11th Maths Monthly Test-4 ( Combinatorics and Mathematical Induction )

St. Britto Hr. Sec. School - Madurai

11th Maths Monthly Test-4 ( Combinatorics and Mathematical Induction )-Aug 2020

Time : 2.00 Hours

Part - A

Generic 2 - Answer All Question

A box contains 5 different red and 6 different white balls. In how many ways 6 balls be selected so that are atleast 2 balls of each colour.

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 atleast one boy and one girl

Let N denote the number of days. If the value of N! is equal to the total number of hours in N days then find the value of N?

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?

Four children are running a race.
(i) In how many ways can the first two places be filled?
(ii) In how many different ways could they finish the race?

There are 15 persons in a party and if each 2 of them shakes hands with each other, how many handshakes happen in the party?

In rating 20 brands of cars, a car magazine picks a first, second, third, fourth and fifth best brand and then 7 more as acceptable. In how many ways can it be done?

A coin is tossed 8 times,
(i) How many different sequences of heads and tails are possible?
(ii) How many different sequences containing six heads and two tails are possible?

In how many ways 5 boys and 4 girls can be seated in a row so that no two girls are together.

How many strings can be formed from the letters of the word ARTICLE, so that vowels occupy the even Places?

Part - B

Generic 3 - Answer All Question

If the letter of the word 'RACHIT' are arranged in all possible ways as listed in dictionary, then what is the rank of the word 'RACHIT'?

If \({n!\over 3!(n-4)!}and {n!\over 5!(n-5)!}\) are in the ratio 5 : 3 find the value of n.

Out of 18 points in a plane, no three are in the same line except five points which are collinear. Find the number of lines that can be formed joining the points.

How many 4 - digit even numbers can be formed using the digits 0, 1, 2, 3 and 4, if repetition of digits are not permitted?

A box contains two white balls, three black balls and four red balls. In how many ways can three balls be drawn from the box, if atleast one black ball is to be included in the draw?

Part - C

Generic 5 - Answer All Question

n2 < 2ⁿ, for all natural numbers n \(\ge\) 5.

Prove that using the Mathematical induction \(sin (α) + sin\left(\alpha+{\pi\over 6}\right)+ sin\left(\alpha+{2\pi\over 6}\right)+...+ sin\left(\alpha+{(n-1)\pi\over 6}\right)={sin\left[\alpha+{(n-1)^\pi\over 12}\right]\times sin\left(n\pi\over12\right)\over sin\left(\pi\over 12\right)}\)

If 22 P_r+1:20 P_r+2=11 : 52 find r.

If ⁿP_r =ⁿP_r+1and ⁿC_r = "C_r-1, find the values of nand r.

By the principle of mathematical induction, prove that, for all integers n≥1,
1²+2²+3²+...n² =\(\frac { n(n+1)(2n+1) }{ 6 } \).

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11th Maths Monthly Test-4 ( Combinatorics and Mathematical Induction )

St. Britto Hr. Sec. School - Madurai

11th Maths Monthly Test-4 ( Combinatorics and Mathematical Induction )-Aug 2020

Part - A

Generic 2 - Answer All Question

A box contains 5 different red and 6 different white balls. In how many ways 6 balls be selected so that are atleast 2 balls of each colour.

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 atleast one boy and one girl

Let N denote the number of days. If the value of N! is equal to the total number of hours in N days then find the value of N?

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?

Four children are running a race. (i) In how many ways can the first two places be filled? (ii) In how many different ways could they finish the race?

There are 15 persons in a party and if each 2 of them shakes hands with each other, how many handshakes happen in the party?

In rating 20 brands of cars, a car magazine picks a first, second, third, fourth and fifth best brand and then 7 more as acceptable. In how many ways can it be done?

A coin is tossed 8 times, (i) How many different sequences of heads and tails are possible? (ii) How many different sequences containing six heads and two tails are possible?

In how many ways 5 boys and 4 girls can be seated in a row so that no two girls are together.

How many strings can be formed from the letters of the word ARTICLE, so that vowels occupy the even Places?

Part - B

Generic 3 - Answer All Question

If the letter of the word 'RACHIT' are arranged in all possible ways as listed in dictionary, then what is the rank of the word 'RACHIT'?

If \({n!\over 3!(n-4)!}and {n!\over 5!(n-5)!}\) are in the ratio 5 : 3 find the value of n.

Out of 18 points in a plane, no three are in the same line except five points which are collinear. Find the number of lines that can be formed joining the points.

How many 4 - digit even numbers can be formed using the digits 0, 1, 2, 3 and 4, if repetition of digits are not permitted?

A box contains two white balls, three black balls and four red balls. In how many ways can three balls be drawn from the box, if atleast one black ball is to be included in the draw?

Part - C

Generic 5 - Answer All Question

n2 < 2n, for all natural numbers n \(\ge\) 5.

Prove that using the Mathematical induction \(sin (α) + sin\left(\alpha+{\pi\over 6}\right)+ sin\left(\alpha+{2\pi\over 6}\right)+...+ sin\left(\alpha+{(n-1)\pi\over 6}\right)={sin\left[\alpha+{(n-1)^\pi\over 12}\right]\times sin\left(n\pi\over12\right)\over sin\left(\pi\over 12\right)}\)

If 22 Pr+1:20 Pr+2=11 : 52 find r.

If nPr =nPr+1and nCr = "Cr-1, find the values of nand r.

By the principle of mathematical induction, prove that, for all integers n≥1, 12+22+32+...n2 =\(\frac { n(n+1)(2n+1) }{ 6 } \).

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Tamilnadu Stateboardszs - Class

Four children are running a race.
(i) In how many ways can the first two places be filled?
(ii) In how many different ways could they finish the race?

A coin is tossed 8 times,
(i) How many different sequences of heads and tails are possible?
(ii) How many different sequences containing six heads and two tails are possible?

n2 < 2ⁿ, for all natural numbers n \(\ge\) 5.

If 22 P_r+1:20 P_r+2=11 : 52 find r.

If ⁿP_r =ⁿP_r+1and ⁿC_r = "C_r-1, find the values of nand r.

By the principle of mathematical induction, prove that, for all integers n≥1,
1²+2²+3²+...n² =\(\frac { n(n+1)(2n+1) }{ 6 } \).