x² - 2x + 2 = 0

x³ - 2x² + 2x = 0

x⁴ - 2x³ + 2x² = 0

3) + 2) - 1)

x⁴ - x³ - x² + 4x - 2 = 0

x⁴ - x³ + x² + 2 = 2x² - 4x + 4

= 2(x² - 2x + 2)

= 0

999/5 = 199

999/7 = 142

Not divisible by 35
So exclude

199/7 = 28

142/5 = 28

199 + 142 - 28 - 28 = 285

105*2 = 210

210*8 + 105 = 1785

1785*1 + 210 = 1995

1995 + 1785 = 3780

See

1 digit = No such number is possible

2 digit =

10x + y = 6(x + y)

4x = 5y

x = 5, y = 4

54


3 digit = [100/6] = 16

So sum of digits must be greater than 16.

You can check from 17 to 27(maximum possible for 999)
Or

By observation can see 17*6 = 102 and 27*6 = 162
No Number can satisfy the given condition.

So only one solution

54

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A cube is placed inside a cone of radius 20 cm and height
10 cm, one of its face being on the base of the cone and
vertices of opposite face touching the cone. What is the
length (in cm) of side of the cube?

[20 - x/√2]/x = 20/10

20 - x/√2 = 2x

20 = x[2 +1/√2]

20√2= x [2√2+1]

x= 20√2/[1+2√2]

The radius of base of a solid cylinder is 7 cm and its height
is 21 cm. It is melted and converted into small bullets.
Each bullet is of same size. Each bullet consisted of two
parts viz. a cylinder and a hemisphere on one of its base.
The total height of bullet is 3.5 cm and radius of base is 2.1
cm. Approximately how many complete bullets can be
obtained?

7*7*21/(2.1*2.1*2.8
.
250/3 = 83 ??

49*21 = n( 2.1² *1.4 + 2/3 *2.1³)
490/2.1 = n( 2.8)
1000/12 =n
83 = n

Let the equal sides are a
Unequal side b
Perimeter 200
2a+b=200

Triangles inequality
2a>b
Put b from the above
2a>200-2a
A>50
A Will be maximum integer at b=2
2a=200-2
A=99
So 50<a<99
49 values