Hi bobbym,

The answers 110 and 111 are correct. Well done!

112. The sum of the first n natural numbers is 10. The sum of their cubes is _____________
(a) 20
(b) 100
(c) 1000
(d) 30

113. If -33(mod 9) = x, then the value of x is _____________
(a) 3
(b) 4
(c) 5
(d) -3

Hi bobbym,

The answer 112 is correct. Well done! Regarding 113,

114. The curved surface area of a hollow cylinder is __________
(a)

(b)

(c)

(d)

115.

= _____________
(a)

(b)

(c)

(d)

Hi bobbym,

The

The answer 115 is correct. Well done!

116. If f(x) = x + 5, g(x) = x², then f o g = ____________
(a) (x + 5)²
(b) x² + 5
(c) x² + x
(d) x² - 5

117. The Greatest Common Divisor of

is ____________
(a)

(b)

(c)

(d)

Hi bobbym,

The answers 116 and 117 are correct. Well done!

118. If the straight line 7x - 5y = k passes through the point (1,1), then k is _________
(a) 2
(b) -2
(c) 1
(d) -1

119. A die is rolled once, the probability of getting an even number is _____________
(a) 1/3
(b) 2/3
(c) 1/2
(d) 1/4

Hi bobbym,

The answers 118 and 119 are correct. Well done!

120. Which is a linear equation?
(a) 2x + 3y = 5
(b) 2x + 3y > 5
(c) 2x + 3y < 5
(d) 2x < 3y

121. The tangents at the ends of a diameter are _______________
(a) perpendicular
(b) parallel
(c) equal
(d) None of these

Hi bobbym,

The answers 120 and 121 are correct. Well done!

122. If the sum of 1 + 2 + 3 + ....... + 10 is 55, then the sum of 1³ + 2³ + 3³ + ....... + 10³ is _________________
(a) 55²
(b) 55³
(c) 3024
(d) 3026

123. The common ratio of the Geometric Progression 64, 16, 4, .... is ___________
(a) 4
(b) -4
(c) 1/4
(d) -1/4

Hi bobbym,

The answers 122 and 123 are correct. Well done!

124. If (ax + b) divides p(x), then the remainder is _______________
(a) p(-a)
(b) p(a)
(c) p(-b/a)
(d) p(b/a)

125. The nature of the roots of the equation x² + 9 = 0 is ___________
(a) unreal or imaginary
(b) real, unequal, and irrational
(c) real, unequal, and rational
(d) real, equal, and rational

Hi!

#124 brings to mind a generalization of the remainder theorem:

The theorem is simple to prove: Let the divisor be called D(x).
Then  P(x)                R(x)
         ___  = Q(x) + ____   and multiplying through by D(x) gives
         D(x)               D(x)

         P(x) = Q(x)D(x)  + R(x)

Therefore for the values of x that make D(x)=0 we have P(x)=Q(x)*0 + R(x); that is, P(x)=R(x).
And this works for real and COMPLEX values of x.

D(x) = (x-3)(x+1) = x^2-2x-3 divided into P(x) gives R(x)=-2x+3 so P(3)=-2(3)+3=-3 and P(-1)=-2(-1)+3=5.

D(x) = (x+i)(x-i) = x^2+1 divided into P(x) gives R(x)=5x-8 so P(i)=5i-8 and P(-i)=-5i-8.

Let D(x) = x^2-4x+13 which has roots 2+3i and 2-3i.
Then P(x) divided by D(x) has remainder 3x-8.  so P(2+3i) = R(2+3i) = 3(2+3i)-8 = -2+6i.
We also have P(2-3i) = R(2-3i) = 3(2-3i)-8 = -2-6i.

So to evaluate P(x) at a+bi we form D(x) = (x-(a+bi))(x-(a-bi)) = x^2-2ax+(a^2+b^2).
Then dividing P(x) by D(x) we get the corresponding remainder R(x).  Since this is a linear
expression P(a+bi) = R(a+bi) and P(a-bi) = R(a-bi) are complex conjugates.

This gives us a way to evaluate polynomials at complex values WITHOUT having to work with
complex values in the powers of x above the first power (which is involved in R(x)).  It keeps it
all real until the very end when evaluating R(x).

Of course there is not much advantage to evaluating with real roots this way since we can just
divide P(x) by x-a and look at the remainder.  But we could do just ONE division to obtain the
evaluations for two (or more) real roots using this generalization.

Also if R(x)=0 then D(x), each factor of D(x), and Q(x) are all factors of P(x), which is the factor
theorem with higher degree divisors.

Have a blessed day! 

Oops!  It was number 124 than brought this to mind, not number 23.

Last edited by noelevans (2012-09-26 10:47:09)

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