Algebra Class Quiz on Sequences and Series

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Questions and Answers

Given the binomial expansion, what is the coefficient of the term T5?

1120 (correct)

In the given expression for T5, what is the exponent of 'x' after simplification?

4 (correct)

Which statement about the sequence 2, 6, 18, 54, 160,... is correct?

It is an arithmetic sequence.

It is not a geometric sequence because the last term is incorrect. (correct)

It is a geometric sequence with a common ratio of 4.

It is a geometric sequence with a common ratio of 3.

What is the common difference in the sequence x, 2x + y, 3x + 2y, ...?

x + y Signup and view all the answers

Is the sequence x, 2x + y, 3x + 2y, ... an arithmetic progression?

Yes, because the common difference is constant. Signup and view all the answers

In the arithmetic sequence 1, 5, 9, 13, 17, ..., what is the common difference?

4 Signup and view all the answers

The sum of the first 10 terms of the arithmetic sequence 1, 5, 9, 13, 17,... is:

190 Signup and view all the answers

Which of the given statements correctly identifies an arithmetic sequence?

x, 2x + y, 3x + 2y,... Signup and view all the answers

Using the given values, what is the correct sum of the arithmetic sequence?

190 Signup and view all the answers

If the formula for the sum of an arithmetic sequence is $S_n = \frac{n}{2}[2a_1 + (n - 1)d]$, what does $a_1$ represent?

The first term in the sequence Signup and view all the answers

How many distinct arrangements are possible for the letters in the word 'PUPIAN'?

360 ways Signup and view all the answers

What is the key concept used to solve the arrangement problem of the word 'PUPIAN'?

Permutation with repetition Signup and view all the answers

What is the correct formula for simple interest?

$I = PRT$ Signup and view all the answers

If an investment of $70,000 earns $15,000 in interest at a 3.8% annual rate, what is the time it took in years?

5.64 years Signup and view all the answers

In the simple interest formula, if the interest (I) and the time (T) are constant, how does the principal (P) affect the interest rate (R)?

As P increases, R decreases Signup and view all the answers

What is the decimal form of 3.8%?

0.038 Signup and view all the answers

A right triangle has sides of length 9 and 40. What is the area of the triangle?

180 Signup and view all the answers

Jandrik travels 15 km west and then 36 km south. What is the straight-line distance in kilometers from his starting position?

39 Signup and view all the answers

If $a$ and $b$ are real numbers such that $a + b = 15$ and $ab = 45$, what is the value of $a^2 + b^2$?

135 Signup and view all the answers

An isosceles right triangle has a leg length greater or equal to 13 cm. What is the minimum possible area of this triangle?

84.5 sq.cm. Signup and view all the answers

There are 4 distinct Model A cars and 4 distinct Model B cars. If they are arranged in a row such that no two cars of the same model are adjacent, how many possible arrangements are there?

1152 Signup and view all the answers

A line segment is drawn from the center of a polygon perpendicular to one of its sides. What is this line segment called?

Apothem Signup and view all the answers

What is the result of 1/2 * 9 * 40?

180 Signup and view all the answers

Given that $\sqrt{15^2+36^2}= x$, what is value of $x$?

39 Signup and view all the answers

Seph invests a total of Php 500,000 in two businesses, a computer shop and a coffee shop. If the computer shop yields 5% monthly interest and the coffee shop yields 3%, how much should be invested in each to earn Php 18,000 monthly?

Computer shop: Php 150,000, Coffee shop: Php 350,000 Signup and view all the answers

In the given problem, if 'x' represents the amount invested in the computer shop and 'y' represents the amount invested in the coffee shop, which equation represents the total monthly interest earned?

$0.05x + 0.03y = 18,000$ Signup and view all the answers

Based on the problem, what is the first step in solving for the amount invested in each business using the system of linear equations?

Solving for 'y' in terms of 'x' in the first equation. Signup and view all the answers

After substituting the value of 'y' in the second equation, how is the value of 'x' determined?

By dividing both sides by $0.02$. Signup and view all the answers

What does the equation `x + y = 500,000` represent in the context of Seph's investments?

The total amount invested in both businesses. Signup and view all the answers

If Seph had invested Php 100,000 more in the computer shop, what would be the new value of x, given that the total investment remains at Php 500,000?

Php 250,000 Signup and view all the answers

In the step `0.05x + 15,000 - 0.03 = 18,000`, how is the value 15,000 derived?

By multiplying 0.03 by 500,000. Signup and view all the answers

What is the purpose of using the subtraction property of equality in the solution?

To isolate y by subtracting x from both sides of the equation. Signup and view all the answers

Which of the following sets of numbers does NOT represent a Pythagorean triple?

7-21-22 Signup and view all the answers

Given a right triangle in the second quadrant where sin θ = $\frac{7}{25}$, what is the value of sec θ?

-$\frac{25}{24}$ Signup and view all the answers

A piggy bank contains 35 coins in total, consisting of 10-centavo and 25-centavo coins. If the total value of the coins is P5.90, how many 10-centavo coins are in the piggy bank?

19 Signup and view all the answers

If a right triangle has a sine value of $\frac{7}{25}$ and is located in the second quadrant, what is the value of tan θ?

$-\frac{7}{24}$ Signup and view all the answers

Given a right triangle with sin θ = $\frac{7}{25}$, what is the length of the side adjacent to angle θ?

24 Signup and view all the answers

If a piggy bank has P5.90 consisting of 10 and 25-centavo coins, and the total number of coins is 35, what is the total value of the 25-centavo coins?

P 4.00 Signup and view all the answers

What is the value of cos θ if sin θ = $\frac{7}{25}$ and the triangle is in the second quadrant?

$-\frac{24}{25}$ Signup and view all the answers

In the context of Pythagorean triples, if 'a' is an odd number, how are 'b' and 'c' typically related?

b and c are consecutive numbers Signup and view all the answers

What is the area of a triangle with side lengths 13, 14, and 15?

84 Signup and view all the answers

A triangle has an area of 84, with a semi-perimeter of 21. What is the radius of the inscribed circle?

4 Signup and view all the answers

If $z = 2 + 3i$, what is the magnitude of $z^2$, denoted as $|z^2|$?

13 Signup and view all the answers

How many diagonals does a regular icosagon have?

170 Signup and view all the answers

Given a random variable X following a uniform distribution on the interval [0,10], what is the probability that $3 < X < 7$?

0.4 Signup and view all the answers

What is the semi-perimeter of a triangle with sides of length 13, 14, and 15?

21 Signup and view all the answers

What is the value of $i^2$?

-1 Signup and view all the answers

Which of the following formulas is used to calculate the number of diagonals in a polygon?

$D = \frac{n(n-3)}{2}$ Signup and view all the answers

Study Notes

PUPCET 2025 Mock Exam - Mathematics

Mathematics exam for PUP (Philippine University of the Philippines) students
Prepared by the Public Information and Affairs Committee, Academic and Research Committee, and Volunteers
Exam questions cover Systems of Linear Equations (Grade 8) and Arithmetic Sequences.

Question 1 - Investment Decisions

Seph invests Php 500,000 in a computer shop (5% monthly interest) and a coffee shop (3% monthly interest)
He wants a total monthly interest of Php 18,000
Correct answer: Computer shop: Php 150,000, Coffee shop: Php 350,000

Question 2 - Arithmetic Sequence

Fourth term of an arithmetic sequence is 2
Seventeenth term is 7
Determine the general term of the sequence
Correct answer: an = 5n+6 / 13

Question 3 - Binomial Expansion

Find the fifth term in the expansion of (2x + y)⁸
Correct answer: 1120x⁴y⁴

Question 4 - True/False Statements

Statement 1: 2, 6, 18, 54, 160 is a geometric sequence => False
Statement 2: x, 2x + y, 3x + 2y... is an arithmetic sequence => True
Statement 3: 1, 5, 9, 13, 17... The sum of the first 10 terms is 175 => False

Question 5 - Letter Arrangement

Determine the number of ways to arrange the letters in the word PUPIAN
Correct answer: 360 ways

Question 6 - Simple Interest

Mx. Cruz wants to earn Php 15,000 in interest on a PHP 70,000 investment at 3.8% annual interest
Determine the time length needed to reach the goal
Correct answer: 5.64 years

Question 7 - Equation of a Line

Determine the slope-intercept form of a line passing through (5, 4) and (6, 9)
Correct answer: y=5x-21

Question 8 - Compound Interest

Joshua invests Php 14,300 at a 6% interest rate compounded monthly for 4 years
How much money will Joshua have?
Correct answer: Php 18,167.99

Question 9 - Consecutive Odd Integers

Find three consecutive odd integers.
Sum of the smallest and largest: 62. Sum of all three: 93
Correct answer: 29, 31, and 33

Question 10 - Normal Distribution

In normal distribution, the mean, median, and mode are all equal and located at the center of the spread.

Question 11 - Number Pattern

Determine the missing number in the sequence 1, 2, 7, 34, 203, 1420, …
Correct answer: 11359

Question 12 - Divisible Polynomials

Given the polynomial 2x⁴ - 9x³ + mn² + nx + 54 = 0 is divisible by x² + 4x + 3
Find m+n
Correct answer: -23

Question 13 - Factorization

Factorize 4x² - 4y² + 40x - 28y + 51
Correct answer: (2x+2y + 17)(2x -2y+3)

Question 14 - Positive Factors

Find the sum of all positive factors of 756
Correct answer : 2240

Question 15 - Intersection of Lines

Find the point of intersection of line A: y = 4x + 8 and line B: y = 7x + 5
Correct answer: (1,12)

Question 16 - Area

Find the area of a triangle with side lengths 9, 40, and 41
Correct answer: 180

Question 17 - Positional Change

Determine the distance between an original position after going 15 km west then 36 km south
Correct answer : 39

Question 18 - Algebraic Calculation

Given 'a' + 'b' = 15 and ab = 45, find a² + b²
Correct answer: 135

Question 19 - Right Angled Triangle

Given ∆ABC, find the minimum value of its area , where dist(B,C) >= 13cm and ∠ACB= 90°
Correct answer: 84.5 cm2

Question 20 - Arrangement and Models (Permutation)

Determine the arrangements of 4 models of type A and 4 models of type B when they cannot be placed next to each other, lining up in a row.
Correct answer: 1152

Question 21 - Line Segment

Identify the geometric term for a line segment drawn from the center of a polygon perpendicular to a side.
Correct answer: Apothem

Question 22 - Pythagorean Triples

Which is not a Pythagorean Triple?
Correct answer: 7-21-22

Question 23 - Trigonometry in Second Quadrant

Given sin θ = 7/25 in the second quadrant, determine sec θ
Correct answer: -25/24

Question 24 - Coin Values

Determine the number of 10-centavo and 25-centavo coins in a piggy bank.
There are 35 coins with a total value of P5.90
Correct answer: Nineteen 10-centavo coins and sixteen 25-centavo coins

Question 25 - Reciprocal Difference

Find the two positive numbers given their reciprocal difference and one is 3 times the other.
Correct answer: 4 and 12

Question 26 - Age Problem

Heather's age. One-half of her age in two years plus one-third of her age three years ago is 20.
Correct answer : 24

Question 27 - Mixture of Rice

Find the kg of rice that sells for P10 per kilo to mix with rice that sells for P7.50 to make a 100-kilo mixture that sells for P8.50.
Correct answer: 40

Question 28 - Travel Time and Displacement

Ji leaves home for PUP Exam at 12 mph. 20 minutes later her mother brings his test permit. Mother drives at 36 mph.
Correct answer: 6 miles

Question 29 - Probability of Fire Occurrence

Given the probability of dangerous fires and the probability of fires producing smoke.
Calculate the probability that a fire occurs when there is smoke
Correct answer: 0.09

Question 30 - Meal Option Combination

Calculate the possible number of meals offered with specified choices for side item, juice, and pasta.
Correct answer: 32

Question 31 - Polynomial Roots

Find the roots of the polynomial
x³ — 6x² — 11x + 6 = 0
Correct Answer: x = 1, 2, 3.

Question 32 - Logarithmic Equations

Solve for x in the logarithmic equation log₃(x − 1) + log₃(x + 1) = 2
Correct answer: √10

Question 33 - Radius of Inscribed Circle

Find the radius of the circle inscribed in a triangle with sides 13, 14, and 15.
Correct answer: 6

Question 34 - Complex Number

Find the magnitude of z^2 if z = 2 + 3i
Correct answer: 13

Question 35 - Diagonals in a Regular Polygon

Find the number of diagonals in an icosagon
Correct answer: 170

Question 36 - Uniform Distribution

Given a uniform distribution X on [0,10], find P(3 < X < 7)
Correct answer: 0.4

Question 37 - Poisson Distribution

Given X ~ Poisson(λ = 4), find the variance
Correct answer: 4

Question 38 - Standard Deviation Scaling

If the standard deviation of a dataset is 10, and every value is multiplied by 5, what is the new standard deviation?
Correct answer: 50

Question 39 - Probability of Union

Find P(A∪B) given P(A) = 0.4, P(B) = 0.6, P(A∩B) = 0.2
Correct answer: 0.8

Question 40 - Geometric Distribution

In a geometric distribution with p = 0.25, find the probability of the first success occurring on the 4th trial.
Correct answer: 0.140625

Question 41 - Project Completion Time

Sarah completes a project in 20 days and David completes the same project in 15 days. If they work together, how many days will it take to complete the project?
Correct answer: 60/7 days

Question 42 - Rocket Distance

What distance does a rocket traveling at 3 miles per second cover in 1 hour?
Correct answer: 10,800 miles

Question 43 - Irrational Numbers

Which of the following is NOT an irrational number?
Correct answer: 7/3

Question 44- Two-Digit Numbers

If 't' is the tens digit and 'u' is the units digit of a two-digit number, which expression represents the number?
Correct answer: 10t + u

Question 45 - Age Calculation

Jack is four years older than Jill. Three years ago, Jack's age was two years less than twice Jill's age. What is the equation to find Jill's current age?
Correct answer: 2(x - 3) - 2 = x + 1

Question 46 - Approximation

Estimate the value of 0.889 × 55 / 9.97
Correct answer: 4.9

Question 47 - Sphere in Cube

Find the volume of the largest sphere that can fit inside a cube with a volume of 216 cm³.
Correct answer: 36π cm³

Question 48 - Lines Intersection

Determine the point of intersection of Line A (y = 4x + 8) and Line B (y = 7x + 5)
Correct answer: (1,12)

Question 49 - Algebraic Simplification

Simplify the expression √256/(x² - 4) if x = 4
Correct answer: 2

Question 50 - Fraction of Money Spent

Kelly spent half her money on a DVD and 1/6 on popcorn. What fraction of her money did she spend in total?
Correct answer: 2/3

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