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Detailed Solution

Daigonal of sq=4+4√2=4(1+√2)

Side=4(1+√2)/√2=2√2(1+√2)

?A={2√2(1+√2)}²-2.π .2²

={8(1+2.1.√2+2)}-8π

=8(3+2√2)-8π

=8(3+2×1.414)-8×3.14

=8(3+2.828)-8×3.14

=8(5.828)-8×3.14

=46.828-25.12

=21.708. Cm²

In the bigger triangle, all angles are 60 degree each.(Isoceles and angle sum property)

In smaller traingle

A=180-70-60=50 degree.

B=180-50-90=40 degree.

x=180-40-60 =80 degree.

Extend AB and DE such that they intersect at FAF

AF=7+5=12

EF=8-3=5

AE²= 12²+5² =169= 13²

So AE=13m

Adding resistance in series is straight addition...adding resistance in parallel is by adding their recipricals and inverting to find the total.

So for the two in parallel...

1/R = 1/40 + 1/40 = 2/40 =1/20

Inverting gives...

R=20

So adding the series ..

Total resistance = 20 + 20 = 40 ohms

So the current flowing is 24V/40 ohms = 0.6 amps

After the bulb blows the total resistance is 20 + 40 = 60 ohms

So the new current is 24V/60 ohms = 0.4 amps.

So bulb A had 0.6 amps previously, but now has 0.4 amps...so it dims.

Bulb B had a half share of 0.6 amps which was 0.3 amps, but now it has 0.4 amps ...so it brightens.

On the 10th birth day mother was of 30 year older (40-10 ) . Now - she is 60 years and my age is 30 years.

Solution

Total 100 students

60 likes Maths means 40 dislikes maths

50 likes Science means 50 dislike Science.

10 dislikes both

That means 10 is part of 40 who dislike maths

Also 10 is part of 50 who dislike science.

Intersection is 10

Total students who dislike either maths or science. 100-(10+30+40) =20

Step-by-Step Solution

1. Use substitution:Let x = y + 25.This means:

• y + 20 = x - 5

• y + 30 = x + 5

2. Rewrite the equations with :The given equation becomes:

x^2 = 2023

The expression we need to find becomes: (x - 5)(x + 5)

3.Recall that (a - b)(a + b) = a^2 - b^2.Applying this to our expression: (x - 5)(x + 5) = x^2 - 5^2

(x - 5)(x + 5) = x^2 - 25

(x - 5)(x + 5) = 2023 - 25 (Since x^2 = 2023)

(y + 20) (y + 30) = 1998

4. Final Answer:The value is 1998.

Solution

Let the base = x

If x² + 4² = 8²

then x² = 8² - 4²

x² = 64 - 16

x² = 48

√x² = √48

x = 6.9 ft (base)

The boy sitting upright at 6 ft from the stem wouldn't be safe.

Solution

The Hour hand can move one resolution dot for every 12 min (60/5). So in the next 12 min it will align to hour position of 11:00 (upside down). Hence it is now 10:48

Solution

5/2(h)=20 cm2

h=8cm

Area of shaded=1/2(10)11-1/2(5)8

= 35cm2 ans

Solution:

If the ABCD trapezium area is 54cm*2, then

(AB+CD)×AD/2=54

at this condition AB=AD=(1/2)DC AB=X and we have

equality

(X+2X)×X/2=54

(3X*2/2)=54

X*2=54×2÷3=36

X=V36=6cm

So the trapezium sides length are AB=AD=6cm and CD=12cm

The circle radius is R=6÷2=3cm

Answer: the circle radius is 3cm

Solution

Area of big square+Area of small square

9^2+6^2=81+36=117cn^2

Area of big triangle =15×9/2=135/2=67.5cm^2

Area of two small triangles =1/2×6×6+1/2×3×9

18+13.5=31.5cm^2

Therefore,The Area of shaded region=117-(67.5+31.5)

117-99=18cm^2 Ans

Solution

2n + 2(n+3)= 26 , 4n = 20 , n = 5

But DC = n+3 = 5+3 = 8

Also DC = 2CE , 8 = 2CE , CE = 4

BC= AD = n = 5

Therefore Area of Triangle BCE

= 5×4/2 = 10 sq cm

Solution

c² = a² + b² for equal lines a=b

√32 = 2a²

2a² = 2√16

a²= √16

a =4

Therefore perimeter (P)

P = 4×4

= 16 units

Solution

In Left Triangle => 180-70-70=40

Exterior Angle of the same Triangle => 180-40=140

Remaining angle of Quadrilateral => 360-140-140=80

X => 180°-80°= 100° (ASP)

SOLUTION

1st square side=√144=12

2nd square side=6

3rd square side=3

Top square side=X

X²==9²+9²

X²=81+81

X²=162

Area=X²

Area=162 cm²

Answer:

Perimeter = (3*5)+(9*3)+6

= 15+27+6 = 48 cm.

Question

Answer

𝒶 + 4𝒷 = 34

𝒶 + 𝒷 = 19

𝒶 + 4𝒷 - 𝒶 - 𝒷 = 34 - 19

= 3𝒷 = 15

𝒷 = ¹⁵∕₃ = 5

𝒶 + 𝒷 - 𝒷 = 19 - 5

= 𝒶 = 14

Angles in small isosceles triangle ACE are 20°, 80° and 80° ( Angle Sum Property)

Angle BEC= 100° (Linear Pair)

Angles in larger isosceles triangle EBC are 100°, 40 ° and 40°.( Angle Sum Property)

So x = 180° - 20° - 40° = 120°.

25(On the left half of the circle they all divide by 3 to give the answer opposite )

Buy 36 apples @ $2 =$72 + 12 apples free ( as per deal buy 3 get 1 free )

= 48 apples

Pay $ - 72

Answer is 1

Let, the weight of empty jar = x, and weight of liquid= y

For the first jar,

weight of empty jar + weight of liquid, i.e.,

x + y = 120 [Equation 1]\

And, for the second case,

weight of half filled second jar + empty second jar, i.e. {(x/2)+y} + y = 150

or, (x/2) + 2y = 150,

or, x + 4y = 300. [Equation 2]

\Now comparing Equation 1 with Equation 2, we can find,

3y = 180, or, y = 60

Therefore, Weight of the empty jar is 60g.

Three friends are on a road trip, and they rent a triple room for a night. When they get to the hotel, they pay the fee of $30 and go up to their room. The porter brings up their bags and refunds them $5 because the hotel is running a weeknight special. The three friends each keep one of the dollars and give the porter a $2 tip. Later, they sit down to work out their expenses for the weekend and find they have a problem. They each paid $10 for the room and each got $1 back, making their contributions $9. Then they gave the porter a $2 tip. However, 9 times 3 is 27, plus 2 is $29. Where did the extra dollar go?

Answer: They didn’t spend $29, they spent $27: $25 on the room, and $2 for the porter’s tip. The three remaining dollars were returned to them!

x^(64/x) = 4

x^(1/x) = 4^(1/64)

x^(1/x) = 4^(4/256)

x^(1/x) = (4⁴)^(1/256)

x^(1/x) = 256^(1/256) [Note: 4⁴ =256]

Hence x = 256

Amazing Number Coincidence! Can you find another?

Infinite a,b,c if bc-ab-ac=1 where a,b,c are integers > 0

Why 2WhyThe numbers 220 and 284 are called a pair of amicable numbers because each number is equal to the sum of the proper divisors of the other number.

A proper divisor is a positive factor of a number, excluding the number itself.

For 220, the proper divisors are: 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110. If you add these up, the sum is 284.

For 284, the proper divisors are: 1, 2, 4, 71, and 142. If you add these up, the sum is 220.

So, each number is the sum of the proper divisors of the other number, which is the definition of amicable numbers. This concept has been known since ancient times and is a special case in number theory.

called pair of amicable numbers ?

एक बैट और एक गेंद का मूल्य कुल मिलाकर 1.10 डॉलर है।

बैट का भाव गेंद से 1.00 डॉलर अधिक है।

गेंद का मूल्य क्या है?

एक बैट और एक गेंद का मूल्य कुल मिलाकर 1.10 डॉलर है।

बैट का भाव गेंद से 1.00 डॉलर अधिक है।

गेंद का मूल्य क्या है?

उत्तर है 5 सेंट।

As per L’Hopital’s rule it is in an indeterminate form meaning it has no value as is and is undefined.

0⁰ is one of the 7 indeterminate forms along with 0/0, ထ/ထ, ထ - ထ, ထx0, ထ^0, and 1^ထ

Which is not a leap year

1900,1976,2000,2024

A year is 365.2422 days so adding a day every 4 years is almost sufficient, but then there should be exactly 365,25 days. Of that reason you have to “skip” some leap years. If it was exactly 364.24, you could just skip every 25th leap year, which is by every turn of century, but that is too much skipping, since it’s slightly longer with the added digits (365.2422 instead of 365.24). Therefore the rule is every year divisible by 4 is leap year, except years divisible by 100 i.e not those divisible by 400.

2000 is divisible by 100 but also 400, so it is actually a leap year.

1900 is divisible by 100 but not by 400, so NOT a leap year

Brainiac-The Brain ticklers

The middle triangle is equilateral, so each angle is 60 degrees, the supplement of 180 -(60+56) = 64 . Likewise, the supplement of 180 - (60+36) = 84

angle a = 180 - (56 + 84) = 40

Since ab = ac, then < b = < c = (180 - 40)/2 = 70 degrees.

The red angle = 180 - (64 + 70) = 46 degrees

The plural words of Syllabus, locus, focus,index, formula and axis are Syllabi, loci, foci, indices, formulae, axes respectively.

Answer

Find Area of Shaded Region

Answer

You can use the angles to show all 3 triangles are similar . Thus all the angles are proportional. using the 2x2 square and its adjacent triangle, you can establish the ratio for all the triangles there to be 2:1. Now imagine an 5th larger triangle of height 6 and base 3 (as it would be also be similar to the others) with the hypotenuse as the side of the green square. You can then determine the side length of the square with the Pythagorean theorem - thus the side of the square is root 45. And area is then 45.

<D=600 (Complete Angle)

& AD=DC (Given)

⇒ Angle A=Angle C=600

⇒Triangle ADC is equilateral

⇒AC=6 cm

⇒Area of Triangle ADC=15.57cm2 approx. (Heron’s Formula)

⇒Area of Triangle ABC=34.85cm2 approx. (Heron’s Formula)

⇒ Required Area= Area of Triangle ABC- Area of Triangle ADC

=34.85-15.57= 19.28 cm2

Three Red Balls =30 gm

One Red Balls =10 gm

One Red Ball+Two White Balls=30+30=60gm

10gm+Two White Balls=60gm

Two White Balls=50gm

One White Ball=25gm

25. (On the left half of the circle they all divide by 3 to give the answer opposite )

Solution

The Hour hand can move one resolution dot for every 12 min (60/5). So in the next 12 min it will align to hour position of 11:00 (upside down). Hence it is now 10:48

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Detailed Solution

Solution

Question

Answer

Let, the weight of empty jar = x, and weight of liquid= y

For the first jar,

weight of empty jar + weight of liquid, i.e.,

x + y = 120 [Equation 1]\

And, for the second case,

weight of half filled second jar + empty second jar, i.e. {(x/2)+y} + y = 150

or, (x/2) + 2y = 150,

or, x + 4y = 300. [Equation 2]

\Now comparing Equation 1 with Equation 2, we can find,

3y = 180, or, y = 60

Therefore, Weight of the empty jar is 60g.

**Answer**: They didn’t spend $29, they spent $27: $25 on the room, and $2 for the porter’s tip. The three remaining dollars were returned to them!

x^(64/x) = 4

Amazing Number Coincidence! Can you find another?

Why 2WhyThe numbers 220 and 284 are called a pair of amicable numbers because each number is equal to the sum of the proper divisors of the other number.

called pair of amicable numbers ?

Which is not a leap year

1900,1976,2000,2024

Brainiac-The Brain ticklers

Answer

Answer

25.

(On the left half of the circle they all divide by 3 to give the answer opposite )

Answer: They didn’t spend $29, they spent $27: $25 on the room, and $2 for the porter’s tip. The three remaining dollars were returned to them!