Mathematics Exam

1. Let f(x) = {x + 1}/{x- 1} and g(x) = √x+2 be two followings.
Which one of the following is true about the combination values of f and g at x=2?

A. (f.g)(2) = 6
B. (^f/_g)(2) = (²/₃)
C. (f+g)(2) = 3
D. (f-g)(2) = -1

2. Which one of the following defines an exponential function

A. f:R → (0, ∞) defined by f(x)=b^x, where b>0, and b ≠ 1.
B. f:R → (0, ∞) defined by f(x)=x^b, where b>0, and b ≠ 1.
C. f:R → (∞, 0) defined by f(x)=b^x, where b>0, and b ≠ 1.
D. f:R → (∞, 0) defined by f(x)=x^b, where b>0, and b ≠ 1.

3. Let f be a differentiable function at a number (a) for a ∈ Domain f. In the expression ((f(a+h)-f(a))/h) for h ≠ 0 as h gets closer and closer to zero from both direction, the expression becomes closer to f‘(a) which is equal to

A. The gradient of the normal line to the graph of ( f ) at a point ( (a, f(a)) ).
B. The gradient of the vertical line to the graph of ( f ) at a point ( (a, f(a)) ).
C. The gradient of the secants line to the graph of ( f ) at a point ( (a, f(a)) ).
D. The gradient of the tangent line to the graph of ( f ) at a point ( (a, f(a)) ).

4. The following figure, O is the center of the circle and the line segments AB and AC are intersecting the circle at D and E respectively.

If m(∠ACD) = 30° and m(∠BEC) = 50°, then which one of the following is equal to m(∠BAC)?

A. 70°
B. 80°
C. 20°
D. 30°

5. p and q be propositions. Which one of the following propositions is a tautology?

A. (p ⇒ ¬q) ⇔ (p ∧ q)
B. (¬p ⇒ q) ⇒ (p ∧ q)
C. (p ∧ q) ⇒ (p ∨ q)
D. (p ∨ q) ⇒ (p ∧ q)

6. 180 individuals can complete a job in 15 days by working 10 hours a day, how many individuals are needed to complete the job in 10 days if they work 12 hours a day?

A. 200
B. 144
C. 225
D. 220

7. The measure of each interior angle of a regular polygon is 135°, then which one of the following is the number of lines of symmetry of the polygon?

A. 7
B. 6
C. 9
D. 8

8. Consider the following graph of a given exponential function f(x).

Which one of the following functions represents the graph shown above?

A. f(x) = 2^{x-2}
B. f(x) = (¹/₂)^x-2
C. f(x) = (2)^x - 2
D. f(x) = (¹/₂)^x - 2

9. What is the perimeter of a regular hexagon of radius 6cm?

A. 18√3 cm
B. 36 cm
C. 18 cm
D. 36√3 cm

10. Which one of the following is a quantitative data?

A. Measuring the heights of each student in the given class.
B. Determining the tastes of foods in a restaurant.
C. Identifying the social status of individuals.
D. Checking the quality of a new product.

11. Which one of the following is true about the integral of a function f(x) with respect to a variable x?

A. ∫f ’(x) is the set of all anti-derivatives of f. B. ∫f ’(x) is the set of all derivatives of f. C. ∫f (x) is the set of all anti-derivatives of f. D. ∫f(x) is the set of all derivatives of f.

12. One of the following is NOT a valid reason for saving?

A. For betting in football
B. For vacation
C. To maximize intereston
D. For the time of retirement

13. Shaded region in the following figure is the graph of a relation R.

A coordinate plane showing a shaded region R bounded by two lines:

Line 1: x - y + 1 = 0 | Line 2: x + y = 0

The intersection occurs at (-1/2, 1/2). Region R is shaded to the left of this intersection.

Which one of the following is the domain and range of R respectively?

A. (−∞, −1/2] and ℝ )
B. ℝ and (−∞, −1/2)
C. ℝ and (−∞, −1/2)
D. (−∞, −1/2) and ℝ

14. Which one of the following statements is the polynomial division theorem?

A. Let f be a non-constant polynomial with integer coefficients. If the rational number c/d in its lowest terms is a root of f, then c is a factor of the constant term of f and d is a factor of the leading coefficient.
B. If f(x) and d(x) are polynomials such that d(x) ≠ 0 and degree(d(x)) ≤ degree(f(x)), then there exist unique polynomials q(x) and r(x) such that f(x) = d(x)q(x) + r(x), where r(x) = 0 or degree(r(x)) degree(d(x)).
C. If f is a non-constant polynomial and c is a real number, then the remainder when f(x) is divided by x - c is f(c).
D. If f is a non-constant polynomial and c is a real number, then x - c is a factor of f(x) if and only if f(c) = 0.

15. Which one of the following is equal to cot(1755°)?

A. √3
B. -√3
C. 1
D. -1

Mathematics Exam: Part one

Exam Results