Algebra Flashcards
To deepen the understanding of algebraic techniques
How many solutions are there to the equation 3x³ - 9x² + 3x + 9 = 0, x ∈ ℝ?
2 marks
There is one solution.
x² + 1 has no solutions, such thereas their only exists x = - 3
Given the polynomial x³ + 2x² - ax is divisible by (x - a), what is the value of a, a ∈ ℝ{0}?
3 marks
a = - 2
Using synthetic division, a equals polynomial in the form a³ + 2a² = 0
If log(∛x) + log2 - 3 = log(x²), what is the value of x? Evaluate to 3 decimal places.
3 marks
x = 0.024
To combine the x terms, consider index laws.
Given9ˣ - 27× 3ˣ = 0, what is the value of x?
3 marks
x = 3
Use the method involving ‘hidden quadratics’
Let f(x) = x⁻¹. The point X(0.5, 2) lies along f. The graph, f, is reflected in the y-axis, dilated from the x-axis by a factor of 2, and then translated 3 units horizontally left. What is the final image of X?
3 marks
X(-0.25, 4)
Given g(t) = ln(t-4)+2, what is the inverse function? And what is its range and domain?
3 marks
g⁻¹(x) = eˣ ⁻ ² + 4
⇒ Domain {x: x ∈ ℝ}
⇒ Range {y ∈ ℝ: y > 4}
The simultaneous linear equations ax - 3y = 5 and 3x - ay = 8 have no solutions for…
2 marks
a = - 3
Create a pseudocode algorithm which uses bisection method to estimate the solution in the form f(x) = 0
5 marks
Define bisection(f(x), a, b, max)
If f(a) × f(b) > 0
Then Return “Invalid Interval”
i ← 0
While i < max
mid ← (a + b) ÷ 2
If f(mid) = 0
Then Return mid
Else If f(a) × f(mid) < 0
b ← mid
Else
a ← mid
i ← i + 1
EndWhile
Return mid
State the period, amplitude, range, and x-intercepts for the function V(x) = 3π - 4sin(8πx) + 1, x ∈ [0 < x < 1]
4 marks
What is the product of the solutions of the equation cos(3x)y -cos(3x)= sin(3x) over the period [-π, π]?
3 marks
If the trigonometric function f(x) = -asin(ax)+3 goes through 4 periods between -3 and 7, what is the range of f(x)?
3 marks
