Sequences and Series Test

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Sequences and Series Test - 62

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Weekly Quiz Competition

Question 1
1 / -0

$$40280625, 732375, 16275, 465, 18.6, 1.24,?$$

A
$$0.248$$

B
$$0.336$$

C
$$0.424$$

D
$$0.512$$

E
$$0.639$$

Solution

$$40280625, 732375,16275,465,18.6,1.24$$
$$\cfrac{40280625}{732375}=55$$
$$\cfrac{732375}{16275}=45$$
$$\cfrac{16275}{465}=35$$
$$\cfrac{465}{18.6}=25$$
$$\cfrac{18.6}{1.24}=15$$
$$\cfrac{1.24}{x}=5$$
$$x=0.248$$
Question 2
1 / -0

$$435, 354, 282, 219, 165, ?$$

A
$$103$$

B
$$112$$

C
$$120$$

D
$$130$$

E
$$none\ of\ these$$

Solution

$$\therefore 165-45=x$$
$$x=120$$.
Question 3
1 / -0

For a positive integer n, let $$f_n(\theta)=(2\cos \theta+1)(2\cos \theta -1)(2\cos 2\theta -1)(2\cos (2^2)\theta -1)......(2\cos (2^{n-1})\theta -1)$$. Which one of the following hold(s) not good?

A
$$f_2(\pi/6)=0$$

B
$$f_3(\pi/8)=-1$$

C
$$f_4(\pi/32)=1$$

D
$$f_5(\pi/128)=1+\sqrt{2}$$

Solution

$$\begin{array}{l} f\left( \theta \right) =\left( { 2\cos \theta +1 } \right) \left( { 2\cos \theta -1 } \right) \left( { 2\cos 2\theta -1 } \right) \left( { 2\cos \left( { { 2^{ 2 } } } \right) \theta -1 } \right) ......\left( { 2\cos \left( { { 2^{ n-1 } } } \right) \theta -1 } \right) . \\ =\left( { 2\cos \theta +1 } \right) +\left( { 2\cos \theta -1 } \right) ....\left( { 2\cos { 2^{ n-1 } } \theta +1 } \right) \\ Now, \\ { f_{ n } }\left( \theta \right) =2\cos { 2^{ n } } \theta +1 \\ { f_{ 2 } }\left( { \frac { \pi }{ 6 } } \right) =2\cos \left( { 4\times \frac { \pi }{ 6 } } \right) +1 \\ =2\cos \frac { { 2\pi } }{ 3 } +1=0 \\ { f_{ 3 } }\left( { \frac { \pi }{ 8 } } \right) =2\cos \left( { 8\times \frac { \pi }{ 8 } } \right) +1=-1 \\ { f_{ 4 } }\left( { \frac { \pi }{ { 32 } } } \right) =2\cos \left( { 16\times \frac { \pi }{ { 32 } } } \right) +1=1 \\ { f_{ 5 } }\left( { \frac { \pi }{ { 128 } } } \right) =2\cos \left( { 32\times \frac { \pi }{ { 128 } } } \right) +1=1+\sqrt { 2 } \\ Hence,\, the\, option\, D\, is\, the\, correct\, answer. \end{array}$$
Question 4
1 / -0

If $$x=\sqrt 1+\dfrac{1}{{1}^{2}}+\dfrac{1}{{2}^{2}}+\sqrt 1+\dfrac{1}{{2}^{2}}+\dfrac{1}{{3}^{2}}+\sqrt 1+\dfrac{1}{{2019}^{2}}+\dfrac{1}{{2020}^{2}}$$ then $$4040(2020-x)$$

A
$$1$$

B
$$2$$

C
$$-1$$

D
$$-2$$

Solution
Question 5
1 / -0

If $$a_{k}=\dfrac{1}{K(K+1)}$$ for $$K=1, 2, 3,.....n,$$ then $$\left(\sum_{k=1}^{n}{a_{k}}\right)^{2}=$$

A
$$\dfrac{n}{n+1}$$

B
$$\dfrac{n^{2}}{(n+1)^{2}}$$

C
$$\dfrac{n^{4}}{(n+1)^{4}}$$

D
$$\dfrac{n^{6}}{(n+1)^{6}}$$

Solution

$$\begin{array}{l} { a_{ k } }=\dfrac { 1 }{ { k\left( { k+1 } \right) } } =\dfrac { 1 }{ k } -\dfrac { 1 }{ { k+1 } } \\ { \left( { \sum _{ k=1 }^{ n }{ { a_{ x } } } } \right) ^{ 2 } } \\ { \left( { \sum _{ k=1 }^{ n }{ \left( { \dfrac { 1 }{ k } -\dfrac { 1 }{ { k+1 } } } \right) } } \right) ^{ 2 } } \\ =\left( { 1-\dfrac { 1 }{ 2 } } \right) +\left( { \dfrac { 1 }{ 2 } -\dfrac { 1 }{ 3 } } \right) \, \, \, \, \, \left( { \dfrac { 1 }{ n } \dfrac { { -1 } }{ { n+1 } } } \right) \\ ={ \left( { 1-\dfrac { 1 }{ { n+1 } } } \right) ^{ 2 } } \\ ={ \left( { \dfrac { { n+1-1 } }{ { n+1 } } } \right) ^{ 2 } } \\ =\dfrac { { { n^{ 2 } } } }{ { { { \left( { n+1 } \right) }^{ 2 } } } } \end{array}$$

Hence, this is the answer.
Question 6
1 / -0

If $${ x }_{ 1 },{ x }_{ 2 },{ x }_{ 3 },......,{ x }_{ n }\quad $$ are the roots of $$\quad { x }^{ n }+ax+b=0$$. then the value of $$\quad \left( { x }_{ 1 }-{ x }_{ 2 } \right) \left( { x }_{ 1 }-{ x }_{ 3 } \right) \left( { x }_{ 1 }-{ x }_{ 4 } \right) ......\left( { x }_{ 1 }-{ x }_{ n } \right) $$

A
$${ nx }_{ 1 }+b$$

B
$${ nx }_{ 1 }^{ n-1 }+a$$

C
$${ nx }_{ 1 }^{ n-1 }$$

D
$${ x }_{ 1 }^{ n-1 }$$

Solution
Question 7
1 / -0

$$\dfrac { { C }_{ 1 } }{ { C }_{ 0 } } +2\dfrac { { C }_{ 2 } }{ { C }_{ 1 } } +3\frac { { C }_{ 3 } }{ { C }_{ 2 } } +......+\dfrac { n{ C }_{ n } }{ { C }_{ n-1 } } $$ equals

A
$${ n2 }^{ n-1 }$$

B
$$\dfrac { { n2 }^{ n-1 } }{ { 2 }^{ n }-1 } $$

C
$$\dfrac { n(n+1) }{ 2 } $$

D
$$\dfrac { (n+1)(n+2) }{ 2 } $$

Solution
Question 8
1 / -0

The value of $$2+\dfrac{1}{2+\dfrac{1}{2+.....\infty}}$$ is :

A
$$1-\sqrt{2}$$

B
$$1+\sqrt{2}$$

C
$$1 \pm \sqrt{2}$$

D
$$none\ of\ these$$

Solution

$$\begin{array}{l} 2+\dfrac { 1 }{ { 2+.... } } \\ x=2+\dfrac { 1 }{ x } \\ x-2-\dfrac { 1 }{ x } =0 \\ { x^{ 2 } }-2x-1=0 \\ \dfrac { { x=+2\pm \sqrt { 4+4 } } }{ 2 } =\dfrac { { 2\pm 2\sqrt { 2 } } }{ 2 } \\ =1\pm \sqrt { 2 } \end{array}$$
Question 9
1 / -0

If $$\left( 1+x+{ x }^{ 2 } \right) ={ a }_{ 0 }+{ a }_{ 1 }x+{ a }_{ 2 }{ x }^{ 2 }+........{ a }_{ 2n }{ x }^{ 2n }$$, then the value of $${ a }_{ 0 }+{ a }_{ 3 }+{ a }_{ 6 }+.....$$ is

A
$${ a }_{ 1 }+{ a }_{ 4 }+{ a }_{ 7 }+.....$$

B
$${ a }_{ 2 }+{ a }_{ 5 }+{ a }_{ 8 }+.....$$

C
$${ 3 }^{ n-1 }$$

D
All(A,B,C)
Question 10
1 / -0

The sum of the infinite series $$\cot { ^{ 1 } } \left( \dfrac { 7 }{ 4 } \right) +\cot { ^{ -1 }\left( \dfrac { 19 }{ 4 } \right) } +\cot { ^{ -1 }\left( \dfrac { 39 }{ 4 } \right) } +\cot { ^{ -1 }\left( \dfrac { 67 }{ 4 } \right) } +........\infty $$ is:

A
$$\\ \dfrac { \pi }{ 4 } -\cot { ^{ -1 } } (3)$$

B
$$\\ \dfrac { \pi }{ 4 } -\tan { ^{ -1 } } (3)$$

C
$$\\ \dfrac { \pi }{ 4 } +\cot { ^{ -1 } } (3)$$

D
$$\\ \dfrac { \pi }{ 4 } +\tan { ^{ -1 } } (3)$$

Solution

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Re-Attempt Weekly Quiz Competition

Sequences and Series Test - 62

Result

Sequences and Series Test - 62

Score

-

Rank

-

SHARING IS CARING

Question 1

$$0.248$$

$$0.336$$

$$0.424$$

$$0.512$$

$$0.639$$

Solution

Question 2

$$103$$

$$112$$

$$120$$

$$130$$

$$none\ of\ these$$

Solution

Question 3

$$f_2(\pi/6)=0$$

$$f_3(\pi/8)=-1$$

$$f_4(\pi/32)=1$$

$$f_5(\pi/128)=1+\sqrt{2}$$

Solution

Question 4

$$1$$

$$2$$

$$-1$$

$$-2$$

Solution

Question 5

$$\dfrac{n}{n+1}$$

$$\dfrac{n^{2}}{(n+1)^{2}}$$

$$\dfrac{n^{4}}{(n+1)^{4}}$$

$$\dfrac{n^{6}}{(n+1)^{6}}$$

Solution

Question 6

$${ nx }_{ 1 }+b$$

$${ nx }_{ 1 }^{ n-1 }+a$$

$${ nx }_{ 1 }^{ n-1 }$$

$${ x }_{ 1 }^{ n-1 }$$

Solution

Question 7

$${ n2 }^{ n-1 }$$

$$\dfrac { { n2 }^{ n-1 } }{ { 2 }^{ n }-1 } $$

$$\dfrac { n(n+1) }{ 2 } $$

$$\dfrac { (n+1)(n+2) }{ 2 } $$

Solution

Question 8

$$1-\sqrt{2}$$

$$1+\sqrt{2}$$

$$1 \pm \sqrt{2}$$

$$none\ of\ these$$

Solution

Question 9

$${ a }_{ 1 }+{ a }_{ 4 }+{ a }_{ 7 }+.....$$

$${ a }_{ 2 }+{ a }_{ 5 }+{ a }_{ 8 }+.....$$

$${ 3 }^{ n-1 }$$

All(A,B,C)

Question 10

$$\\ \dfrac { \pi }{ 4 } -\cot { ^{ -1 } } (3)$$

$$\\ \dfrac { \pi }{ 4 } -\tan { ^{ -1 } } (3)$$

$$\\ \dfrac { \pi }{ 4 } +\cot { ^{ -1 } } (3)$$

$$\\ \dfrac { \pi }{ 4 } +\tan { ^{ -1 } } (3)$$

Solution

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