Home » Ask & Discuss » Mathematics. » Algebra « Back to Discussion

Algebra

Blazing goIITian

Joined:	13 Aug 2008
Post:	1313

14 Feb 2009 15:58:15 IST

0 People liked this

1354

CMI sample paper 2 Questions :::

None

Q 1. Which is the smallest number with exactly 12 divisors?? (If n is a positive integer, 1 and n are also counted as divisors of n)

(A) 72 (B) 2¹¹ (C) 12 (D) 48 (E) None of these

Answer : (E) correct number is 60 ( how do u find this? )

Q 2. ( Subjective) If (1 + x)ⁿ = c₀ + c₁x + c₂x² + .... + c_nxⁿ, show that

c₀² + 2c₁² + ... + (n+1) c_n² =

Share this article on:

Comments (7)

Rahul Duggal

Scorching goIITian

Joined: 10 May 2008

Posts: 289

14 Feb 2009 18:52:29 IST

Like 1 people liked this

Ans 1

12 = 4*3 = 2 * 2 * 3 -------------(1)

Since we know for a number like 30 we can calculate number of divisors by writing 30 as 3*2*5 .

Number of divisors is (3⁰+1)(2⁰+1)(5⁰+1) = 8

Also we know from (1) the number has to have 3 prime factors. let p, q, r be the powers of the prime factors. to keep number minimum we choose 2,3,5 as the prime factors as they are the three smallest primes.

so required number is

(2^p)(3^q)(5^r)

also (p+1) (q+1) (r+1) = 12

For smallest number p = 2, q = 1, r = 1 (Since 2 must have highest power 2, and 3 and 5 should have lower powers to keep number as small as possible)

So our number is 60

Decoder

Blazing goIITian

Joined: 1 Apr 2007

Posts: 1084

14 Feb 2009 19:06:37 IST

Like 0 people liked this

@mirka... gud ques. ...decoder the chef is back...see how it goes... will look very easy...it is actually sum { (r+1)Cr.Cr }write it as sum{ r.Cr.Cr} + 2nCn ....as sum Cr^2 is 2nCn...(bino-bino series)...now by reversing techiniques....replace r by n-r...and then add both equaltion ...we get ... n/2 sum{Cr^2}so total sum is ...2nCn { n/2 +1 }or (2n-1)! (n+2) / n! .(n-1)!...cheers..!!!

Decoder

Blazing goIITian

Joined: 1 Apr 2007

Posts: 1084

14 Feb 2009 19:07:18 IST

Like 1 people liked this

Kevin Arnold

Cool goIITian

Joined: 15 Jan 2009

Posts: 73

14 Feb 2009 19:11:49 IST

Like 1 people liked this

(1+x)ⁿ=summation (C_rx^r)

x(1+x)ⁿ=summation (C_rx^r+1)

xn(1+x)^n-1+(1+x)ⁿ=summ.(r+1C_rx^r)--------2)

(1+x)ⁿ/xⁿ = summ.(C_r(1/x)^r)------------1)

now multiplying the 2 eqn.s we have the reqd. series as the coefficient of x⁰

which is the given result...

Rahul Duggal

Scorching goIITian

Joined: 10 May 2008

Posts: 289

30 Jul 2009 16:28:04 IST

Like 0 people liked this

wrong post

Bipin Dubey

Forum Expert

Joined: 23 Jan 2007

Posts: 7942

30 Jul 2009 18:46:19 IST

Like 2 people liked this

$(1+x)^{n}=C_{0}+C_{1}x+C_{2}x^{2}+....+C_{n}x^{n}$

$x(1+x)^{n}=C_{0}x+C_{1}x^{2}+C_{2}x^{3}+....+C_{n}x^{n+1}$

Differentiate both sides wrt x :

$(1+x)^{n}+nx(1+x)^{n-1}=C_{0}+2C_{1}x+3C_{2}x^{2}+....+(n+1)C_{n}x^{n}$ ............ (1)

Consider : $(x+1)^{n}=C_{0}x^{n}+C_{1}x^{n-1}+C_{2}x^{n-2}+....+C_{n}$ .......... (2)

Multiply (1) and (2) : coefficient of xⁿ RHS is the required sum of series, so find coefficient of xⁿ in LHS

LHS = (1+x)²ⁿ + nx(1+x)^2n-1

coefficient of xⁿ = ²ⁿC_n + n^2n-1C_n-1

Mirka

Blazing goIITian

Joined: 13 Aug 2008

Posts: 1313

30 Jul 2009 22:00:17 IST

Like 0 people liked this

wow... Thanks sir!!

Quick Reply

Reply

	Some HTML allowed.
	Keep your comments above the belt or risk having them deleted.
	Signup for a avatar to have your pictures show up by your comment
	If Members see a thread that violates the Posting Rules, bring it to the attention of the Moderator Team