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QUESTION 1Determine the domain and range of(a)(b)(c)(d)
QUESTION 1 Solution:The domain of y excludes values of x that will make thedenominator zero. Thus, the domain isTo solve for the range, we first solve for x in terms of y: x 5y y x 1 x 5 Therefore, the x 7 range is xy 7y x 5 xy x 7y 5 x y 1 7y 5 7y 5 x y 1
QUESTION 2 Solution:Recall: for functions F and G:and
QUESTION 2 Solution:QUESTION 2 Alternative Solution:SUBSTITUTE a value of x and test which choice will givethe same value.Para madali, let x = 0.
QUESTION 2 Solution:QUESTION!Which will give a value of 3 at x = 0?(a) (c)(b) (d)Astig, ‘di ba?
QUESTION 3Which of the following is a linear function? (a) (b) (c) (d)
QUESTION 3 Solution: Recall that a linear function is a polynomial function wherein the highest power of the independent variable is 1. Is LINEAR, so the answer is (a) (a) Is QUADRATIC because of (b) the terms 3x2 WAIT! This is (c) also linear! cannot be a linear function since x and are (d) in the denominatorThe answers are BOTH (a) & (c)! Weh, ‘di nga?!
QUESTION 4What is the equation of the linear function ywhose graph passes through the point (2, 4) andhas the given slope m = 5/7? (a) (c) (b) (d)
QUESTION 4 Solution:We use the slope-intercept formSTRATEGY: Substitute x = 2, y = 4 and m = 5/7 thensolve for b. Hence, the equation of the line is or
QUESTION 4 Solution:QUESTION 4 Alternative Solution:Check the choices! Which among the choices… CLUE: 5 ang nasa unahan1. Has slope 5/7? ng x at 7 ang nasa denominator2. Has a value y = 4 when x = 2?
QUESTION 5Determine the distance from the point ( 2, 9) tothe line 3x + 4y = 2.
QUESTION 5 Solution: "No choice" Solution: We have NO CHOICE but use the following formula for the distance D of a point (x0, y0) from a line with equation Ax + By + C = 0:Before doing anything, rewrite 3x + 4y = 2 as 3x + 4y 2 = 0Then, substitute the values A = 3, B = 4, C = 2, x0 = 2, and y0 = 9.
QUESTION 6If ax2 + bx + c = 0, where a, b and c are realnumbers and a ≠ 0, which of the followingstatements is true about the discriminant D?(a) If D < 0, the two roots are real and equal.(b) If D < 0, the two roots are imaginary and unequal.(c) If D > 0, the two roots are real and unequal.(d) If D < 0, the two roots are imaginary and equal.
QUESTION 6 Solution:Recall: the solutions or ROOTS of the quadratic equationax2 + bx + c = 0, where a, b and c are real numbers anda ≠ 0, can be solved using the QUADRATIC FORMULA: The DISCRIMINANT D of ax2 + bx + c = 0 is the value INSIDE THE SQUARE ROOT; i.e.,
QUESTION 6 Solution:The DISCRIMINANT D determines the type orNATURE of solutions or roots a quadratic equationwith real coefficients has.
As an ASIDE…Some "UPCAT-level" problems that can be solvedusing the discriminant:
QUESTION 7Determine the radius of the circle whoseequation is (a) 2 y (b) 3 (c) 4 r (d) 5 x
QUESTION 7 SolutionThe CENTER-RADIUS FORM of the equation of acircle with radius r and center at (h, k) isTo write x2 + y2 8x + 6y = 0 in center-radiusform, complete the square: The radius is
QUESTION 11 SolutionBy definition, the LOGARITHM of a positive number x tothe base b, denoted by logb x, is the POWER y of bequal to x; i.e., Example: log3 9 = 2 since 32 = 9. Simple, ‘di ba?CHALLENGE: What is the value of ?
QUESTION 12Solve for all possible values of x in the equation (a) 3 and 2 (b) 2 and 3 (c) 6 and 9 (d) 9 and 6
QUESTION 12 SolutionA property of logarithm is that Shortest solution: SUBSTITUTE the choices to the original equation!
QUESTION 13Solve for q in the equation (a) (c) (b) (d)
QUESTION 14Faye is 5 greater than twice the age of Luigi. 5years from now, Faye will be twice as old asLuigi. How old is Faye 3 years ago? (a) 41 (c) 39 (b) 38 (d) 37
QUESTION 14 SolutionAGE PROBLEM:Let x = Luigi’s age 2x+5 = Faye’s age Age 5 years Age now from nowLuigi x x+5 (2x + 5) + 5 =Faye 2x + 5 2x + 10
QUESTION 15Paolo can finish compiling the books in library in 25minutes. Kevin can finish it in 25 minutes whileCarmela took her 50 minutes. How many minuteswill it take them if they were to compile the booksaltogether? (a) 10 (c) 20 (b) 25 (d) 33
QUESTION 15 SolutionWORK PROBLEM:Let x = no. of min they can finish the job together No. of Rate per EQUATION: minutes minute Paolo 25 1/25 Kevin 25 1/25Carmela 50 1/50Together x 1/x
QUESTION 16There are 570 students in a school. If the ratio offemale to male is 7:12, how many male studentsare there? (a) 300 (c) 380 (b) 370 (d) 390
QUESTION 16 Solution 570 students in the ratio 7:12 MALES FEMALESOne block =
QUESTION 17When each side of a square lot was decreased by3m, the area of the lot was decreased by 105 sq.m. What was the length of each side of the originallot? (a) 18 (c) 20 (b) 19 (d) 21
QUESTION 17 Solution Let x = length of the side of the square EQUATION: Length of a Area sideOriginal x x2 New x 3 (x 3)2
QUESTION 18The difference of 2/3 of an even integer and one-half of the next consecutive even integers is equalto 5. What is the odd integer between these twoeven integers? (a) 26 (c) 36 (b) 27 (d) 37
QUESTION 18 SolutionLet x = 1st even integer x + 2 = 1st even integerEQUATION: The ODD integer in between is the one AFTER 36, which is 37
QUESTION 19Find the average of all numbers from 1 to 100 thatend in 8. (a) 53 (c) 51 (b) 52 (d) 45
QUESTION 19 SolutionThe average looks like this:The numerator is actually a sum of an ARITHMETICPROGRESSION with first term a1 = 8 and tenth terma10 = 98, given by The average is then 530/10 = 53
As an ASIDE…FACT: The average of the first n terms of anarithmetic progression is just actually theAVERAGE of the FIRST AND LAST TERM!
QUESTION 20A tank is 7/8 filled with oil. After 75 liters of oil aredrawn out, the tank is still half-full. How manyliters can the tank hold? (a) 200 (c) 240 (b) 220 (d) 260
CAPACITYQUESTION 20 Solution = 25(8) = 200 L 25 L 25 L 75 L drawn out 25 L 25 L 25 L 25 L 25 L 25 L7/8 full 1/2 full
QUESTION 21Two new aquariums are being set up. Each onestarts with 150 quarts of water. The first fills at therate of 15 quarts per minute. The second one fillsat the rate of 20 quarts per minute. When wouldthe first tank contain 6/7 as much as the secondtank? (a) After 7 min (c) After 9 min (b) After 8 min (d) After 10 min
QUESTION 21 SolutionLet x = no. of minutesEQUATION:
QUESTION 22In a classroom, chairs are arranged so that eachrow has the same number. If Ana sits 4th from thefront and 6th from the back, 7th from the left and3rd from the right. How many chairs are there? (a) 49 (b) 64 (c) 81 (d) 100
QUESTION 22 Solution FRONTLEFT anna RIGHT NO. OF CHAIRS: 9 X 9 = 81 BACK
QUESTION 23A circle with radius of 5 m and a square of 10 m arearranged so that a vertex of the square is at thecenter of the circle. What is the area common tothe figures?
QUESTION 23 Solution The area common to the figures is 10 m equal to ¼ the area of the circle:10 m 5m 5m
QUESTION 24How many liters of 20% chemical solution must bemixed with 30 liters of 60% solution to get a 50%mixture? (a) 5 L (b) 10 L (c) 15 L (d) 20 L
QUESTION 24 SolutionLet x = no. of L of 20% chemical sol’n % Amount of Vol (L) concen- chemical tration Sol 1 x 20% 0.2x Sol 2 30 60% 30(0.6) = 18 mixture (x + 30) 50% 0.5(x + 30)
QUESTION 25A URent-A-Car rents an intermediate-size car at adaily rate of 349.50 Php plus 1.00 Php per km. abusiness person is not to exceed a daily rentalbudget of 800.00 Php. What mileage will allow thebusiness person to stay within the budget? (a) 300 (c) 400 (b) 350 (d) 450
Rules of CountingThe Fundamental Principle of Counting:If an operation can be performed in n1 ways, and for each of these a second operation can be performed in n2 ways,then the two operations can be performed in n1n2 ways.Extension: The Multiplication RuleIf an operation can be performed in n1 ways, andfor each of these a second operation can beperformed in n2 ways, a third operation in n3ways,…, and a kth operation in nk ways, then the koperations can be performed in n1n2n3…nk ways.
Permutations Rules of CountingPERMUTATION – based on arrangement of objects, with order being consideredPermutation of n objects: n(n – 1)(n – 2)… (3)(2)(1) = n! (n factorial)Permutation of n objects taken r at a time: n! n Pr n r !Permutation of n objects with repetition: n! n1 ! n2 !...nk !
Combinations Rules of CountingCombination – based on arrangement of objects, without considering orderCombination of n objects taken r at a time: n n! n Cr r r! n r !
QUESTION 27The basketball girls are having competition forinter-colleges. There are 15 players but the coachescan choose only five. How many ways can fiveplayers be chosen from the 15 that are present?(a) 3,103 (c) 3,000(b) 2,503 (d) 3,003
QUESTION 27 SolutionSince order is NOT important in choosing the five players out of 15, we use the Combination rule with n = 15 and r = 5:
QUESTION 28A coach must choose first five players from a teamof 12 players. How many different ways can thecoach choose the first five?(a) 790 (c) 800(b) 792 (d) 752
QUESTION 30What is the perimeter of the triangle defined bythe points (2 , 1), (4 , 5) and (2 , 5)?
QUESTION 30 SolutionWe can use the DISTANCE FORMULA to compute theperimeter of a triangle in the Cartesian plane(i.e., the sum of the lengths of the sides of thetriangle).Kaya lang, ‘di ko na realize na easy lang ang casesa problem kasi RIGHT TRIANGLE na! (See the boardfor the solution) :p
QUESTION 32If arcs AB and CD measure 4s - 9o and s + 3orespectively and angle X is 24o, find the value of s.See figure below.(a) 6 (c) 18(b) 12 (d) 20
QUESTION 33How many possible chords can you form given 20points lying on a circle?(a) 380 (c) 382(b) 190 (d) 191
QUESTION 33 SolutionThe number of chords can be obtained using the Combination Rule with n = 20, r = 2
QUESTION 34Which of the following sets of numbers cannot bethe measurements of the sides of a triangle? (a) 1, 2, 2 (b) (c) 3, 4, 5 (d) 1, 2, 3
QUESTION 34 SolutionUse the TRIANGLE INEQUALITY:The sum of the lengths of any two sides of a triangle isgreater than the length of the third side.(d) 1, 2, 3 1 + 2 = 3 – should be GREATER!
QUESTION 35The figure shows a square inside a circle that is inside thebigger square. If the diagonal of the bigger square isunits, what is the area of the shaded region?
As an ASIDE…The Pythagorean Theorem and Special Right Triangles
QUESTION 35 Solution Note that: •The side of the larger square is 2 (special right triangle) 2 •The side of the square is the diameter of the circle, so the radius of the circle is 1
QUESTION 35 Solution Note that: •The diagonal of the smaller square is also 2. s •If s is the side of the smaller square, then 2 •The area of the shaded area is then
QUESTION 36Which of the following statements is NOT true about thefigure? Parallel lines a and b are intersected by line xforming the angles 1, 2, 3, 4, 5 and 6.(a) Angles 1 and 6 are congruent.(b) Angles 1 and 5 are supplementary with each other.(c) Angles 3 and 4 are congruent.(d) Angles 2 and 4 are supplementary with each other.
QUESTION 36 Solution (a) Angles 1 and 6 are congruent. (alt. ext.) (b) Angles 1 and 5 are supplementary with each other. (ext.) (c) Angles 3 and 4 are congruent. (alt. int.) (d) Angles 2 and 4 are NOT supplementary with each other – they are CONGRUENT (corresponding angles)
QUESTION 38How many sides does a polygon have if the sum ofthe measurements of the interior angles is 1980o? (a) 11 (b) 12 (c) 13 (d) 14
QUESTION 38 SolutionThe sum of the interior angles of a triangle is given by http://www.mathopenref.com/polygoninteriorangles.html
QUESTION 39An ore sample containing 300 milligrams of radioactivematerial was discovered. It was known that the material hasa half-life of one day. Find the amount of radioactivematerial in the sample at the beginning of the 5th day.(a) 9.375 mg(b) 18.75 mg(c) 37.5 mg(d) 75
QUESTION 39 SolutionThis can be solved using a geometric progression withfirst term a1 = 300 common ratio r = ½, and n = 5 days
QUESTION 40A survey of 60 senior students was taken and the followingresults were seen: 12 students applied for UST and UP only,6 students applied for ADMU only, 29 students applied forUST, 2 students applied for UST and ADMU only, 10 studentsapplied for UST, ADMU and UP, 33 students applied for UPand only 1 applied for ADMU and UP only. How many of thesurveyed students did not apply in any of the threeuniversities (UP, UST, ADMU)?(a) 0 (b) 8 (c) 14 (d) 20
QUESTION 40 SolutionUsing Venn Diagram 12 – UP & UST onlyUP UST 6 – ADMU only 12 2 – UST and ADMU only 10 10 – all three 1 2 1 – UP and ADMU only 6 ADMU
BRIEF TIPS AND TRICKS1. READ EACH QUESTION CAREFULLY.2. Take each solution one step at a time. Some seemingly difficult questions are really just a series of easy questions.3. Remember thy formulas and important facts (especially in Geometry)4. Answer the easy items first. If you can’t solve a problem right away, SKIP it and proceed to the next.
BRIEF TIPS AND TRICKS4. Try the PROCESS OF ELIMINATION. A little guessing might work.5. Employ the EASIEST way as possible (e.g., substitution, shortcuts, tricks, etc.)6. Use your scratch paper wisely…7. If you still have time, CHECK your answers, ESPECIALLY your shaded ovals!8. RELAX…. Don’t panic!
PRACTICE PROBLEMS!1. If x + y = 4 and xy = 2, find the value of x2 + y22. If 1/3 of the liquid contents of a can evaporates on the first day and 3/4 of the remaining contents evaporates on the second day, what is the fractional part of the original contents remaining at the end of the second day?3. What is the smallest three-digit number that leaves a remainder of 1 when divided by 2, 3, or 5?4. The average of 4 numbers is 12. What is the new average if 10 is added to the numbers?5.
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