SAT Math Practice — Set 2 (Elite / Highest-Difficulty)

1

Module 1 · Nested Radicals

What is the value of $x$ that satisfies $\sqrt{x+\sqrt{x+11}}=3$?

Square both sides: $x+\sqrt{x+11}=9$, so $\sqrt{x+11}=9-x$. Squaring again: $x+11 = 81-18x+x^2$, giving $x^2-19x+70=0$, which factors as $(x-5)(x-14)=0$. Check both: for $x=14$, $9-x=-5<0$, impossible for a square root, so it's extraneous. Only $x=5$ is valid — always check nested-radical solutions against the sign requirement $9-x\ge0$.

2

Module 1 · Rational Equations

Solve for $x$: $\dfrac{x}{x-2}+\dfrac{3}{x+2}=\dfrac{8}{x^2-4}$.

Multiply every term by $(x-2)(x+2)$: $x(x+2)+3(x-2)=8$, giving $x^2+5x-14=0$, which factors as $(x+7)(x-2)=0$. Since $x=2$ is excluded from the domain (it makes denominators zero), the only valid solution is $x=-7$.

3

Module 1 · Logarithmic Equations

Solve for $x$: $\log_2 x + \log_2(x-2) = 3$.

Combine logs: $\log_2[x(x-2)]=3$, so $x(x-2)=2^3=8$, giving $x^2-2x-8=0$, i.e. $(x-4)(x+2)=0$. Since $x-2$ must be positive for the log to be defined, $x=-2$ is rejected; $x=4$ is the valid solution.

4

Module 1 · Ellipses

An ellipse has semi-major axis $a=5$ and semi-minor axis $b=3$. What is the distance between its two foci?

For an ellipse, $c^2=a^2-b^2=25-9=16$, so $c=4$. The distance between the two foci is $2c=8$, not just $c$ — a very common oversight.

5

Module 1 · Piecewise Functions

A function is defined as $f(x)=x^2+1$ for $x\le2$ and $f(x)=kx-3$ for $x>2$. For what value of $k$ is $f$ continuous at $x=2$?

At $x=2$, the first piece gives $f(2)=2^2+1=5$. For continuity, the second piece must also equal 5 as $x\to2$: $2k-3=5$, so $k=4$.

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Module 1 · Circle-Line Intersection

A circle is given by $x^2+y^2=25$ and a line by $y=x+1$. What is the sum of the $y$-coordinates of the two intersection points?

Substitute: $x^2+(x+1)^2=25 \Rightarrow 2x^2+2x-24=0 \Rightarrow x^2+x-12=0 \Rightarrow (x+4)(x-3)=0$. So $x=-4,y=-3$ or $x=3,y=4$. The sum of the $y$-values is $-3+4=1$.

7

Module 1 · Weighted Averages

A class of 20 students has an average score of 85, and another class of 30 students has an average score of 92. What is the combined average score of all 50 students?

Combined average $=\dfrac{20(85)+30(92)}{50}=\dfrac{1700+2760}{50}=\dfrac{4460}{50}=89.2$. Simply averaging $85$ and $92$ to get $88.5$ ignores the different group sizes — a very common weighted-average trap.

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Module 1 · Inverse Functions

If $f(x)=\dfrac{2x+3}{x-1}$, what value of $x$ satisfies $f(x)=5$? (This value equals $f^{-1}(5)$.)

Set $\dfrac{2x+3}{x-1}=5$. Cross-multiplying: $2x+3=5x-5$, so $8=3x$, giving $x=\dfrac{8}{3}$.

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Module 1 · Polynomial Roots

The polynomial $x^3-4x^2+kx-6$ has $x=3$ as a root. What is the value of $k$?

Substitute $x=3$: $27-36+3k-6=0$, so $3k-15=0$, giving $k=5$.

10

Module 1 · Midpoint Formula

A segment has midpoint $(4,-1)$ and one endpoint $(1,3)$. What is the sum of the coordinates of the other endpoint?

Using the midpoint formula: $\dfrac{1+e_x}{2}=4 \Rightarrow e_x=7$; $\dfrac{3+e_y}{2}=-1 \Rightarrow e_y=-5$. The sum of coordinates is $7+(-5)=2$.

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Module 1 · Exponential Decay

A radioactive sample decays according to $N=N_0\left(\frac{1}{2}\right)^{t/h}$, where $h$ is the half-life. If $\dfrac{N}{N_0}=\dfrac{1}{8}$, what is the value of $\dfrac{t}{h}$ (i.e., how many half-lives have elapsed)?

Since $\dfrac{1}{8}=\left(\dfrac{1}{2}\right)^3$, matching exponents gives $\dfrac{t}{h}=3$.

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Module 2 · Trigonometric Identities

What is the simplified value of $\sec^2\theta - \tan^2\theta$ for any angle $\theta$ where both are defined?

This is a direct consequence of the Pythagorean identity $1+\tan^2\theta=\sec^2\theta$, which rearranges to $\sec^2\theta-\tan^2\theta=1$ for all valid $\theta$.

13

Module 2 · Rational Expressions

Simplify: $\dfrac{x^2-9}{x^2-5x+6}$

Factor both: numerator $=(x+3)(x-3)$; denominator $=(x-2)(x-3)$. Canceling the common factor $(x-3)$ (with $x\ne3$) gives $\dfrac{x+3}{x-2}$.

14

Module 2 · Systems with Parameters

For what value of $k$ does the system $kx+3y=5$, $4x+6y=12$ have no solution?

The lines are parallel (no solution) when the coefficient ratios match but the constants don't: $\dfrac{k}{4}=\dfrac{3}{6}=\dfrac{1}{2}$, so $k=2$. Checking the constants, $\dfrac{5}{12}\ne\dfrac{1}{2}$, confirming the lines are distinct and parallel — genuinely no solution.

15

Module 2 · Circle Theorems

An inscribed angle in a circle measures $40^\circ$. What is the measure of the central angle that intercepts the same arc?

The Inscribed Angle Theorem states a central angle is always twice the inscribed angle subtending the same arc: $2\times40^\circ=80^\circ$.

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Module 2 · Optimization (Calculus)

A closed-top box with a square base has volume $x^2h=32$. Its surface area is $S=x^2+4xh$. What is the minimum possible surface area?

Substitute $h=\dfrac{32}{x^2}$ into $S$: $S(x)=x^2+\dfrac{128}{x}$. Taking the derivative, $S'(x)=2x-\dfrac{128}{x^2}=0 \Rightarrow x^3=64 \Rightarrow x=4$. Then $S(4)=16+32=48$, which is a minimum since $S''(x)>0$ there.

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Module 2 · Conditional Probability

Given $P(A\cap B)=0.15$ and $P(B)=0.30$, what is $P(A\mid B)$?

$P(A\mid B) = \dfrac{P(A\cap B)}{P(B)} = \dfrac{0.15}{0.30}=0.5$.

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Module 2 · Systems (Word Problem)

A jar contains 40 coins, all nickels, dimes, and quarters, worth a total of $6.00. The number of dimes is twice the number of nickels. How many quarters are in the jar?

Let $n$ = nickels, $d=2n$ = dimes, $q$ = quarters. Then $n+2n+q=40 \Rightarrow q=40-3n$. Value: $5n+10(2n)+25q=600 \Rightarrow 25n+25q=600 \Rightarrow n+q=24$. Substituting $q=40-3n$: $n+40-3n=24 \Rightarrow n=8$. So $d=16$ and $q=40-24=16$.

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Module 2 · Cubic Equations

What is the real solution to $x^3=27$?

Taking the real cube root of both sides: $x=\sqrt[3]{27}=3$. (The equation also has two complex roots, but only $x=3$ is real.)

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Module 2 · Relative Speed

Two trains are 350 miles apart and travel toward each other at 60 mph and 80 mph, respectively. How many hours until they meet?

Their speeds add when moving toward each other: combined rate $=60+80=140$ mph. Time $=\dfrac{350}{140}=2.5$ hours.

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Module 2 · Trigonometric Equations

What is the sum of all solutions to $2\sin^2\theta - \sin\theta - 1 = 0$ for $\theta$ in $[0^\circ, 360^\circ)$?

Factor as a quadratic in $\sin\theta$: $(2\sin\theta+1)(\sin\theta-1)=0$, so $\sin\theta=1$ or $\sin\theta=-\dfrac12$. In $[0^\circ,360^\circ)$: $\sin\theta=1$ gives $\theta=90^\circ$; $\sin\theta=-\dfrac12$ gives $\theta=210^\circ,330^\circ$. Sum: $90+210+330=630^\circ$.

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Module 2 · Exponential-Linear Equations

What integer value of $x$ satisfies $2^x = 3x+4$?

This equation mixes an exponential and a linear expression, so it's typically solved by testing values or graphing. Testing $x=4$: left side $2^4=16$; right side $3(4)+4=16$. They match, so $x=4$ is the (integer) solution.

Module 1

Module 2

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Answer Key — SAT Math Practice — Set 2 (Elite / Highest-Difficulty)