UPCAT Mathematics Reviewer Notes

Compiled UPCAT Questions

This is a compilation of UPCAT review questions by former UP Proctors and Examiners, based on student feedback. It provides an idea of the UPCAT content and approach.

Importance and Purpose:

• The material aims to provide quality resources to build trust.

• The goal is to help students pass the UPCAT and other college entrance tests, encouraging enrollment in the Review Masters UPCAT Review Program.

• It helps students understand potential questions and topics in the UPCAT, promoting preparedness.

• Review Masters claims to have helped approximately 3000 students pass UP in the past 7 years.

How to Use This Material:

• Print the compiled UPCAT Questions reviewer.

• Allocate 125 minutes for the test, simulating actual exam conditions.

• Check answers using the provided answer key.

• Visit the Facebook page or Ka-TOURS group for question clarification.

• Consider the online Review Portal for an automated test version.

Essential Factor for Passing:

The most essential factor to pass UP is having a strong Desire and Determination (DaD).

Mathematics Questions and Solutions

Sample Questions and multiple choices:

$\frac{2}{5} - \frac{1}{3} = ?$

• A. $\frac{1}{15}$

• B. $\frac{3}{2}$

• C. $\frac{1}{2}$

• D. $\frac{8}{15}$

$\frac{3}{1} - \frac{5}{16} = ?$

• A. $\frac{13}{5}$

• B. $\frac{3}{5}$

• C. $\frac{12}{3}$

• D. $\frac{1415}{16}$

$2 - \frac{2}{7} = ?$

• A. $\frac{7}{7}$

• B. $\frac{-7}{2}$

• C. $\frac{-7}{+2}$

• D. $\frac{7+2}{2}$

Simplification and Algebra

Simplify:$\frac{3}{11} \times \frac{2}{9} \times \frac{8}{7} \times \frac{5}{7} \times \frac{4}{5} \times \frac{3}{3} = ?$

• A. $\frac{60}{99}$

• B. $\frac{20}{27}$

If $\frac{3}{20} + \frac{4}{3} + \frac{1}{2} = x$, then $x = ?$

• $x = \frac{41}{2}$

If $0.875 = \frac{?}{32}$, then the missing number is:

• The missing number is 28

If $\frac{5}{24} = \frac{m}{0.625}$, then $m = ?$

• $m = 0.13$

Exponents and Simplification

Simplify: $\frac{x^m}{x^n} = x^{m-n}$

Simplify: $\frac{x^y}{x} = x^{y-1}$

Simplify: $\frac{m^3}{m^{-2}} = m^{3-(-2)} = m^5$

Simplify: $\frac{n^1}{n^{-3}} = n^{1-(-3)} = n^4$

Simplify: $\frac{y^4}{y} = y^{4-1} = y^3$

Simplify: $\frac{y^3}{y^2} = y^{3-2} = y$

Simplify: $\frac{y}{y} = 1$

Simplify: $\frac{y}{\frac{1}{2}} = y \cdot 2 = 2y$

Simplify: $\left(\frac{2}{3}\right)^2 = \frac{2^2}{3^2} = \frac{4}{9}$

Simplify: $\left(\frac{2}{3}\right)^3 = \frac{2^3}{3^3} = \frac{8}{27}$

Simplify: $(2^3)(2^1) = 2^{3+1} = 2^4 = 16$

Simplify: $(4^2)(4^1) = 4^{2+1} = 4^3 = 64$

Simplify: $(125)^{\frac{2}{3}} = (5^3)^{\frac{2}{3}} = 5^{3 \times \frac{2}{3}} = 5^2 = 25$

Simplify: $(625)^{\frac{1}{4}} = (5^4)^{\frac{1}{4}} = 5^{4 \times \frac{1}{4}} = 5^1 = 5$

Factoring and Algebraic Manipulation

Factor $a^2 - b^2 = (a+b)(a-b)$

Factor $c^2 - \frac{1}{2} = (c+\frac{1}{\sqrt{2}})(c-\frac{1}{\sqrt{2}})$

Simplify $\frac{a}{b} - \frac{a}{c} = \frac{ac - ab}{bc}$

Geometry and Word Problems

Triangle Vertices: (0, 0), (3, 4), and (6, 0)

• This triangle is an isosceles triangle.

Telephone Pole: 9 meters tall, wire extends to a point 12 meters from the base.

• Length of the wire: 15 meters (Pythagorean theorem).

Wheel Diameter: 6 cm, rolls a distance of $108\pi$ cm

• Number of complete revolutions: 18.

Triangle Angles Ratio: 2:3:4

• The degree measure of the smallest angle: 40.

Large Box vs. Smaller Box: Length, width, and height are thrice those of a smaller box.

• Volume of the large box: 27 times that of the smaller box.

Line Segment: One endpoint at -12, midpoint at -4

• The other endpoint is at 4.

Rectangle Rug: Width is 4 feet, perimeter is 20 feet

• The length of the rug: 6 feet.

If AC = 18, BE = 13, DE = 3 and D is the midpoint of CE, what is AB?

• AB = 11

Advanced Algebra and Problem Solving

If $x = y + 4$ and $x = 20 - y$, then what is the value of $x^2 - y^2$?

• $x^2 - y^2 = 80$

Sum of squares of two numbers is 225, and the square of their sum is 441. What is the reciprocal of their product?

• The reciprocal of their product is $\frac{1}{108}$

If $\log<em>{10}x + \log</em>{10}8 = \log_{10}16$, then x is equal to:

• $x = 2$

The solution set of $3y + 2x \geq 2$ does NOT include points in quadrant:

• Quadrant III

Find the fourth term of the arithmetic progression x, y, 2y-x, …

• The fourth term is $3y-2x$

For any real number x, $(x-3)^5 – (x - 2)(x-3)^4 =$

• $-(x-3)^4$

Which of the following has the same value as $i^62$?

• -1

The product of $(-x+1)$ and $(-x^{m-1})$ is:

• $x^{2m}$

If y is an integer and y + 3 > 0, what is the least possible value of y?

• -2

Miscellaneous Problems

If $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 0$, find the value of $xy + yz + zx$.

Compiled UPCAT Questions

This is a compilation of UPCAT review questions by former UP Proctors and Examiners, based on student feedback. It aims to provide insights into the UPCAT content, format, and difficulty level. By understanding the types of questions asked and the areas emphasized, students can better prepare for the exam.

Importance and Purpose:

• The material aims to provide quality, reliable resources to build trust among students and educators.

• The primary goal is to assist students in passing the UPCAT and other college entrance tests, thereby encouraging enrollment in the Review Masters UPCAT Review Program. This highlights the program's effectiveness and value proposition.

• It helps students understand potential questions and topics covered in the UPCAT, promoting preparedness and reducing anxiety.

• Review Masters claims to have helped approximately 3000 students pass UP in the past 7 years, showcasing their track record and expertise in UPCAT preparation.

How to Use This Material:

• Print the compiled UPCAT Questions reviewer for ease of access and focused study sessions.

• Allocate 125 minutes for the test, simulating actual exam conditions to build stamina and time management skills.

• Check answers using the provided answer key to identify areas of strength and weakness.

• Visit the Facebook page or Ka-TOURS group for question clarification, fostering a collaborative learning environment.

• Consider the online Review Portal for an automated test version, providing a digital and interactive learning experience.

Essential Factor for Passing:

The most essential factor to pass UP is having a strong Desire and Determination (DaD). This underscores the importance of motivation and perseverance in achieving academic success.

Mathematics Questions and Solutions

Sample Questions and multiple choices:

$\frac{2}{5} - \frac{1}{3} = ?$

• A. $\frac{1}{15}$

• B. $\frac{3}{2}$

• C. $\frac{1}{2}$

• D. $\frac{8}{15}$

$\frac{3}{1} - \frac{5}{16} = ?$

• A. $\frac{13}{5}$

• B. $\frac{3}{5}$

• C. $\frac{12}{3}$

• D. $\frac{1415}{16}$

$2 - \frac{2}{7} = ?$

• A. $\frac{7}{7}$

• B. $\frac{-7}{2}$

• C. $\frac{-7}{+2}$

• D. $\frac{7+2}{2}$

Simplification and Algebra

Simplify:$\frac{3}{11} \times \frac{2}{9} \times \frac{8}{7} \times \frac{5}{7} \times \frac{4}{5} \times \frac{3}{3} = ?$

• A. $\frac{60}{99}$

• B. $\frac{20}{27}$

If $\frac{3}{20} + \frac{4}{3} + \frac{1}{2} = x$, then $x = ?$

• $x = \frac{41}{2}$

If $0.875 = \frac{?}{32}$, then the missing number is:

• The missing number is 28

If $\frac{5}{24} = \frac{m}{0.625}$, then $m = ?$

• $m = 0.13$

Exponents and Simplification

Simplify: $\frac{x^m}{x^n} = x^{m-n}$

Simplify: $\frac{x^y}{x} = x^{y-1}$

Simplify: $\frac{m^3}{m^{-2}} = m^{3-(-2)} = m^5$

Simplify: $\frac{n^1}{n^{-3}} = n^{1-(-3)} = n^4$

Simplify: $\frac{y^4}{y} = y^{4-1} = y^3$

Simplify: $\frac{y^3}{y^2} = y^{3-2} = y$

Simplify: $\frac{y}{y} = 1$

Simplify: $\frac{y}{\frac{1}{2}} = y \cdot 2 = 2y$

Simplify: $\left(\frac{2}{3}\right)^2 = \frac{2^2}{3^2} = \frac{4}{9}$

Simplify: $\left(\frac{2}{3}\right)^3 = \frac{2^3}{3^3} = \frac{8}{27}$

Simplify: $(2^3)(2^1) = 2^{3+1} = 2^4 = 16$

Simplify: $(4^2)(4^1) = 4^{2+1} = 4^3 = 64$

Simplify: $(125)^{\frac{2}{3}} = (5^3)^{\frac{2}{3}} = 5^{3 \times \frac{2}{3}} = 5^2 = 25$

Simplify: $(625)^{\frac{1}{4}} = (5^4)^{\frac{1}{4}} = 5^{4 \times \frac{1}{4}} = 5^1 = 5$

Factoring and Algebraic Manipulation

Factor $a^2 - b^2 = (a+b)(a-b)$

Factor $c^2 - \frac{1}{2} = (c+\frac{1}{\sqrt{2}})(c-\frac{1}{\sqrt{2}})$

Simplify $\frac{a}{b} - \frac{a}{c} = \frac{ac - ab}{bc}$

Geometry and Word Problems

Triangle Vertices: (0, 0), (3, 4), and (6, 0)

• This triangle is an isosceles triangle. Students should be familiar with the properties of isosceles triangles, such as having two equal sides and two equal angles.

Telephone Pole: 9 meters tall, wire extends to a point 12 meters from the base.

• Length of the wire: 15 meters (Pythagorean theorem). This problem reinforces the application of the Pythagorean theorem in real-world scenarios.

Wheel Diameter: 6 cm, rolls a distance of $108\pi$ cm

• Number of complete revolutions: 18. This question tests the understanding of circumference and its relation to the distance traveled by a rolling wheel.

Triangle Angles Ratio: 2:3:4

• The degree measure of the smallest angle: 40. This problem assesses the ability to work with ratios and apply them to geometric figures.

Large Box vs. Smaller Box: Length, width, and height are thrice those of a smaller box.

• Volume of the large box: 27 times that of the smaller box. This question explores the concept of volume scaling in three dimensions.

Line Segment: One endpoint at -12, midpoint at -4

• The other endpoint is at 4. This problem tests the understanding of midpoints and their properties on a number line.

Rectangle Rug: Width is 4 feet, perimeter is 20 feet

• The length of the rug: 6 feet. This question assesses the ability to apply the formula for the perimeter of a rectangle to find the length.

If AC = 18, BE = 13, DE = 3 and D is the midpoint of CE, what is AB?

• AB = 11

Advanced Algebra and Problem Solving

If $x = y + 4$ and $x = 20 - y$, then what is the value of $x^2 - y^2$?

• $x^2 - y^2 = 80$ This problem requires solving a system of linear equations and then evaluating an algebraic expression.

Sum of squares of two numbers is 225, and the square of their sum is 441. What is the reciprocal of their product?

• The reciprocal of their product is $\frac{1}{108}$ This challenging problem involves algebraic manipulation and problem-solving skills.

If $\log<em>{10}x + \log</em>{10}8 = \log_{10}16$, then x is equal to:

• $x = 2$ This question tests the understanding of logarithmic properties and equation solving.

The solution set of $3y + 2x \geq 2$ does NOT include points in quadrant:

• Quadrant III. This problem assesses the ability to interpret inequalities and their graphical representation on the Cartesian plane.

Find the fourth term of the arithmetic progression x, y, 2y-x, …

• The fourth term is $3y-2x$ This problem requires recognizing patterns in arithmetic progressions and finding subsequent terms.

For any real number x, $(x-3)^5 – (x - 2)(x-3)^4 =$

• $-(x-3)^4$ This problem tests algebraic simplification skills and the ability to factor expressions.

Which of the following has the same value as $i^62$?

• -1. This question tests the understanding of imaginary numbers and their properties.

The product of $(-x+1)$ and $(-x^{m-1})$ is:

• $x^{2m}$ This problem involves exponent rules and algebraic manipulation.

If y is an integer and y + 3 > 0, what is the least possible value of y?

• -2. This question assesses the