Mathematics Questions and Answers

\(\frac{a - b}{a + b}\) - \(\frac{a + b}{a - b}\) = \(\frac{(a - b)^2}{(a + b)}\) - \(\frac{(a + b)^2}{(a - b0}\)

applying the principle of difference of two sqrt. Numerator = (a - b) + (a + b) (a - b) - (a + b)

= (a = b + a = b)(a = b - a = b)

2a(-2b) = -4ab

= \(\frac{-4ab}{(a + b)(a - b)}\)

= \(\frac{-4ab}{a^2 - b^2}\)

Sum of interior angles of a polygon = (2n - 4) x 90

where n is the no. of sides

sum of interior angles of the quad. = ({2 x 4} -4) x 90

(8 - 4) x 90 = 4 x 90o

= 360o

(p + 10)o + (p - 30)o + (2p - 45)o + (p + 15)o = 360o

p + 10o + p - 30o + 2p - 45o + p + 15 = 360o

5p = 360o + 50o

= 82o

surface area of formula = πr\(^2\) 

If the radius is reduced then let its radius be x.

Its area is πx\(^2\) .

x : r = 2 : 5,

so \(\frac{x}{r}\) =  \(\frac{2}{5}\)

 → 5x = 2r

Hence, x = 0.4r.

Hence the area of the new circle is π (0.4r)\(^2\)  = 0.16π r\(^2\) .

The ratio of the two areas is

0.16πr\(^2\) : πr\(^2\) 

 = 0.16 : 1 = 16 : 100

= 4 : 25.

Soln. a*b = ab + a + b,

a ♦ b = a + b + 1

a*c = ac + a + c

(a*b) ♦ (a*c) = (ab + a + b + ac + a + c + 1)

= ab + ac + 2a + b + c + 1

Let the cost of living = y.

The new cost of living = \(y + \frac{15y}{100} = 1.15y\)

The food bill now = \((1 - \frac{90}{100})(1.15y)\)

= \(1.035y\)

The fractional increase in food bill = \((1.035 - 1) \times 100% = 3.5%\)

= \(\frac{35}{1000} = \frac{7}{200}\)