Math & Puzzles - Mixed Problems

In number theory, a deficient number (where the sum of proper divisors is less than the number) is also known as a defective number. 'Comply' and 'Faulty' are not standard mathematical terms for this classification.

The pattern is n^3 + 1: 1^3+1=2, 2^3+1=9, 3^3+1=28, 4^3+1=65. The next term is 5^3+1 = 125 + 1 = 126. 125 is a distractor as it is a perfect cube (5^3) without the addition.

The expression (a-x)(b-x)...(z-x) contains the term (x-x) because the sequence includes every letter of the alphabet subtracted by x. Since (x-x) = 0, the entire product becomes 0 regardless of the value of other terms.

The sequence follows the pattern n^2 + n: 1^2+1=2, 2^2+2=6, 3^2+3=12, 4^2+4=20. The next term is 5^2+5=30. Alternatively, the differences are 4, 6, 8, so the next difference is 10.

The sequence follows prime number positions in the alphabet: a(1), b(2), c(3), e(5), g(7), k(11). The 7th letter of the alphabet is 'g'. 'h', 'i', and 'd' do not fit the prime number sequence.

The pattern is n^2 * (n-1) for n=1, 2, 3, 4, 5. 1^20=0 (though sequence starts at 1, likely n^2 + n^3 logic or similar). Correct logic: 1^21=1, 2^23=12, 3^24=36, 4^25=80, 5^26=150. Thus, 80 is the missing number.

The logic is (sqrt(a) + sqrt(b))^2. For 25+36: (5+6)^2 = 121. For 49+49: (7+7)^2 = 196. For 64+49: (8+7)^2 = 15^2 = 225. Other options do not fit this pattern of adding the square roots before squaring.

If log4(x) = 12, then x = 4^12. We need log2(x/4). x/4 = 4^12 / 4^1 = 4^11. Since 4 = 2^2, 4^11 = (2^2)^11 = 2^22. Therefore, log2(2^22) = 22.

To find 20% of 2, multiply 0.20 by 2, which equals 0.4. 20 and 4 are whole numbers far exceeding the value, and 0.04 represents 2% of 2.

Log_x(1/8) = -3/2 means x^(-3/2) = 1/8, so x^(3/2) = 8. Squaring both sides: x^3 = 64, therefore x = 4^(1/3) = 4. But we must check if x=1/4 works: (1/4)^(-3/2) = (4)^(3/2) = 8, which is correct.

The slope of ax+by=c is -a/b. The slope of bx-ay=c is b/a. The product of slopes is (-a/b) × (b/a) = -1, which means the lines are perpendicular to each other.

The volume of water displaced is V = π * r^2 * h = π * 2^2 * 0.6 = 2.4π. The volume of the ball (sphere) is (4/3)π * R^3. Setting 2.4π = (4/3)π * R^3 gives R^3 = 1.8. The cube root of 1.8 is approximately 1.216, which rounds to 1.2.

The number of ways to choose 2 teachers from 5 is 5C2 = 10. The number of ways to choose 4 students from 10 is 10C4 = 210. Multiplying these together (10 * 210) gives 2100 total possible committees.

Greece joined the European Union in 1981, later than Ireland (1973), France (founding member 1957), and Britain (1973). Among the given options, Greece was the last to join the EU.

The penalty spot in soccer is 12 yards (approximately 11 meters) from the goal-line. One-sixth of 12 yards equals 2 yards, which is 6 feet.

Abraham Lincoln is famously known for his iconic beard (specifically a chin curtain). While he had a mole, he did not have a mustache, and he is not primarily associated with scars or noserings in popular culture.

In the American West, 'Boot Hill' is a common name for cemeteries, referring to the fact that many people died with their boots on. Therefore, the item of clothing is a boot. Shoes, skirts, and shirts are not associated with this specific graveyard terminology.

At the time of the question (likely referring to Gordon Brown or similar context), the hint points to a specific PM. However, brown is a common eye color. Note: If referring to Boris Johnson or David Cameron, eyes were blue, but 'Brown' is the stored answer matching the surname of PM Gordon Brown.

In Shakespeare's 'Macbeth', Macbeth uses a dagger to murder King Duncan in his sleep. While swords are common in the play, the specific murder of Duncan is famously associated with the 'air-drawn dagger' soliloquy and the physical daggers used.