 2
(c) 3abc + ac 2 (d) 3abc – a 2 c
(ii) If roots are in GP, the value of a c is
3
(a)
Êaˆ
Á ˜ (b)
Êb
ˆ Êd
ˆ
Ëb¯
Á
Ë
˜ (c) Á ˜ (d)
Êb
ˆ
d ¯ Ëb
¯ Á
Ë
˜
d ¯
(iii) If roots are in HP and a = b = c = 1, the value of d is
(a) 1, 2 (b) 1 (c) 2 (d) –2
Passage V
Let x 1
, x 2
, x 3
,…, x n
are distinct real numbers in HP.
(i) Which of the following is true?
(a) x 1
x 4
> x 2
x 3
(b) x 1
x 4
< x 2
x 3
(c) x 1
x 2
> x 3
x 4
(d) x 1
x 2
< x 3
x 4
(ii) Which of the following is true?
(a) x 5
+ x 8
> x 6
+ x 7
(b) x 5
+ x 8
< x 6
+ x 7
(c) x 5
+ x 6
> x 8
+ x 7
(d) x 5
+ x 6
< x 8
+ x 7
(iii) x 1
x 2
+ x 2
x 3
+ x 3
x 4
+ x 1
x 5
is
(a) 4x 1
x 5
(b) 2x 1
x 5
(c) 3x 1
x 5
(d) 5x 1
x 5
3