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11th Maths Monthly Test - 1( Binomial Theorem, Sequences and Series)

St. Britto Hr. Sec. School - Madurai

11th Maths Monthly Test - 1( Binomial Theorem, Sequences and Series)-Aug 2020

Time : 2.30 Hours

Part - A

Multiple Choice Question - Answer All Question

Expansion of \(log(\sqrt \frac{1+x}{1-x})\) is:

\(x+\frac{x^3}{3}+\frac{x^5}{5}+...\)
\(1.\frac{x^2}{2}+\frac{x^4}{4}+...\)
\(1-x+\frac{x^2}{2}+\frac{x^3}{5}+...\)
\(x-\frac{x^2}{3}+\frac{x^3}{3}+...\)

The remainder when 52⁴⁰ is divided by 17 is

1
3
5
6

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

¹⁰C₆
26
¹⁰C₆2⁶
¹⁰C₆2¹⁰

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

If \(\Sigma n=210\) then \(\Sigma { n }^{ 2 }\)=

2870
2160
2970
none of these

The value of \({ 9 }^{ \frac { 1 }{ 3 } }\) ,\({ 9 }^{ \frac { 1 }{ 9 } }\)\({ 9 }^{ \frac { 1 }{ 27 } }\) ,\(\infty \)is

1
3
9
none of these

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

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

The sum of an infinite GP is 18. If the first term is 6, the common ratio is

\(\frac { 1 }{ 3 } \)
\(\frac { 2 }{ 3 } \)
\(\frac { 1 }{ 6 } \)
\(\frac { 3 }{ 4 } \)

If ⁿC₁₀ > ⁿC_r for all possible F, then a value of n is

10
21
19
20

Part - B

Generic 2 - Answer All Question

If a, b, c are in A.P., show that (a-c)² = 4(b² - ac).

Write the first 6 terms of the exponential series e^-2x

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

Write down the series whose r^th term is \(\frac{1}{3}r\). Is it an arithmetic series?

Find the \(\sqrt [ 3 ]{ 126 } \) approximately to two decimal places.
Find the last two digits of the number 3⁶⁰⁰

Find the sum of the first n terms of the series \({1\over 1+\sqrt{2}}+{1\over\sqrt{2}+\sqrt{3}}+{1\over\sqrt{3}+\sqrt{4}}+...\)

Find the middle terms in the expansion of (x +y)⁷.

Find seven numbers A₁, A₂, ... , A₇ so that the sequence 4, A₁, A₂, ... , A₇, 7 is in arithmetic progression and also 4 numbers G₁, G₂, G₃, G₄ so that the sequence 12, G₁, G₂, G₃, G₄, is in geometric progression.

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.

Part - C

Generic 3 - Answer All Question

Find the co-efficient of xⁿ in the series 1 + (a+bx) + \(\frac { (a+bx)^2}{ 2! } +\frac { (a+bx)^{ 3 } }{ 3! } \)

A man repays an amount of Rs.3250 by paying Rs.20 in the first month and then increases the payment by Rs.15 per month. How long will it take him to clear the amount?

Evaluate the following:
\(\frac { 1 }{ \sqrt [ 3 ]{ 128 } } \)correct to 4 places of decimals

Expand \({1\over (3+2x)^2}\)in powers of x. Find a condition on x for which the expansion is valid.

Find \(\sum_{1}^{\infty}{\frac{1}{(k+1)(k+2)}}\).

If the roots of the equation (q - r) x² + (r - p)x + p - q = 0 are equal, then show that p, q and r are in A.P.
Find the coefficient of the term involving x³² and x^-17in the expansion of \((x^{4}-\frac{1}{x^{3}})^{15}\).

Sum the series: (1 + x) + (1 + x + x²) + (1 + x + x² +x³) + ... up to n terms

If x so large prove that \(\sqrt { { x }^{ 2 }+25 } -\sqrt { { x }^{ 2 }+9 } =\frac { 8 }{ x } \) nearly.

Find all the values of x≠0 in(-\(\pi\),\(\pi\)) satisfying the equation 8¹+cosx+cos2x+...=4³

Part - D

Generic 5 - Answer All Question

If three consecutive coefficients in the expansion of (1+x)ⁿ are in the ratio 6:33:110,find n.

Find the sum of the series \(1+\frac { 2 }{ 5 } +\frac { 3 }{ { 5 }^{ 2 } } +\frac { 5 }{ { 5 }^{ 3 } } +\)

Find the Co-efficient of x⁶ and the co -efficient of x² in \(\left( { x }^{ 2 }-\frac { 1 }{ { x }^{ 3 } } \right) ^{ 6 }\)

Find \(\sqrt {4+x^2}-\sqrt {4-x^2}\) when x is small.

If \(\alpha ,\beta \)are the roots of the equation x²-px+q=0, then prove that \(\log { (1+px+q{ x }^{ 2 }) } =(\alpha +\beta )x=\frac { { \alpha }^{ 2 }+{ \beta }^{ 2 } }{ 2 } { x }^{ 2 }+\frac { { \alpha }^{ 2 }+{ \beta }^{ 2 } }{ 3 } { x }^{ 3 }-....\infty \)
Find the sum to n terms of the series 1 - 5 + 9 - 13+ ......

Show that \(\frac { 5 }{ 1\times 3 } +\frac { 5 }{ 2\times 7 } +\frac { 5 }{ 3\times 5 } +...=\frac { 15 }{ 4 } \)

If x=0.001, prove that \(\frac { { \left( 1-2x \right) }^{ \frac { 2 }{ 3 } }{ \left( 4+5x \right) }^{ \frac { 3 }{ 2 } } }{ \sqrt { 1-x } } \) =8.01 up to two places of decimals

The product of three increasing numbers in GP is 5832. if we add 6 to the second number and 9 to the third number, then resulting number form an AP. Find the numbers in GP

if the binomial co-efficients of three consecutive terms in the expansion of ( a + xⁿ) are in the radio 1:7:42 then find n

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