If X = {x : x < 7} and Y = {y: y is a factor of 24} are subsets of μ = {1, 2, 3...10}, find X ∩ Y.
Step 1: Identify elements in X. Since μ = {1,2,3,...,10} and x < 7, X = {1,2,3,4,5,6}. Step 2: Identify elements in Y. Factors of 24 within μ: 1, 2, 3, 4, 6, 8 (since 12 > 10). Step 3: Find intersection X ∩ Y by listing common elements: 1,2,3,4,6. Thus, X ∩ Y = {1,2,3,4,6}.
Evaluate 0.42 ÷ 2.5 / 0.5 × 2.05, leaving the answer in standard form.
Step 1: Perform division left to right: 0.42 ÷ 2.5 = 0.168. Step 2: Next division: 0.168 ÷ 0.5 = 0.336. Step 3: Multiplication: 0.336 × 2.05 = 0.6888. Step 4: Convert to standard form: 0.6888 = 6.888 × 10⁻¹.
The set of numbers {1, -1} is closed under which operation?
Step 1: Test addition: 1 + 1 = 2 (not in set), 1 + (-1) = 0 (not in set). Not closed. Step 2: Test subtraction: 1 - (-1) = 2 (not in set). Not closed. Step 3: Test multiplication: 1 × 1 = 1, 1 × (-1) = -1, (-1) × (-1) = 1 (all in set). Closed. Step 4: Test division: 1 / (-1) = -1 (in set), but (-1) / 0 undefined if considered, but mainly 1/1=1, but set has no 0. Primarily, multiplication works consistently.
Solve for x: 2x² - 5x - 3 = 0.
Step 1: Factor the quadratic: Look for factors of 2*(-3)=-6 that add to -5: -6 and 1. Rewrite: 2x² - 6x + x - 3 = 0. Step 2: Group: 2x(x - 3) + 1(x - 3) = 0. Step 3: Factor: (2x + 1)(x - 3) = 0. Step 4: Set each to zero: 2x + 1 = 0 → x = -1/2; x - 3 = 0 → x = 3.
The sum of the first n terms of an A.P. is S_n = n² + n. Find the 5th term.
Step 1: Recall a_n = S_n - S_{n-1} for n ≥ 2. Step 2: Compute S_5 = 5² + 5 = 25 + 5 = 30. Step 3: Compute S_4 = 4² + 4 = 16 + 4 = 20. Step 4: Subtract: a_5 = 30 - 20 = 10.
The roots of the equation x² - 6x + 8 = 0 are
Step 1: Find two numbers that multiply to 8 and add to 6: 2 and 4. Step 2: Rewrite: x² - 2x - 4x + 8 = 0. Step 3: Group: x(x - 2) - 4(x - 2) = 0. Step 4: Factor: (x - 2)(x - 4) = 0. Roots: x = 2, x = 4.
Simplify (2x - 3y)² - (x - y)².
Step 1: Recall a² - b² = (a - b)(a + b), where a = 2x - 3y, b = x - y. Step 2: Compute a - b = (2x - 3y) - (x - y) = x - 2y. Step 3: Compute a + b = (2x - 3y) + (x - y) = 3x - 4y. Step 4: Multiply: (x - 2y)(3x - 4y) = x*3x + x*(-4y) + (-2y)*3x + (-2y)*(-4y) = 3x² - 4xy - 6xy + 8y² = 3x² - 10xy + 8y².
The value of log₅ 25 is
Step 1: Express 25 as power of 5: 25 = 5². Step 2: By definition, log₅ (5²) = 2.
The gradient of the line passing through (2,3) and (4,7) is
Step 1: Identify points: (x₁,y₁) = (2,3), (x₂,y₂) = (4,7). Step 2: Apply formula m = (y₂ - y₁)/(x₂ - x₁) = (7 - 3)/(4 - 2) = 4/2 = 2.
The area of a triangle with vertices (1,1), (5,1), (3,4) is
Step 1: List coordinates: A(1,1), B(5,1), C(3,4). Step 2: Apply shoelace formula: 1/2 |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)| = 1/2 |1(1 - 4) + 5(4 - 1) + 3(1 - 1)| = 1/2 |-3 + 15 + 0| = 1/2 × 12 = 6.
Solve the inequality 3x - 2 > 7.
Step 1: Add 2 to both sides: 3x > 9. Step 2: Divide by 3: x > 3.
The solution to the system: x + y = 5, 2x - y = 1 is
Step 1: Add the equations: (x + y) + (2x - y) = 5 + 1 → 3x = 6 → x = 2. Step 2: Substitute into first: 2 + y = 5 → y = 3.
The matrix [[1,2],[3,4]] times [[5],[6]] is
Step 1: First row: 1×5 + 2×6 = 5 + 12 = 17. Step 2: Second row: 3×5 + 4×6 = 15 + 24 = 39. Result: [[17],[39]].
The binomial expansion of (x + 2)^4 the constant term is
Step 1: General term T_{k+1} = C(4,k) x^{4-k} × 2^k. Step 2: For constant term, set 4 - k = 0 → k = 4. Step 3: T_5 = C(4,4) × 2^4 = 1 × 16 = 16.
The value of sin 30° + cos 60° is
Step 1: sin 30° = 1/2 = 0.5. Step 2: cos 60° = 1/2 = 0.5. Step 3: Sum: 0.5 + 0.5 = 1.
The circumference of a circle with diameter 14 cm is
Step 1: Use formula C = π d. Step 2: Substitute π = 22/7, d = 14: C = (22/7) × 14 = 22 × 2 = 44 cm.
In a right triangle, tan θ = 3/4, then sin θ =
Step 1: tan θ = opposite/adjacent = 3/4. Step 2: Hypotenuse = √(3² + 4²) = √(9 + 16) = √25 = 5. Step 3: sin θ = opposite/hypotenuse = 3/5.
The area of a circle with radius 7 cm is
Step 1: Use formula A = π r². Step 2: Substitute π = 22/7, r = 7: A = (22/7) × 49 = 22 × 7 = 154 cm².
The volume of a cylinder with radius 5 cm and height 10 cm is
Step 1: Use formula V = π r² h. Step 2: Substitute r = 5, h = 10: V = π × 25 × 10 = 250π cm³.
The angle in a semicircle is
Step 1: Recall theorem: angle subtended by diameter at circumference is right angle. Step 2: Thus, 90°.
The mean of numbers 1,2,3,4,5 is
Step 1: Sum = 1 + 2 + 3 + 4 + 5 = 15. Step 2: Number of terms = 5. Step 3: Mean = 15 / 5 = 3.
The median of 2,4,6,8,10 is
Step 1: List in order: 2,4,6,8,10. Step 2: Since odd number of terms (5), median is middle (3rd) term: 6.
The mode of 1,2,2,3,3,3,4 is
Step 1: Count frequencies: 1 (1 time), 2 (2 times), 3 (3 times), 4 (1 time). Step 2: Highest frequency: 3 (3 times).
The probability of drawing a red ball from a bag with 3 red and 7 blue is
Step 1: Total balls = 3 + 7 = 10. Step 2: Favorable = 3 red. Step 3: P = 3/10.
The range of 5,7,3,9,1 is
Step 1: Order: 1,3,5,7,9. Step 2: Max = 9, min = 1. Step 3: Range = 9 - 1 = 8.
The variance of 1,2,3 is
Step 1: Mean = (1+2+3)/3 = 2. Step 2: Deviations squared: (1-2)²=1, (2-2)²=0, (3-2)²=1. Step 3: Average = (1+0+1)/3 = 2/3.
In a pie chart, a sector of 72° represents
Step 1: Full circle = 360°. Step 2: Proportion = 72 / 360 = 0.2. Step 3: Percentage = 0.2 × 100% = 20%.
The number of ways to arrange 3 books on a shelf is
Step 1: For first position: 3 choices. Step 2: Second: 2 remaining. Step 3: Third: 1 left. Step 4: Total = 3 × 2 × 1 = 6.
The combination C(5,2) is
Step 1: C(5,2) = 5! / (2! × (5-2)!) = 120 / (2 × 6) = 120 / 12 = 10. Step 2: Or: (5×4)/ (2×1) = 20/2 = 10.
The standard deviation is the square root of
Standard deviation = √variance. By definition in statistics.
Solve 2^x = 8.
Step 1: Express 8 as power of 2: 8 = 2^3. Step 2: Thus, x = 3.
The limit as x->0 of sin x / x is
lim_{x→0} (sin x / x) = 1. Standard limit in calculus.
The derivative of x² is
d/dx (x²) = 2x. Apply power rule: d/dx (x^n) = n x^{n-1}.
The integral of 3x dx is
Step 1: ∫ 3x dx = 3 ∫ x dx. Step 2: ∫ x dx = x²/2 + C. Step 3: 3 × (x²/2) + C = (3x²/2) + C.
The probability of at least one head in two coin tosses is
Step 1: Total outcomes: 4 (HH, HT, TH, TT). Step 2: No heads: TT (1 outcome). Step 3: At least one head: 1 - 1/4 = 3/4.
The HCF of 12 and 18 is
Step 1: 18 ÷ 12 = 1 remainder 6. Step 2: 12 ÷ 6 = 2 remainder 0. Step 3: HCF = 6.
The LCM of 4 and 6 is
Step 1: Prime factors: 4 = 2², 6 = 2 × 3. Step 2: Highest powers: 2² × 3 = 4 × 3 = 12.
Convert 1011 binary to decimal.
Step 1: Positions from right: 2⁰=1 (1), 2¹=2 (1), 2²=4 (0), 2³=8 (1). Step 2: 8 + 0 + 2 + 1 = 11.
The number 0.25 in binary is
Step 1: 0.25 × 2 = 0.5 (bit 0). Step 2: 0.5 × 2 = 1.0 (bit 1). Step 3: Result: 0.01 binary.
The sum of 1 + 1/4 + 1/16 is
Step 1: Common denominator 16: 1 = 16/16, 1/4 = 4/16, 1/16 = 1/16. Step 2: Sum = (16 + 4 + 1)/16 = 21/16. Step 3: 21 ÷ 16 = 1.3125.
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