NMC Sample Problems: Grade 9. One root of the cubic polynomial x + x 4x + is. What is the sum of the other two roots of this polynomial?. A pouch contains two red balls, three blue balls and one green ball. What is the probability of drawing a red ball and the green ball if two balls are drawn randomly? 6. A line of slope passes through the points (, 4) and (, b). Find the value b. 8 6 4 4. Find the solution of the system of equations { x + 6y = x + y = 8. (, ) (, ) (, ) (, ). If f(x) = x, what is f(f(x))? x 4 9x 9x 4 x 4 6. There is an isosceles triangle which is also a right triangle. Its area is 0 square inches. Find the length of the base of the triangle. inches inches inches 4 0 inches. Suppose that a, b, c, d are positive integers satisfying a b = c necessarily true? a : b = c : d a : b = c : d ad = bc None of the above d. Which of the following is not a+b b = c+d d 8. What is the twenty-first term in the arithmetic sequence, 9, 6,,? 4 49 6 6 9. What is the sum of the first terms of the arithmetic sequence, 9, 6,,? 660 86 99 of 8
0. Solve the equation x + 4 = x for x.,,. From a group of twenty people, how many ways can the officers: President, Vice-president, Secretary, and Treasurer be chosen? 4 4 0C 4 0P 4. (Refer to Problem #) Now they have chosen the four officers. The President wants two officers and two non-officers to join him on a walk. How many ways are there to choose his four fellow walkers? 60 0 440 880. A ball is thrown up into the air from a patio in the two-story building which is at 0 feet high. The ball will be h feet above the ground at t seconds, and h = 0 + 8t 6t. Find the maximum height in feet which the ball can reach. 86 6 6 4. A square is inscribed in a circle of radius r. What fraction of the area of the circle lies outside the square? r π π π πr π πr. A newly discovered element Yourelementium, has a half-life of 8 months. How much of a 0 g sample of this new element will remain after 4 months? (Note: The half-life of a chemical substance is the time required for half of a given substance to disintegrate.) 0 g g. g. g 6. If you convert 9 to Base Ten, what number do you have? 80 9. A right triangle has two legs of length and. What is the length of the altitude to the hypotenuse? (Note: Figure not drawn to scale!) 60 6 of 8
8. How many diagonals does an nonagon(a nine sided polygon) have? 44 6 4 9. Farmer Bob has 400 ft of fencing to enclose a rectangular field. What is the maximum possible area (in ft ) that he can enclose? 0000 6000 0000 40000 0. Let the three roots of the equation x x 4x = 0 be a, b and c. What is a b c+a bc +ab c? 8. If Alice s ipad crashes, she will be angry. Which of the following statement is also true? If Alice is angry, then her ipad has crashed. If Alice is not angry, then her ipad has not crashed. If Alice s ipad does not crash, then she will be angry. Either Alice is not angry or her ipad has crashed. None of the above. What is the sum of the coefficients of the expression (a + b + c) 4 when expanded and simplified? 6 6 6 96. How many ways can I arrange the six letters of the word LIGHTS such that no vowel occupies an odd-numbered position? 60 0 60 0 4. Alice, Bob and Chris must clean their dorm. If it takes Alice hour to clean it alone, Bob hours to clean it alone, and Chris. hours to clean it alone, how many hours does it take the three of them working together to finish the task? hours. hours 0.6 hours 0. hours. (Refer to Problem #4) Suppose all three begin the task at the same time. However, Alice, Bob and Chris pull the fire alarm 0 minutes after they begin. What fraction of the task remains? 6 8 8 of 8
6. Let a = + + +.... Then a is a real number. Find a. +. Alice has two bags of marbles: Bag A contains 8 red and blue marbles, and Bag B contains red and blue marbles. She randomly select a bag and then draw one marble at random. It is blue. What is the probability that it came from Bag B? Not enough information 0 8. How many positive integral divisors does the number 6 have? 6 8 9 9. Compute φ(0), where φ(n) is the number of positive integers less than and relatively prime to n. 440 608 04 0. How many ordered triples of positive integers (x, y, z) satisfy x + y + z? 8 4 44. What is the coefficient of xy z in the expansion of (x + y + z) 4? 4 6. What is the probability of rolling a sum of with three standard six-sided dice? 6 8 6. A square with vertices (0, 0), (, 0), (0, ) and (, ) is rotated around the y-axis. Compute the volume of the resulting figure. 4π 8π π 6π 4. What is the sum of all prime factors of 0? 49 0 064 06. Solve the inequality 4 x 8 for x. 4 x x < 4 or x > x 4 x 4 or x 4 of 8
6. What is the set of points equidistant from (, ) and (6, 0). circle line parabola hyperbola. The area of triangle ABC is 8. If the coordinates of A and B are (, ) and (, ) respectively, what is the locus of point C? y = x + x = 6 or x = 6 y = 8 or y = 4 y = or y = 8. In New-Kid s-town, required bicycle licenses consist of two letters followed by three digits. (For example, LV, AA000, MP906, ZZ999, and so on.) How many distinct license numbers are available? 66, 000 68, 600 68, 40 00, 600 9. Given that x and x + are reciprocal of each other, what is x? 4 40. The length of the shortest side of a right triangle is. If the lengths of all three sides are integers, what is the perimeter of the triangle? 4 98 4. Consider point P = (, ) on the Cartesian plane. If you first reflect P about the line y = x and then rotate it about the origin by 90 counterclockwise, what are the coordinates of the new position of P? (, ) (, ) (, ) (, ) 4. For what real values of x does the expression below represent a real number? x + 4 x x < 4 x < x 4 x 4. What is the value of (0 + 06) 0 06? 0 06 4 of 8
44. Find the area of parallelogram ACDB with vertices A = (0, 0), B = (, ), C = (4, 4), D = (, ). 0 9 8 4. Consider functions f whose domain is given by {,, } and whose values lie in the set of integers {0,,,, 4}. How many different functions f can be formed? 60 00 4 46. A chemist mixes distilled water with a 90% solution of sulfuric acid to produce a 0% solution. If liters of distilled water is used, how much 0% solution is produced? (Answer in liters using decimals.) 0.... 4. Evaluate the sum of the first values n++ n starting from n = (up to n = ): that is the value 4 + + + 4 + + 64 + 6. 8 48. What is the sum of the measures of the interior angles of an hexagon(a six sided polygon)? (Answer in degrees.) 60 480 600 0 49. A 6-faced die is tossed twice. What is the probability that the same numbers are shown in both toss? 6 8 6 0. An average mountain bald eagle is 8 inches long, weighs 0 ounces and has a wingspan of 00 inches. Assume that wingspan is directly proportional to the fraction weight length. If I am a mountain bald eagle that is feet, 6 inches long and weighs pounds and 0 ounces, what is my wingspan in inches? (Hint: foot= inches and pound=6 ounces) 90 00 0 0. How many ordered pairs (x, y) of positive integers x and y satisfy the equation x + y = 80?. What is the area of a regular hexagon(a six sided polygon) inscribed in a circle with radius? (Answer in fraction.) 6 of 8
. If a fair coin is tossed times, what is the probability of getting at least heads ( at least heads means or more heads)? 4. How many ways are there to distribute 9 (indistinguishable) balls into different homerooms if each homeroom must receive at least one ball?. In a pizza shop, it is possible to have on a pizza a choice of any combination of the following eight ingredients: bacon, mushrooms, onion, pepperoni, peppers, salami, sardines, sausage. How many different varieties of pizza can be ordered if the pizza shop orders all pizzas with at least three toppings of any combination of the eight ingredients? of 8
KEYS [] [] [9] [4] [] [6] [0] [44] [] [] [] [4] [4] [8] [] [46] [] [9] [] [4] [6] [0] [4] [48] [] [] [] [49] [8] [] [6] [0] [9] [] [] [] (pairs) [0] [4] [8] [] [] [] [9] [] [] [6] [40] [4] 8 [] [] [4] [] 9 [4] [8] [4]