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11th Maths Monthly Test-1 ( Combinatorics and Mathematical Induction )

St. Britto Hr. Sec. School - Madurai

11th Maths Monthly Test-1 ( Combinatorics and Mathematical Induction )-Aug 2020

Time : 2.30 Hours

Part - A

Multiple Choice Question - Answer All Question

If p(n):49ⁿ + 16ⁿ + \(\lambda \) is divisible by 64 for n \(\in \) N is true, then the least negative integral value of \(\lambda \) is

-3
-2
-1
-4

The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines.

6
9
12
18

The number of different signals which can be give from 6 flags of different colours taking one or more at a time is

1958
1956
16
64

In an examination there are three multiple choice questions and each question has 5 choices. Number of ways in which a student can fail to get all answer correct is

125
124
64
63

If ⁿC₄,ⁿC₅,ⁿC₆ are in AP the value of n can be

14
11
9
5

Everybody in a room shakes hands with everybody else. The total number of shake hands is 66. The number of persons in the room is

11
12
10
6

If ⁿC₁₀ = ⁿC₆, then ⁿC₂ =

16
4
120
240

If 10 lines are drawn in a plane such that no two of them are parallel and no three are concurrent, then the total number of points of intersection are

45
40
10!
210

The number of ways in which the following prize be given to a class of 30 boys first and second in mathematics, first and second in physics, first in chemistry and first in English is

30⁴\(\times\) 29²
30³\(\times\) 29³
30²\(\times\) 29⁴
30\(\times\)29⁵

If 7 points out of 12 are in the same straight line then the number of triangles formed is

35
21
220
185

Part - B

Generic 2 - Answer All Question

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?

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

Find the total number of subsets of a set with [Hint: ⁿC₀+ⁿC₁+ⁿC₂+...+ⁿC_n=2ⁿ].
n elements.

If the letters of the word FUNNY are permuted in all possible ways and the strings thus formed are arranged in the dictionary order, find the rank of the word FUNNY
What is the unit digit of the sum 2! + 3! + 4! + ...+ 22! ?

In a village, out of the total number of people, 80 percentage of the people own Coconut groves and 65 percent of the people own Paddy fields. What is theminimum percentage of people own both?

How many chords can be drawn through 20 points on a circle?

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.

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

Find the value of n if (n+1)! =20 (n-1)!

Part - C

Generic 3 - Answer All Question

A polygon has 90 diagonals. Find the number of its sides?

If ¹⁵C_2r-1=¹⁵C_2r+4, find r.

Prove that n!(n+2)=n!+(n+1)!

Find the sum of all 4-digit numbers that can be formed using the digits 1, 2, 4, 6, 8.

Let p(n) be the statement "3ⁿ>n". If p(n) is true, prove that p(n+1) is true.
In a plane, there are 37 straight lines, of which 13 pass through the point A and 11 pass through the point B. Besides, no three lines pass through one point, no line passes through the points A and B and no two are parallel. Find the number of points of intersection of the straight lines.

A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of: exactly 3 girls

By using the digits 0, 1, 2, 3, 4 and 5 (repetition not allowed) numbers are formed by using any number of digits. Find the total number of non-zero numbers that can be formed.

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?

How many different selections of 5 books can be made from 12 different books if,
(i) Two particular books are always selected?
(ii) Two particular books are never selected?

Part - D

Generic 5 - Answer All Question

Prove that for any natural number n, aⁿ - bⁿ is divisible by a-b, where a > b.

By the principle of mathematical induction, prove that, for n\(\in \)N, cosα+cos(α+β)+cos(α+2β)+...+ cos(α+(n-1)β) = \(\left( \alpha +\frac { (n-1)\beta }{ 2 } \right) \times \frac { sin\left( \frac { n\beta }{ 2 } \right) }{ sin\left( \frac { \beta }{ 2 } \right) } \).

How many automobile license plates can be made, if each plate contains two different letters followed by three different digits?

In how many ways can the letters of the word PERMUTATIONS be arranged if vowels are all together.

Prove by the principle of mathematical induction if x and y are any two distinct integers, then xⁿ - yⁿ is divisible by x - y. [OR] xⁿ - y ⁿ is divisible by x - y, where x - y \(\neq\)o.

How many strings of length 6 can be formed using letters of the word FLOWER if (i) either starts with F or ends with R? (ii) neither starts with F nor ends with R?
By the principle of mathematical induction, prove that for n > 1,
\(1^2 + 3^2 + 5^2 + ... + (2n-1)^2 = {n(2n-1)(2n+1)\over 3}\)

By the principle of mathematical induction, prove that for n > 1,
\(1^2+2^2+3^2+L+n^2>{n^3\over 3}\)

n² - n is divisible by 6, for each natural number n \(\ge\) 2.

Find r if ⁵P_r = 2⁶P_r-1

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