11th Maths Monthly Test - 3( Binomial Theorem, Sequences and Series)

St. Britto Hr. Sec. School - Madurai

11th Maths Monthly Test - 3( Binomial Theorem, Sequences and Series)-Aug 2020

Time : 2.00 Hours

Part - A

Generic 2 - Answer All Question

Write the first 6 terms of the exponential series \({ e }^{ \frac { 1 }{ 2 } x }\)

If n is a positive integer and R is a nonnegative integer. prove that the co-efficients of x^rand x^n-r Expansion of (1+x)ⁿ are equal

Find \(\sum_{k=1}^{n}{1\over k(k+1)}.\)

Find the 5^th term in the sequence whose first three terms are 3, 3, 6 and each term after the second is the sum of the two terms preceding it.

In the binomial expansion of (a+b)ⁿ the coefficients of the 4^th and 13^th terms are equal to each other, find n.

Show that the sum of (m + n)^th and (m - n)^th term of an A.P is equal to twice the m^th term.

Find the last two digits of the number 3⁶⁰⁰

Prove that if a, b, c are in HP, if and only if \({a \over c}={a-b\over b-c}.\)

The number of bacteria in a certain culture doubles every hour. If there were 30 bacteria present in the culture originally, how many bacteria will be present at the end of 2^nd hour, 4^th hour and n^th hour?

Find the middle term in \({ \left( x-\frac { 1 }{ 2y } \right) }^{ 10 }\)

In the binomial expansion of (1+a)^m+n,Prove that the coefficients of a^m and aⁿ are equal.

Write the first 6 terms of the sequences whose n^th term an is given below:\({ a }_{ n }=\begin{cases} 1 \quad if\ n=1 \\ 2 \quad if\ n=2\\ { a }_{ n-1 }+{ a }_{ n-2 }\quad if\ n>2 \end{cases}\)

Write the n^th term of the following sequences
\(\frac { 1 }{ 2 } ,\frac { 2 }{ 3 } ,\frac { 3 }{ 4 } ,\frac { 4 }{ 5 } ,\frac { 5 }{ 6 } \)

Write the n^th term of the following sequences
2,2,4,4,6,6,....

Part - B

Generic 3 - Answer All Question

If two G.M's g₁,g₂ and g₃ and one A.M be inserted between two numbers then show that 2A=\(\frac { { g }_{ 1 }^{ 2 } }{ { g }_{ 2 } } +\frac { { g }_{ 2 }^{ 2 } }{ { g }_{ 1 } } \)

If a and b are distinct integers, prove that aⁿ - bⁿ , is the factor of aⁿ -bⁿ , whenever n is a positive integer [Hint: write aⁿ =(a-b+b)]ⁿ

Find the A.P in which the sum of any number of terms is always there times the square of the number of these terms

Find 3 numbers in AP where sum is 15 and sum of their reciprocals is \(\frac{71}{105}.\)

Write the first 4 terms of the logarithmic series of log (1 - 2x)

Expand the following in ascending powers of x and find the condition on x for which the binomial expansion is valid.
\(\frac { 2 }{ { \left( 3+4x \right) }^{ 2 } } \)

If n is a postive integer, show that 9ⁿ⁺¹ - 8n - 9 is always divisible by 64

The sum of two members is\(\frac { 13 }{ 6 } \) .An even number A.M.S are being inserted between them and their sum exceeds their number by 1.Find the number of A.M.S inserted.

Prove that the product of the 2^nd and 3^rd terms of an arithmetic progression exceeds the product of the first and fourth by twice the square of the difference between the 1^stand 2^nd.

Sum up to n terms the series:
7 + 77 + 777 + 7777 + ...

Show that \({ x }^{ n }=1+n\left( 1-\frac { 1 }{ x } \right) +\frac { n(n+1) }{ 1.2 } \left( 1-\frac { 1 }{ x } \right) ^{ 2 }+...\)

Write nth term of the Sequence \(\frac { 3 }{ { 1 }^{ 2 }{ 2 }^{ 2 } } ,\frac { 5 }{ { 2 }^{ 2 }{ 3 }^{ 2 } } ,\frac { 7 }{ { 3 }^{ 2 }{ 4 }^{ 2 } } \) as a difference of two terms

Find the Constant term of \(\left( { 2x }^{ 3 }-\frac { 1 }{ { 3x }^{ 2 } } \right) ^{ 5 }\)

Write the first 6 terms of the sequences whose n^th terms are given below and classify them as arithmetic progression, geometric progression, arithmetic-geometric progression, harmonic progression and none of them. \(\frac { 1 }{ 2^{ n+1 } } \)

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11th Maths Monthly Test - 3( Binomial Theorem, Sequences and Series)

St. Britto Hr. Sec. School - Madurai

11th Maths Monthly Test - 3( Binomial Theorem, Sequences and Series)-Aug 2020

Part - A

Generic 2 - Answer All Question

Write the first 6 terms of the exponential series \({ e }^{ \frac { 1 }{ 2 } x }\)

If n is a positive integer and R is a nonnegative integer. prove that the co-efficients of xr and xn-r Expansion of (1+x)n are equal

Find \(\sum_{k=1}^{n}{1\over k(k+1)}.\)

Find the 5th term in the sequence whose first three terms are 3, 3, 6 and each term after the second is the sum of the two terms preceding it.

In the binomial expansion of (a+b)n the coefficients of the 4th and 13th terms are equal to each other, find n.

Show that the sum of (m + n)th and (m - n)th term of an A.P is equal to twice the mth term.

Find the last two digits of the number 3600

Prove that if a, b, c are in HP, if and only if \({a \over c}={a-b\over b-c}.\)

The number of bacteria in a certain culture doubles every hour. If there were 30 bacteria present in the culture originally, how many bacteria will be present at the end of 2nd hour, 4th hour and nth hour?

Find the middle term in \({ \left( x-\frac { 1 }{ 2y } \right) }^{ 10 }\)

In the binomial expansion of (1+a)m+n,Prove that the coefficients of am and an are equal.

Write the first 6 terms of the sequences whose nth term an is given below:\({ a }_{ n }=\begin{cases} 1 \quad if\ n=1 \\ 2 \quad if\ n=2\\ { a }_{ n-1 }+{ a }_{ n-2 }\quad if\ n>2 \end{cases}\)

Write the nth term of the following sequences \(\frac { 1 }{ 2 } ,\frac { 2 }{ 3 } ,\frac { 3 }{ 4 } ,\frac { 4 }{ 5 } ,\frac { 5 }{ 6 } \)

Write the nth term of the following sequences 2,2,4,4,6,6,....

Part - B

Generic 3 - Answer All Question

If two G.M's g1,g 2 and g3 and one A.M be inserted between two numbers then show that 2A=\(\frac { { g }_{ 1 }^{ 2 } }{ { g }_{ 2 } } +\frac { { g }_{ 2 }^{ 2 } }{ { g }_{ 1 } } \)

If a and b are distinct integers, prove that an - bn , is the factor of an -bn , whenever n is a positive integer [Hint: write an =(a-b+b)]n

Find the A.P in which the sum of any number of terms is always there times the square of the number of these terms

Find 3 numbers in AP where sum is 15 and sum of their reciprocals is \(\frac{71}{105}.\)

Write the first 4 terms of the logarithmic series of log (1 - 2x)

Expand the following in ascending powers of x and find the condition on x for which the binomial expansion is valid. \(\frac { 2 }{ { \left( 3+4x \right) }^{ 2 } } \)

If n is a postive integer, show that 9n+1 - 8n - 9 is always divisible by 64

The sum of two members is\(\frac { 13 }{ 6 } \) .An even number A.M.S are being inserted between them and their sum exceeds their number by 1.Find the number of A.M.S inserted.

Prove that the product of the 2nd and 3rd terms of an arithmetic progression exceeds the product of the first and fourth by twice the square of the difference between the 1st and 2nd.

Sum up to n terms the series: 7 + 77 + 777 + 7777 + ...

Show that \({ x }^{ n }=1+n\left( 1-\frac { 1 }{ x } \right) +\frac { n(n+1) }{ 1.2 } \left( 1-\frac { 1 }{ x } \right) ^{ 2 }+...\)

Write nth term of the Sequence \(\frac { 3 }{ { 1 }^{ 2 }{ 2 }^{ 2 } } ,\frac { 5 }{ { 2 }^{ 2 }{ 3 }^{ 2 } } ,\frac { 7 }{ { 3 }^{ 2 }{ 4 }^{ 2 } } \) as a difference of two terms

Find the Constant term of \(\left( { 2x }^{ 3 }-\frac { 1 }{ { 3x }^{ 2 } } \right) ^{ 5 }\)

Write the first 6 terms of the sequences whose nth terms are given below and classify them as arithmetic progression, geometric progression, arithmetic-geometric progression, harmonic progression and none of them. \(\frac { 1 }{ 2^{ n+1 } } \)

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If n is a positive integer and R is a nonnegative integer. prove that the co-efficients of x^rand x^n-r Expansion of (1+x)ⁿ are equal

Find the 5^th term in the sequence whose first three terms are 3, 3, 6 and each term after the second is the sum of the two terms preceding it.

In the binomial expansion of (a+b)ⁿ the coefficients of the 4^th and 13^th terms are equal to each other, find n.

Show that the sum of (m + n)^th and (m - n)^th term of an A.P is equal to twice the m^th term.

Find the last two digits of the number 3⁶⁰⁰

The number of bacteria in a certain culture doubles every hour. If there were 30 bacteria present in the culture originally, how many bacteria will be present at the end of 2^nd hour, 4^th hour and n^th hour?

In the binomial expansion of (1+a)^m+n,Prove that the coefficients of a^m and aⁿ are equal.

Write the first 6 terms of the sequences whose n^th term an is given below:\({ a }_{ n }=\begin{cases} 1 \quad if\ n=1 \\ 2 \quad if\ n=2\\ { a }_{ n-1 }+{ a }_{ n-2 }\quad if\ n>2 \end{cases}\)

Write the n^th term of the following sequences
\(\frac { 1 }{ 2 } ,\frac { 2 }{ 3 } ,\frac { 3 }{ 4 } ,\frac { 4 }{ 5 } ,\frac { 5 }{ 6 } \)

Write the n^th term of the following sequences
2,2,4,4,6,6,....

If two G.M's g₁,g₂ and g₃ and one A.M be inserted between two numbers then show that 2A=\(\frac { { g }_{ 1 }^{ 2 } }{ { g }_{ 2 } } +\frac { { g }_{ 2 }^{ 2 } }{ { g }_{ 1 } } \)

If a and b are distinct integers, prove that aⁿ - bⁿ , is the factor of aⁿ -bⁿ , whenever n is a positive integer [Hint: write aⁿ =(a-b+b)]ⁿ

Expand the following in ascending powers of x and find the condition on x for which the binomial expansion is valid.
\(\frac { 2 }{ { \left( 3+4x \right) }^{ 2 } } \)

If n is a postive integer, show that 9ⁿ⁺¹ - 8n - 9 is always divisible by 64

Prove that the product of the 2^nd and 3^rd terms of an arithmetic progression exceeds the product of the first and fourth by twice the square of the difference between the 1^stand 2^nd.

Sum up to n terms the series:
7 + 77 + 777 + 7777 + ...

Write the first 6 terms of the sequences whose n^th terms are given below and classify them as arithmetic progression, geometric progression, arithmetic-geometric progression, harmonic progression and none of them. \(\frac { 1 }{ 2^{ n+1 } } \)