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#### By using1,3,5,7,9,11,13,15 Add five times and get 30

If there will be leniency of factorials use, then the answers will be
$3!+1+9+3+11=30$

$3!+1+1+7+15=30$

#### When the length of each side of a cube is increased by 3 cm, its volume is increased by 2457Â cm3. Find its side. How much will its volume decrease, if length of each side of it is reduced by 20%?

When length of each side of a cube is increased by 3 cm, its volume is increased by 2457

$(a+3)^3-a^3=2457\;\;\;\;\;\;(\because Volume\;of \;cube=a^3)$

$a^3+3^3+3\times a^2\times (3)+3\times a \times 3^2-a^3=2457$

$27+9\times a^2+27\times a=2457$

$a^2+3a-270=0$

$a^2+18a-15a-270=0$

$a(a+18)-15(a+18)=0$

$(a-15)(a+18)=0$

$a=(-18) \;\;\;\;{not possible}$

$So, a=15\;cm$

If side reduced by 20% then  it becomes 12 cm

then volume reduced by

$=15^3-12^3=3375-1728=1647 cm^3$

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#### Which of the following sets have closure property with respect to addition and multiplication? 1. 1 2. 2,-1 3.1,-1?

For this question go by options is the best approach

Option 1) 1

As only 1 element so it definitely holds closure property.

Option 2)  2,-1

$2\times-1=-2;\;\;\;\;2+(-1)=1$

Both multiplication and addition are not contained in the given set so this set does not hold closure property.

Option 3) 1,-1

$1\times-1=-1;\;\;\;\;1+(-1)=0$

The addition is not contained in the given set so this set does not hold closure property.

#### Which of the following sequences are A.P.?If they are A.P. find the common difference.is 0.3,0.33,0.333,......

The given sequence is 0.3,0.33,0.333,0.3333

$a_{2}-a_{1}=0.33-0.3=0.03$

$a_{3}-a_{2}=0.333-0.33=0.003$

$a_{3}-a_{2}\neq a_{2}-a_{1}$

Means common difference is not same

So, this is not A.P.

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#### What is the contribution of the Thales of Miletus?

$Thales \;has \; been \; credited \; with \; the \; discovery \; of \;five \;geometric \; theorems:$

(1) Opposite angles formed by intersecting straight lines are equal

(2) Angle inscribed inside a semicircle is a right angle

(3) A circle is bisected by its diameter

(4) Angles in a triangle opposite two sides of equal length are equal

(5) A triangle is determined if its base and the two angles at the base are given

#### In the given figure,AB//QRand PA=2cm and PB=3cm,then find the ratio of the areas of the?PQR and?PAB

PA = 2cm;    AQ = 3cm

PQ = PA+AQ= 2+3= 5cm

$\frac{Area\; \Delta\; PQR}{Area \;\Delta\; PAB}=\frac{PQ^2}{PA^2}=\frac{5^2}{2^2}=\frac{25}{4}$

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#### (C) Find out Median from the following seriesSizeFrequency 0-10 02 10-20 18 20-30 30 30â€“40 45 40â€“50 35 50-60 30 60-70 06 70-80 03?

 Data Range Frequency Cumulative Frequency 0-10 02 02 10-20 18 20 20-30 30 50 30-40 45 95 40-50 35 130 50-60 30 160 60-70 06 166 70-80 03 169 Total 169

First, add the total number of frequencies.

Here, F=169

$\frac{F}{2}=\frac{169}{2}=84.5$

Hence, the value 84.5 lies in the class interval 30-40 from the cumulative frequency table.

So, 30-40 is the median class.

#### Trignometry identites

Some of the important trigonometry identities are

$\sin^2 \theta+\cos^2 \theta=1$

$\csc^2 \theta-\cot^2 \theta=1$

$\sec^2 \theta-\tan^2 \theta=1$

$\sin(90-\theta)=\cos \theta$

$\cos(90-\theta)=\sin \theta$

$\tan(90-\theta)=\cot \theta$

$\cot(90-\theta)=\tan \theta$

$\sec(90-\theta)=\csc \theta$

$\csc(90-\theta)=\sec \theta$

$\sin(A+B)=\sin A \cos B+\cos A \sin B$

$\sin(A-B)=\sin A \cos B-\cos A \sin B$

$\cos(A+B)=\cos A \cos B-\sin A \sin B$

$\cos(A-B)=\cos A \cos B+\sin A \sin B$

$\tan(A+B)=\frac{\tan A+\tan B}{1-\tan A \tan B}$

$\tan(A-B)=\frac{\tan A-\tan B}{1+\tan A \tan B}$

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#### Express the rational number with numerators 15 and denominator (-10)Â with: 1)numerator 75, 2)denominator -40

1)numerator 75

$Rational\; number=\frac{15\times5}{-10\times 5}=\frac{75}{-50}$

2)denominator (-40)

$Rational\; number=\frac{15\times4}{-10\times4}=\frac{60}{-40}$