Laws of Exponents: 1. aman =am+n 2. n m-n m
a
a
a
=
3.( )
am n =amn 4.( )
n n n b a ab = 5. n n n b a b a = Laws of Radicals: 1. n an =a 2. n ab =n an b 3. n n nb
a
b
a
=
4. m n a =mna Laws of Logarithms:1. logbMN=logbM+logbN
2. log M-log N N M logb = b b 3. log MN logbM b =N Important Properties: 1: a0 =1 provided
a
≠
0
2: -n na
1
a
=
3: n m( )
n m n ma
a
a
=
=
4:a
logab=
b
or elnb =b 5: am =an implies that m = n6: logbM=N implies that M=bN 7: logbM=logbN implies that M = N 8:
a
log
N
log
N
log
b b a=
9: logbb=1 provided b>0,b≠1 10: logb1=0 provided b >0,b ≠1 QUADRATIC EQUATIONGenerally, an equation is said to be of quadratic form if it has the form ax2n + bxn + c = 0, where n is an integer or a fraction; such as x4 – 5x2 + 6 = 0 and y-3 + y-3/2 + 6 = 0 Quadratic Formula: a 2 ac 4 b b X 2 − ± − =
The expression b2 – 4ac is called the discriminant
1. When b2 – 4ac > 0, the roots are real and unequal.
2. When b2 – 4ac = 0, the roots are real and equal (or quadratic equation is a perfect
trinomial)
3. When b2 – 4ac < 0, the roots are imaginary and unequal (complex conjugates)
The roots: a 2 ac 4 b b X 2 1 − + − = a 2 ac 4 b b X 2 2 − − − = Sum of the roots, x1 + x2 = - b/a
Product of the roots, x1 ∙ x2 = c/a THE BINOMIAL THEOREM
BINOMIAL EXPANSION TRIANGLEPASCAL’S
(x + y)0 = 1 (x ≠ -y) 1 1 (x + y)1 = x + y 1 1 (x + y)2 = x2 + 2xy + y2 1 2 1 (x + y)3 = x3 + 3x2y + 3xy2 + y3 1 3 3 1 (x + y)4 = x4 + 4x3y + 6x2y2 +4xy3 + y4 1 4 6 4 1 (x + y)5 = x5 + 5x4y + 10x3y2 +10x2y3 + 5xy4 + y5 1 5 10 10 5 1 (x + y)6 = x6 + 6x5y + 15x4y2 +20x3y3 + 15x2y4 + 6xy5 + y6 1 6 15 20 15 1
This array of numbers is called the Pascal’s Triangle. Any lower row is formed by adding any two adjacent numbers of the upper row and place 1 at both ends so as to form a triangle. Pascal’s Triangle is used to easily recall the numerical coefficients of the expansion of the powers of a binomial. But for large powers of a binomial, Pascal’s Triangle becomes inconveniently to use. For such, use Binomial Theorem.
The rth term of (x + y)n = n(n-1)(n-2)…(n-r+2) xn-r+1yr-1 (r – 1)!
LOWEST COMMON MULTIPLE (LCM)
The lowest common multiple (LCM) of several natural numbers is the smallest natural number of which each of the given numbers is a factor. It mat be found by taking the product of all the different prime factors of the numbers, each taken the greatest number of times that
it occurs in any of those numbers.
Example: Find the lowest common multiple of 24, 10, 18, and 25. Solution:
24 = 2x2x2x3, 10 = 2x5, 18 = 2x3x3, 25 = 5x5 LCM = 2x2x2x3x3x5x5 = 1800
HIGHEST COMMON FACTOR (HCF)
The highest common factor (HCF) of several natural numbers is the largest natural number which is a factor of all the given numbers. It may be found by taking the product of all the different prime factors common to the given numbers, each taken the smallest number of times that it occurs in any of those numbers. If the given numbers have no prime factors in common, the HCF is defined to be 1, in this case the numbers are said to be relatively prime.
Example: Find the highest common factor of 24, 30, 18 and 150. Solution:
24 = 2x2x2x3, 30 = 2x3x5, 18 = 2x3x3, 150 = 2x3x5x5 HCF = 2x3 = 6
PROGRESSION
Arithmetic Progression (A. P.)
- a sequence of terms in which each term after the first is obtained by adding a fixed number to the preceding term.
- a sequence of terms in which any two consecutive terms has a common difference.
That is, the sequence a1, a2, a3 are in arithmetic progression if and only if: a2 – a1 = a3 – a2
Let: a1 = first term of an A. P. an = nth term of an A. P. d = common difference n = number of terms Sn = the sum of n terms Then,
an = a1 + (n – 1)d Sn = n/2 (a1 + an) Sn = n/2 [ 2a1 + (n – 1)d]
Arithmetic Mean
The arithmetic mean between two numbers is the number which when placed between the two numbers, forms with them an arithmetic progression.
In general, for n terms, arithmetic mean (AM) = a1 + a2 + a3 + … + an n Geometric Progression (G.P.)
- a sequence of terms in which each term after the first is found by multiplying the preceding term by a fixed number called common ratio. - The sequence a1, a2, a3 are in G.P. if and only if:
a2/a1 = a3/a2 = r The nth term, an
an = a1rn-1
Sum of the first n terms in G.P. Sn = a1 1-r n
1-r
where a1 = first term r = common ratio n = number of terms Infinite Geometric Progression
The sum of terms in geometric progression can be found if the common ratio | r |<1, -1 < r < 1 r a S − = 1 1 Geometric Mean
The term in between the first and last terms of the geometric sequence. Let x = geometric mean,
a1, x2, a2→ geometric progression Then, x/a1 = a2/x →common ratio
Solving for x: x2 = a 1a2
x = a1a2 → geometric progression
Harmonic Progression
That is, a1, a2, a3…an are in harmonic progression If 1/a1, 1/a2, 1/a3…1/an form an arithmetic progression
Harmonic Mean
Let x = harmonic mean between a and b a, x, b → in H. P.
1/a, 1/x, 1/b → in A. P. Then,
1/x – 1/a = 1/b – 1/x → common difference Solving for x:
2/x = 1/a + 1/b 2/x = a + b/ab x = 2ab/(a + b)
Harmonic Mean (HM) = _ n 1/a1 + 1/a2 + 1/a3 + … + 1/an
RATIO, PROPORTION AND VARIATION 1. RATIO
The ratio of a number a to another number b is the fraction a/b usually as a:b (read a “is to” b). Where a is called antecedent and b is called consequent.
2. PROPORTION
Proportion is a statement that two ratios are equal. Usually written as a:b = c:d or a/b = c/d. Where a and d are called the extremes and b and c are called the means. D is called the fourth proportional to a, b, and c. If the means of a proportion are equal, as in a/x = x/b, the number b is called the third proportional to a and x, while the number x is called the mean
proportional between a and b. It is obvious that the mean proportional between a and b is
equal to their geometric mean. A proportion may be altered in four different ways summarized in the tabular form below.
Basic Proportion Transformation by Transformed Form
Alternation a:c = b:d
a:b = c:d Inversion b:a = d:c
Addition (a+b):b = (c+d):d
Subtarction (a-b):b = (c-d):d
3. VARIATION
i. Direct Variation (also direct proportion)
The Five Statements Below Have Same Meaning
As x increases y increase proportionately y is proportional to x
y is directly proportional to x y varies as x
y varies directly as x
In symbols the above statements mean,
y α x
In mathematical terms, y = kx
where k is called the constant of proportionality or also called the constant of variation
ii. Inverse Variation (also indirect variation)
The following Statements Below Have Same Meaning
As x decreases y increase (and vice versa) y is inversely proportional to x
y varies indirectly as x
In symbols the above statements mean,
y α 1/x
In mathematical terms,
y = k/x, (x not equal to zero) Examples
1. Boyle’s Law: “When the temperature of a confined gas is held constant, the pressure of the gas varies inversely as its absolute pressure.”
2. Ohm’s Law: “The current is directly proportional to the impressed emf and inversely to the resistance
iii. Joint Variation
y varies jointly as x and w
In symbols,
y α xw
Mathematically, Y = kxw; Warning: Not y = k(x+w) EQUATION OF THE HIGHER DEGREE
• Rational integral term – a constant, or a positive integral power of any variable, or the product of such qualities.
Ex. 5, 2x4, - 6y3, 15x2y5
• Degree of a term
-the term Cxn, where C is a constant and n is a positive integer is said to be of degree n in terms of x.
-the term Cxnyn is said to be of the degree n in terms of x, degree p in terms of y, and degree n+p in terms of x and y.
• Polynomial Function
-An algebraic sum of rational integral term
-A series of a power of a base where he exponents are positive integer -Also called rational integral function.
f(x) = a0xn + a1xn-1 + a2xn-2 + …+ an polynomial function in degree n where: n = non-negative integer
a0, a1, a2,….,an are any constants a0 ≠ 0
• Zero of a function
any value of the unknown x, that will make a function f(x) equal to zero also called root of f(x) = 0
• Fundamental theorem
every equation f(x) = 0 has at least one root
there exist at least one number either real or complex which will satisfy f(x) =0
• Number of roots
every equation f(x) = 0 of degree n, has n roots and no more if root of order k is counted as k roots.
• Multiple roots
consider that the roots r1,r2, r3, …, rn of f(x) = 0 are equal
if f(x) is exactly divisible by x – r but not by (x – r)2, then x – r is simple root of f(x) = 0. If f(x) = 0 is exactly divisible (x –r)2 but not by (x – r
1)3, then r1 is called double root of f(x) = 0.
In general, if f(x) = 0 is divisible by (x – r)k but not by (x – r
1)k+1, then r1 is a k – fold root of order k.
• Multiplicity
the number of times that r appears as roots of f(x) = 0
• Theorem of complex roots
complex roots always occur in pair.
That is if b = bi is root, then a – bi is also a root (-conjugate zeros) where a, b are real but b ≠ 0
• Theorem on quadratic surd roots
surd root always occur in pair
That is, if a +√bi is a root, then a –√b is also a root where a and b are rational and √b is irrational.
• Remainder theorem
if f(x) is divided by (x – r), the remainder is f(r).
• Factor theorem
if r is a root of the equation f(x), then (x – r) is factor of f(x).
• Converse of factor theorem
if (x – r) is a factor of f(x), then r is a root of f(x) = 0
• Depressed equation
if r is a root of the equation f(x) = 0, then (x – r) is a factor of f(x). Thus, f(x) = (x – r). The equation Q(x) = 0 is called depressed equation of f(x) = 0, and the roots of Q(x) = 0 are the remaining roots of f(x) = 0
• Descartes’ rule of signs
Let f(x) be a polynomial with real coefficients.
a) The number of positive real roots of f(x) is either equal to the number of variations in sign of f(x), or that number diminished by a positive integer.
b) The number negative with real zeros of f(x) is either equal to the number of variations in sign of f(-x) or that number diminished by positive even integer.
• Relationship Between Coefficients and Zeros of Polynomial Given an integral rational function:
f(x) = a0xn + a1xn-1 + a2xn-2 + … + an-1x + an
The coefficients for the polynomial function in terms of its zeros can be given as: -a1/ a0= sum of roots
a2 / a0 = sum of product of the roots taken two at a time -a3/ a0 = sum of product of the roots taken three at a time (-1)n a
n/ a0 = product of all roots Supplementary Problems
1. Determine m so that x3 – 2x2 + mx + 8 shall be divisible by x + 3. Ans. m = 9 2. If 3x4 = kx3 + x2 – 16 + 4 is divided by (x – 2), for what value of k will the
remainder be 8 ? Ans. k = 2
3. For what value of k will x = 3 be a factor of x3 + 7x2 + kx – 12? Ans. k = 88 4. Find the remainder when 149x1592 – 375x375 + 10 is divided by x + 1. Ans.
1877
5. One of the roots of 3x3 – mx2 + 23x – 14 = 0 is 2. Determine the value of m. Answer: m = 14
6. 2x3 –x2 + mx + n is to be exactly divisible by x2 – 2x – 8. Determine the values of m and n. Ans. m = -2, n = -24
8. What is the sum and product of the roots of the equation 3x4 – 2x2 + 8x – 6 = 0?
Answer: sum = 0; product = -2
9. Given the equation x3 – 4x2 + 3x – 5 = 0. from the equation whose roots are : a) negative of the roots of the given equation
b) thrice the roots of the given equation
c) the roots of the given equation diminished by 2. Answer:
a) x3 + 4x2 + 3x + 5 = 0 b) x3 + 12x2 + 27x – 135 = 0 c) x3 + 2x2 – x – 7 = 0
10. The sum of the roots of 2x3 + mx2 – 5x - 3 = 0 equals twice the product of the roots. Determine m. Ans. m = -6
11. The product of the two roots of the equation x3 + 5x + 12 = 0 is 4. Find the third root. Ans. -3
12. The sum of the roots 3x2 – 2mx2 + 4 = 0 is 6. Find m. Ans. m = 9
MISCELLANEOUS QUESTIONS
1. A set of elements that is taken without regard to the order in which the elements are arranged is called:
a. combination b. sequence c. permutation d. series 2. When a logarithm is expressed as an integer plus a decimal (between 0
and 1), the integer is called the
a. Briggs Logarithm c. Napierian Logarithm
b. Characteristic d. Mantissa
3. Any positive integers as 1, 2, 3, etc. is also called
a. Real Number c. Natural Number
b. Rational Number d. Irrational Number
4. The set of integers that does not satisfy the closure property under the operation of a. addition b. subtraction c. multiplication d. division
5. An equation which is satisfied by some, but not all, of the values of the variables for which the members of the equation are not defined is called a
a. linear equation c. rational equation b. conditional equation d. irrational condition
6. An equation which is satisfied by all of the values of the variables for which the members of the equation are defined is
a. linear equation c. rational equation b. conditional equation d. irrational equation
7. Two prime numbers which differ by 2 are called prime twins. Which of the following pairs of numbers are prime twins?
a. (1,3) b. (13, 15) c. (7,9) d. (9,11)
8. A relation in which every ordered pair (x,y) has one and only one value of y corresponds to the values of x is called
a. term b. function c. coordinated d. abscissa
9. Tossing a coin is generally called
a. an experiment b. an event c. an outcome d. a trial 10. A polynomial which is exactly divisible by two or more polynomials is called as:
a. least common denominator b. common multiple
c. factors d. binomial
11. The roots of the equation 2x2 – 13x + 20 = 0 are
a. real and equal b. real and unequal
c. complex and equal d. complex and unequal
12. Each of two or more numbers which is multiplied together to form a product is called
a. term b. multiplier c. kilogram d. kilowatt
13. In the SI unit, the small letter k means kilo while the capital letter K means
a. Kilometer b. Kelvin c. Kilogram d. Kilowatt
14. Any number that can be expressed as a quotient of two integers (division of zero excluded) is called
a. irrational number c. rational number b. imaginary number d. odd number
15. A rectangular array of numbers forming m rows and n columns are called as
a. determinants c. elements
b. Pascal’s triangle d. none of the above 16. In the expression n√a, the letter n represents the
a. power b. order c. exponent d. radicand
17. A number of the form a + bi with a and b real constants and i is square root of -1 is called
a. imaginary number c. complex number
b. radical d. compound number
18. Which of the following nonterminating decimals is rational
a. 3.14159265… c. 2.470470…
19. A succession of numbers in which one number is designated as first, another as second, another as third and so on is called a
a. series c. order of numbers
b. arrangement d. sequence
20. An equation in which a variable appears under a radical sign is called a. irradical equation c. irrational equation b. quadratic equation d. linear equation 21. If 1/
4 and – 7/2 are the roots of the quadratic equation Ax2 + Bx + C = 0, what is the value of B?
a. -28 b. -7 c. 4 d. 26
22. If 1/
4 and – 7/2 are the roots of the quadratic equation Ax2 + Bx + C = 0, what is the value of C?
a. -28 b. -7 c. 4 d. 26
23. Radicals can be added if they have the same radicand and
a. exponent b. power c. order d. coefficients
24. If the roots of ax2 + bx + c = 0, are real and equal, then a. b2 – 4ac > 0 c. b2 – 4ac < 0 b. b2 – 4ac = 0 d. b2 – 4ac < 0
25. The sum of the integers between 288 and 887 that are exactly divisible by 15 is
a. 23,700 b. 22,815 c. 21,800 d. 24,150
26. The sum of the prime numbers between 1 and 15
a. 42 b. 41 c. 39 d. 38
27. What is the sum to infinity of the sequence 1 + 1/3 + 1/9 + …
a. 2/5 b. 5/6 c. 2/3 d. 3/2
31. The term free of y in the expansion of is
a. 46 b. 84 c. 47 d. 49
32. If f(x) = x + 2 and g(y) = y + 2, then f[g(2)] equals x – 2
a. 6 b. 5 c. 4 d. 3
33. If x3 + 3x2 + (K + 5)x + 2 – K is divided by x + 1 and the remainder is 3, then the value of K is
a. -2 b. -4 c. -3 d. -5
34. The value of K which will make 4x2 – 4kx + 5k a perfect square trinomial is
a. 6 b. 5 c. 4 d.3
35. If 3x = 9y and 27y = 81z, then is equal to
a. 3/7 b. 3/5 c. 3/4 d. 3/8
.
36. A certain work can be done in as many days as there are men in the group. If the number of men in the group is reduced by 3, the job will be delayed by 4 days. The number of men originally in the group is
a. 8 men b. 10 men c. 12 men d. 14 men
37. How many terms in the progression 3, 5, 7, 9, … must be taken in order that the sum is 2,600?
a. 53 b. 52 c. 51 d. 50
38. The other form of logaN = b is
a. N = ab b. N = ba c. N = a/b d. N = ab
39. In the quadratic equation Ax2 + Bx + C = 0, the product of the root is:
a. C/A b. –B/A c. –C/A d. B/A
40. Two students were solving a problem that would reduce it to a quadratic equation. The first student committed an error in the constant term and found the roots to be 5 and 7 while the second student made an error in the first degree term and gave the roots as 2 and 16. if you were to check their solutions, the right equation is:
a. x2 + 12x + 35 = 0 b. x2 + 18x + 32 = 0 c. x2 + 7x – 14 = 0 d. x2 – 12x + 32 = 0
41. Determine the value of k so that the equation x2 + (k-5)x + k – 2 = 0 is a perfect trinomial square.
Ans. 3 or 11
42. The expression x4 + ax3 + 5x2 + bx + 6 when divided by (x – 2) leaves the remainder 16, and when divided by (x – 1) leaves the remainder 10. Find the values of a and b.
Ans. a = - 11/3, b = 5/3
43. Given the equation x4 + x2 + 1 = 0. Which of the following is not a root?
a. 1 /120° b. 1 /135° c. 1 /240° d. 1 /300°
44. The area of a square field exceeds another square by 56 square meters. The perimeter of the larger field exceeds one half of the smaller by 26 meters. What are the sides of each field?
Ans. larger field, 9m or 25/3m; smaller field, 5m or 11/3m
45. The sum of the areas of two unequal square lots is 5,200 square meters. If the lots were adjacent to each other, they would require 320 meters of fence to enclose the combined area formed by them. Find the dimensions of each lot. Ans. 60m and 40m or 68m and 24m BINOMIAL EXPANSION
1. Solve the following equations:
A) Find the value of x: (a + b)x = (a2 + 2ab + b2)x-1 B) Find the roots of the equation 4x4 + 1 = 0
2. Without expanding, find the term involving x4 of (3x2 – 2x-1)8. Ans. 90,720x4 3. A) Expand to 4 terms (x2/3 – ½)x13
B) Find the 9th term in the expression of (x2 + ½)13
C) Write the first four terms and the last term of the expansion of (3/x – x/3)65
D) Find the term independent of y in the expansion of (y2 – y-1)9.
Ans. 84
4. Which of the following has no middle term?
a. (x + y)3 b. (a – b)4 c. (u + v)6 d. (x – y)8 5. Find the middle term of (x2 – 2y)10
Ans. -8,064x10y5
6. If the middle term in the expansion of (x + 2y)n is kx4y4, find k and n.
Ans. n = 8, k = 1,120
7. If the rth term of (x2 – 2y3)n is Cx8y12, find the value of C. Ans. 1120 8. Without expanding, find the 10th term of the expansion of (S – 2t2)14 9. In the expansion of (x2 + 1/x)12
find: a) the 6th term b) the middle term c) the term involving x6 d) the term free of x
10. Find the sixth term in the expansion of (x/2 + y)9
Ans. 63/8x4y5
11. Find the term containing x26 in the expansion of (x-2 + x3)12
Ans. 66x26
12. The term containing x9 in the expansion of (x3 + 1/x)15
13. Find the coefficient of the expansion of (x2 + y)10 containing x10y5
a. 149 b. 252 c. 105 d. 10,818
AGE PROBLEMS
1. A father is twice older than his son and the sum of their ages is 48. How old is each?
a. 8, 40 b. 12, 36 c. 16, 32 d. none of these
2. Maria is 36 years old, Maria was twice as old as Anna was when Maria was as old as Anna is now. How old is Anna now? Ans. 24 years old
3. The sum of the ages of the father and his son is 99. If the age of the son is added to the inverted age of the father, the sum is 72. If the inverted age of the son is subtracted from the age of the father, the difference is 22. What are their ages? Ans. 74 & 25
4. Maria is 24 years old now. Maria was twice as old as Ana when Maria was as old as Anna is now. How old is Anna now? Ans. 16 years old
5. A father is twice as old as his son and the sum of their ages is 48. How old is each?
a. 8, 40 b. 12, 36 c. 16, 32 d. nota
6. Pedro is as old as Juan was when Juan is twice as old as Pedro was. When Pedro will be as old as Juan is now, the difference between their ages is 6 years. Find the age of each now. Ans. Juan, 24 years old and Pedro, 18 years old
7. I am three times as old as you were when I was as old as you are now. When you got to be my age together our ages will be 84. How old are we now?
a. 24 & 36 b. 24 & 8 c. 16 & 8 d. 18 & 27
8. The sum of the ages of two boys is four times the sum of the ages of a certain number of girls. Four years ago, the sum of the ages of the girls was one eleventh of the sum of the ages of the boys and eight years hence, the sum of the ages of the girls will be one half that of the boys. How many girls are there? Ans. 4 girls
9. In a family, there are 8 children, two of them are twins. The youngest is 3 years old and the eldest is 21 years old. Their ages are in arithmetic progression. There are three children younger than the twins. How old are the twins? Ans. 12 years
10. The sum of the ages of two men equals 99. If the inverted age of the elder is added to the age of the younger, the sum is 108. However, if the age of the younger is inverted and subtracted from the age of the older, the difference is 44. Find the age of the older man.
a. 67 b. 32 c. 53 d. 46
INTEGER AND DIGIT PROBLEMS
1. Separate 132 into 2 parts such that the larger divided by the smaller the quotient is 6 and the remainder is 13. What are the parts? Ans. 17 and 115
2. A number of two digits divided by the sum of the digits the quotient is 7 and the remainder is 6. If the digits of the number are interchanged, the resulting number exceeds three times the sum of the digits by 5. What is the number? Ans. 83
3. Six times the middle of a three digit number is the sum of the other two. If the number is divided by the sum of the digits, the answer is 51 and the remainder is 11. If the digits are reversed, the number becomes smaller by 198. Find the number. Ans. 725
4. Find the number such that their sum multiplied by the sum of their squares is 65, and their difference multiplied by the difference of their squares is 5. Ans. 2 and 3
5. Three numbers are in the ratio 2:5:8. If their sum is 60, find the numbers. Ans. 8, 20, 32 6. The sum of the digits of a three-digit number is twelve. The sum of the squares of the hundreds’ digit and the tens’ digit is equal to the square of the units’ digits. If the hundreds’ digit is increased by two, the digits will be reversed. Find the number. Ans. 345
7. April 1978. The square of a number increased by 16 is the same as 10 times the number. Find the number. Ans. 8, 2
8. The sum of the digits of a three-digit number is 12. The middle digit is equal to the sum of the other two digits and the number shall be increased by 198 if its digits are reversed. Find the number. Ans. 264
9. Find three consecutive odd integers such that twice the sum of the first and the second integers plus four times the third is equal to 60. Ans. 5, 7, 9
10. The sum of the digits of a three-digit number is 12. The sum of the squares of the hundreds digit and the tens digit is equal to the square of the units digit. If the hundreds digit is increased by 2 and the units digit is decreases by 2, the digits of the original number will be reversed. Find the number. Ans. 345
11. The excess of the sum of the fifth and the seventh parts over the difference of the half and third parts of number is 259. What is the number? Ans. 1470
12. The sum of the digits of the three-digit number is 6. The middle digit is equal to the sum of the two other digits and the number shall be increased by 99 if the digits are reversed. Find the number. Ans. 132
MIXTURE PROBLEMS
1. The tank of a car contains 50 liters of alcogas 25% of which is pure alcohol. How much of the mixture must be drawn off which when replaced by pure alcohol will yield a 50-50% alcogas?
a. 16 2/3 b. 15 1/3 c. 14 d. 20
2. How much silver and how much copper must be added to 20kg of an alloy containing 10% silver and 25% copper to obtain an alloy containing 36% silver and 38% copper?
a. 14kg, 12kg b. 16kg,14kg c. 12kg,10kg d. 16kg, 18kg 3. A tank full of alcohol is emptied of one third of its content and then filled up with water and mixed. If this is done six times, what fraction of the volume (original) of alcohol remains? Ans.64/729
4. How much tin and how much iron must be added to 50 kilograms of an alloy containing 10 percent tin and 25 percent iron to obtain an alloy containing 25 percent tin and 50 percent iron? Ans. 27.5 kg(tin), 52.5 kg (iron)
5. How many liters of water must be added to 45 liters of solution which is 90% alcohol in order to make the resulting solution 80% alcohol? Ans. 5.63L
6. A 40-gram solution of acid and water is 20% acid by weight. How much pure acid must be added to this solution to make it 30% acid? Ans. 5.71 grams
7. How much water must be evaporated form 80 liters of 12% solution of salt in order to obtain a 20% solution of salt? Ans. 32 L 39. How many liters of water must be added to 100 liters of 85% sulfuric acid solution to produce 60% sulfuric acid solution? Ans. 41.67 L 8. A certain solution should contain 8% alcohol. If it was mistakenly mixed to contain 6% alcohol, how much must be drawn from a 5-liter tank and replaced by 10 percent alcohol solution to provide the proper concentration? Ans. 2.5 L
9. A certain amount of 80% sugar solution added to another amount of 40% sugar solution yields a solution that contains 14 kg of sugar. Had the amount been reversed, the solution would have contained 16 kg sugar. How much of the 80% solution was there? Ans. 10 kg
10. Ten liters of 25% salt solution and 15 liter of 35% salt solution are poured into a drum originally containing 30 liters of 10 % salt solution. What is the percent concentration of salt in the mixture?
a.19.55% b. 22.15% c. 27.05% d. 25.72%
RATE AND MOTION PROBLEMS
1. A motorist is traveling from town A to town B at 60 kph and returns from town B to town A at 30 kph. His average velocity for the roundtrip is
2. At the recent Olympic games in Montreal, Canada, a team which participated in 1600 meters relay event had the following individual speed. First runner, 24 kph, second runner, 20 kph, third runner, 22 kph and fourth runner 23 kph. What was the team’s speed. Ans. 22.149 kph
3. A troop of soldiers marched 15 km, going to the concentration camp after they were forced to surrender, at the same time that the victorious general who is supervising the “march” rode from the rear of the troop to the front and back at once to the rear. If the distance covered by the victorious general is 25 km. and both the troop and general traveled at uniform rate, how long is the troop? Ans. 8 km.
4. Two cyclists are practicing on a circular tract of circumference 276 meter. Starting at the same instant and from the same place, when they run in opposite directions they pass each other every 6 seconds and when they run the same direction the faster passes the slower at every 23 seconds. Determine their rates. Ans. F= 29 m/s, S= 17m/s
5. Two cars A and B start at the same point and at the same time and travel in opposite directions, car B traveling 20 km/hr slower than A. If they are 420 kilometers apart after 3 hours, find the rate of each. Ans. 60 kph, 80 kph
6. Two cars A and B are to race around a 1,500-meter circular track. If they will start at the same point and travel opposite directions, they will meet for the first time in 3 minutes. But if they will travel in the same direction, with the same starting point, car A will reach the starting point with car B trailing behind by 500 meters. What should be the rates of each?
Ans. 300 m/min, 200 m/min
7. A one kilometer long caravan of men is walking at a constant rate. A man from the rear ends walk towards the head and back to the rear at the instant when the caravan has covered a distance of one kilometer. Find the total distance traveled by the man. Ans. 2.414 km
8. A boy started one hour and twenty minutes earlier than a man. If the man ran at 6 kph faster than the boy and overtook the boy in 40 minutes, find the rate of each. Ans. 3 kph for the boy and 9 kph for the man
9. A man walked 24 km in time T. During the first part of this time, he walked at 6 kph and the last part at 4 kph. Had he reversed his rates, he would have walked two km more. Find the time. Ans. 5 hrs.
10. A traffic check counted 390 cars passing a certain spot on one day and 430 cars at the same spot on the second day. On the first day, there were three times as many cars going east and half as many going west on the second day. What was the total number of west bound cars for the two days?
Ans. 280 east, 540 west
11. A man is sent to deliver an important package ant travels by car 75 kilometers per hour from point A to B and then by airplane to point C against a wind blowing 40 kilometers
per hour. The airplane can fly 280 kilometers per hour in still air. If the package carrier takes 3 2/3 hours in going from A to C and 3 1/6 hours for the return trip, what is the total distance of travel covered by the man? D = 605 km. t1 = 1.5 hr., t 2 = 5/3 hr.
12. A motorcycle messenger left the rear of a motorized troop 8 kilometers long and rode to the front of the troop, returning at once to the rear. How far did he ride, if the troop traveled 15 kilometers during this time and each traveled at a uniform rate? Ans. 25kms.
13. An army officer made the first part of the trip on a plane which flew at the rate of 210 kilometers per hour. At the landing field, he was met by a jeep which took him the rest of the way to his destination at a rate of 40 kilometers per hour. The trip required 3 hours and 15 minutes. On his return trip, the jeep traveled at the rate of 50 kilometers per hour and the plane which he took flew at the rate of 200 kilometers per hour. The return journey required the same amount of time, but this included a minute which he spent waiting for the plane to take off. Find the total distance that he flew and the total distance that he traveled by jeep.
Ans. 532km (by plane) and 28 2/3km (by jeep)
14. A BMW car drives from A toward C at 30 miles/hr. Another car starting from B at the same time, drives towards A at 20 mi/hr. If AB = 20 miles, find when the cars will be nearest together. Ans. 24 min.
15. Two boats started their voyages is in a straight line towards each other. One has an average navigational speed of 30 km/h and the other one has an average of 20kph. Assuming that they can not avoid a collision, how long will it take before the collision occurs? How far would each boat have traveled before the collision? Ans. 4 hrs, 120 km, 80 km
16. A man traveling 40 km finds that by traveling one more km per hour, he would made the journey in 2 hours less time. How many kilometer per hour did he actually travel?
a. 4 b. 8 c. 18 d. 6
PROGRESSION
1. Two numbers differ by 40 and their arithmetic mean exceeds their positive geometric mean by 2. The numbers are
a. 45, 85 b. 64, 104 c. 81, 121 d.100, 140
2. A sets out to walk at the rate of four km per hour. After he had been walking for 2-3/4 hours, B sets out to overtake and went 4-1/2 km the first hour, 4-3/4 km the second hr., 5 km the third hr and so on gaining 1 quarter of a km. every hour. In how many hours would B overtake A? Ans. 8 hours
3. A besieged fortress is held by 5700 men who have provisions for 66 days. If the garrison loses 20 men each day, for how many days can the provisions hold out? Ans. 76 days
4. The sum of three numbers in arithmetic progressions is 60. If the numbers are increased by 2, 1, and 28, respectively, the new numbers will be in geometric progression. Find the arithmetic progression.
5. Three numbers are in arithmetic progression. Their sum is 15, and the sum of their squares is 83. Find the numbers.
6. A 20-liter container is filled with pure acid. Five liters are drawn off and replaced with water; then 5 liters of the mixture drawn off and replaced with water, and so on until 5 drawings and 5 replacements have been made. Find the amount of acid in the final mixture. 7. A rubber ball is dropped from a height of 27 meters. Each time that it hits the ground it bounces to a height 2/3 of that from which it fell. Find the distance that it travels up to the time that it hits the ground for the 5th time. The total distance traveled by the ball until it comes to rest.
8. A man wishes to buy a piece of land worth 150,000 pesos. If it were possible for him to save one centavo on the first day, two centavos on the second day, 4 centavos on the third day and so on, in how many days would he save to be able to buy the land? Ans. about 24 days
9. The sum of the two numbers is 20 and their positive geometric mean is one greater than one half of their arithmetic mean. Find their difference.
10. A man cuts a piece of paper 0.03 mm thick into three equal parts. Then he cuts each of these parts into three equal parts again and the process is repeated 10 times after which he piles together the pieces of paper. How thick is the pile?
11. A rich man called his seven sons. He had with him a number of pebbles, each pebble representing a gold bar. To his first son, he gave half the pebbles that he had and one pebble more. To his second son, he gave half the remaining pebbles and one pebble more. He did the same to each to his five other sons and then found out that he had one pebble left. How many pebbles were there initially? Ans. 382
12. A man receives a salary of P36,000 per annum for the first year and a 10% raise every year for ten years. What is his salary during the fifth year? Ans. P52,707.60 13. A car running at 25 kilometers per hour can cover a certain distance in 8 hours. By how many kilometers per hour must its rate be increased in order to cover the same distance in three hours less? Ans. 15km/hr
14. Find the harmonic mean 7, 1, 5, 2, 6 and 3
a. 2.36 b. 2.46 c. 2.56 d.2.66
15. The 8th term of an AP is 3 while its 84th term is 273. Find the 35th term. 16. Find the sum to infinity of 1/3, 1/27,1/243… Ans. 3/8
17. In the series 1.01, 1.0, .099, .098… Find the 80th term. 18. Find the sum of 3+0.4+0.05+0.004+0.0005+… Ans. 38/11
19. An arithmetic progression starts with 1, has 9 terms, and the middle term is 21. Determine the sum of the first 9 terms.
20. A pendulum swings 24 inches for the first time. It is swinging 11/12 of its previous swing. What would be the total distance traveled when the pendulum
stopped?
a. 246 b.264 c.288 d. 312
21. A small line truck hauls poles from substation stockyard to pole sites along a proposed distribution line. The truck can handle only one pole at a time. The first pole site is 150 meters from the substation and the poles are to be 50 meters apart. Determine the total distance traveled by the line truck, back and forth, after returning from delivering the 30th pole.
a. 35.0km b. 30.0km c. 37.5km d. 40.0km
22. Two positive numbers may be inserted between 3 and 9 such that the first three are in geometric progression, while the last three are in arithmetic progression. What is the sum of these two positive integers?
a.1.25 b. 12.25 c. 11.25 d.6.25
23. A man piles 150 logs in layers so that the top layer contains 3 logs and each lower layer has one more log than the layer above. How many logs are at the bottom? Ans. 17 logs 24. A body falls 16.1 meters during the first second, 48.3 meters during the second, 80.5 meters during the third second and so on. How long will it take the body to reach the ground if it was released at an altitude of 15,000 meters? Ans. 30.5 seconds
25. The 18th and the 52nd terms of an arithmetic progression are 3 and 173, respectively. The 25th term is
a. 38 b. 35 c. 28 d. 25
25. Find the number of terms of a geometric progression in which the first term is 48, the last term is 384 and the sum of the terms is 720. Ans. 4 terms
26. The sum of three numbers in A.P. is 27. If the first number is increased by 2, the second by 7, and the third by 20, the resulting numbers will be in G.P. Find the original numbers.
a. 3, 9, 15 b. 4, 9, 14 c. 5, 9, 13 d. 6, 9, 12
1. If 4 men can plow 12 hectares in 8 hours, how many men are needed to plow 24 hectares in 24 hours? Ans. 6 men
2. A garden can be cultivated by 8 boys in 5 days. The same job can be done 5 men in 6 days. How long will it take to finish the job if A) the 8 boys and 5 men will work together? B) only 6 boys ands 3 men will work together? C) two days after the 5 men were working the 8 boys arrived to help?
3. A, B and C can do a piece of work in 10 days, A and B can do it in 12 days, A and C in 20 days. How many days would it take each to do the work alone? Ans. 30, 20 ,60
4. A boat’s crew rowing at half their usual rate can negotiate 2km. down a river and back in one hour and 40 minutes. At their usual rate in still water, they would have gone over the same course in 40 min. Find their rate of rowing in still water. Ans. 32/5 km/hr
5. Two pipes running simultaneously can fill a tank in 3 hours and 20 minutes. If both pipes run for 2 hours and the first is then shut off, it requires 2 hours more for the second to fill the tank. How long does it take each pipe to fill it alone?
6. A and B can do a job in 12 days. A and C can do the same job in 18 days while B and C can do it in 24 days. How will it take A, B and C to do the job together?
7. A man can finished a certain job in three-fourths the time that the boy can; the boy can finish the same job in two-thirds the time that a girl can; and the man and the girl working only jointly can finish the job in 4 hours. How long will it take to finish the job if they all work together? Ans. 8/3 hr.
8. The intake pipe to a reservoir is controlled by a valve which automatically closes when the reservoir is full and opens again when four-fifths of the water had been drained off. The intake pipe can fill the reservoir in 4 hours and the outlet pipe can drain it in 10 hours. If the outlet pipe remains open, how much time elapses between the two instants that the reservoir is full? Ans. 13.3 hr.
9. Two brothers washed the family car in 24 minutes. Previously, when each had washed the car alone, the younger boy took 20 min. longer to do the job than the older boy. How long did it take the older boy to wash the car alone? Ans. 40 minutes 10. A swimming pool holds 54 cubic meters of water. It can be drained at a rate of one cubic meter per minute faster than it can be filled. If it takes 9 minutes longer to fill it than to drain it, find the drainage rate. Ans. 3 cu.m/min
11. One input pipe can fill a tank alone in 8 hrs. another input pipe can fill it alone in 6 hours and a drain pipe can empty the full tank in 10 hours. If the tank is empty and all the pipes are wide open, how long will it take to fill the tank? Ans.5.22 hours
12. A steel company has three blast furnaces of varying sizes. If furnaces A, B, C are used full time, 800 metric tons of steel are produced per day. If A and B are used half time and C full time, 545 metric tons are produced. If A is not used, B is used full time, and C half dime, 410 metric tons are produced. How many metric tons per day does each furnace produce?
13. Three observation planes A, B, and C, working together, can map the region in 4 hours. Planes A and B can map the region in 6 hours, planes B and C can map it in 6 hours and 40 minutes. How long would it take each of the planes working alone to map this region?
14. A pump discharging 9 gpm requires 36 hours to fill a tank. If the pump is replaced by one that will discharge 16 gpm, how long will it take to fill the tank?
a. 64 hr b. 16 hr. c. 20.25 hr d. 40.5 hr
15. Two pipes running simultaneously can fill a swimming pool in 6 hours. If both pipes run for 3 hours and the first pipe is then shut off, it requires 4 hours more for the second to fill the pool. How long does it take each pipe running separately to fill the pool? Ans. 8 & 24 16. A man and a boy can dig a trench in 20 days. It would take the boy 9 days longer to dig it alone than it would take the man. How long would it take the boy to dig alone?
a. 45 days b. 16 days c.25 days d. 4 days
17. A job can be done in as many days as there are men in the group. If the number of men is reduced by 3, the job will be delayed by 4 days. How many men are there originally in the group?
a. 6 b. 12 c. 20 d. 30
VENN DIAGRAM
1. A certain part can be defective because it has one or more out of three possible defects; insufficient tensile strength a burr or diameter outside of tolerance limits. In a lot of 500 pcs:
19 have a tensile strength defect 17 have a burr
11 have an unacceptable diameter 12 have tensile strength and burr defects 7 have tensile strength and diameter defects
5 have burr and diameter defects 2 have all three defects
a. how many have four defects
b. how many pcs have only a burr defect
c. how many pcs have exactly two defects. Ans. 475, 2, 18
2. During the election, the total number of votes recorded in a certain municipality was 12,400 had 2/5 of the supporters of LABAN candidate stampede away from the pools and ½ of the supporters of GAD candidate behaved likewise, the LABAN candidates majority would
have been reduced by 100. How many votes did the LABAN & GAD candidates actually received? Ans. 7,000, 5,400
3. The President just recently appointed 25 Generals of the Phil. Army of these 14 have already served in the war of Korea, 12 in the war of Vietnam and 10 in the war of Japan. Therefore 4 who have served both in Korea and Japan, 6 have served both in Vietnam and Korea and 3 have served in Japan, Korea, and Vietnam. Ans. 2 generals
4. A survey of 500 T.V. viewers proceed the following result: 285 watch football games
195 watch hockey games 115 watch basketball games 45 watch football and basketball 70 watch football and hockey 50 watch hockey and basketball 50 do not watch any of the 3 games, How many watch the basketball games only?
a.50 b.40 c.30 d.60
CLOCK PROBLEMS
1. At what time between 7 and 8 o’clock are the hands of the clock are A) at right angles B) straight line C) coinciding
2. The time is 3:00 o’clock and the hands of the clock are at right angles to each other. What is the nearest time of the clock such that the hands of it will be at right angles again? Ans. 32.72 minutes
3. How long will it be from the time the hour hand and the minute hand of a clock are together until they will be together again? Ans. 1 hr. and 5.45 min
4. At what time between 4 and 5 o’clock do the hands of the clock coincide? Ans. 4:21.82 o’clock
5. It is exactly 3 o’clock. In how many seconds will the angle formed by the hour hand and the minute hand be twice the angle formed by the hour and the second hand? Ans. 22.4 seconds
6. It is now between 9 and 10 o’clock. In 4 minutes, the hour hand will be exactly
opposite the position occupied by the minute hand 3 minutes ago. What is the time now? Ans. 9:20
7. How many minutes after 2:00 o’clock will the hands of the clock extend in the opposite directions for the first time?
a. 40.636 b. 41.636 c. 43.636 d. 42.636
8. A student left his home to attend a party one morning at past 6 o’clock and returned at past 3 o’clock. He noticed the hands of the wall clock have exchanged position. What exact time did he arrive?
a. 3:26.07 b. 3:34.62 c. 3:31.47 d. 3:32.19
9. How many times in one complete day will the hour and the minute hands coincide with each other?
a. 24 b. 23 c. 22 d. 25
RATIO, PROPORTION AND VARIATION
1. The kilowatt that can be transmitted safely by a shaft varies directly as the number of revolution it makes per minute and the cue of its diameter. If a shaft 3 centimeters in diameter making 200 revolutions per minute can safely transmit 60 kilowatt, what kilowatt can be safely transmitted by a 2-centimeter shaft making 300 revolutions per minute?
2. The time required for an elevator to lift a weight varies directly with the weight and distance through which it is lifted and inversely as the power of the motor. If it takes 30 seconds for a 10 HP motor to lift 100 lbs through a height of 50 ft, what size of motor is required to lift 800 lbs in 40 sec through the height of 40 ft? Ans. 48 HP
3. Eight men can excavate 15m3 of drainage open canal in 7 hrs. Three men can backfill 10m3 in 4 hrs. How long will it take 10 men to excavate and back fill 20 m3 in the project? Ans. 9.87 hrs.
4. A man is sent to deliver an important package and travels by car 75 kilometers per hour from point A to B and then by airplane to point C against a wind blowing 40 kilometers per hour in still air. If the package carrier takes 3 2/3 hours in going from A to C and 3 1/6 hours for the return trip, what is the total distance of ravel covered by the man?
5. A sphere 30 cm in diameter is divided into two segments. One of which is two times as high as the other. Find the volume of the bigger segment.
6. Two flywheels are connected by a belt. The radius of the flywheels are 30 in and 50 in. The small flywheel has a speed of 350 rpm. Determine the velocity of the belt in ft/sec. What would be the angular velocity of the larger flywheel?
7. A cylindrical tin can has its height equal to the diameter of its base. Another cylindrical tin can with the same capacity has its height equal to twice the diameter of its base. Find the ratio of the amount of tin required for making the two cans with covers. Ans. 0.9524
8. The diameters of two spheres are in the ratio 2:3 and the sum of their volumes is 1,260 cubic meters. Find the volume of the larger sphere.
9. If the square root of x varies directly as y and inversely as the square of z and if x = 16 when y = 24 and z = 2, find z when x = 9 and y =2. Ans. 2/3
10. If a:b = 2:3, b:c= 4:5, what is a:b:c?
a. 2:3/4:5 b. 8:12:16 c. 8:12:16 d. 6:9:12
ANALYTIC GEOMETRY
Distance Between Two Points P1 (x1,y1) and P2 (x2,y2)
y P2(x2,y2) d P1(x1,y1) o x d = √ (x2 – x1)2 + (y2 – y1)2
Area of Polygon (Non-overlapping) of n-sides Given Vertices Given vertices (x1, y1), (x2, y2), ……… (xn, yn) oriented counterclockwise
x1 x2 x3 …………x1 A = y1 y2 y3 …………y1 + + + A = ½ [ (x1y2 + x2y3 + x3y4 ……. + xny1) – (y1x2 + y2x3 + y3x4 ……. + ynx1)] Division of Line Segment
Let P(x, y) be a point on the line joining P1(x1, y1) and P2(x2, y2) and located in such a way that segment P1P is a given fraction k of P1P2, that is P1P = kP1P2.
y P2(x2,y2) P(x,y) x = x1 + k (x2 – x1) y = y1 + k (y2 – y1) P1(x1,y1) o x
If k = ½, then formula above becomes a midpoint formula
x0 = ½ (x1 + x2) ; y0 = ½ (y1 + y0) Angle Between Two Concurrent Lines
Let α and β be the inclinations of lines L1 and L2 respectively and let θ be the angle between the two lines
y
L1 θ = β - α ;m1 = tan α, m2 = tan β L2 tan θ = tan (β - α)
θ tan θ = tan β - tan α __ 1 + tan β tan α tan θ = m2 – m1 α β 1 + m1m2
SUPPLEMENTARY PROBLEMS
1. A point P(x, 3) is equidistant from points A(1, 5) and B(-1, 2). Find x. Ans. ¾
2. Find the locus of points P(x, y) such that the distance from P to (3, 0) is twice its distance to (1, 0). Ans. 3x2 – 3y2 – 2x – 5 = 0
3. Find the length of the segment joining the two midpoints of the sides of the triangle if the length of the third side opposite to it is 30 cm. Ans. 15 cm.
4. A line from P(1, 4) to Q(4, -1) is extended to a point R so that PR = 4PQ. Find the coordinate of R. Ans. R(13, -16)
5. Two vertices of a triangle are (0, -8) and (6, 0). If the medians intersect at (9, -3), find the third vertex of the triangle. Ans. (-3, -1)
6. The area of a triangle with vertices (6, 2), (x, 4) and (0, -4) is 26. Find x.Ans. – 2/
3 and 50/3 7. Find the length of the median from A of a triangle ABC given vertices A(1, 6), B(-1, 3) and
C(3, -3). Ans. 6
8. If the midpoint of a segment is (5, 2) and one endpoint is (7, -3), what are the coordinates of the other end? Ans. (3, 7)
9. Given vertices of a triangle ABC :
A(1, 5),B(-1, 1) and C(6, 3). Find the intersection of the median. Ans.(2, 3) 10. Find the inclination of the line 2x + 5y = 10. Ans. 158.2°
Locus – the curve traced by an arbitrary point as it moves in a plane is called locus of a point.
– the locus of an equation is a curve containing only those points whose coordinates satisfy the equation.
EQUATION OF A STRAIGHT LINE
Line – is a locus of points which has constant slope.
Theorems :
• Every straight line can be represented by a first-degree equation. • The locus of an equation of the first degree is always a straight line. General Equation of a Line
Ax + By + C = 0 ; A, B, C are constants ; A and B, not zero at the same time Standard Equation of a Line
1. Two Point Form
Y P2(x2,y2)
P(x,y) By similarity of triangles y – y1 = y2 – y1 (x – x1) P1(x1,y1) x x2 –x1 y 2. Point-Slope Form P(x, y) In (1) replacing y2 – y1 by m, x2 – x1 y – y1 = m (x – x1) θ x m = tan θ 0°≤θ≤ 180° 3. Slope-Intercept Form y = mx + b (0, b)
where m = slope P(x,y)
b = y intercept
4. Two- Intercept Form
x + y = 1 P2(0, b)
a b P(x, y)
where a = x intercept
b = y intercept P1(a, 0)
5. Normal Equation of a Straight Line
Given ρ = normal intercept N(ρcosθ,ρsinθ) = segment from the origin ρ
perpendicular to the required line θ ρsinθ θ = normal angle ρcosθ = inclination of the normal intercept
From the point slope form :
y – y1 = m (x – x1) where x1 = ρcos θ, y1 = ρsin θ mL = -1 / tan θ
y - ρsin θ = (-1/ tan θ ) (x–ρcos θ) Simplifying, xcosθ + ysinθ = ρ
Reduction to Normal Form : Given the line Ax + By + C = 0 The normal form is :
A x + B y + C = 0 (1) y x y x y x
±√ A2 + B2 ±√ A2 + B2 ±√ A2 + B2
Note : The sign of the radicand must be chosen such that the last term will become negative since ρ > 0.
Special Cases of a Straight Line Α A. Equation of the x – axis: y = 0
Equation of a horizontal line : y = b where b is a constant B. Equation of the Y-axis : x = 0
Equation of a vertical line : x = a where a is a constant SUPPLEMENTARY PROBLEMS:
Find the equations of the line/s satisfying the given conditions.
1. Passing through (1, -2) and perpendicular to the line through (2, -1) and (-3, 2)
Ans. 5x – 3y – 11 = 0
2. With x intercept of 5 and passing through (3, 4) Ans. 2x + y – 10 = 0
3. Passing through (-3, 4) and with equal intercepts Ans. x – y + 7 = 0 and x + y – 1 = 0 4. Making an angle of 45° with the x-axis and passing through (2, 3) Ans. x – y – 1 = 0
5. With slope -12/
5 crosses the first quadrant and forms with the axes a triangle with perimeter of 15. Ans. 5x + 12y – 3 = 0
6. Passing through (7, -4) and at a distance of 1 unit from the point (2, 1)
Ans. 4x + 3y – 16 = 0 ; 3x + 4y – 5 = 0
7. Passing through the midpoint of the segment joining the points (1, 3) and (5, 1) and parallel to the line 2x – y + 5 = 0 Ans. 2x – 3y – 5
8. Find the value of parameter k so that the line 3x – 5ky + 5 = 0 a) will pass through (0, 1)
b) will be parallel to x + 2y = 5 c) will be perpendicular to 4x + 3y = 2 d) has the y-intercept equal to 3
Ans. a) 1 b) –6/
5 c) 4/5 d) 1/3
9. Find the equations of the lines parallel to the line x + 2y – 5 = 0 and passing at a distance 2 from the origin Ans. x + 2y + 2√5 = 0 and x + 2y - 2√5 = 0
10. Find the equation of the perpendicular bisector of the segment joining (2, 5) and (4, 3).
Ans. x – y + 1 = 0
11. Given vertices of a triangle ABC, A(2, 0); B(3, -2) and C(7, 5)
a) find the equation of the median from A b) find the equation of the altitude from B c) the intersection of medians from B to C
Ans. a) x – 2y – 2 = 0 b) x + y – 1 = 0 c) (4, 1)
12. Find the normal intercept and the normal angle of line 5x+12y–39 = 0
Ans. ρ =3, θ = 67.38°
y Distance Between Parallel Lines L1
Let the parallel lines be given by the equations : L1 : Ax + By + C1 = 0 L2
L2 : Ax + By + C2 = 0
The distance between the two lines is given by the formula
d = C2 – C1_
√ A2 + B2 Sample Problems:
1. Find the distance from point (3, -1) to the line 3x – 4y – 3 = 0 Solution :
Here, A = 3, B = -4, C = -2 P0(x0, y0) ↔ (3, -1) Using the formula d = Ax0 + By0 + C = 3(3) + (-4)(-1) – 3
√ A2 + B2 +√ 32 + 42
d = 2 units (the point (3, -1) and the origin are on the opposite side of the line)
2. Find the distance between parallel lines 8x + 15y + 18 = 0 and 8x + 15y + 1 = 0. Solution :
A = 8, B = 15, C2 = 18, C1 = 1
D = C2 – C1 = 18 – 1 = 1 unit Answer √ A2 + B2
√ 82 + 152 Distance from a Point to a Line
The directed distance from a point P(x0,y0) to a line Ax + By + C = 0 is given by the formula: d = Ax0 + By0 + C
±√ A2 + B2
where the sign of the radical is chosen to be the opposite that of C. Remarks:
1. If d > 0, the origin and P lie on opposite sides of the given line. 2. If d < 0, the origin and P lie on the same side of the line. Notes:
Regardless of the location of the point P0(x0, y0), the distance being always positive the formula can be expressed using the absolute value as:
d =Ax0 + By0 + C √ A2 + B2
Line Through the Intersection of Two Lines Let Ax + By + C = 0 and
Dx + Ey + F = 0 be two intersecting lines, where A, B, C, D, E and F are constants and A = B ≠ 0, E = F ≠ 0.
The equation of the family of lines passing through the intersection of the two given lines is given by,
(Ax + By + C) + k (Dx + Ey + F) = 0 where k is an arbitrary constant.
INTERCEPT OF A CURVE
• intercept – directed distance from the origin to the point where the curve crosses the
x-axis
To find the x intercept of a curve, set y = 0, then solve for x.
• y-intercept – the directed distance from the origin to the point where the curve crosses
the y-axis
to find the y- intercept of a curve, set x = 0, then solve for y. SYMMETRY
• If the equation of a curve does not change upon replacement of y by –y, then the locus is symmetric with respect to the x-axis.
f(x, -y) = f(x,y) =0
• If an equation of a curve does not change upon replacement of x by –x, then the locus is symmetric with respect to the y-axis
f(-x,y) = f(x,y) = 0
• If an equation of a curve does not change upon replacement of x by –x and y by –y, then the locus is symmetric with respect to the origin.
f(-x, -y) = f(x, y) = 0
ASYMPTOTE - a straight line which the curve f(x, y) = 0 approaches indefinitely near as its tracing point approaches to infinity.
• To find the vertical asymptote, solve the equation for y in terms of x and set the linear factors of the denominator equal to zero.
• To find the horizontal asymptote, solve the equation for x in terms of y and set the linear factors of the denominator equal to zero.
CIRCLE
Circle is the locus of a point which moves so that it is always equidistant from a
fixed point.
Note: fixed point is called the center Fixed distance is called the radius
Equation of a Circle
In normal form
Consider a circle of radius r with center at C(u, k) Let P(x, y) be a point in the circle
y
P(x,y) By Pythagorean Theorem
r y–k (x – h)2 + (y – k)2 = r2 → standard form
C(h,k)
Center at the origin C(0, 0) x2 + y2 = r2
x 0
General Form
Expanding the form (x – h)2 + (y – k)2 = r2 becomes x2 + y2 – 2xh – 2ky + h2 + k2 – r2 = 0
This is of the form:
x2 + y2 + Dx + Ey + F = 0 → general form
where D, E, F are constants not all zero at a time. Note: By equation of coefficients:
-2h = D ; h = -½ D → abscissa of center -2k = E ; k = -½ E → ordinate of center h2 + k2 – r2 = F ; r = √(h2 + k2 – F)
Radical Axis of Two Circles
Consider the two non-concentric circles x2 + y2 +D 1x + E1y + F1 = 0 x2 + y2 +D 2x + E2y + F2 = 0 The equation: x2 + y2 +D 1x + E1y + F1 + k (x2 + y2 +D2x + E2y + F2) = 0
represent a circle for any value of k except for k = -1
if k = -1, the equation of the family of circles above becomes: (D1 – D2) x + (E1 – E2) y + (F1 – F2) = 0
This represents a straight line called the RADICAL AXIS of two circles. Properties of the Radical Axis
A. If two circles intersect at two distinct points, their radical axis is the common chord of the circles.
Common Chord
Condition for Orthogonality
The two non-concentric circles : x2 + y2 + D
1x + E1y + F1 = 0 x2 + y2 + D
2x + E2y + F2 = 0,
meet at right angles (orthogonal) if : D1D2 + E1E2 = 2(F1 + F2)
B. If two circles are tangent, their radical axis is the common tangent to the circles at their point of tangency.
Radical Axis
C. The radical axis of two circles is perpendicular to their line of centers. Radical Axis
AB – line of centers A B
D. All tangents drawn to two circles from a point on their radical axis have equal lengths. y Radical Axis
P
T1 & T2 are points of
T1 tangency
T2
PT1 = PT2
x
Supplementary Problems
1. Find the center and radius of the circle whose equation is x2 + y2 – 4x –6y –12 = 0 (ECE Board Problem – Oct 1981) Ans: C(2, 3) r = 5
2. Find the area of the circle whose equation x2 + y2 = 6x – 8y (ECE Board Problem – Mar. 1981) Ans: 25π sq. units
3. Find the equation of the circle whose center is at (3, -5) and whose radius is 4 units.
Ans: (x – 3)2 + (y + 5)2 = 16
For Problems 4 – 9, determine the equation of the circle given the following conditions 4. Passes through the point (2, 3), (6, 1) and (4, -3) Ans: x2 + y2 – 10y = 0
5. Center on the y – axis, and passes through the origin and point (4, 2). Ans: x2 + y2 – 10y
= 0
6. Passes through the points of intersection of the circles x2 + y2 = 5, x2 + y2–x + y = 4, and through the point (2, -3) Ans: x2 + y2 –2x + 2y –3 = 0
7. Center on the line x – 2y –9 = 0 and passes through the points (7, -2) and ( 5, 0)
Ans: x2 + y2 – 10x + 4y +25 = 0
8. Circumscribe the triangle determine by the lines x – u – 8 = -y and y = -1.
Ans: x2 + y2 –8x + 2y + 8 = 0
9. Given the endpoints of the diameter (5, 2) (-1, 2) Ans: x2 + y2 – 4x – 4y – 1 = 0
10. Find the equation of the line tangent to the circle x2 + y2 – 8x – 8y + 7 = 0 at the point (1, 0) Ans: 3x + 4y – 3 = 0 y x 0 y x 0 y x 0
PARABOLA
The locus of a point that moves in a plane such that its distance from a fixed point equals its distance from a fixed line.
Notes:
• Fixed point is called focus • Fixed line is called directrix
• Axis – the line passing through the focus and perpendicular to the directrix • Vertex – The midpoint of the segment of the axis from the focus to the directrix.
• Latus rectum – a segment passing through the focus and perpendicular to the axis of the
parabola.
• Focal distance – distance from vertex to focus = a Standard Equations of Parabola
A. Vertex at V(h,k), Vertical Axis (x-h)2 = 4a(y-k)
if a is positive (+a) --- concave upward if a is negative(-a) --- concave downward
Notes:
1. Equation of axis : x=h y axis 2. Focus : F(h,k + a)
3. End of Latus Rectum L F R L(h-2a, k+a) R(h+2a, k+a) 4. Equation of Directrix V(h,k) y = k-a directrix 0 x B. Vertex at V(h,k), Horizontal Axis
(y-k)2 = 4a(y-k)
if a is positive (+a) --- concave to the right if a is negative (-a) --- concave to the left
Notes:
1. Equation of the axis: y=k directrix 2. Focus: F(h+a, k) y L 3. Ends of Latus Rectum:
L(h+a, k+2a) V F axis R(h+a, k-2a) (h,k)
4. Equation of Directrix
x = h-a R
0 a>0 x
C. Vertex at the Origin, Vertical Axis x2 = 4ay
if a is positive (+a) --- concave upward if a is negative (-a) --- concave downward
Notes: y
1. Axis : the y axis axis
2. Focus: F(0,a)
3. Latus Rectum: /4a/ F
Ends: L(-2a,a) L(-2a,a) R(2a,a) R(2a, a)
4. Equation of directrix
y = -a V(0,0)
directrix D. Vertex at the Origin, Horizontal Axis
y2 = 4ax
if a is positive (+a) --- concave to the right if a is negative (-a) --- concave tot he left
Notes: y
1. Axis : the x-axis directrix
2. Focus: f(a,0) L 3. Latus Rectum = 4a Ends: L(a,2a) V F R(a,-2a) x axis 4. Equation of Directrix x = -a R Remarks:
1. The vertex and focus always lie on the axis of the parabola. 2. Focus is always located on the concave side of the parabola.
General Equations of Parabola 1. Vertical Axis
• Ax2 + Dx + Ey + F = 0, E or A must not be zero
2. Horizontal Axis
