11th Maths Weekly Test -2 ( Combinatorics and Mathematical Induction )

St. Britto Hr. Sec. School - Madurai

11th Maths Weekly Test -2 ( Combinatorics and Mathematical Induction )-Aug 2020

Time : 1.30 Hours

Part - A

Generic 2 - Answer All Question

If ⁿP_r = 11880 and ⁿC_r = 495, Find n and r.

Evaluate: ⁶P₅.

Find the number of ways of forming a committee of 5 members out of 7 Indians and 5 Americans, so that always Indians will be the majority in the committee.

Prove that ²⁴C₄ + \(\sum _{ r=0 }^{ 4 }{ { { (28-r) }_{ C } }_{ 3 } } \) =²⁹C₄.

A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected, if the team has no girls

Evaluate \(\frac { n! }{ r!(n-r)! } \) when for any n and r = 3.

Evaluate \(\frac { n! }{ r!(n-r)! } \) when n=6,r= 2

7 relatives of a man comprises 4 ladies and 3 gentlemen, his wife has also 7 relatives 3 of them are ladies and 4 gentlemen. In how many ways can they invite a dinner party of 3 ladies and 3 gentlemen, so that there are 3 of men's relative and 3 of the wives relatives?

How many words can be formed using the letter A thrice, the letter B twice and the letter C thrice?

There are 15 candidates for an examination. 7 candidates are appearing for mathematics examination while the remaining 8 are appearing for different subjects. In how many ways can they be seated in a row so that no two mathematics candidates are together?

Part - B

Generic 5 - Answer All Question

By the principle of mathematical induction, prove that for n > 1
\(1^3 +2^3 +3^3 + .. +n^3=\left[n(n+1)\over 2\right]^2\)

By the principle of mathematical induction, prove that, for all integers n \(\ge\)1,1+2+3+....+n=\({n(n+1)\over2}\) .

Using the mathematical induction, show that for any natural number n > 2,
\(\left(1-{1\over 2^2} \right)\left(1-{1\over 3^2} \right)\left(1-{1\over 4^2} \right)...\left(1-{1\over n^2} \right)={n+1\over 2n}\)

Using the mathematical induction, show that for any natural number n > 2
\({1\over 1+2}+{1\over 1+2+3}+{1\over 1+2+3+4}+...+{1\over 1+2+3..+n}={n-1\over n+1}\)

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11th Maths Weekly Test -2 ( Combinatorics and Mathematical Induction )

St. Britto Hr. Sec. School - Madurai

11th Maths Weekly Test -2 ( Combinatorics and Mathematical Induction )-Aug 2020

Part - A

Generic 2 - Answer All Question

If nPr = 11880 and nCr = 495, Find n and r.

Evaluate: 6P5.

Find the number of ways of forming a committee of 5 members out of 7 Indians and 5 Americans, so that always Indians will be the majority in the committee.

Prove that 24C4 + \(\sum _{ r=0 }^{ 4 }{ { { (28-r) }_{ C } }_{ 3 } } \) =29C4.

A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected, if the team has no girls

Evaluate \(\frac { n! }{ r!(n-r)! } \) when for any n and r = 3.

Evaluate \(\frac { n! }{ r!(n-r)! } \) when n=6,r= 2

7 relatives of a man comprises 4 ladies and 3 gentlemen, his wife has also 7 relatives 3 of them are ladies and 4 gentlemen. In how many ways can they invite a dinner party of 3 ladies and 3 gentlemen, so that there are 3 of men's relative and 3 of the wives relatives?

How many words can be formed using the letter A thrice, the letter B twice and the letter C thrice?

There are 15 candidates for an examination. 7 candidates are appearing for mathematics examination while the remaining 8 are appearing for different subjects. In how many ways can they be seated in a row so that no two mathematics candidates are together?

Part - B

Generic 5 - Answer All Question

By the principle of mathematical induction, prove that for n > 1 \(1^3 +2^3 +3^3 + .. +n^3=\left[n(n+1)\over 2\right]^2\)

By the principle of mathematical induction, prove that, for all integers n \(\ge\)1,1+2+3+....+n=\({n(n+1)\over2}\) .

Using the mathematical induction, show that for any natural number n > 2, \(\left(1-{1\over 2^2} \right)\left(1-{1\over 3^2} \right)\left(1-{1\over 4^2} \right)...\left(1-{1\over n^2} \right)={n+1\over 2n}\)

Using the mathematical induction, show that for any natural number n > 2 \({1\over 1+2}+{1\over 1+2+3}+{1\over 1+2+3+4}+...+{1\over 1+2+3..+n}={n-1\over n+1}\)

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If ⁿP_r = 11880 and ⁿC_r = 495, Find n and r.

Evaluate: ⁶P₅.

Prove that ²⁴C₄ + \(\sum _{ r=0 }^{ 4 }{ { { (28-r) }_{ C } }_{ 3 } } \) =²⁹C₄.

By the principle of mathematical induction, prove that for n > 1
\(1^3 +2^3 +3^3 + .. +n^3=\left[n(n+1)\over 2\right]^2\)

Using the mathematical induction, show that for any natural number n > 2,
\(\left(1-{1\over 2^2} \right)\left(1-{1\over 3^2} \right)\left(1-{1\over 4^2} \right)...\left(1-{1\over n^2} \right)={n+1\over 2n}\)

Using the mathematical induction, show that for any natural number n > 2
\({1\over 1+2}+{1\over 1+2+3}+{1\over 1+2+3+4}+...+{1\over 1+2+3..+n}={n-1\over n+1}\)