Trial lesson

HIGH SCHOOL PRE-CALCULUS QUIZ CONTEST 2019
SAMPLE QUESTIONS

(1) (a) The quadrant in which the point with polar coordinates

�
−2, −3π

4

�
lies is

. (Answer: First)

(b) The polar coordinates (r, θ) of the point with polar co-
ordinates
�
−1, π

3

�
where r > 0 and 0 ≤ θ ≤ 2π are

. (Answer:
�
1,4π

3

�
)

(2) (a) The center of the circle (x + 1)2+ (y − 2)2= 5 lies in the

quadrant. (Answer: Second)
(b) The radius of the circle centered at (1, −3) and touching
the x-axis is

. (Answer: 3)
(3) If P(t) denotes the point on the unit circle with coordinates

� 5

13, −12

13

�
then what are the coordinates of the point on the

unit circle that corresponds to P(π−t)? (Answer:
�
−5

13, −12

13

�
)

(4) (a) What is the domain of the function

f(x) =
3
�

5 − |x − 2|
?

(Answer: (−3, 7))
(b) What is the range of the quadratic function

f(x) = −(x + 3)2− 2?

(Answer: (−∞, −2])
(5) (a) The number of points in which the graph of y = 3−|x − 1|
meets the x-axis is

. (Answer: 2)
(b) The number of points in which the graph of the func-

tion f(x) =
�
x2+ x − 2, x ≤ −2
x − x2, x > −2
meets the x-axis is

. (Answer: 3)
(c) The number of points in which the graph of the func-
tion f(x) = −3x+ 1 meets the x-axis is

.
(Answer: 1)
1

2 HIGH SCHOOL PRE-CALCULUS QUIZ CONTEST 2019 SAMPLE QUESTIONS

(d) The number of points in which the graph of the function
f(x) = sin 2x, 0 ≤ x ≤ 2π meets the x-axis is

.
(Answer: 5)
(6) (a) Every polynomial function of degree n has n real zeros,
some of which may be repeated (True or False). (Answer:
False)
(b) The equation sin2x − sin x − 6 = 0 has no solutions (True
or False). (Answer: True)
(7) (a) If f(x) =√x and (f ◦ g)(x) =
√

x2+ 1 then (g ◦ f)(x) =

. (Answer: x + 1)
(b) If g(x) = sin x and (f ◦ g)(x) = esin xthen (g ◦ f)(x) =

. (Answer: sin ex)
(8) What is the radius of the circle given by the equation

x2+ y2− 6x = 0?

(Answer: 3)

(9) If 2x= 1 then what is the value of 23x−1? (Answer:1

2)

(10) If f is an odd function such that f(−2) = 1 and g is an even
function such that g(1) = 2 then g(f(2)) =

.
(Answer: 2)

(11) What is the value of sin(arccos(−5

13))? (Answer: 12

13)

(12) (a) The function f(x) = |x − 3| is an even function (True or
False). (Answer: False)
(b) The function f(x) = x cos x+sin x is an odd function (True
or False). (Answer: True)
(13) (a) A polynomial with real coefficients has zeros i and 1−i, and
1. What is the smallest possible degree of the polynomial?
(Answer: 5)
(b) A polynomial has zeros i and 1 − i, and 1. What is the
smallest possible degree of the polynomial? (Answer: 3)

(14) (a) Iflog a

log b= 4 then what is the value of logb2 a3? (Answer:

6)
(b) For any positive real number x, logx2√x =

.

(Answer:1

4)

(15) If x− 1 is a factor of the polynomial x3+k2x2−kx− 3 find the
values of k? (Answer: −1 and 2)
(16) If 210+ 45= 2xthen what is the value of x? (Answer: 11)

HIGH SCHOOL PRE-CALCULUS QUIZ CONTEST 2019 SAMPLE QUESTIONS 3

(17) What is the period of the function f(x) = tan 5x. (Answer:
π
5)

(18) (a) tan 48◦tan 138◦=

. (Answer: −1)
(b) For any x such that −1 ≤ x ≤ 1, the value of cos(sin−1x+
cos−1x) is

. (Answer: 0)
(19) If P(x) = (x2+ x − 2)(x2− 4)(x + 2) what is the multiplicity
of the zero −2? (Answer: 3)
(20) What is the maximum number of positive zeros of the polyno-
mial x7− 13x6− 6x5− 7x4+ 11x3+ 3x2− 6x − 5. (Answer:
3)
(21) If x �= 1 and x3= 1 then what is the value of x+x2? (Answer:
−1)
(22) What is the value of sin2 3π

8+ sin2 π

8? (Answer: 1)
(23) The value of eln 3+3 ln 2is

. (Answer: 24)
(24) (a) If one of sin t and cos t is positive and the other negative
then the possible values of t satisfy
(i) 0 < t < π

2.

(ii) 0 < t < π.
(iii)π

2< t < π.

(iv) π < t <3π

2.

(v)π

2< t < 3π

2.

(vi)3π

2< t < 2π.

(Answer: (iii) and (vi))
(b) If 0 ≤ t ≤ 2π and tan t is negative then what is the sign of
sin 2t? (Answer: Negative)
(25) A quadratic equation has integer coefficients and leading coef-
ficient in the equation is 1. If one of the roots of the quadratic
equation is 2 +
√

3 then the constant term in the equation is

. (Answer: 1)