Math 3 – Hard Level – 75 Questions

Explanation: f(3) = 2*3^2 – 3*3 + 5 = 18 – 9 + 5 = 14.

Explanation: Let integers be n and n+1. n^2 + (n+1)^2 = 365 → 2n^2 + 2n + 1 = 365 → 2n^2 + 2n – 364 = 0 → n^2 + n – 182 = 0 → (n+14)(n-13) = 0 → n = 13 or n = -14.

Explanation: (x+2)(x-1) = 60 → x^2 + x – 2 = 60 → x^2 + x – 62 = 0 → solve quadratic: x = (-1 ± √(1 + 248))/2 = (-1 ± 15.748)/2 → x ≈ 7.374 or -8.374. Only positive length x ≈ 7.374.

Explanation: a_15 = 3*15 + 2 = 45 + 2 = 47.

Explanation: Slope m = (13-5)/(6-2) = 8/4 = 2.

Explanation: Solve 3x – 7 = 2x + 8 → 3x – 2x = 8 + 7 → x = 15.

Explanation: Solve 4y – 9 = x → y = (x + 9)/4 → g^-1(7) = (7 + 9)/4 = 16/4 = 4.

Explanation: For a right triangle, Pythagoras: x^2 + 7^2 = 24^2 or x^2 + 24^2 = 7^2. Only plausible: x^2 + 7^2 = 24^2 → x^2 = 576 – 49 = 527 → x = √527 ≈ 22.96.

Explanation: Multiplying by 2 does not change roots. Original equation: x^2 -5x + 6 = 0 → (x-2)(x-3)=0 → x=2 or 3.

Explanation: Sum = (n-2)*180 → (n-2)*180 = 1260 → n-2 = 7 → n = 9.

Explanation: f(3) = 2*(3)^2 – 3*3 + 5 = 18 – 9 + 5 = 14.

Explanation: Sum formula: n(n+1)/2 = 210 → n(n+1) = 420 → n^2 + n – 420 = 0 → n = 20 (positive root).

Explanation: Solve 2x + 3 = -x + 6 → 3x = 3 → x = 1 → y = 2*1 + 3 = 5. Intersection at (1,5).

Explanation: Let integers be x and x+2 → x(x+2)=288 → x^2 + 2x – 288 = 0 → x = 16 (positive root) → integers 16 and 18.

Explanation: Volume = πr^2h = 3.14*25*10 = 785 m^3.

Explanation: Substitute y = 8 – x into 3x – 2y = 12 → 3x – 2(8 – x) = 12 → 3x – 16 + 2x = 12 → 5x = 28 → x = 5.6.

Explanation: Substitute the three points to form equations. Solving yields a = 1, b = -2, c = 3. So a = 1.

Explanation: 125 = 5^3 → 5^x = 5^{3(y-1)} → x = 3(y – 1) = 3y – 3.

Explanation: A = (1/2)bh = 60 and b = 2h → (1/2)(2h)(h) = 60 → h^2 = 60 → h = √60 ≈ 7.75, b ≈ 15.5.

Explanation: C = 2πr = 31.4 → r = 5 → A = πr^2 = 3.14*25 = 78.5 cm².

Explanation: g(3) = 3 – 4 = -1. Then f(-1) = 2(-1)² – 3(-1) + 1 = 2 + 3 + 1 = 6.

Explanation: Complete the square: h(x) = 4(x² – 3x + (9/4)) = 4(x – 1.5)². So k = 1.5.

Explanation: Let shorter leg = x. Then longer leg = x + 7. By Pythagoras: x² + (x + 7)² = 13² → 2x² + 14x + 49 = 169 → 2x² + 14x – 120 = 0 → x = 6. So legs are 6 cm and 13 cm.

Explanation: Slope = (k – 5)/(6 – 2) = 1.5 → (k – 5)/4 = 1.5 → k – 5 = 6 → k = 11.

Explanation: Let efficiency E = k / weight → 30 = k / 3000 → k = 90000. For 4500 lb, E = 90000 / 4500 = 20 mpg.

Explanation: 125 = 5³, so 5^(2x + 1) = 5³ → 2x + 1 = 3 → x = 1.

Explanation: From x – y = 1 → x = y + 1. Substitute into 2x + 3y = 14 → 2(y + 1) + 3y = 14 → 5y + 2 = 14 → y = 2.4 → x = 3.4 → x + y = 5.8.

Explanation: Let width = w → length = 2w + 5 → perimeter = 2(2w + 5 + w) = 50 → 6w + 10 = 50 → w = 6.67 → length = 18.34 → area ≈ 122.23 cm².

Explanation: f(2 + h) = 3(2 + h)² – 4(2 + h) + 2 = 3(h² + 4h + 4) – 8 – 4h + 2 = 3h² + 8h + 6. Then f(2) = 3(4) – 8 + 2 = 6 → difference = 3h² + 8h.

Explanation: P(12) = 500(1.08)¹² ≈ 500 × 2.518 = 1259.

Explanation: Roots are x = -2 and x = 5 → x² = 4 and 25 → sum = 29.

Explanation: For a parabola f(x) = ax² + bx + c, vertex at x = -b/(2a) = 1 → f(1) = 2(1)² – 4(1) + 1 = -1.

Explanation: 7² + 24² = 49 + 576 = 625 = 25² → right triangle → largest angle = 90°.

Explanation: The function shifts left by 3 units (x + 3) and down by 4 units (-4).

Explanation: Cross-multiply → 5(x + 1) = 3(2x – 5) → 5x + 5 = 6x – 15 → x = 20.

Explanation: Solve x – y = 1 → y = x – 1. Substitute into 3x + 2(x – 1) = 12 → 5x – 2 = 12 → x = 14/5.

Explanation: Total distance = 60t + 40t = 100t. Total time = 2t. Average speed = 100t / 2t = 50 mph.

Explanation: 125 = 5³ → 2x – 1 = 3 → 2x = 4 → x = 2.

Explanation: A = πr² = 100π → r² = 100 → r = 10 → C = 2πr = 20π.

Explanation: g(2) = 2² – 1 = 3 → f(3) = 3(3) + 2 = 11.

Explanation:
Solve 2x + 3 > 7 → x > 2.
Solve x – 1 < 4 → x < 5. Combine: 2 < x < 5.

Explanation: Substitute y = 4 → x² + 16 = 25 → x² = 9 → x = ±3.

Explanation: Numbers that leave a remainder of 2 when divided by 5 follow the pattern: 5k + 2.
Possible numbers: 2, 7, 12, 17, etc.
15 does not fit the pattern.

Explanation: Total balls = 10. Not green = 3 red + 2 blue = 5.
P(not green) = 5/10 = 1/2.

Explanation: A right shift of 2 means replace x with (x – 2):
g(x) = (x – 2)² – 6(x – 2) + 5 = x² – 10x + 21.

Explanation:
Convert from logarithmic to exponential form:
2⁵ = x → x = 32.

Explanation:
|3x – 6| = 9 → 3x – 6 = 9 or 3x – 6 = -9.
Solve both: x = 5 or x = -1.

Explanation:
Above y = 2x – 3 means y > 2x – 3.
Below y = -x + 4 means y < -x + 4. So the system is: y > 2x – 3 and y < -x + 4.

Explanation:
Original line: 3x + 4y = 12 → y = -3/4x + 3.
Slope = -3/4 → perpendicular slope = 4/3.
Through origin: y = (4/3)x.

Explanation:
Sum of first 5 scores = 5 × 80 = 400.
Add 100 → total = 500.
New mean = 500 ÷ 6 = 83.33.

Explanation:
Substitute y = x + 1 → x² + (x + 1)² = 25 → 2x² + 2x + 1 = 25 → 2x² + 2x – 24 = 0 → x² + x – 12 = 0.
Factor: (x + 4)(x – 3) = 0 → x = -4 or x = 3.

Explanation:
g(2) = 2 + 4 = 6 → f(6) = 2(6²) – 3 = 72 – 3 = 69.

Explanation:
The circle’s standard form is (x – h)² + (y – k)² = r².
So center = (3, -2) and radius = √49 = 7.

Explanation:
Use exponential growth formula: P = 50,000(1.08)³ = 50,000 × 1.2597 = 62,985.

Explanation:
Adding a constant to each value shifts the mean but does not affect the standard deviation.
New standard deviation = 4.

Explanation:
The expression changes sign at x = -5 and x = 3.
Since it’s less than 0 between these roots, the inequality holds for -5 < x < 3.

Explanation:
Inverse variation: y = k/x.
k = xy = 6(2) = 12 → y = 12/8 = 1.5.

Explanation:
In a 30-60-90 triangle, sides are in the ratio 1 : √3 : 2.
Opposite 30° → (1/2) of the hypotenuse → 10 × 1/2 = 5.

Explanation:
For no solution, the lines are parallel → same slope but different intercepts.
Slope₁ = -2/3.
For 4x + ky = 12 → y = (-4/k)x + 12/k → -4/k = -2/3 → k = 6.

Explanation:
Vertical stretch by 3 → multiply output by 3: 3[(x – 4)² + 1].
Then shift down 2 → 3[(x – 4)² + 1] – 2 = 3(x – 4)² + 1.

Explanation:
Rewrite in exponential form: 2⁴ = x + 3 → 16 = x + 3 → x = 13.

Explanation:
Cross-multiply → 3(x + 1) = 2(x – 2) → 3x + 3 = 2x – 4 → x = -7.

Explanation:
Complete the square:
x² – 6x + y² + 8y = 0 → (x – 3)² – 9 + (y + 4)² – 16 = 0 → (x – 3)² + (y + 4)² = 25.
Center = (3, -4).

Explanation:
Vertex of a parabola: t = -b/(2a) = -32/(2 × -16) = 1.
Substitute: h(1) = -16(1)² + 32(1) + 48 = 64 feet.

Explanation:
Use inclusion-exclusion: P(M ∪ S) = P(M) + P(S) – P(M ∩ S) = 60 + 45 – 25 = 80%.

Explanation:
Number of doublings in 24 hours = 24 ÷ 6 = 4.
Population = 500 × 2⁴ = 500 × 16 = 8000.

Explanation:
The region lies between the two lines: above y = 2x − 1 and below y = −x + 5.
Thus, the solution set is the intersection of these two half-planes.

Explanation:
Center = (−4, 3).
Distance to origin = √[ (−4)² + (3)² ] = √(16 + 9) = √25 = 5.

Explanation:
Subtracting 4 from x shifts right by 4 units; subtracting 2 shifts down by 2 units.

Explanation:
Mean = total / 5 → total = 80 × 5 = 400.
If lowest score is 60 → sum of others = 400 − 60 = 340.

Explanation:
From x − y = 1 → x = y + 1.
Substitute: 2(y + 1) + 3y = 12 → 2y + 2 + 3y = 12 → 5y = 10 → y = 2.

Explanation:
Set equal: x² − 4x + 3 = −x + 3 → x² − 3x = 0 → x(x − 3) = 0 → x = 0 or x = 3.

Explanation:
sin²θ + cos²θ = 1 → (3/5)² + cos²θ = 1 → 9/25 + cos²θ = 1 → cos²θ = 16/25 → cos θ = 4/5.

Explanation:
g(2) = 2² − 3 = 1.
Then f(g(2)) = 2(1) + 1 = 3.

Explanation:
Original speed = 180 ÷ 3 = 60 mph.
New speed = 60 + 20 = 80 mph.
Time = 300 ÷ 80 = 3.75 hours.