Summation of Combination Terms MCQ Quiz in தமிழ் - Objective Question with Answer for Summation of Combination Terms - இலவச PDF ஐப் பதிவிறக்கவும்

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Latest Summation of Combination Terms MCQ Objective Questions

Top Summation of Combination Terms MCQ Objective Questions

Summation of Combination Terms Question 1:

Comprehension:

Direction: Consider (1+x+x2)2n=r=04nar.x, where a0, a1, a2, ....a4n are real numbers and n is positive integer on the basis of above information, answer the following question.

The correct statement is

  1. ar = an - r, 0 ≤ r ≤ n
  2. an - r = an + r, 0 ≤ r ≤ n
  3. ar = a2n - r, 0 ≤ r ≤ 2n
  4. ar = a4n - r, 0 ≤ r ≤ 4n

Answer (Detailed Solution Below)

Option 4 : ar = a4n - r, 0 ≤ r ≤ 4n

Summation of Combination Terms Question 1 Detailed Solution

Calculation:

Given,

The equation is (1+x+x2)2n

We are asked to determine the correct relationship between the coefficients ar in the expansion of (1+x+x2)2n.

The expansion of (1+x+x2)2n is given by:

(1+x+x2)2n=r=04narxr

From the structure of the binomial expansion of (1+x+x2)2n, we can observe that the coefficients ar follow a symmetry. Specifically, the coefficients on opposite ends of the expansion are equal. That is:

ar=a4nr,0r4n

This symmetry implies that:

a0=a4n

a1=a4n1

a2=a4n2

and so on.

Therefore, the correct statement is:

ar=a4nr,0r4n

Hence, the correct answer is Option (4) 

Summation of Combination Terms Question 2:

Comprehension:

Direction: Consider (1+x+x2)2n=r=04nar.x, where a0, a1, a2, ....a4n are real numbers and n is positive integer on the basis of above information, answer the following question.

The value of a4n-1 is

  1. 2n
  2. 2n2 + 4n
  3. 2n + 3
  4. 2n2 + 3n

Answer (Detailed Solution Below)

Option 1 : 2n

Summation of Combination Terms Question 2 Detailed Solution

Calculation:

Given,

The equation is (1+x+x2)2n

We need to find the value of a4n1, which is the coefficient of x4n1 in the expansion of (1+x+x2)2n.

We first replace x by1x in the given expression to get the equation:

(1+x+x2)2n=r=04narxr

We then modify the expression as follows:

(1+x+x2)2n=r=04narxr=r=04narx4nr

By comparing the coefficients, we obtain:

ar=a4nr

Thus, the value of a4n1 is equal to the coefficient of x  in the expansion of (1+x+x2)2n, which is 2nC1.

Hence, the value of a4n1 is 2n.

Hence, the correct answer is Option (1) 

Summation of Combination Terms Question 3:

Comprehension:

Direction: Consider (1+x+x2)2n=r=04nar.x, where a0, a1, a2, ....a4n are real numbers and n is positive integer on the basis of above information, answer the following question.

The value of a2 is

  1. 4n+1C2
  2. 3n+1C2
  3. 2n+1C2
  4. n+1C2

Answer (Detailed Solution Below)

Option 3 : 2n+1C2

Summation of Combination Terms Question 3 Detailed Solution

Calculation:

Given,

The equation is (1+x+x2)2n

We need to find the value of a2, which is the coefficient of x2 in the expansion of (1+x+x2)2n.

The equation is expanded as follows:

(1+x+x2)2n=1+2nC1(x+x2)+2nC2(x+x2)2+

We are interested in the coefficient of x2.

From the binomial expansion, we have:

a2=2nC1+2nC2

a2=2n+1C2

Hence, the correct answer is Option (3) 

Summation of Combination Terms Question 4:

The value of 1n1(x+3)(x+4) is

  1. nn+2
  2. 2nn+1
  3. n4(n+4)
  4. n2(n+2)

Answer (Detailed Solution Below)

Option 3 : n4(n+4)

Summation of Combination Terms Question 4 Detailed Solution

Given:

Series is 1n1(x+3)(x+4)

Concept Used:

1(a)(b)=1(a)1(b), where(a<b)

Calculation:

1n1(x+3)(x+4)=1(1+3)(1+4)+1(2+3)(2+4)+..........+1(n+3)(n+4)

14.5+15.6+..........+1(n+3)(n+4)

1415+1516..........+1(n+3)1(n+4)

141(n+4)

n+444.(n+4)

The value of 1n1(x+3)(x+4)=n4.(n+4)

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