NMC Sample Problems: Grade 10

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1 NMC Sample Problems: Grade 0. Burger Queen advertises, Our French fries is % larger than MacTiger s fries at a price % less than MacTiger s. For the same size, by how much, in percentage, are Burger Queen s fries cheaper than MacTiger s? 0% % 0% 0% %. Simplify a a a, where a is a positive real number. a 9/ a / a / a / Answer:. Let f() =. Find the value that f() cannot assume How many factors does 80 have? Answer:. What is the sum of the positive factors of 80? What is the least positive integer with eactly positive factors? 60 7 of 8

2 NMC Sample Problems Grade 0 7. Find CD in the figure below if AB = 6, BD =., AC = and AD bisects BAC. A B D C If < 0 and 0 < y <, which of the following is true? < y < y < y < y y < y < y < < y Answer: 9. The Fibonacci sequence,,,,, 8,, starts with two s, and each term afterwards is the sum of its two predecessors. Which one of the ten digits is the last to appear in the units position of a number in the Fibonacci sequence? Solve + 6 =. = = 7 = = 7, = 0. What are the -intercepts of + y =? = 0, = 0, =, =,. In the figure below AC and BD are perpendicular, AD = 6, and BC =. Find AB + CD. D A C B Cannot be determined of 8

3 NMC Sample Problems Grade 0. If 7 t = 6, then what is the value of 9 t? 6 / 6 8. For f() = +, define f () = f(), f () = f(f ()), f () = f(f ()), and so on. Evaluate f 009 (f 00 ()). 8 9 Answer:. What is the sum of ? If, y, and z are positive integers such that + y + z = 7. What is the greatest value of + y + z? Solve the equation sin sin = 0, [0, ]., 7 6, 6, 8. When the mean, median, and mode of the list 0,,,,,, are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real value of? Answer: 9. What is the sum of all numbers such that ( + 9 0) 0 0+ =? Suppose that α and β are nonzero real solutions of the equation + = 0. Find the sum α + β. 0 of 8

4 NMC Sample Problems Grade 0. For an equilateral triangle with a side length 9, find the area of the circumscribing circle Simplify the following epression Arrange the numbers a = 6 0, b = 7, and c = from smallest to greatest. a < b < c a < c < b b < a < c b < c < a. Let a point Q be on the line segment connecting A(0, ) and B(, 0). Find the maimum area of the rectangle OP QR. A R Q O P B 0 0 Answer:. Find the constant term in the epansion of ( + ) If yz = 8,, y, and z are positive, find the minimum value of + y + z Find tan(arccos ) in terms of. Answer: of 8

5 NMC Sample Problems Grade 0 8. Find the period of f() = sin( ). 9. Find the phase shift of f() = cos( + ) How many zeros does 0! 9 end in? 7 8. What is the eact value of 8 + 6? Write your answer as a common fraction Suppose that solves 6 + a = 0. Find the value of a. 7 8 Answer:. Find the shaded area. 0 6 Answer: of 8

6 NMC Sample Problems Grade 0. For a graph of y = f() shown below, find the number of real numbers satisfying f(f()) =. y y = f() Answer:. Consider the circle ( ) + (y ) = 8. Find an equation of the line tangent to the circle at the point (, ). y = + y = + y = + 6 y = + 6 y = + Answer: 6. If a function f() satisfies f() + f( ) = for all real numbers, what is f(0)? - Answer: 7. Jennifer is visiting her mother living in a town 60 miles from her house. She drove 0 mph to her mother s house and 0 mph on the way back. What is her average speed for the trip? 6 mph mph Jim parked his car in a parking lot at a randomly chosen time between :0 PM and :00 PM. Eactly half an hour later he drove his car out of the parking lot. What is the probability that he left the parking lot after :00 PM? Answer: 9. Two numbers, and y are selected at random from the interval (0, ). What is the probability that y +? /7 /7 / /9 /9 Answer: 0. How many non-negative integer solutions are there to the equation + y + z + w = 0? of 8

7 NMC Sample Problems Grade 0. How many ways can I select 0 cans of soda from a cooler containing large quantities of Coke, Pepsi, Dr. Pepper, and Sprite? A die is tossed times. Compute the probability that a comes up eactly times (out of trials) Answer:. Square ABCD has sides of length 0 units. Isosceles triangle GDC has base DC, and the area common to triangle GDC and square ABCD is 80 square units. Find the length of the altitude to DC in GDC Answer:. In the circle below, AB and CD are chords intersecting at E (Not drawn to scale). If AE =, BE =, and CE = 7, what is the length of DE? A C E D 8 7/ 9/ 0 / B. John forgot to write down a very important phone number. All he remembers is that it started with 866 and that the net set of digits involved, and 9 with one of these numbers appearing twice. If he guesses a phone number, what is the probability that he gets the number correct? / /6 /7 /6 /6 Answer: 6. For a real number find the sum of the maimum and minimum of y = ( ) ( + + ). 0 Answer: 7 of 8

8 NMC Sample Problems Grade 0 7. What is the sum of ? monkeys find an incredibly large ship filled with bananas stranded on the shore. The k th monkey takes k 0 of the bananas that remain on the ship. If every monkey took a positive whole number of bananas, what is the smallest number of bananas that the 0 th monkey will take? Answer: 9. Jisu and Junho leave their apartment building at the same time. The building is located on a circular walkway. Jisu and Junho walk, in opposite directions, around the circle. Jisu takes minutes to walk around the circle, and Junho takes. minutes to walk around the circle. At some random time between 9 and minutes after they leave, a photographer comes and, standing inside the circle, takes a picture of of the circle, with the apartment building directly in the center of the picture. What is the probability that neither Jisu nor Junho are in the picture? What is the remainder when 7 is divided by? 6 Answer:. What is the hundreds digit of ? Answer:. How many integers between and 00 (inclusive) are divisible by or? Answer: 700. How many pairs of positive integers (a, b) eist that satisfy the equation a + b = a? Answer: Infinitely many 8 of 8

CBSE Board Class X Summative Assessment II Mathematics

CBSE Board Class X Summative Assessment II Mathematics Board Question Paper 2014 Set 2 Time: 3 hrs Max. Marks: 90 Note: Please check that this question paper contains 15 printed pages. Code number given