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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 14,413

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

Character is who you are when no one is looking.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,380

Hi;

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 14,413

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)

Character is who you are when no one is looking.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,380

Hi;

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

**Online**

**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 14,413

Hi bobbym,

The

The answer 115 is correct. Well done!

116. If f(x) = x + 5, g(x) = x[sup]2[/sup], then f o g = ____________

(a) (x + 5)[sup]2[/sup]

(b) x[sup]2[/sup] + 5

(c) x[sup]2[/sup] + x

(d) x[sup]2[/sup] - 5

117. The Greatest Common Divisor of

is ____________(a)

(b)

(c)

(d)

Character is who you are when no one is looking.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,380

Hi;

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

**Online**

**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 14,413

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

Character is who you are when no one is looking.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,380

Hi;

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

**Online**

**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 14,413

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

Character is who you are when no one is looking.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,380

Hi;

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

**Online**

**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 14,413

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) 55[sup]2[/sup]

(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

Character is who you are when no one is looking.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,380

Hi;

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

**Online**

**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 14,413

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 x[sup]2[/sup] + 9 = 0 is ___________

(a) unreal or imaginary

(b) real, unequal, and irrational

(c) real, unequal, and rational

(d) real, equal, and rational

Character is who you are when no one is looking.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,380

Hi ganesh;

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

**Online**

**noelevans****Member**- Registered: 2012-07-20
- Posts: 236

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 (2012-09-25 12:47:09)*

Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).

LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 14,413

Hi bobbym and noelevans,

The answers 124 and 125 are correct. Well done, bobbym!

Thanks for the information, noelevans!

126. The point which satisfies the linear inequation x + y ≤ 8 is ________________

(a) (4,5)

(b) (4,3)

(c) (5,4)

(d) (3,8)

127. The angle in the semi-circle is ________________

(a) acute angle

(b) obtuse angle

(c) right angle

(d) straight angle

Character is who you are when no one is looking.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,380

Hi;

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

**Online**

**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 14,413

Hi bobbym,

The answer 126 is correct. Well done!

The

128. If a line is drawn parallel to one side of a triangle the other two sides are divided in __________

(a) same

(b) the same ratio

(c) parallel

(d) perpendicular

129. The co-ordinate of the mid-point of the line segment joining the points A(-3,2) and B(7,8) is ____________

(a) (5,5)

(b) (-5,5)

(c) (-2,5)

(d) (2,5)

Character is who you are when no one is looking.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,380

Hi;

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

**Online**

**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 14,413

Hi bobbym,

The answers 128 and 129 are correct. Well done!

130. If sin θ = tan θ, then the value of θ is _______________

(a) 45 degrees

(b) 90 degrees

(c) 1 degree

(d) 0 degrees

131. The variance of 5 scores is 16. If each one of them is divided by 2, then the standard deviation of the new scores is ________________

(a) 4

(b) 8

(c) 2

(d) 16

Character is who you are when no one is looking.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,380

Hi;

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

**Online**

**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 14,413

Hi bobbym,

The answers 130 and 131 are correct. Well done!

132. The n[sup]th[/sup] term of a sequence is 7n - 3. Its common difference is _____________

(a) 7

(b) 4

(c) 7n

(d) -7

133. The Least Common Multiple of

is ____________________(a) x[sup]6[/sup]

(b) x[sup]10[/sup]

(c) x[sup]7[/sup]

(d) x[sup]23[/sup]

Character is who you are when no one is looking.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,380

Hi;

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

**Online**

**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 14,413

Hi bobbym,

The answers 132 and 133 are correct. Well done!

134. The mid-point of the line segment joining the points (8,2) and (4,6) is ____________

(a) (2,2)

(b) (6,4)

(c) (4,4)

(d) (2,-2)

135. The slope of y = 5x - 5 is ____________

(a) 5

(b) -5

(c) 1

(d) -1

Character is who you are when no one is looking.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,380

Hi;

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

**Online**