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QUESTION: 1

If in the expansion of (1 + x)^{m} (1 – x)^{n}, the co-efficients of x and x^{2} are 3 and – 6 respectively, then m is [JEE 99,2 ]

Solution:

(1+x)^{m} (1−x)^{n} = (1 + mx + m(m−1)x^{2}/2!)(1 − nx + n(n−1)x^{2}/2! )

= 1 + (m−n)x + [(n^{2} − n)/2 − mn + (m2 − m)/2] x^{2}

Given m − n = 3 or n = m − 3

Hence (n^{2} − n)/2 − mn + (m^{2 }− m)/2 = 6

⇒ [(m−3)(m−4)]/2 − m(m−3) + (m^{2} − m)/2 = −6

⇒ m2 − 7m + 12 − 2m2 + 6m + m^{2} − m + 12 = 0

⇒ − 2m + 24 = 0

⇒ m = 12.

QUESTION: 2

For 2 £ r £ n,

Solution:

QUESTION: 3

Find the largest co-efficient in the expansion of (1 + x)^{n}, given that the sum of co-efficients of the terms in its expansion is 4096. [REE 2000 (Mains)]

Solution:

We know that, the coefficients in a binomial expansion is obtained by replacing each variable by unit in the given expression.

Therefore, sum of the coefficients in (a+b)^n

= 4096=(1+1)^{n}

⇒ 4096=(2)^{n}

⇒ (2)^{12}=(2)^{n}

⇒ n=12

Here n is even, so the greatest coefficient is nCn/2 i.e., 12C6

QUESTION: 4

In the binomial expansion of (a - b)^{n}, n ³ 5, the sum of the 5th and 6th terms is zero. Then equals.

Solution:

It is given that T6+T5=0.

Hence nC4a^{n}−4b^{4}−nC5a^{n}−5b^{5}=0

nC4a^{n}−4b^{4}= nC5a^{n}−5b^{5}

nC4a = nC5b

n!b/(n−4)!4! = n!b/(n−5)!5!

= a/(n−4) = b/5

Therefore a/b=(n−4)/5

QUESTION: 5

Find the coefficient of x^{49} in the polynomial [REE 2001 (Mains), 3] where C_{r} = ^{50}C_{r}

Solution:

QUESTION: 6

The sum , (where = O if P < q) is maximum when m is [JEE 2002 (Scr.), 3]

Solution:

QUESTION: 7

(a) Coefficient of t^{24} in the expansion of (1 + t^{2})^{12} (1 + t^{12}) (1 + t^{24}) is [JEE 2003 (Scr.), 3]

(b) Prove that :

[JEE 2003 (Mains),2]

Solution:

(1+x)^{12} (1+x^{12}) (1+x^{24})

= [C_{0} + C_{1}x^{2} + C_{2}x^{4} + C_{3}x^{6} + C_{4}x^{8} +.......C_{12}x^{24}) (1 + x^{12} + x^{24} + x^{36})

= x^{24}(^{12}C_{0} + ^{12}C_{6} + ^{12}C_{12})

= 1 + ^{12}C_{6} + 1

= ^{12}C_{6} + 2

QUESTION: 8

^{n_1}C_{r} = (k^{2} – 3). ^{n}C_{r + 1}^{, }if k Î [JEE 2004 (Scr.)]

Solution:

Formula,

(n - 1)!/{r! × (n - 1 - r)!} = (k² - 3) × n!/(r + 1)!(n - r - 1)!

or, (n - 1)!/r! = (k² - 3) × n!/(r + 1)!

or, (n - 1)!/r! = (k² - 3) × n(n - 1)!/(r + 1)r!

or, 1/1 = (k² - 3) × n/(r + 1)

or, (r + 1)/n = (k² - 3)

we know, r and n are integers so, (r + 1)/n (0, 1]

So, 0 < (r + 1)/n ≤ 1

or, 0 < k² - 3 ≤ 1

or, 3 < k² ≤ 4

or, √3 < k ≤ 2 , -2 ≤ k < -√3

Hence, k (√3, 2]

QUESTION: 9

^{n-1}C_{r} = (k^{2} - 3). ^{n}C_{r + 1} [JEE 2004 (Scr.)]

Solution:

(n - 1)!/{r! × (n - 1 - r)!} = (k² - 3) × n!/(r + 1)!(n - r - 1)!

or, (n - 1)!/r! = (k² - 3) × n!/(r + 1)!

or, (n - 1)!/r! = (k² - 3) × n(n - 1)!/(r + 1)r!

or, 1/1 = (k² - 3) × n/(r + 1)

or, (r + 1)/n = (k² - 3)

we know, r and n are integers so, (r + 1)/n (0, 1]

so, 0 < (r + 1)/n ≤ 1

or, 0 < k² - 3 ≤ 1

or, 3 < k² ≤ 4

or, √3 < k ≤ 2 , -2 ≤ k < -√3

hence, k (√3, 2]

QUESTION: 10

The value of

is, where = ^{n}C_{r } [JEE 2005 (Scr.)]

Solution:

QUESTION: 11

The number of seven digit integers, with sum of the digits equal to 10 and formed by using the digits 1, 2 and 3 only, is [JEE 2009]

Solution:

QUESTION: 12

For r = 0, 1, ...., 10 let A_{r}, B_{r}, C_{r} denote, respectively, the coefficient of x^{r} in the expansions of (1 + x)^{10}, (1 + x)^{20} and (1 + x)^{30}. Then is equal to [JEE 2010]

Solution:

QUESTION: 13

Let a_{n} denote the number of all n-digit positive integers formed by the digits 0, 1 or both such that no consecutive digits in them are 0. Let b_{n} = the number of such n-digit integers ending with digit 1 and c_{n} = the number of such n-digit integers ending with digit 0. [JEE 2012]

Which of the following is correct ?

Solution:

QUESTION: 14

Let a_{n} denote the number of all n-digit positive integers formed by the digits 0, 1 or both such that no consecutive digits in them are 0. Let b_{n} = the number of such n-digit integers ending with digit 1 and c_{n} = the number of such n-digit integers ending with digit 0. [JEE 2012]

The value of b_{6} is

Solution:

### CBSE Previous Year Questions - Real Numbers

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