A 12 m long wire is cut into two pieces, one of which is bent into a circle and the other into a square enclosing the circle. What is the radius of the circle?

Correct Answer: 4

Explanation:

Let 2+?2+?2..... = t
2+?t = t
t - 2 = ?t
t = 4

Correct Answer: a/b = c/d

Explanation:

(4a+7b)(4c-7d) = (4a-7b)(4c+7d)

(4a+7b)/(4a-7b) = (4c+7d)/(4c-7d)

Using componendo and dividendo

(4a+7b)+(4a-7b) / (4a+7b)-(4a-7b)

= (4c+7d)+(4c-7d) / (4c+7d)-(4c-7d)

Or 8a/14b = 8c/14d

Or a/b = c/d

Correct Answer: xy/x+Y

Explanation:

Correct Answer: Each fraction is equal to 1/2 or -1

Explanation:

a/(b+c) = b/(c+a) = c/(a+b)

Taking reciprocal and adding 1 to each ratio we get;

(b+c)/a + 1 = b/(c+a) + 1 = c/(a+b) + 1

Or (a+b+c)/a = (a+b+c)/b = (a+b+c)/c

So this can only be equal when a=b=c or a+b+c = 0

When a=b=c we get

a/(b+c)= ½When a+b+c = 0 we get

b+c = -a

So a/(b+c) = -1

So the ratios are ½ or -1

Correct Answer: cos2A

Correct Answer: 20 cm

Explanation:

Let there be 2 circles with centre O1 and OAB is the common chord

Since both passes through the center of each other as shown in figure

So O1O is the radius of both

Let O1O = r = AO1= AO

AX = AB / 2 = 5?3 cm (since OX perpendicular to chord bisects it)

AOO1 forms an equilateral triangle with on side = radius = r

Sin 60 = ?3/2 = AX / AO = 5?3/r

So r = 10 cm

So diameter = 20 cm

 

Correct Answer: 2m ?m^2/n

Explanation:

Mean of m observations is n

Mean of n-m observations is m

So total = nm+ (n-m)m

Total observations = n

Mean = Total / Total observations = (2mn-m^2)/n = 2m ?m^2/n

 

Correct Answer: 1

Correct Answer: 5?3

Correct Answer: y