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

St. Britto Hr. Sec. School - Madurai

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

Time : 1.00 Hours

Part - A

Multiple Choice Question - Answer All Question

\(\left(1+\frac{1}{\lfloor2}+\frac{1}{\lfloor4}+\frac{1}{\lfloor6}+...\right)^2-\left(1+\frac{1}{\lfloor3}+\frac{1}{\lfloor5}+\frac{1}{\lfloor7}+...\right)^2=\)

1
2
e
2e

The coefficient of x³² in the expansion of \((x^4-\frac{1}{x^3})^{15}\)

15C₄
15C₃
15C₅
15C₆

Match List - I with List II.

List I List II

1 e^x a \(x-\frac { { x }^{ 2 } }{ 2 } +\frac { { x }^{ 3 } }{ 3 } -\frac { { x }^{ 4 } }{ 4 } +...\) for all values of x satifying|x|<1

2 log(1+x) b 1,1,2,3,5,...

3 (1+x)ⁿ c \(1+\frac { x }{ 1! } +\frac { { x }^{ 2 } }{ 2! } -\frac { { x }^{ 3 } }{ 3! } +\frac { { x }^{ 4 } }{ 4! } +...1+nx+\frac { n(n-1) }{ 2! } \)

4 Fibonacci
sequence d \({ x }^{ 2 }+\frac { n(n-1)(n-2) }{ 3! } { x }^{ 3 }+...\)

The Correct match is

i ii iii iv

c a d b
i ii iii iv

b a d c
i ii iii iv

d c b a
i ii iii iv

d a b c

The sequence\(\frac { 1 }{ \sqrt { 3 } } ,\frac { 1 }{ \sqrt { 3 } +\sqrt { 2 } } \frac { 1 }{ \sqrt { 3 } +2\sqrt { 2 } } \)...from an

AP
GP
HP
AGP

The first and last term of an A. P. are 1 and 11. If the sum of its terms is 36, then the number of terms will be

5
6
7
8

AM, GM, HM denote the Arithmetic mean, Geometric mean and Harmonic mean respectively the relationship between this is:

AM < GM < HM
AM ≤ GM ≤ HM
AM>GM>HM
AM≥GM≥HM

1 - 2x + 3x² - 4x³ + ..., Ixl< 1 is:

(1-x)^-2
(1+x)^-2
(1-x)²
(1+x)²

If (1 + x²)² (1 + x)ⁿ = a₀+ a₁x + a₂x² + ... + xⁿ⁺⁴ and if a₀, a₁, a₂are in AP, then n is

1
2
3
4

3 log 2 + \(\frac{1}{4}-\frac{1}{2}(\frac{1}{4})^2+\frac{1}{3}(\frac{1}{4})^2\)+ . . . =

log8
log10
log2
log4

In the expansion of (2x + 3)⁵ the coefficient of x² is:

720
1080
810
5

If an A.P the sum of terms equidistant from the beginning and end is equal to

first term
second term
sum of first and last term
last term

The value of 2 + 4 + 6 + + 2n is

\(\frac { n\left( n-1 \right) }{ 2 } \)
\(\frac { n\left( n+1 \right) }{ 2 } \)
\(\frac { 2n\left( 2n-1 \right) }{ 2 } \)
n(n + 1)

The series 1+4x+8x²+\(\frac { 32 }{ 3 } { x }^{ 3 }+.....+\infty \quad is\)

e^x
e^4x
e^2x
e^8x

The sum of the series C₀²- C₁² + C₂² .....+ (- 1)ⁿC²_n where n is an even integer is

2nC_n
(-1)ⁿ2nC_n
(-1)ⁿ2nC_n-1
(-1)^n/2nC_n/2

The sum of the series \(\frac { 1 }{ { log }_{ 2 }^{ 4 } } +\frac { 1 }{ { log }_{ 4 }^{ 4 } } +\frac { 1 }{ { log }_{ 8 }^{ 4 } } +.....+\frac { 1 }{ { log }_{ { 2 }^{ n } }^{ 4 } } is\)

\(\frac { n(n+1) }{ 2 } \)
\(\frac { n(n+1)(2n+1) }{ 2 } \)
\(\frac { n(n+1) }{ 4 } \)
none of these

Sum of n terms of the series \(\sqrt { 2 } +\sqrt { 8 } +\sqrt { 18 } +\sqrt { 32 } ..is\)

\(\frac { n(n+1) }{ 2 } \)
2n(n+1)
\(\frac { n(n+1) }{ \sqrt { 2 } } \)
1

The n^th term of the sequence \(\frac { 1 }{ 2 } ,\frac { 3 }{ 4 } ,\frac { 7 }{ 8 } ,\frac { 15 }{ 6 } \).+.....is

2ⁿ-n-1
1-2^-n
2^-n+n-1
2^n-1

The value of n for which \(\frac{a^{n+1}+b^{n+1}}{a^n+b^n}\) is the arithmetic mean of a and b is

1
2
4
0

The coefficient of x⁶ in (2 + 2x)¹⁰ is

¹⁰C₆
26
¹⁰C₆2⁶
¹⁰C₆2¹⁰

The Co-efficient of x³ in \(\sqrt { \frac { 1-x }{ 1+x } } ,\left| x \right| <1\quad is\quad \)

\(\frac { 1 }{ 2 } \)
\(\frac { 3 }{ 8 } \)
\(\frac { -3 }{ 8 } \)
\(\frac {- 1 }{ 2 } \)

In the series \(\frac{1}{1+\sqrt 2}+\frac{1}{\sqrt 2+\sqrt 3}+\frac{1}{\sqrt 3+\sqrt 4}+...\) some of first 24 number is:

4
\(\sqrt 24\)
\(\frac{1}{\sqrt 24}\)
\(\frac{1}{\sqrt 25-\sqrt 24}\)

The coefficient of x⁵ in the series e^-2x is

\(\frac { 2 }{ 3 } \)
\(\frac { 2 }{ 3 } \)
\(\frac { -4 }{ 15 } \)
\(\frac { 4 }{ 15 } \)

The larget coefficients in the expansion of (1 + X)²⁴ is

24C₂₄
24C₁₃
24C₁₂
24C₁₁

The sum of 40 terms of an A.P whose first term is 2 and common difference 4 will be

3200
1600
200
2800

The value of \(1-\frac { 1 }{ 2 } \left( \frac { 2 }{ 3 } \right) +\frac { 1 }{ 3 } { \left( \frac { 2 }{ 3 } \right) }^{ 2 }-\frac { 1 }{ 4 } { \left( \frac { 2 }{ 3 } \right) }^{ 2 }+....is\)

\(log\left( \frac { 5 }{ 3 } \right) \)
\(\frac { 3 }{ 2 } log\left( \frac { 5 }{ 3 } \right) \)
\(\frac { 5 }{ 3 } log\left( \frac { 5 }{ 3 } \right) \)
\(\frac { 2 }{ 3 } log\left( \frac { 2 }{ 3 } \right) \)

If the first, second and last term of an A. P. are a, b and 2a respectively, then its sum is

\(\frac { ab }{ 2(b-a) } \)
\(\frac { ab }{ b-a } \)
\(\frac { 3ab }{ 2(b-a) } \)
none of these

\(\sqrt \frac{1-2x}{1+2x}\) is approximately equal to:

1- 2x-x²
1 + 2x+ x²
1+ 2x
1-2x+x²

The last term in the expansion of (2 +\(\sqrt { 3 } \))⁸

81
27
\(\sqrt { 3 } \)
3

The series for log \(\left( \frac { 1+x }{ 1-x } \right) is\)

\(x+\frac { { x }^{ 3 } }{ 3 } +\frac { { x }^{ 5 } }{ 5 } +...+\infty \)
\(2\left[ x+\frac { { x }^{ 3 } }{ 3 } +\frac { { x }^{ 5 } }{ 5 } +...+\infty \right] \)
\(\frac { { x }^{ 2 } }{ 2 } +\frac { { x }^{ 4 } }{ 4 } +\frac { { x }^{ 6 } }{ 6 } +...+\infty \)
\(2\left[ \frac { { x }^{ 2 } }{ 2 } +\frac { { x }^{ 4 } }{ 4 } +\frac { { x }^{ 6 } }{ 6 } +...+\infty \right] \)

The HM of two positive numbers whose AM and GM are 16,8 respectively is

10
6
5
4

The sum up to n terms of the series \(\sqrt { 2 } +\sqrt { 8 } +\sqrt { 18 } +\sqrt { 32 } +\).....is

\(\frac { n(n+1) }{ 2 } \)
2n(n+)
\(\frac { n(n+1) }{ \sqrt { 2 } } \)
1

The value of the series\(\quad \frac { 1 }{ 2 } +\frac { 7 }{ 4 } +\frac { 13 }{ 8 } +\frac { 19 }{ 6 } +\).....is

14
7
4
6

Sum of the binomial coefficients is

2n
n²
2ⁿ
n+17

The remainder when 38¹⁵ is divided by 13 is

12
1
11
5

The n^th term of the sequence 1, 2, 4, 7, 11,... is

n³+3n²+2n
n3-3n²+3n
\(\frac { n(n+1)(n+2) }{ 3 } \)
\(\frac { { n }^{ 2 }-n+2 }{ 2 } \)

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