NMC Sample Problems: Grade 11

Transcription

1 NMC Sample Problems: Grade. Which one of the following functions has different domain? x x 0 x + x x + x x(x +). In the expansion of (x + y), what is the coefficient of x y? Find the equation of the circle centered at (, 0) passing through the point (0, ). (x ) + y = (x + ) + y = x + (y ) = (x ) + y = x + (y ) =. What is the solution set of the following equation: ln(x x) = ln(x )?(Here, ln x = log e x) {0} {} {} {, } {0, }. Evaluate log /. 6. Find the sum of the maximum and minimum values of the expression 0 sin(x + ) Find the sum of all solutions of the equation x x = Find the polar coordinates (r, θ) for the point with rectangular coordinates (x, y) = (, ). (, π ) (, π ) (, π) (, π ) (, π). If 0 x =, then which of the following is 0x? If csc x = 0 and 0 < x < 0, then what is cos x? 0 of 0 0

2 NMC Sample Problems Grade. Find the equation of the straight line which passes through the point (0, 0) and is perpendicular to x + y + = 0. x y = 0 x + y = 0 x y = 0 x y + = 0 x + y = 0. Find the vertex point of the parabola x y + = 0. (, 0) (, 0) (0, ) (0, ) (, ). Find the rectangular coordinates (x, y) for the point with polar coordinates (r, θ) = (, π ). (, ) (, ) (, ) (, ) (, ). When cos θ = and tan θ > 0, find sin θ Let x and y be real numbers satisfying the following two conditions: Find x + y. x y = and x y = Find the sum What is the remainder when! +! +! + + 0! is divided by 0? 6 0. How many seven digit numbers contain the digit pattern 0 exactly once? Let α, β and γ be three roots of x + = 0. Find the product p(α)p(β)p(γ), where p(x) = x + x x +. 0 of

3 NMC Sample Problems Grade 0. Find the area of the following triangle. 0. Suppose a, b, c and d are integers, and x = + is a solution of the equation x + ax + bx + cx + d = 0. What is the sum a + b + c + d? 6 6. What is the remainder when x is divided by x +? Find the value of the product: ( ) ( ) ( ) ( ) Which one is the number of the real solutions of the equation e x sin x = 0? 0 (no solutions) (two solutions) (four solutions) 6 (six solutions) (infinitely many solutions). A ball is thrown vertically upward and its height after t seconds is h(t) = t t feet. Find the maximum height reached by the ball. 6 feet feet feet feet 0 feet 6. The function f(x) = x x = b. Find a + b. x has one horizontal asymptote, say y = a, and one vertical asymptote, say 0. If sin x + cos x =, then what is the value of sin x cos x? of

4 NMC Sample Problems Grade. Find the remainder when is divided by Evaluate ( + ) 00 i. 0 + i i +i 0. Let a 00, a,, a and a 0 be the coefficients of the expansion of the expression (x + ) 00, that is (x + ) 00 = a 00 x 00 + a x + + a x + a 0 Find the sum of even coefficients, a 00 + a + + a + a Find the minimum value of sin x cos x Find the solution set of the equation ( x+) x+ =. { ± } { ± } { ± } { ± } {} (No solution). If sin x + cos x =, then what is the value of sin x + cos x? 6 6. There are red ball, blue balls, and white balls in a bag. If James takes out four balls at once, then what is the probability that he takes out at least two white balls? A committee composed of Alice, Mark, Ben, Connie and Francisco is about to select two representatives randomly. What is the probability that Ben is not included in the selection? 0 6. Find the sum of all integers between to 000 (inclusive) that are not multiples of of

5 NMC Sample Problems Grade. Evaluate the following(ln x = log e x): ( ) ( ln + ln ) + ln ( ) ( ) + + ln 000 ln ln ln 0 ln 0 0. How many functions which are one-to-one are there from the set X = {a, b, c} to the set Y = {,,, }? Let z = a + i be a complex number, where a is a positive real number (i = ). If the real part of z equals that of z, then what is a? What is the average of the numbers in the set {,,,,, } given by an arithmetic progression?. The rational function f(x) = ax+b x has the inverse function f (x) = x+. Find a + b. 0. Find cos B + cos A of the following triangle. A B C. An employee of a computer store is paid a base salary $, a month plus a % commission on all sales over $,000 during the month (For example, if the gross sales during the month is $000, then the commission is $0). How much must the employee sell in a month to earn a total of $, for the month? $6,00 $,00 $,00 $0,0 $0,0. A speedboat takes a half hour longer to go miles up a river than to return. If the boat cruises at 0 miles per hour in still water, what is the rate of the current? (The units are miles/hour.) ( ) ( ) 6 of

6 NMC Sample Problems Grade. Which one of the following is the greatest? Suppose that a, b are integers and is a solution of the equation x + x + ax + b = 0. Find a + b Harry performed the calculation: =. It turns out Harry s calculation is correct in base b. Find the value b. 0. Suppose that f(x) is a polynomial with integer coefficients, having 00 and 00 as zeros. Which of the following could possibly be the value of f(0)?. A polynomial f(x) = x + ax + bx + c satisfies f() =, f() = and f() =. What is the remainder when f(x) is divided by x? Evaluate The domain of the function is {x x > c}. What is the value of c? f(x) = log 0 (log (log (log x))) What is the solution set of the following equation: ln x x+ = ln x? (Here, ln x = log e x) {0} {} {,, } {, } {0, }. What is the number of real solutions of the equation sin x π x = 0? (one solutions) (two solutions) (three solutions) (four solutions) (infinitely many solutions) 6 of

7 NMC Sample Problems Grade. Let a 00, a 0,, a and a 0 be the coefficients of the expansion of the expression (x+) 00, that is (x + ) 00 = a 00 x 00 + a 0 x a x + a 0 What is the sum of even coefficients a 00 + a 0 + a 06 + a a + a 0 of all even indices There are red ball, blue balls, and white balls in a bag. If James takes out five balls at once, then what is the probability that he takes out at least three white balls? 6 6. Suppose that a, b are integers and + is a solution of the equation x + ax + bx + = 0. Find a + b A polynomial f(x) of degree whose coefficients are all integers satisfies f() = and f( ) =. What is the remainder when f(x) is divided by x?. Let f be a function satisfying the following equation: f(x) = f(x + ) for all real numbers x Suppose further f(x) > 0 for all x. What is the value of f(π+) f(π )?. The sides a, b, c of a right triangle satisfy the following equation Find the length of the hypotenuse? a + b + c =. 60. Find sin A+sin B+sin C cos A+cos B+cos C of the following triangleȧ B C 6. Find the remainder when is divided by. 6. If log = a, then what is log 6 in terms of a? 6. Let a and b be integers which satisfy + 0 = a + b. Find a + b. of

8 NMC Sample Problems Grade 6. Let f be a function satisfying the following equation: Find f(00). f(x) = πxf( x) for all real numbers x. 6. How many ordered pairs (x, y) of real numbers are there which satisfy the following equations? x 00 + y 00 = x 0 + y 0 = x 0 + y The sides of a right triangle form an arithmetic progression. Find the ratio of the shorter leg to the hypotenuse. of

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