IIT JEE , AIEEE Forum - Mathematics , Algebra

joyfrancis

1) if

,

be the roots of x2-3x+a=0 and

,

the roots of x2-12x+b=0 the numbers

,

and

form an increasing G.P then

a)a=3,b=12

b)a=12,b=3

c)a=2,b=32

d)a=4,b=16

2) the nth term of the series 3,root3,1,........is 1/243 then n is?

3) if 1/b-a + 1/b-c + = 1/a + 1/c then a,b,c are in?

4) 2^1/4*4^1/8*8^1/16.16^1/32.........is ?

5) Sum of 1/2,1/3,1/6.......9 terms

6) If H is the Harmonic mean between P and Q then (H/P) + (H/Q) = ?

Avinash_Bhat

1)

and

are the roots of x² - 3x + a = 0. SO :

+

= 3

= a

and

are the roots of x² - 12x + b = 0. SO :

+

= 12

= b

,

are in G.P. Let

= p ;

= pq ;

= pq² ;

= pq³

ie, p + pq = p(1 + q) = 3 --------- (1)

pq² + pq³ = pq²(1 + q) = 12 ---------- (2)

Solving (1) and (2) to get q = 2 and p = 1.

SO : a = p²q = 2 & b = p²q⁵ = 32

Avinash_Bhat

3)

1/(b - a) + 1/(b - c) = 1/a + 1/c

ie, 1/(b - a) - 1/c = 1/a - 1/(b - c)

=> (c - b + a)/(bc - ac) = (b - c - a)/(ab - ac)

=> (a - b + c)/(bc - ac) = - (a - b + c)/(ab - ac)

=> bc - ac = ac - ab

=> b = 2ac/(a + c)

=> a, b, c in H.P.

Avinash_Bhat

6)

H.M. between P and Q = H = 2PQ/(P + Q)

SO : (H/P) + (H/Q) = 2PQ / P(P + Q) + 2PQ / Q(P + Q)

= (2Q + 2P) / (P + Q)

= 2

Avinash_Bhat

4)

2^1/4 * 4^1/8 * 8^1/16 * ..............

= 2^1/4 * 2^2/8 * 2^3/16 * ...............

= 2^{1/4 + 2/8 + 3/16 + ................}

= 2^S

S = 1/4 + 2/8 + 3/16 + .............. -

S/2 = 1/8 + 2/16 + ..............

=> S/2 = 1/4 + 1/8 + 1/16 + ...............(Infinite G.P.)

=> S = 1

SO : 2^S = 2¹ = 2

kishore.subramanian.b

2. T_n=a.r^n-1

The give series is 3,

3,1, ..........,1/243

a=3 r=1/

3 Tn=1/243

hence 1/243=3.(1/

3)^n-1

3^-6=(1/3)^(n-1)/2

i.e. -6= -(n-1)/2

12=n-1

n=13

give me the ratings dude.

Avinash_Bhat

Take a thumbsup, Kishore.

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