Math

Solution

Correct: A

Let E be the expected number. After the first flip, the second flip either matches (game ends in 2 flips) or mismatches (game continues). With probability 1/2 the game ends in 2 flips. With probability 1/2 we are back to the start but have spent 2 flips. Thus E = 1/2·2 + 1/2·(2 + E). Solving gives E = 3.

Solution

Correct: A

Use integration by parts: u = ln x, dv = x^3 dx ⇒ du = dx/x, v = x^4/4. The integral becomes [x^4 ln x / 4]_0^1 - ∫0^1 x^3/4 dx. The boundary term is 0 (limit as x→0+ of x^4 ln x = 0) and the remaining integral is -1/4 · [x^4/4]_0^1 = -1/16.

Solution

Correct: A

By the Law of Cosines: a² = b² + c² - 2bc cos A = 25 + 64 - 2·5·8·cos 60° = 89 - 80·0.5 = 89 - 40 = 49. Thus a = 7.

Solution

Correct: A

The probability is the area under xy < 1/2 in the unit square. For x ≤ 1/2 the entire vertical strip satisfies xy < 1/2 (area 1/2). For x > 1/2 the height is 1/(2x), so integrate 1/(2x) from 1/2 to 1 giving (ln 2)/2. Total area = 1/2 + (ln 2)/2 = (1 + ln 2)/2.

Solution

Correct: B

Take ln: ln y = x^2 ln x. Differentiate implicitly: y'/y = 2x ln x + x^2·1/x = 2x ln x + x. At x = 1, y = 1 and ln x = 0, so y' = 1·(0 + 1) = 1.

Solution

Correct: C

Total ways C(8,3)=56. Favorable: choose 2 blue from 5 and 1 red from 3: C(5,2)·C(3,1)=10·3=30. Probability=30/56=15/28.

Solution

Correct: A

Use Taylor series: sin x = x - x^3/6 + O(x^5). Then (sin x - x)/x^3 = (-x^3/6)/x^3 = -1/6.

Solution

Correct: C

Square both sides: (sin θ + cos θ)^2 = 1.44 ⇒ sin²θ + 2 sin θ cos θ + cos²θ = 1.44 ⇒ 1 + sin 2θ = 1.44 ⇒ sin 2θ = 0.44.

Solution

Correct: A

Integrate pdf: ∫0^0.5 2x dx = [x^2]_0^0.5 = 0.25.

Solution

Correct: A

Intersection: x^2 = 2x - x^2 ⇒ 2x^2 - 2x = 0 ⇒ x=0,1. Area = ∫0^1 (2x - x^2 - x^2) dx = ∫0^1 (2x - 2x^2) dx = [x^2 - 2x^3/3]_0^1 = 1 - 2/3 = 1/3.

Solution

Correct: D

Binomial: n=4, k=2, p=1/6. P=C(4,2)(1/6)^2(5/6)^2=C(4,2)·25/6^4.

Solution

Correct: A

Integration by parts: u=ln x, dv=dx ⇒ du=dx/x, v=x. ∫ ln x dx = x ln x - ∫ x·dx/x = x ln x - x + C.

Solution

Correct: D

Angles 30° and 60°. Smaller is 30°, sin 30°=1/2.

Solution

Correct: B

The region where distance ≥0.25 is a smaller square of side 0.5 centered, area 0.25. So probability distance <0.25 is 1-0.25=0.75.

Solution

Correct: A

Notice y=(x-1)^4. Minimum at x=1, y=0.

Solution

Correct: A

C(4,2)=6 ways to choose 2 aces, C(52,2)=1326 total. Probability=6/1326=1/221.

Solution

Correct: A

Use identity sin^2 x = (1 - cos 2x)/2. Integral becomes 1/2 ∫0^π (1 - cos 2x) dx = 1/2 [x - sin 2x/2]_0^π = 1/2·π = π/2.

Solution

Correct: A

cos 2θ = (1 - tan^2 θ)/(1 + tan^2 θ) = (1 - 9/16)/(1 + 9/16) = (7/16)/(25/16) = 7/25.

Solution

Correct: A

P(X=4)=q^{k-1}p=0.8^3·0.2.

Solution

Correct: A

Chain rule: y' = e^(x^2)·2x. At x=0, y'=0.

Solution

Correct: A

Hypergeometric: C(15,4) ways to choose girls, C(10,2) for boys, total C(25,6).

Solution

Correct: A

Integration by parts: u=x, dv=e^x dx ⇒ du=dx, v=e^x. ∫ x e^x dx = x e^x - ∫ e^x dx = x e^x - e^x + C.

Solution

Correct: A

Side opposite 30° is half hypotenuse: a = c sin A = 10·sin 30° = 5.

Solution

Correct: A

Poisson pmf: P(X=k)=λ^k e^{-λ}/k!. For k=2: 3^2 e^{-3}/2 = 9e^{-3}/2.

Solution

Correct: A

∫1^e dx/x = [ln x]_1^e = ln e - ln 1 = 1 - 0 = 1.

Solution

Correct: A

cos θ = √(1 - 0.6^2) = 0.8, tan θ = sin/cos = 0.6/0.8 = 3/4 = 0.75.

Solution

Correct: A

Binomial with n=3, k=2, p=4/10=0.4. P=C(3,2)(0.4)^2(0.6).

Solution

Correct: A

Standard limit form: lim (1 + a/x)^x = e^a. Here a=2, so e^2.

Solution

Correct: A

Parabola opens downward, vertex at x = -b/(2a) = 4/2 = 2. f(2) = -4 + 8 - 3 = 1.

Solution

Correct: A

Width of interval is 3, total width 10, probability = 3/10 = 0.3.