UIL MATHEMATICS MAGIC : BOOK 3 - Ram Materials · PDF fileUIL MATHEMATICS WORKBOOK 3 1 UIL...

UIL MATHEMATICS WORKBOOK 3 1

UIL MATHEMATICS MAGIC : BOOK 3 BASIC MATHEMATICS 7. Otis Hot has a rectangular shape pool that is 20' long, 12' wide, and 4' deep. How many gallons of water will it take to fill the pool without spilling over? (A) 1870 (B) 3590 (C) 4155 (D) 6231 (E) 7181 1 cubic foot = 1728 cubic inches Volume = 20(12)4 = 960 cu. ft. Volume = 960(1728) cubic inches

Number of gallons = 960(1728)

231 = 7181

13. Victor has three more quarters than dimes. The total value of the dimes and quarters together is $13,00. How many dimes does he have? (A) 29 (B) 37 (C) 32 (D) 35 (E) 38 Let x = the number of dimes x + 3 = the number of quarters. .10x + .25(X + 3) = 13.00 10x + 25(x + 3) = 1300 10x + 25x + 75 = 1300 35x = 1225 ; x = 35 16. In distributing milk at a summer camp it is found that a quart of milk will fill either 3 large glass tumblers or 5 small glass tumblers. How many small glass tumblers can be filled with one large glass tumbler?

(A) 3

5 (B) 1

2

5 (C) 1

2

3 (D) 2 (E) 2

1

3


Let L = large tumbler and S = small tumbler

1 qt = 3L , therefore L = 1

3 qt

1 qt = 5S, therefore S = 1

5 qt

1

3 ÷

1

5 =

1

3

5

1

!"#

$%&

= 5

3 = 1

2

3

ALGEBRA 1 2. Find the distance between the x-intercept and the y-intercept of the function 2x - 3y + 6 = 0. (A) 1 (B) 5 (C) 5 (D) 2 3 (E) 13 To find the x-intercept, let y = 0 and solve for x. 2x - 3(0) + 6 = 0 ; 2x = - 6 ; x = - 3 To find the y-intercept, let x = 0 and solve for y. 2(0) - 3y + 6 = 0 ; - 3y = - 6 ; y = 2. If you connect the origin, x-intercept, and y-intercept you will have a right triangle whose legs are 2 and 3 units long. Use the Pythagorean Formula to find the distance between the x-intercept and the y-intercept. d 2 = 2 2 + 3 2 ; d 2 = 13 ; d = 13 5. R and S are the roots of 2x 2 - 3x - 5 = 0. Find R 2 + 2RS + S 2 .

(A) 2 1

4 (B) 3

2

5 (C) 4

2

3 (D) 5

3

5 (E) 6

1

4

R 2 + 2RS + S 2 = (R + S) 2 which represents the square of the sum of


the roots. If Ax 2 + Bx + C = 0, then - B

A is the sum of the roots and

!B

A

"#$

%&'2

is the square of the sum of the roots.

!!32

"#$

%&'

(

)*

+

,-

2

= 9

4 = 2

1

4

10. The T BAG wants to mix 90 cents a pound tea with some $1.50 a pound tea to make 20 pounds of a special blend of tea that sells for $1.20 a pound. How many pounds of the 90 cents a pound tea will the T BAG need? (A) 9 (B) 10 (C) 12 (D) 14 (E) 15 Let x = the number of pounds of the 90 cents a pound tea 90(x) + 150(20 - x) = 120(20) 90x + 3000 - 150x = 2400 - 60x = - 600 x = 10

22. If 1

a +

1

b =

1

c, then c = __________.

(A) 1

ab (B) ab (C)

a + b

ab (D)

ab

a + b

2ab

a + b

1

a +

1

b =

1

c

b

ab +

a

ab =

1

c

a + b

ab =

1

c


c = ab

a + b

26. What is the nature of the roots of the equation 2x 2 - 3x + 5 = 0? (A) imaginary (B) real and unequal (C) real and equal (D) equal to zero (E) both negative Use the discriminant : b 2 - 4ac If the discriminant is greater than 0 there are two unequal real roots. If the discriminant is equal to 0 there is one real root. If the discriminant is less than 0 there are two imaginary roots. b 2 - 4ac = (- 3) 2 - 4(2)(5) = - 31 The roots are imaginary.

27. Cindy and Sandra can complete a job in 6 2

3 minutes when they work

together. Cindy can do the job by herself in 10 minutes. How much time would Sandra need to complete the job by herself? (A) 20 min (B) 30min (C) 15 min (D) 24 min (E) None of these

Both

A(alone) +

Both

B(alone) = 1

Let x = Sandra's time

62

3

10 +

62

3

x = 1

10x62

3

10+

62

3

x

!

"

###

$

%

&&&

= 10x[1]

62

3

!"#

$%&

x + 62

3

!"#

$%&

10 = 10x


20

3

!"#

$%&

(10) = 10x - 62

3x

!"#

$%&

200

3 =

10

3x

x = 3

10

200

3

!"#

$%&

= 20

You should know the following : Cevian : A line segment joining the vertex of a triangle to any point on the opposite side. Centroid : The point of intersection of the three medians of a triangle. Circumcenter : The center of the circumscribed circle of a triangle. Circumradius : The radius R of the circumscribed circle of a triangle ABC.

2R = a

sinA =

b

sinB =

c

sinC (Law of Sines)

Incenter : The center of an inscribed circle of a triangle. The point of intersection of the three angle bisectors of a triangle. Orthocenter : The point of intersection of the three altitudes of a triangle. 7. Two similar triangles have their sides in the ratio of 1:3. The longest and shortest sides of the larger triangle are 21 and 9 respectively. Find the length of the other side of the larger triangle if the perimeter of the smaller triangle is 15. (A) 5 (B) 12 (C) 15 (D) 18 (E) None of these The sides of the larger triangle are 9, a, and 21 (from smallest to largest). The sides of the smaller triangle are x, y, and z.

1

3 =

x

9 ; 3x = 9 ; x = 3


1

3 =

y

21 ; 3y = 21 ; y = 7

Since x + y + z = 15, then 3 + 7 + z = 15 and z = 5.

1

3 =

5

a ; a = 15

12. If the perpendicular bisector of the segment with endpoints A(1, 2) and B(2, 4) contains the point (4, c), then the value of c is ___________.

(A) 7 (B) 7

4 (C) - 7 (D) 4 (E) - 4

Draw a segment with endpoints A and B. Draw a perpendicular bisector of segment AB with point C(4, c). Draw segments connecting point C to both A and B. The lengths of segments CA and CB should be equal. Use the distance formula and the fact that CA = CB to find c. 4 !1)

2+ (c ! 2)

2 = (4 ! 2)2+ (c ! 4)

2 9 + c 2 - 4x + 4 = 4 + c 2 - 8c + 16

4c = 7 ; c = 7

4

15. The distance between the centers of the circles x 2 + 2x + y 2 - 4y = 6 and x 2 - 6x + y 2 + 4 = - 1 is __________. (A) ` 20 (B) 4 (C) 4 2 (D) 5 (E) 5 2 Write the standard equation for the first circle. x 2 + 2x + 1 + y 2 - 4y + 4 = 6 + 1 + 4 (x + 1) 2 + (y - 2) 2 = 11 ; Center is (- 1, 2) Write the standard equation for the second circle. x 2 - 6x + 9 + y 2 + 4y + 4 = - 1 + 9 + 4


(x - 3) 2 + (y + 2) 2 = 12 ; Center is (3, - 2) Use the distance formula to find the distance between the two centers. d = (!1! 3)

2+ (2 ! !2)

2 = (!4)2+ 4

2 = 32 = 4 2 22. The distance from the center of a circle to a chord is 5. If the length of the chord is 24, what is the length of the radius of the circle? (A) 13 (B) 7 (C) 5 (D) 10 (E) 17 Sketch a circle with a chord. Draw the radius that is perpendicular to the chord. Note : The radius will bisect the chord so that the two segments formed are 12 units long. Label the segment going from the center of the circle to the chord as 5. Draw a radius from the center to one endpoint of the chord. Call this radius R. Use the Pythagorean Theorem. R 2 = 5 2 + 12 2 R = 5

2+12

2 = 13 37. In an isosceles trapezoid ABCD, AB = CD = 13, BC = 24 and AD = 14. Find the length of the altitude. (A) 5 (B) 7 (C) 13 (D) 15 (E) None of these Sketch trapezoid as described in the problem. Draw altitude. You will notice that the altitude will be a leg of a right triangle whose hypotenuse is 13 and other leg is 5. Use the Pythagorean Theorem to find the altitude. 13 2 = 5 2 + h 2 h 2 = 169 - 25 h = 144 = 12 39. The hypotenuse c and one leg a of a right triangle ABC differ by 1. The square of the other leg is __________. (A) 2c + 1 (B) 2c - 1 (C) c + 1 (D) c - 1 (E) None of these Draw a right triangle ABC, with hypotenuse c, and legs b and c - 1.


Substitute this information in the Pythagorean Formula. c 2 = (c - 1) 2 + b 2 c 2 = c 2 - 2c + 1 + b 2 2c - 1 = b 2 ; b 2 = 2c - 1 ALGEBRA 2 Be familiar with the following : Arithmetic, Geometric and Harmonic Means Let a

k ≥ 0, k = 1, 2, 3, ...,n then the following are known :

(A) Arithmetic Mean = a1+ a

2+ ...+ a

n

n

Example A : Find the arithmetic mean of 4 and 10.

Solution : 4 +10

2 = 7

Example B : Find the arithmetic mean of 8, 22, and 60.

Solution : 8 + 22 + 60

3 = 30

(B) Geometric Mean = a

1a2...a

n

n Example A : Find the geometric mean between 8 and 18. Solution : 8(18) = 144 = 12 Example B : Find the geometric mean of 4, 8, and 16. ` Solution : 4(8)(16)3 = 512

3 = 8


(C) Harmonic Mean = 1

H = 1

n

1

a1

+1

a2

+ ...+1

an

!

"#$

%& or

1

H =

1

n

1

akk=1

n

!

(i) H2

= 2a1a2a1+ a

2

Example A : Find the harmonic mean of 3 and 9.

Solution : 2(3)(9)

3+ 9 =

54

12 = 4.5

(ii) H3 = 3a

1a2a3

a1a2+ a

1a3+ a

2a3

Example A : Find the harmonic mean of 2, 8, and 12.

Solution : 3(2)(8)(12)

2(8) + 2(12) + 8(12) =

576

136 = 4.24

------------------------------------------------------------------------------------------------------------ Given a positive integer n > 1, let ! (n) [sigma of n] denote the sum of all the positive divisors (factors) of n. If n = p

1

k1( ) p2k2( )(p3)(p4 ) is the prime factorization

of n > 1 where pi, 1 ≤ i ≤ 4 are distinct prime factors, then

! (n) = p1k1+1!1

p1!1

p2k2+1!1

p2!1

(p3 + 1)(p

4 + 1)

A positive integer n is either deficient, perfect or abundant. Deficient ! (n) < 2n Ex. 1 - 5, 7 - 11, 13 - 17, 19 Note : 10 = 1 + 2 + 5 = 8 ; 8 < 2(10) Perfect ! (n) = 2n Ex. 6, 28, 496, 8128 Note : 28 = 1 + 2 + 4 + 7 + 14 + 28 = 56 ; 56 = 2(28)


Abundant ! (n) > 2n Ex. 12, 18, 20, 30 Note : 12 = 1 + 2 + 3 + 4 + 6 + 12 = 28 ; 28 > 2(12) Note : A multiple of an abundant number is abundant. ------------------------------------------------------------------------------------------------------------ Let the sequence a

1, a

2, a

3, ..., a

n, ... be defined recursively as :

a1 = a

1, a

2 = a

2, a

n = a

n!1 + a

n!2 where n ≥ 3

The 9th term of the function above is 5a

5 + 3a

4.

Example A : The 9th term of 1, 3, 4, 7, 11, ... is __________.

Solution : 5(11) + 3(7) = 55 + 21 = 76

Example B : The 9th term of 2, 5, 7, 12, 19 + ... is __________. Solution : 5(19) + 3(12) = 95 + 36 = 131

Example C : The 9th term of 1, 4, 5, 9, 14, ... is __________. Solution : 5(14) + 3(9) = 70 + 27 = 97 ------------------------------------------------------------------------------------------------------------ The sum of the first nine terms of the sequence 2 + 4 + 6 + 10 + 16 + ... is

__________. The sum of the first nine terms of the sequence is equal to : 14(5th term) + 6(4th term) Solution : 14(16) + 6(10) = 224 + 60 = 284 Example A : The sum of the first nine terms of the sequence 1 + 5 + 6 + 11 + 17 + ... is __________. Solution : 14(17) + 6(11) = 238 + 66 = 304 Example B : The sum of the first nine terms of the sequence 1 + 3 + 4 + 7 + 11 + ... is __________.


Solution : 14(11) + 6(7) = 154 + 42 = 196 Example C : The sum of the first nine terms of the sequence 2 + 5 + 7 + 12 + 19 + ... is __________. Solution : 14(19) + 6(12) = 338 Example D : The sum of the first nine terms of the sequence 1 + 4 + 5 + 9 + 14 + ... is _________. Solution : 14(14) + 6(9) = 196 + 54 = 250 ------------------------------------------------------------------------------------------------------------ Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, ... where F

0 = F

1 = 1 and

Fn

= Fn!1

+ Fn!2

Lucas sequence is 1, 3, 4, 7, 11, 18, 29, ... where L

0 = 1, L

1 = 3 and

Ln

= Ln!1

+ Ln!2

where n ≥ 2 ------------------------------------------------------------------------------------------------------------

The Golden Mean is denoted by ! and ! = 1+ 5

2 = 1.618

------------------------------------------------------------------------------------------------------------ The "nint" function is the "nearest integer function" and is denoted by {x] which means the nearest integer to the number x. Ex. [12.7] = 13 and [7.2] = 7 ------------------------------------------------------------------------------------------------------------

Fn

= ! n

5

"

#$

%

&' and L

n = ! n"# $%

Example A : Find the value of F

10 + F

11.

(A) 123 (B) 144 (C) 155 (D) 164 (E) 177

Solution : 1.61810

5

!

"#

$

%& + 1.618

11

5

!

"#

$

%& = [54.9921] + [88.9772] =


55 + 89 = 144 Example B : Find the value of L

x if L

9 - L

8 = L

x.

(A) 18 (B) 29 (C) 38 (D) 43 (E) 55

Solution : 1.6189

5

!

"#

$

%& + 1.618

8

5

!

"#

$

%& = [33.9877] + 21.0060] =

34 + 21 = 55 ADVANCED MATHEMATICS Be familiar with the following :

(A) e x = 1 + x + x2

2! +

x3

3! + ... +

xn

n! + ....

(B) sin x = x - x3

3! +

x5

5! -

x7

7! + ... +

(!1)nx2n+1

(2n +1)! + ...

(C) cosh x = 1 + x2

2! +

x4

4! + ... +

x2n

(2n)! + ... =

ex+ e

! x

2

(D) e ! x = 1 - x + x2

2! -

x3

3! + ... + (- 1) n

xn

n! + ... (let x = - x in e x in (A))

(E) cos x = 1 - x2

2! +

x4

4! -

x6

6! + ... + (- 1) n

x2n

(2n)! + ... =

d

dx(sin x) from (B)

(F) sinh x = x + x3

3! +

x5

5! + ... +

x2n+1

(2n +1)! + ... =

ex! e

! x

2 =

d

dx (cosh x) from (C)

Example A : Find the digit in the hundred thousandth place of


2 - 8

6 +

32

120 -

128

5040 + ...

(A) 0 (B) 2 (C) 5 (D) 7 (E) 9

Solution : 2 - 8

6 +

32

120 -

128

5040 + ... =

2 - 23

3! +

25

5! -

27

7! + ...

Note : sin x = x - x3

3! +

x5

5! -

x7

7! + ... +

(!1)nx2n+1

(2n +1)! + ...

Set your calculator to radian mode, then find sin 2. sin 2 = .909297426... The hundred thousandth digit is 9. Example B : Find the hundred thousandth digit of

1 + 3 + 9

2 +

27

6 +

81

24 + ...

(A) 5 (B) 8 (C) 0 (D) 6 (E) 3

Solution : 1 + 3 + 9

2 +

27

6 +

81

24 + ... =

1 + 3 + 32

2! +

33

3! +

34

4! + ...

Note : e x = 1 + x + x2

2! +

x3

3! + ... +

xn

n! + ....

Find e 3 . e 3 = 20.08553692,,, The hundred thousandth digit is 3. Example C : Find the ten thousandth digit of

1 + 25

2 +

625

24 +

15625

720 + ...


(A) 9 (B) 2 (C) 0 (D) 8 (E) 4

Solution : 1 + 25

2 +

625

24 +

15625

720 + ...

1 + 52

2! +

54

4! +

56

6! + ...

Note : cosh x = 1 + x2

2! +

x4

4! + ... +

x2n

(2n)! + ... =

ex+ e

! x

2

Find e5+ e

!5

2.

e5+ e

!5

2 = 74.20994852...

The ten thousandth digit is 9. 3. Let f be a function such that it is continuous on [a, b], it is differentiable on (a, b), and f(a) = f(b) = 0. There exists a number c in (a, b) such that f'(c) = 0. This theorem is known as : (A) Fundamental Theorem of Algebra (B) Intermediate-value Theorem (C) Mean-value Theorem (D) Rolle's Theorem (E) Fundamental Theorem of Calculus Fundamental Theorem of Algebra : Every polynomial equation having complex coefficients and degree ≥ 1 gas at least one complex root Intermediate-value Theorem : If f is continuous on a closed interval [a, b] and c

is any number between f(a) and f(b) inclusive, then there is at least one number x in the closed interval such that f(x) = c.

Mean Value Theorem : Let f(x) be differentiable on the open interval (a, b) and continuous on the closed interval [a, b]. Then there is at least one point

c in (a, b) such that f'(c) = f (b) ! f (a)

b ! a.

Rolle's Theorem : Let f be differentiable on the open interval (a, b) and continuous on the closed interval [a, b]. Then if f(a) = f(b), then there is at least one point c in (a, b) where f'(c) = 0. Fundamental Theorem of Calculus : If f is continuous on the closed interval


[a, b] and F is the antiderivative (indefinite) of f on [a, b], then

f (x)dxa

b

! = F(b) - F(a).

The answer is D, the Rolle's Theorem. 14. A bridge is to be built across a small lake from point A to point B. Bill Der, the surveyor, determines that the line of sight distance from him to point B and from him to point A are 456.2 ft. and 429.8 ft., respectively. Bill measures the angle between the two lines of sight to be 48.7°. Determine the length o the bridge (nearest tenth). (A) 495.5 ft (B) 443.0 ft (C) 391.4 ft (D) 366.1 ft (E) 313.4 ft Sketch drawing depicting information given. Use the Laws of Cosines formula, c 2 = a 2 + b 2 - 2abcosC x 2 = 456.2 2 + 429.8 2 - 2(456.2)(429.8)cos48.7° x = 366.1 15. The distance between the point P with coordinates (a, 2a) and the line 3x - 4y + 5 = 0 is a. The value of a is : (A) 2 (B) 1 (C) 4 (D) 3 (E) None of these Use the following formula to find the distance between a point and a line :

D = Ax

1+ By

1+ C

A2+ B

2

a = 3(a) ! 4(2a) + 5

32+ (!4)

2 ; a =

!5a + 5

25 ;

a = !5a + 5

5 ; |-5a + 5| = 5a

- 5a + 5 = 5a or - 5a + 5 = - 5a 10a = 5 or 5 = 0


a = 1

2 or 5 ! 0

The answer is E.

17. Evaluate : limx!"

3x2+ 2x !1

4x3! 5x + 6

(A) 3

4 (B)

1

2 (C) 0 (D)

1

6 (E) None of these

When finding the limit of a rational function as x approaches infinitiy : (A) the limit is equal to 0 if the degree of the numerator is less than the degree of the denominator (B) the limit is equal to the ratio of the leading coefficient in the numerator and the leading coefficient in the denominator, if the degrees in the numerator and the denominator are equal. (C) the limit is equal to ! if the degree in the numerator is greater than the degree in the denominator

limx!"

3x2+ 2x !1

4x3! 5x + 6

= 0 (Note : the degree in the numerator is less than

the degree in the denominator. 19. A triangle has sides of 4" and 8" with an included angle of 42°. Its area to the nearest tenth of a sq. in. is : (A) 21.4 (B) 10.7 (C) 23.8 (D) 11.9 (E) None of these

Area = 1

2abSinC =

1

2(4)(8)sin 42 = 10.7

UIL MATHEMATICS MAGIC : BOOK 3 - Ram Materials · PDF fileUIL MATHEMATICS WORKBOOK 3 1 UIL...

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