Share
Comprehensive Mathematics Exam Solutions for Trigonometry, Algebra, and Calculus
Published on September 11, 2026
Algebra
Step-by-step solutions to a variety of mathematical problems including trigonometric equations, joint variation, quadratic equations, sequence formulas, and proofs.
Question
solve the attached questions
Answer
QUESTION 1
a) Solve
- Let
. The equation becomes . - Factor the quadratic:
. - Solve for
: or . - For
: . - For
: and .
Answer:.
b) Joint variation:
(i) Find
(ii) Find
(iii) Find
QUESTION 2
a) Solve
b) Solve
Rearrange:
Using the quadratic formula
Rearrange:
Using the quadratic formula
c) Solve
Since cosine is even,
Answer:
Since cosine is even,
Answer:
QUESTION 3
a) Solve
b) Arithmetic sequence:
The general formula is
(To find the total number of terms
The general formula is
(To find the total number of terms
QUESTION 4
a) Prove
Using
Using
b) Prove
This is the fundamental Pythagorean identity derived from a right triangle where
This is the fundamental Pythagorean identity derived from a right triangle where
QUESTION 5
a)
Equation:
Equation:
b) Sum of infinite progression
This is a geometric series with
Sum
This is a geometric series with
Sum