Q22. Express 103(97) as a difference of squares.

A22. 103(97) = (100 + 3)(100 - 3)

= 100^{2} - 3^{2} = 9991

Q23. The shaded area between a circle of radius *r* and an ellipse is π*r*^{2}/2. Find x.

A23. Shaded area = π*r*^{2}/2 = π*r*^{2} - πab, where a = x and b =*r*.

π*r*^{2}/2 = π*r*^{2} - πx*r*

*r*/2 = *r* - x

x = *r*/2.

Q24. What is the perimeter of the triangle defined by A( 0,0 ) , B( 4,0 ) and C( 4,4 ) ?

A24. Perimeter = AB + BC + CA

= 4 + 4 + 4√2 = 8 + 4√2

Q25. A cone-shaped cup and a cylinder have the same height and the same diameter. How many

cups of water from the cone are needed to fill the cylinder ?

A25. 3

Q26. The Pythagorean triple that includes 11 also includes two consecutive integers. Find them.

A26. 11^{2} + x^{2} = (x+1)^{2}

121 + x^{2} = x^{2} + 2x +1

x = 60

The full triple is 11, 60 and 61.

Q27. Find the distance between the plane and the horizon:

A27. d^{2} + 6100^{2} = 6110^{2}.

d = 349 km.

Q28. The cost of two pizzas of the same thickness is proportional to area. A pizza 10 cm in diameter costs $5. How much should a 20 cm one cost ?

A28. (20/10)^{2}($ 5) = $ 20.

Q29. Factor a^{3} + b^{3} + c^{3} -3abc

A29. (a + b + c)(a^{2} + b^{2 }+ c^{2}- ab -ac -bc)

Q30. Why can't a Pythagorean triple include 3 odd numbers ?

A30. The square of any odd number is odd, and the sum of any two odd numbers is even. Since the square root of an even number is also even, then c = √(a^{2} +b^{2}) would not be odd.

Q31. Find the next number in the sequence

2, 12, 36, 80, 150 , ...

A31. 2 = 1^{2} + 1^{3}

12 = 2^{2} + 2^{3}

36 = 3^{2} + 3^{3}

sixth term = 6^{2} + 6^{3} = 252.

Q32. One way of testing whether 10277 is prime is to test its divisibility by all primes < y.

What should y be ?

A32. √10277 = 101.4. Incidentally 10277 = 43* 239.

Q33. How many real roots does the equation x^{4} + x^{3} + x^{2} + x = 0 have ?

A33. Just two. x^{4} + x^{3} + x^{2} + x = 0.

x^{3}( x + 1) + x (x + 1) = 0.

(x + 1 )( x^{3} + x) = 0.

x(x+1)(x^{2}+1) = 0.

so x = 0 or -1.

Q34. Express the area of the triangle in terms of angles A and B and its base d.

A34.

(continued)

xtanA = tanB (d - x)

xtanA = dtanB - xtanB

x(tanA + tanB) = dtanB

Q35. When reflected in y = x, the equation of a certain line remains unchanged. If this line passes through (2 , 2), what is its equation ?

A35. For the equation of a line to be unaffected after such a reflection, its slope must be -1.

y = mx + b

2 = -1(2) + b

b = 4

y = -x + 4

Q36. When expanded, ( 3x + 2 )^{6} has seven terms. What is the coefficient of the seventh term ?

A36. Using Pascal's triangle:

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

*1 6 15 20 15 6 1*

The above line applies to a power of 6 so that:

(3x)^{6} + *6*(3x)^{5}(2) + *15*(3x)^{4}(2^{2}) + 20(3x)^{3}(2^{3}) + *15*(3x)^{2}(2^{4}) + *6*(3x)(2^{5}) + 2^{6}

The middle term's coefficient is 4320.

Q37. Why are sewer caps round and not square-shaped?

A37. The round cover sits on a lip that is smaller than the cover. Because its diameter is constant, the cover cannot fall through. Keep in mind that the diagonal of a square is about 1.41 times as long as one of its sides. As a result if the side of a square-

cover was lined up with the diagonal of the hole, it would slide through in spite of its lip.