Find x if 2x + 5 = 19
Solve for the variable.
Solution
2x = 14 ⟹ x = 7.
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Algebra
Find the roots of x² − 5x + 6 = 0
Solve using factorization or quadratic formula.
Solution
(x−2)(x−3)=0 ⟹ x=2 or x=3.
Solve the system:
{ 3x − 2y = 11, 4x + y = 17 }
Find ordered pair (x, y).
Solution
From 4x + y = 17 ⇒ y = 17−4x. Substitute: 3x − 2(17−4x) = 11 ⇒ 3x − 34 + 8x = 11 ⇒ 11x = 45 ⇒ x = 45/11. Then y = 17 − 4(45/11) = (187−180)/11 = 7/11.
Solve the inequality: 5 − 2x < 3x + 10
Give your answer in interval notation.
Solution
5−2x < 3x+10 ⇒ −5 < 5x ⇒ x > −1. Interval: (−1, ∞).
Solve for x: 3x+1 = 27
Express 27 as 3k.
Solution
27 = 3³ ⇒ x+1 = 3 ⇒ x = 2.
Sequence defined by a₁=2, an+1=3an+1. Find a closed form.
Solve the linear non-homogeneous recurrence.
Solution
Let bn=an+½ ⇒ bn+1=3bn, b₁=2.5 ⇒ bn=2.5·3^{n−1}. Hence an=2.5·3^{n−1}−½.
Geometry
Find the area of a triangle with base 10 and height 7.
Use A = ½bh.
Solution
A = ½·10·7 = 35.
A right triangle has legs 6 and 8. Find the radius of the inscribed circle.
In a right triangle, r = (a + b − c)/2.
Solution
c=10. r=(6+8−10)/2=2.
In cyclic quadrilateral ABCD, ∠A = 70°, ∠B = 60°. Find ∠C.
Opposite angles in a cyclic quad sum to 180°.
Solution
∠C = 180° − ∠A = 110°.
Distance between P(−3, 4) and Q(5, −2)?
Use √[(Δx)² + (Δy)²].
Solution
Δx=8, Δy=−6 ⇒ d=√(64+36)=√100=10.
Number Theory
Find gcd(84, 60).
Use prime factors or Euclid’s algorithm.
Solution
84=2²·3·7, 60=2²·3·5 ⇒ gcd=2²·3=12.
Compute 7100 mod 5.
Reduce base modulo 5.
Solution
7≡2 (mod 5) ⇒ 2¹⁰⁰ ≡ 0? No. Since 2⁴≡1 ⇒ 100 multiple of 4 ⇒ ≡1.
Find all integer solutions of 14x + 21y = 7.
Simplify first.
Solution
Divide by 7 ⇒ 2x+3y=1 ⇒ x=1−3t, y=−1+2t for t∈ℤ.
Smallest positive n such that sum of digits of 9n equals 9.
Use divisibility by 9 and simple search logic.
Solution
Let 9n have digit sum 9 ⇒ 9n ≡ 0 (mod 9) always; first is 9, digit sum 9 ⇒ n=1.
Calculus
Find d/dx (3x³ − 7x + 4).
Differentiate term by term.
Solution
9x² − 7.
Compute d/dx (sin(x²)).
Apply chain rule.
Solution
cos(x²)·2x.
Evaluate ∫ 2x·e^{x²} dx.
u-substitution recommended.
Solution
Let u=x² ⇒ du=2x dx ⇒ ∫ e^{u} du = e^{x²} + C.
Compute limx→0 (1 − cos x)/x².
Use Taylor or l’Hôpital.
Solution
≈ (1 − (1 − x²/2))/x² → 1/2.
Probability
Flip a fair coin 3 times. Probability of exactly 2 heads?
Use binomial formula.
Solution
C(3,2)/2³ = 3/8.
From a standard deck, probability the first card is a heart or an ace?
Use inclusion–exclusion.
Solution
P=13/52 + 4/52 − 1/52 = 16/52 = 4/13.
Roll a die until a 6 appears. Expected number of rolls?
Geometric distribution with p=1/6.
Solution
E = 1/p = 6.
Ways to distribute 5 identical balls into 3 distinct boxes?
Stars and bars.
Solution
C(5+3−1, 3−1) = C(7,2) = 21.
Logic
Is (P ⇒ Q) logically equivalent to (¬P ∨ Q)?
State yes/no and justify.
Solution
Yes. Their truth tables coincide.
Show among 13 people, two share a birth month.
Name the principle used.
Solution
Pigeonhole principle with 12 months.
On an island of knights (truth-tellers) and knaves (liars), A says “B is a knave.” B says “A and I are the same.” Who is who?
Consider cases for A.
Solution
If A is knight ⇒ B knave. But B says “same,” which would be false ⇒ consistent. So A knight, B knave.
Prove (A∩B)c = Ac ∪ Bc.
Name the law.
Solution
De Morgan’s law: x∉A∩B ⇔ (x∉A)∨(x∉B).