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**Instructions:**
This practice exam consists of 15 questions covering advanced topics in algebra, geometry, trigonometry, statistics, and probability. Questions are designed to challenge Year 9 students at a high difficulty level, incorporating Year 10 concepts such as quadratic inequalities, circle theorems, and conditional probability. Calculators are permitted where indicated. Show all working. Total marks: 150 (10 marks per question). Time allowed: 2 hours.
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Solve the quadratic inequality $2x^2 - 5x - 3 > 0$. Express your solution in interval notation and verify by substituting test points. (Calculator permitted)
The sum of two numbers is 15 and their product is 56. Find the numbers, then form a quadratic equation whose roots are these numbers, and solve it using the quadratic formula. Verify the discriminant.
Expand and simplify $(3x - 2y)^3 + (x + 4y)^2$. Then, factorize the resulting expression if possible, and check by expanding back.
Solve the system of equations:
$3x + 2y = 7$
$x^2 + y^2 = 13$
Graphically verify the solutions on a coordinate plane (describe the intersection points).
Given the function $f(x) = x^3 - 6x^2 + 11x - 6$, factorize it completely and find all roots. Use synthetic division to verify one root and confirm the factorization.
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In $\triangle ABC$, $AB = 8$ cm, $BC = 10$ cm, and $\angle B = 50^\circ$. Calculate the length of $AC$ using the law of cosines. Then, verify using the law of sines and find $\angle C$.
A circle has center $O$ and radius 5 cm. Chord $AB$ subtends an angle of $120^\circ$ at the center. Find the length of chord $AB$ and the distance from $O$ to $AB$. Verify using circle theorems.
Prove that in a right-angled triangle with sides 6 cm, 8 cm, and hypotenuse 10 cm, the altitude to the hypotenuse is 4.8 cm. Then, find the area using two methods and confirm consistency.
Two similar triangles have corresponding sides 3:5. If the smaller triangle has area 27 cm², find the area of the larger triangle. Extend to find the ratio of perimeters and verify with a scale factor example.