Question 01
Easy
What is the domain of $f(x) = \dfrac{1}{\sqrt{x - 3}}$ ?
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Memory Point
SQRT ≥ 0 + DENOM ≠ 0
Square root needs non-negative → denominator can't be zero → combine both!
📖 Quick Example
For $f(x) = \sqrt{x-1}$, we need $x - 1 \geq 0$, so domain is $[1, \infty)$.
For $g(x) = \dfrac{1}{\sqrt{x-1}}$, we need $x - 1 > 0$ (strictly), so domain is $(1, \infty)$.
Question 02
Easy
If $f(x) = 2x + 1$ and $g(x) = x^2$, find $(g \circ f)(3)$.
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Memory Point
g∘f = g(f(x)) — "Outside eats Inside"
Always compute the INNER function first, then plug into OUTER.
📖 Quick Example
$(g \circ f)(x) = g(f(x)) = g(2x+1) = (2x+1)^2$
At $x = 3$: $f(3) = 7$, then $g(7) = 49$.
Question 03
Easy
The graph of $y = f(x)$ is shifted 3 units right and 2 units down. Which is the new equation?
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Memory Point
RIGHT → minus inside UP → plus outside
Horizontal shifts feel "backwards" — right means $f(x - 3)$, not $f(x+3)$!
Question 04
Easy
If $x = 2$ is a zero of $p(x) = x^3 - 3x^2 + ax - 4$, find $a$.
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Memory Point
ZERO → plug in → equals 0
If $x = c$ is a zero, then $p(c) = 0$. Just substitute and solve!
📖 Quick Example
Plug in $x = 2$: $\;8 - 12 + 2a - 4 = 0$
$\Rightarrow 2a - 8 = 0 \Rightarrow a = 4$
Question 05
Easy
When $p(x) = x^3 + 2x - 5$ is divided by $(x - 1)$, the remainder is:
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Memory Point
REMAINDER = p(c) when dividing by $(x - c)$
No long division needed! Just evaluate at $x = c$.
Question 06
Easy
How many real zeros does $f(x) = x^4 + 4x^2 + 4$ have?
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Memory Point
SUBSTITUTE u = x² to simplify even-degree polynomials.
Then check if roots give real $x$ values. Negative $u$ → no real $x$!
📖 Quick Example
Let $u = x^2$: $\;u^2 + 4u + 4 = (u+2)^2 = 0$
$\Rightarrow u = -2 \Rightarrow x^2 = -2$ → No real solutions!
Question 07
Easy
Solve for $x$: $\;2^{x+1} = 32$
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Memory Point
SAME BASE → same exponent
Rewrite both sides with the same base, then set exponents equal.
Question 08
Easy
Simplify: $\;\log_2 8 + \log_2 4$
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Memory Point
log A + log B = log(AB) same base only!
Or just evaluate each: $\log_2 8 = 3$, $\log_2 4 = 2$ → sum = $5$.
Question 09
Easy
Solve: $\;\ln(x - 1) = 0$
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Memory Point
ln(?) = 0 → ? = 1 because $e^0 = 1$
ln and $e^x$ are inverses: apply $e^{\,}$ to both sides to undo ln.
Question 10
Easy
What is $\sin\!\left(\dfrac{5\pi}{6}\right)$ ?
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Memory Point
Reference angle + Quadrant sign
$\frac{5\pi}{6}$ is in Q2 (sin positive). Reference angle = $\pi - \frac{5\pi}{6} = \frac{\pi}{6}$. So $\sin = +\frac{1}{2}$.
Question 11
Easy
Which identity is always true?
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Memory Point
sin²θ + cos²θ = 1 — THE Pythagorean Identity
Memorize this one first. Everything else is derived from it!
A
$\sin\theta + \cos\theta = 1$
B
$\sin^2\theta + \cos^2\theta = 1$
C
$\sin^2\theta \cdot \cos^2\theta = 1$
D
$\tan\theta = \dfrac{\cos\theta}{\sin\theta}$
Question 12
Easy
Solve on $[0, 2\pi)$: $\;\cos\theta = -\dfrac{\sqrt{3}}{2}$
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Memory Point
Negative cos → Q2 and Q3
Reference angle for $\frac{\sqrt{3}}{2}$ is $\frac{\pi}{6}$. In Q2: $\pi - \frac{\pi}{6}$. In Q3: $\pi + \frac{\pi}{6}$.
A
$\theta = \dfrac{\pi}{6},\; \dfrac{5\pi}{6}$
B
$\theta = \dfrac{\pi}{3},\; \dfrac{2\pi}{3}$
C
$\theta = \dfrac{5\pi}{6},\; \dfrac{7\pi}{6}$
D
$\theta = \dfrac{2\pi}{3},\; \dfrac{4\pi}{3}$
Question 13
Easy
The 10th term of the arithmetic sequence $3, 7, 11, \ldots$ is:
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Memory Point
a_n = a₁ + (n−1)d
Find common difference $d$ first. Here $d = 7 - 3 = 4$.
Question 14
Easy
A geometric sequence has $a_1 = 5$ and ratio $r = 2$. Find $S_4$ (sum of first 4 terms).
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Memory Point
S_n = a₁(1 − rⁿ) / (1 − r) when $r \neq 1$
Or just add: $5 + 10 + 20 + 40 = 75$. Quick addition often faster!
Question 15
Easy
The center and radius of $x^2 + y^2 - 4x + 6y - 3 = 0$?
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Memory Point
Complete the square for both $x$ and $y$
Take half the coefficient, square it, add to both sides. Do it for $x$ then $y$ separately.
📖 Quick Example
$(x^2 - 4x + 4) + (y^2 + 6y + 9) = 3 + 4 + 9 = 16$
$(x-2)^2 + (y+3)^2 = 16$ → Center $(2, -3)$, radius $= 4$
A
Center $(2, 3)$, radius $4$
B
Center $(-2, 3)$, radius $4$
C
Center $(2, -3)$, radius $3$
D
Center $(2, -3)$, radius $4$
Question 16
Easy
What is the vertex of $y = -2(x - 3)^2 + 5$?
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Memory Point
Vertex form: y = a(x − h)² + k → vertex (h, k)
Watch the sign! $x - 3$ means $h = +3$, not $-3$.
Question 17
Easy
Evaluate: $\;\displaystyle\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$
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Memory Point
FACTOR → CANCEL → PLUG IN
When direct substitution gives $\frac{0}{0}$, always try factoring first!
📖 Quick Example
$\dfrac{x^2-4}{x-2} = \dfrac{(x-2)(x+2)}{x-2} = x + 2$
As $x \to 2$: $\;x + 2 \to 4$
Question 18
Easy
If $\vec{u} = \langle 3, -1 \rangle$ and $\vec{v} = \langle 2, 4 \rangle$, compute $\vec{u} \cdot \vec{v}$.
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Memory Point
DOT PRODUCT = multiply matching components, then ADD
$\langle a, b \rangle \cdot \langle c, d \rangle = ac + bd$. Result is a NUMBER, not a vector!
C
$\langle 6, -4 \rangle$
Question 19
Easy
Which statement is true for the function $f(x) = \dfrac{x^2 - 1}{x - 1}$ at $x = 1$?
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Memory Point
LIMIT ≠ VALUE — a limit can exist even if $f(c)$ doesn't!
Check: Is $f(1)$ defined? What does the limit approach? These are separate questions.
A
$f(1) = 2$ and the function is continuous at $x = 1$
B
$\lim_{x \to 1} f(x)$ does not exist
C
$\lim_{x \to 1} f(x) = 2$ but $f(1)$ is undefined
Question 20
Easy
What is the sum of the infinite geometric series $\displaystyle\sum_{n=0}^{\infty} \left(\frac{1}{3}\right)^n$ ?
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Memory Point
S∞ = a₁ / (1 − r) only if $|r| < 1$
If $|r| \geq 1$, the series DIVERGES (no finite sum). Always check $|r|$ first!
📖 Quick Example
Here $a_1 = 1$ (when $n = 0$) and $r = \frac{1}{3}$.
$S_\infty = \dfrac{1}{1 - \frac{1}{3}} = \dfrac{1}{\frac{2}{3}} = \dfrac{3}{2}$