Title [MATHS NÂNG CAO] BUỔI 5.pdf ✅

1 | L U Y Ệ N T H I S A T O N L I N E KHÓA SAT NÂNG CAO ADVANCED MATH (PART II) Teacher: Thái Sơn Question 1: x ଶ + y + 7 = 7 20x + 100 − y = 0 The solution to the given system of equations is (x, y). What is the value of x ? Question 2: A right rectangular prism has a height of 9 inches. The length of the prism's base is x inches, which is 7 inches more than the width of the prism's base. Which function V gives the volume of the prism, in cubic inches, in terms of the length of the prism's base? A. V(x) = x(x + 9)(x + 7) B. V(x) = x(x + 9)(x − 7) C. V(x) = 9x(x + 7) D. V(x) = 9x(x − 7) Question 3: The function g is defined by g(x) = x(x − 2)(x + 6)ଶ . The value of g(7 − w) is 0, where w is a constant. What is the sum of all possible values of w ? Question 4: Let the function p be defined as p(x) = (௫ି௖) మାଵ଺଴ ଶ௖ , where c is a constant. If p(c) = 10, what is the value of p(12) ? A. 10.00 B. 10.25 C. 10.75 D. 11.00 Question 5: What is the minimum value of the function f defined by f(x) = (x − 2)ଶ − 4 ? A. -4 B. -2 C. 2 D. 4
2 | L U Y Ệ N T H I S A T O N L I N E Question 6: f(x) = (1.84) ௫ ସ The function f is defined by the given equation. The equation can be rewritten as f(x) = ቀ1 + ௣ ଵ଴଴ቁ ௫ , where p is a constant. Which of the following is closest to the value of p ? A. 16 B. 21 C. 46 D. 96 Question 7: In the xy-plane, the graph of the equation y = −x ଶ + 9x − 100 intersects the line y = c at exactly one point. What is the value of c ? A. − ସ଼ଵ ସ B. -100 C. − ଷଵଽ ସ D. − ଽ ଶ Question 8: For the function q, the value of q(x) decreases by 45% for every increase in the value of x by 1 . If q(0) = 14, which equation defines q ? A. q(x) = 0.55(14)௫ B. q(x) = 1.45(14)௫ C. q(x) = 14(0.55)௫ D. q(x) = 14(1.45)௫ Question 9: f(x) = (x − 10)(x + 13) The function f is defined by the given equation. For what value of x does f(x) reach its minimum? A. -130 B. -13 C. − ଶଷ ଶ D. − ଷ ଶ
3 | L U Y Ệ N T H I S A T O N L I N E Question 10: The rational function f is defined by an equation in the form f(x) = ௔ ௫ା௕ , where a and b are constants. The partial graph of y = f(x) is shown. If g(x) = f(x + 4), which equation could define function g ? A. g(x) = ଺ ௫ B. g(x) = ଺ ௫ାସ C. g(x) = ଺ ௫ା଼ D. g(x) = ଺(௫ାସ) ௫ାସ Question 11: x f(x) 1 a 2 a ହ 3 a ଽ For the exponential function f, the table above shows several values of x and their corresponding values of f(x), where a is a constant greater than 1. If k is a constant and f(k) = a ଶଽ, what is the value of k ? h(x) = −16x ଶ + 100x + 10 The quadratic function above models the height above the ground h, in feet, of a projectile x seconds after it had been launched vertically. If y = h(x) is graphed in the xy-plane, which of the following represents the real-life meaning of the positive x-intercept of the graph? A. The initial height of the projectile B. The maximum height of the projectile C. The time at which the projectile reaches its maximum height D. The time at which the projectile hits the ground
4 | L U Y Ệ N T H I S A T O N L I N E Question 13: For the exponential function f, the value of f(1) is k, where k is a constant. Which of the following equivalent forms of the function f shows the value of k as the coefficient or the base? A. f(x) = 50(2)௫ାଵ B. f(x) = 80(2)௫ C. f(x) = 128(2)௫ିଵ D. f(x) = 205(2)௫ିଶ Question 14: The population of a town is currently 50,000, and the population is estimated to increase each year by 3% from the previous year. Which of the following equations can be used to estimate the number of years, t, it will take for the population of the town to reach 60,000 ? A. 50,000 = 60,000(0.03)௧ B. 50,000 = 60,000(3)௧ C. 60,000 = 50,000(0.03)௧ D. 60,000 = 50,000(1.03)௧ Question 15: If k − x is a factor of the expression −x ଶ + ଵ ଶଽ nkଶ , where n and k are constants and k > 0, what is the value of n ? A. -29 B. − ଵ ଶଽ C. ଵ ଶଽ D. 29 Question 16: p(t) = 90,000(1.06)௧ The given function p models the population of Lowell t years after a census. Which of the following functions best models the population of Lowell m months after the census? A. r(m) = ଽ଴,଴଴଴ ଵଶ (1.06)௠ B. r(m) = 90,000 ቀ ଵ.଴଺ ଵଶ ቁ ௠ C. r(m) = 90,000 ቀ ଵ.଴଺ ଵଶ ቁ ೘ భమ D. r(m) = 90,000(1.06) ೘ భమ Question 17: y = −1.50 y = x ଶ + 8x + a In the given system of equations, a is a positive constant. The system has exactly one distinct real solution. What is the value of a ?