MATH DIAGNOSTIC / COMPREHENSIVE TEST items 26 – 50 College Entrance Test Review Young Einstein Learning Center By Kristine Tan

26. The sum of 3 consecutive ODD integers is 105 26. The sum of 3 consecutive ODD integers is 105. If x is the first EVEN integer, what is the number sentence to represent the sum? x +(x+1)+(x+2)=105 (x+1)+(x+3)+(x+5)=105 x+(x+2)+(x+4)=105 x+(x+3)+(x+5)=105 x = first even integer Thus the distance of the first integer to x is 1 x+1= first odd integer (x+1)+2 = x+3 = 2nd ODD integer (x+3)+2 = x+5 = 3rd ODD integer Therefore the sum is: (x+1)+(x+3)+(x+5) = 105

27. A two-digit integer is represented by m and p 27. A two-digit integer is represented by m and p. If m is the units digit, how can the number be represented? xy yx 10x + y 10y + x If m or x is the units digit = m.1 or x.1 and p or y is the tens digit = 10.p or 10.y Therefore its value is: 10p + m = 10y + x

28. Seven years ago,Luis was six times as old as Melvin 28. Seven years ago,Luis was six times as old as Melvin. In one year, Luis will only be twice as old as Melvin. What is the present age of Luis? 7 yrs ago now In 1 year Luis Melvin 9 yrs.old 19 yrs old 2 yrs. old 12yrs. old 6x 6x + 7 6x + 8 x x + 7 x + 8 6x + 8 = 2(x + 8) x = 8÷4 6x + 8 = 2x + 16 x = 2 6x + 7 = 6(2) + 7 6x – 2x = 16 - 8 = 12 + 7 = 19 4x = 8

29. Kevin can finish fixing the house in 3 days while Rogelio can do the same job in 2 days only. If they will work together, how long will it take them to finish painting the house? 5 days 2.5 days 1.2 days 1 day rate = # of jobs/day Kevin’s rate is 1/3 Rogelio’s rate is ½ Therefore:

30. A boat goes downstream 180 km in 5 hrs and upstream for the same distance in 6 hrs. What is the speed of the boat in still water 3.0 kph 30.0 kph 33.0 kph 36.0 kph Distance Time Rate Downstream Upstream 180 km 5hrs Speed boat + Current Speed boat - Current 180 km 6hrs  

31. A mixture is made from two substances 31. A mixture is made from two substances. Substance A is 20% acidic while substance B is 30% acidic. What is the percent of acidity of the mixture if 100 grams of A and 100 grams of B are combined 20% 25% 30% 50% Substance % Concentration amount Acid present A 20% 100g 20% x 100g = 20g B 30% 30% x 100g = 30g Mixture x 200g 20g + 30g = 50g X = (50g / 200g) x 100% X = 25%

32. A collection of 24 coins worth P5, P1 and 25c amounts to P36. 25 32. A collection of 24 coins worth P5, P1 and 25c amounts to P36.25. If the number of P1 coins is 3 more than thrice the number of P5 coins, how many 25c coins are there? Let # P5 coins = x # P1 coins = 3 + 3x # of 25c = 24 – (x+3+3x) = 24 - 3 - 4x = 21- 4x Therefore: 36.25 = (5x) + 1(3 + 3x) + (0.25)(21-4x) 36.25 = 5x + 3 + 3x + 5.25 – x = 7x + 8.25 36.25 – 8.25 = 7x X = 28÷7 = 4 # of 25c = 21- 4x = 21 – 4(4) = 21- 16 = 5 4 5 10 15

(x+3) = 0 (x-4) = 0 x = -3 x = 4 Therefore domain is:   (x+3) = 0 (x-4) = 0 x = -3 x = 4 Therefore domain is: All real numbers except 3 and -4 All real numbers except -3 and 4 All real numbers except 1 All real numbers

34. What is the value of f(-1) if f(x) = (x-1) / (x+3)(x-4) Therefore we substitute -1 for the value of x f(-1) = (-1-1) / ((-1)+3)((-1)-4) f(-1) = -2 / (2)(-5) f(-1) = -2 / -10 f(-1) = 5 4 5 10 15

35. What is the sum of h(x) + g(x) if h(x) = 5x-1 and g(x) = 3x2 + 6x – 7? We substitute the values of x respectively h(x) + g(x) = (5x-1) + (3x2 + 6x – 7) h(x) + g(x) = 3x2 + 11x - 8 14x4 - 8 3x2 + 11x – 8 3x2 + 11x + 8 14x2 - 8

36. If F(x)= 2x + 3 and G(x) = 5x – 4, what is the composite function G(F(x))?

37. What is the domain of the root function f(x) = x-5 ? DOMAIN The set of all possible values for x, OR The x-coordinates of the points of the graph of the function Since any value of x will make the equation a function Therefore domain = set of all real numbers*BONUS -5 5 x y

38. The degree of polynomial determines the maximum number of roots it may have. How many possible roots does the function f(x) = x3 -2x2 + 5x - 10 Note the word maximum which suggest that A polynomial P(x) of degree n has exactly n roots, real or complex Therefore based on the degree of the given polynomial which 3 Exactly 4 Exactly 3 At most 3 At least 3

39. Which of the following is NOT divisible by (x – 1)? X3 + 3X2 - 4 X2 - 1 X2 + 1 X2 + 4X - 5 If we substitute +1 for the values of x in the given equations: ((1)3 + 3(1)2 - 4)=0 ((1)2 -1)=0 ((1)2 +1)= 2 ((1)2 + 4(1) -5)=0

40. . What are the roots of the function ? ROOTS – also known as the x-coordinates of the x-intercepts, OR the solutions to the equation if h( x ) were equal 0. HOW DO WE GET THE x-intercepts? Let y = 0, meaning let h( x ) = 0. Therefore the correct set of roots of f( x )? D. x = -5, 0 and 3

41. What is the remainder if (x3 + 2x2- 5x + 1) is divided by (x-1)? We use the Remainder Theorem to compute for the remainder. Let x = 1 in the expression (x3 + 2x2- 5x + 1) . x3 + 2x2- 5x + 1 = (1)3 + 2(1)2- 5(1) + 1 Remainder = 1 + 2 -5 + 1 = 4 - 5 Answer: B. -1

42. Where is the center of the graph of the equation (x-6)2 + (y + 5)2 = 4 The standard equation of a circle is: (x - h)2 + (y - k)2 = r2 Wherein (h,k) represent the (x,y) coordinates of the center of the circle, therefore: C. (6, -5) is the center

43. Which is a solution of the inequality -2x > 8? To get the solution set for x, we need to isolate x in this inequality. To isolate x, we multiply the reciprocal of -2 to both sides of the inequality: -5 -4 8 9 Take note: When we multiply by a negative number, the inequality symbol flips.

    -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

45. What is the range of the rational function f(x) = (x+1)/(x-4)? Range = set of all possible y values. To get this, we need to solve for x in terms of y – so that we can see what restrictions must be made for y. D. All real numbers except +1

  g(x) = I(-1)2 – 2I g(x) = I(0)2 – 2I g(x) = I(1)2 – 2I = I1-2I = I -2 I = I 1-2 I = I -1 I = 2 = I -1 I = 1 = 2 = 1 Therefore the Solution set is D.

  (2,7) (-6,3) (-4,-5) (10,11)        

48. What happens to the area of a circle if its radius is doubled?  

49. In arithmetic sequence, the 3rd term is 8 and the 6th term is 23 49. In arithmetic sequence, the 3rd term is 8 and the 6th term is 23. What is the first term? an= a1 + (n-1)d a3= a1 + (3-1)d 8 = a1 + 2d a1 = 8-2d a6= a1 + (6-1)d 23 = a1 + 5d a1 = 23 – 5d 8-2d = 23 -5d 5d-2d = 23-8 3d = 15 ; d = 5 a1 = 8 – (2)(5) B. -2

Or we can make a diagram of the arithmetic sequence: ____ _____ _____ _____ _____ _____

If we forgot the identity above, we can use: 50. What is the value of the function g(x) = cos2x + sin2x if x=45 degrees? According to the Pythagorean Identity, for any angle x, cos2x + sin2x = 1 c. 1 If we forgot the identity above, we can use:

Thank you! 