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Question 1
1 / -0
Find the value of $$'?'$$ in the series: $$12, 17, 15, ?, 18, 23, 21,....$$
Solution
The series alternates the addition of $$5$$ with the subtraction of $$2$$.
So, the number is $$20$$.
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Question 2
1 / -0
Choose the missing number in the series: $$2, 2, 3, 4, 4,$$__$$, 6, 6, 7, 8, 8, 9....$$
Solution
This is a continuous series of adding $$1$$ to the previous number, in which every third number is not repeated.
So, the missing number is $$5$$.
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Question 3
1 / -0
Find the value of $$'x'$$ in the series: $$144, 169, x, 225, 256, 289$$.
Solution
The series is a perfect square in increasing order of $$12, 13, 14, 15...$$
So, the value of $$x$$ is $$196$$.
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Question 4
1 / -0
The value of $$\displaystyle \tan { \alpha } +2\tan { \left( 2\alpha \right) } +4\tan { \left( 4\alpha \right) } +...+{ 2 }^{ n-1 }\tan { \left( { 2 }^{ n-1 }\alpha \right) } +{ 2 }^{ n }\cot { \left( { 2 }^{ n }\alpha \right) } $$ is
Solution
Now, $$\displaystyle { 2 }^{ n }\tan { \left( { 2 }^{ n }a \right) } +{ 2 }^{ n }\cot { \left( { 2 }^{ n }a \right) } $$
$$\displaystyle ={ 2 }^{ n-1 }\left[ \frac { \sin { { 2 }^{ n-1 }a } }{ \cos { { 2 }^{ n-1 }a } } +2\frac { \cos { { 2 }^{ n }a } }{ \sin { { 2 }^{ n }a } } \right] $$
$$\displaystyle ={ 2 }^{ n-1 }\left[ \frac { \cos { { 2 }^{ n }a } \cos { { 2 }^{ n-1 }a } +\sin { { 2 }^{ n }a\sin { { 2 }^{ n-1 }a+\cos { { 2 }^{ n }a } \cos { { 2 }^{ n-1 }a } } } }{ \sin { { 2 }^{ n }a\cos { { 2 }^{ n-1 }a } } } \right] $$
$$\displaystyle ={ 2 }^{ n-1 }\left[ \frac { \cos { { 2 }^{ n-1 }a } \left( 1+\cos { { 2 }^{ n }a } \right) }{ \sin { { 2 }^{ n }a } \cos { { 2 }^{ n-1 }a } } \right] $$
$$\displaystyle ={ 2 }^{ n-1 }\cot { { 2 }^{ n-1 }a } $$
Proceeding in similar way in last, we get
$$\displaystyle \tan { a } +2\cot { 2a } $$
$$\displaystyle =\frac { \sin { a } }{ \cos { a } } +2\frac { \cos { 2a } }{ \sin { 2a } } $$
$$\displaystyle =\frac { \cos { 2a\cos { a } +\sin { 2a } \sin { a } +\cos { 2a\cos { a } } } }{ \sin { 2a\cos { a } } } $$
$$\displaystyle =\frac { \cos { a } \left( 1+\cos { 2a } \right) }{ 2\sin { a } { cos }^{ 2 }a } =\frac { 2{ cos }^{ 2 }a }{ 2\sin { a } } $$
$$\displaystyle =\frac { \cos { a } }{ \sin { a } } =\cot { a } $$
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Question 5
1 / -0
Find the missing number in the pattern: $$2, 5, 8, 11,$$ __$$, 17, 20, 23, 26$$
Solution
The series adds $$3$$ to each number to get the next number.
So, the next number is $$11 + 3 = 14$$
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Question 6
1 / -0
Which number comes next?
$$5, 9, 13, 17, 21, 25, 29,...$$
Solution
The series adds $$4$$ to each number to get the next number.
So, the next number is $$29 + 4 = 33$$
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Question 7
1 / -0
The sum of $$24$$ terms of the following series $$2+4+6.....$$
Solution
$$\overline { 2 } +\overline { 8 } +\overline { 18 } +\overline { 32 } ......24$$terms
$$=\sqrt { 2 } +\sqrt { 8 } +\sqrt { 18 } +\sqrt { 32 } .....24$$terms
$$\\ =\sqrt { 2 } (1+\sqrt { 4 } +\sqrt { 9 } +\sqrt { 16 } +........24$$terms)
$$ =\sqrt { 2 } (1+2+3+4+.....24)$$
Sum of natural number$$=\cfrac { n(n+1) }{ 2 } $$
$$ =\sqrt { 2 } \cfrac { (24)(25) }{ 2 } \\ =300\sqrt { 2 } =300\overline { 2 } $$
Answer$$(C)$$
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Question 8
1 / -0
$$2, 3,$$ __$$, 4, 4, 5, 6, 6, 6, 7, 7, 7...$$ What number should fill the blank?
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Question 9
1 / -0
Fill in the blank: $$62, 66, 63, 66, 64,$$ __$$, 65,....$$
Solution
The series alternates the addition of $$4$$ with the subtraction of $$3$$.
So, the missing number is $$61$$.
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Question 10
1 / -0
Identify the missing integer: $$9, 45,$$ ____$$, 1125, 5625...$$
Solution
This is continuous series multiplied by $$5$$.
$$9 \times 5 = 45$$
$$45 \times 5 = 225$$
$$225 \times 5 = 1,125$$
$$1,125 \times 5= 5,625.$$
So, the missing integer is $$225$$.
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