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JLSS Sample Test: Polytechnic University of the Philippines
(1.) Find the length of the radius of the circle: $x^2 + y^2 + 4x - 6y - 3 = 0$.
In Figure 1, the x-intercepts of a parabola are 2 and 4, and the y-intercept is 8.
If the parabola passes through the point (a, 8), what is the value of a?
The sum of the first n natural numbers is: $\text{Sum} = \dfrac{n(n + 1)}{2} \\[5ex]$
Therefore, the sum of the first x natural numbers is:
$
\text{Sum} = \dfrac{x(x + 1)}{2} \\[5ex]
\dfrac{x(x + 1)}{2} = 171 \\[5ex]
x(x + 1) = 2(171) \\[3ex]
x^2 + x - 342 = 0 \\[3ex]
(x + 19)(x - 18) = 0 \\[3ex]
x + 19 = 0 \hspace{2em}\text{OR}\hspace{2em} x - 18 = 0 \\[3ex]
x = -19 \hspace{2em}\text{OR}\hspace{2em} x = 18 \\[3ex]
\text{Because the sum is a positive number; } x = 18 \\[3ex]
$
Student: Mr. C Teacher: Yes dear Student Student: What if I did not memorize the sum of the first n natural numbers?
Or what if I do not know the formula? Teacher: The sequence is an Arithmetic Sequence.
So, we can derive the formula for the sequence based on the formula for the Sum of Arithmetic
Sequence.
Let's do it.
$
1 + 2 + 3 + 4 + 5 + ... + x \\[3ex]
\text{first term, } a = 1 \\[3ex]
\text{common difference, } d = 2 - 1 = 1 \\[3ex]
\text{number of terms, } n = x \\[3ex]
\text{sum of n terms of an Arithmetic Sequence, } SAS_n = SAS_x \\[3ex]
\implies \\[3ex]
SAS_x = \dfrac{x}{2}[2a + d(x - 1)] \\[5ex]
= \dfrac{x}{2}[2(1) + 1(x - 1)] \\[5ex]
= \dfrac{x}{2}(2 + x - 1) \\[5ex]
= \dfrac{x}{2}(x + 1) \\[5ex]
= \dfrac{x(x + 1)}{2}
$
(4.) Let a and b be the roots of the quadratic equation:
$(x - 2)(x - 3) + (x - 3)(x + 1) + (x + 1)(x - 2) = 0 \\[3ex]$
Evaluate
$
\dfrac{1}{(a + 1)(b + 1)} + \dfrac{1}{(a - 2)(b - 2)} + \dfrac{1}{(a - 3)(b - 3)} \\[6ex]
a.\;\; \dfrac{3}{7} \\[5ex]
b.\;\; \dfrac{4}{15} \\[5ex]
c.\;\; 0 \\[3ex]
d.\;\; \dfrac{1}{4} \\[5ex]
$
Let α, β be the roots of the equation
(replacing a and b respectively in the question).
(5.) Which of the following relations are functions?
$
I.\;\; x^2 + y^2 = 4 \\[3ex]
II.\;\; x - y = 4x \\[3ex]
III.\;\; y = \sqrt{4} - x \\[3ex]
IV.\;\; x = y^2 - y - 6 \\[3ex]
V.\;\; y = x^2 + 5x + 6 \\[3ex]
VI.\;\; x^2 = y^2 \\[5ex]
a.\;\; II, III, IV \\[3ex]
b.\;\; II, III, V \\[3ex]
c.\;\; I, IV, VI \\[3ex]
d.\;\; I, II, III \\[3ex]
$
$
I.\;\; x^2 + y^2 = 4 \\[3ex]
x^2 = 4 - y^2 \\[3ex]
x = \pm\sqrt{4 - y^2}...\text{This is NOT a function} \\[5ex]
II.\;\; x - y = 4x \\[3ex]
y = x - 4x \\[3ex]
y = -3x ...\text{This is a linear function} \\[5ex]
III.\;\; y = \sqrt{4} - x...\text{This is a linear function} \\[5ex]
IV.\;\; x = y^2 - y - 6 \\[3ex]
y^2 = x + y + 6 \\[3ex]
y = \pm \sqrt{x + y + 6} ...\text{This is NOT a function} \\[5ex]
V.\;\; y = x^2 + 5x + 6 ...\text{This is a quadratic function} \\[5ex]
VI.\;\; x^2 = y^2 \\[3ex]
y = \pm\sqrt{x^2} ...\text{This is NOT a function} \\[5ex]
The functions are: b.\;\; II, III, V
$
(6.) A jar contains 12 blue marbles, 8 red marbles, and 20 green marbles.
You pick one marble from the jar.
Find the theoretical probability, P(blue or green)
(8.) You blindly guessed the last five multiple-choice items (4 choices) of a practice test.
What is the probability that you will answer all five questions correctly?
$
n(S) = 4 ...\text{4 choices} \\[3ex]
n(E) = 1 ...\text{1 correct option} \\[3ex]
P(E) = \dfrac{n(E)}{n(S)} = \dfrac{1}{4} ...\text{probability of answering one question correctly} \\[5ex]
\text{The probability of answering five questions correctly} \\[3ex]
= \left(\dfrac{1}{4}\right)^5...\text{Multiplication Rule for Independent Events} \\[5ex]
= \dfrac{1}{1024}
$
(9.) Write the equation of the parabola with vertex (9, −2) and the point (12, 16) in vertex form.
$
a.\;\; y = (x + 9)^2 + 2 \\[3ex]
b.\;\; y = (x - 12)^2 - 16 \\[3ex]
c.\;\; y = 2(x - 9)^2 - 2 \\[3ex]
d.\;\; y = 3(x + 12)^2 + 16 \\[3ex]
$
$
y = a(x - h)^2 + k ...\text{Vertex Form of a Quadratoc Function} \\[3ex]
Vertex = (h, k) = (9, -2) \\[3ex]
h = 9 \\[3ex]
k = -2 \\[3ex]
Point = (x, y) = (12, 16) \\[3ex]
x = 12 \\[3ex]
y = 16 \\[3ex]
\text{Find the leading coefficient, }a \\[3ex]
16 = a(12 - 9)^2 + -2 \\[3ex]
16 = 9a - 2 \\[3ex]
9a = 16 + 2 \\[3ex]
9a = 18 \\[3ex]
a = \dfrac{18}{9} \\[5ex]
a = 2 \\[3ex]
\implies \\[3ex]
y = 2(x - 9)^2 + -2 \\[3ex]
y = 2(x - 9)^2 - 2
$
(10.) A circle with the equation $(x - 2)^2 + (y - 2)^2 = 4$ is on the xy-coordinate plane.
What is the point on the circle opposite the point (0, 2), which is also on the circle?
$
(x - 2)^2 + (y - 2)^2 = 4 \\[3ex]
\text{Compare to the Standard Form of the equation of a circle: } (x - h)^2 + (y - k)^2 = r^2 \\[3ex]
\text{where} \\[3ex]
(h, k) = \text{center of the circle} = (2, 2) \\[3ex]
h = 2 \\[3ex]
k = 2 \\[3ex]
r = \text{radius of the circle} = 2 \\[3ex]
\text{End point 1} = (x_1 - h, k) = (0, 2)...\text{Given} \\[3ex]
(x_1 - 2, 2) = (0, 2) \\[3ex]
x_1 - 2 = 0 \\[3ex]
x_1 = 0 + 2 \\[3ex]
x_1 = 2 \\[3ex]
\text{End point 2} = (x_1 + h, k) \\[3ex]
= (2 + 2, 2) \\[3ex]
= (4, 2)
$
(11.) Which equation represents a hyperbola?
$
a.\;\; y = 16x^2 \\[3ex]
b.\;\; y = 16 - x^2 \\[3ex]
c.\;\; y^2 = 16 - x^2 \\[3ex]
d.\;\; y = \dfrac{16}{x} \\[5ex]
$
$
a.\;\; y = 16x^2...\text{This is a vertical parabola} \\[5ex]
b.\;\; y = 16 - x^2 ...\text{This is a vertical parabola} \\[5ex]
c.\;\; y^2 = 16 - x^2 \\[3ex]
x^2 + y^2 = 16 ...\text{This is a circle} \\[5ex]
d.\;\; y = \dfrac{16}{x} \\[5ex]
xy = 16 ...\text{This is a hyperbola whose axis are the asymptotes.}
$
(12.) Find the equation of the line that passes through the point (−2, −5) and is parallel to the
line $5x - 4y = 2$.
Number of combinations = 3 * 4 * 5 ...Fundamental Counting Principle
= 60 combinations.
(18.) Myra, Mica, Mike, Mikki, and Marla are sitting at a round table when they thought of exchanging seats
and counting all the possible arrangements they can do.
How many arrangements would they count in the end?
Number of permutations around a circular object (round table) refers to the number of distinct ways people
can be arranged when seated around a circular table, where rotations of the same arrangement are considered
identical.
The y is squared, hence this is a horizontal parabola.
The general form of a horizontal parabola is: $(y - k)^2 = 4p(x - h)$ where: p the distance from the vertex to the focus and the distance from the vertex to the directrix.
(h, k) is the vertex
3-digit numbers to be formed from the digits 0, 1, 2, and 3 Restriction:
(1.) No repetition: each digit can be used only once
Given the fact that 012 is the same as 12, we can add another condition:
(2.) The number must not begin with 0
Any of the 3 digits (1, 2, 3) can be the 1st digit: 3 ways
Any of the remaining 2 digits and 0 (3 − 1 + 1 = 3 digits) can be the 2nd digit: 3 ways
Any of the remaining 2 digits (4 − 2 = 2) can be the 3rd digit: 2 ways
$
1, 5, 9, 13, ..., 45 \\[3ex]
\text{first term, } a = 1 \\[3ex]
\text{common difference, } d = 5 - 1 = 4 \\[3ex]
\text{last term = nth term, } p = 45 \\[3ex]
\text{number of terms, } n = ? \\[3ex]
\text{sum of the arithmetic series, } SAS = ? \\[5ex]
p = a + d(n - 1) \\[3ex]
45 = 1 + 4(n - 1) \\[3ex]
4(n - 1) = 45 - 1 \\[3ex]
n - 1 = \dfrac{44}{4} \\[5ex]
n = 11 + 1 \\[3ex]
n = 12 \\[5ex]
SAS = \dfrac{n(a + p)}{2} \\[5ex]
SAS = \dfrac{12(1 + 45)}{2} \\[5ex]
= 6(46) \\[3ex]
= 276
$
(25.) In the parabola with the equation $y = x^2 + 2x - 3$, with an axis of symmetry at $x = -1$, what is the
ordered pair that has the same y-value as the ordered pair (−3, 0)?
1st: 5 cards must be drawn from 52 cards
Number of ways of drawing 5 cards from 52 cards = $_{52}C_5$
2nd: 3 of the 5 cards must be aces
In a deck of 52 cards, there are 4 aces
So, we need to select 3 of those 4 aces
Number of ways of drawing 3 aces from 4 aces = $_4C_3$
3rd: The remaining 2 cards must be non-aces
The 2 non-aces cards are selected from 48 (52 − 4) non-aces
Number of ways of drawing 2 non-aces from 48 non-aces = $_{48}C_2$
∴ The probability of getting 3 aces out of a deck of 52 cards if 5 cards are to be drawn at a time
= $\dfrac{_4C_3 * _{48}C_2}{_{52}C_5}$
(30.) What is the derivative of the trigonometric function $\sec\theta$
(33.) Given that $\displaystyle{\lim_{x \to 3+}} f(x)$, what is the $\displaystyle{\lim_{x \to 3+}} f(x)$?
This question is incomplete. If you have the complete question, please contact me.
A parabola is the set of all points equidistant from a fixed point (known as the focus) and a fixed line (
known as the directrix).
This means that the point, say (x, y) is equidistant from the focus and from the directrix.
In other words, the distance from the point to the focus is the distance from the point to the directrix.
$
\text{distance from the point to the focus} \\[3ex]
\text{Point, }(x, y) \hspace{3em} \text{Focus, } (0, p) \\[3ex]
\text{distance, }d = \sqrt{(x - 0)^2 + (y - p)^2} \\[3ex]
d = \sqrt{x^2 + (y - p)^2} \\[5ex]
\text{distance from the point to the directrix} \\[3ex]
\text{Point, }(x, y) \hspace{3em} \text{Directrix, } y = -p \\[3ex]
d = |y - (-p)| \\[3ex]
d = |y + p| \\[5ex]
d = d \implies \\[3ex]
\sqrt{x^2 + (y - p)^2} = |y + p| \\[3ex]
x^2 + [(y - p)(y - p)] = (y + p)^2 \\[3ex]
x^2 + y^2 - py - py + p^2 = (y + p)(y + p) \\[3ex]
x^2 + y^2 + p^2 - 2py = y^2 + py + py + p^2 \\[3ex]
x^2 - 2py = 2py \\[3ex]
x^2 = 2py + 2py \\[3ex]
x^2 = 4py
$
(37.) Given the series 1, 5, 13, 29, ..., what is the 6th term?
(39.) Ms. Anna would be giving awards to 4 out of her students in section Narra.
If each student can receive at most one award and section Narra is composed of 30 students, how many possible
selections are there?
The question did not specify whether the awards are distinct (in which case, it would be Permutation) or
identical (in which case, it would be Combination).
So, we shall assume that the awards are identical.
