This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Welcome back Mughal — missed you this week. Here are the solutions to the questions: 1) The product of (3+sqrt(5))(3-sqrt(5)) is: Step 1: Use the difference of squares formula, (a+b)(a-b) = a^2 - b^2. Here, a=3 and b=sqrt(5). (3+sqrt(5))(3-sqrt(5)) = 3^2 - (sqrt(5))^2 Step 2: Calculate the squares. 3^2 - (sqrt(5))^2 = 9 - 5 = 4 The product is 4. This is not among the given options (Prime number, Odd number, Rational number, Irrational number). However, 4 is a rational number. The correct option is D) Rational number. 2) The base of common logarithm is: The common logarithm, also known as the decimal logarithm, is the logarithm with base 10. The correct option is C) 10. 3) If A = \\, then P(A) is: Step 1: The power set P(A) of a set A is the set of all subsets of A. If A = \\ (the empty set), its only subset is the empty set itself. P(A) = \\\\ The correct option is B) \\\\. 4) If n(A B) = 50, n(A) = 30 and n(B) = 35, then n(A B) is equal to: Step 1: Use the formula for the union of two sets: n(A B) = n(A) + n(B) - n(A B). Substitute the given values into the formula. 50 = 30 + 35 - n(A B) Step 2: Solve for n(A B). 50 = 65 - n(A B) n(A B) = 65 - 50 n(A B) = 15 The correct option is B) 15. 5) The L.C.M. of 16x^2, 4x and 30xy is: Step 1: Find the LCM of the numerical coefficients: 16, 4, and 30. Prime factorization: 16 = 2^4 4 = 2^2 30 = 2 × 3 × 5 LCM of (16, 4, 30) = 2^4 × 3 × 5 = 16 × 3 × 5 = 240. Step 2: Find the LCM of the variables: x^2, x, and xy. For x: the highest power is x^2. For y: the highest power is y. Step 3: Combine the numerical and variable LCMs. LCM = 240x^2y. The correct option is A) 240x^2y. 6) The factorization of 12x + 36 is: Step 1: Find the greatest common factor (GCF) of 12x and 36. The GCF of 12 and 36 is 12. Step 2: Factor out the GCF. 12x + 36 = 12(x) + 12(3) = 12(x+3) The correct option is D) 12(x+3). 7) Solution of 5x - 10 = 10 is: Step 1: Add 10 to both sides of the equation. 5x - 10 + 10 = 10 + 10 5x = 20 Step 2: Divide both sides by 5. (5x)/(5) = (20)/(5) x = 4 The correct option is B) 4. 8) 60^ = The value of 60^ is sqrt(3)2. The correct option is A) sqrt(3)2. 9) ^2 - ^2 = This is a fundamental trigonometric identity. ^2 - ^2 = 1 The correct option is C) 1. 10) Distance between two points P(1,2) and Q(4,6) is: Step 1: Use the distance formula: d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2). Here, (x_1, y_1) = (1,2) and (x_2, y_2) = (4,6). d = sqrt((4-1)^2 + (6-2)^2) Step 2: Calculate the differences and squares. d = sqrt((3)^2 + (4)^2) d = sqrt(9 + 16) d = sqrt(25) Step 3: Take the square root. d = 5 The correct option is D) 5. 11) If the volume of two similar solids is 125 cm^3 and 27 cm^3, the ratio of their corresponding heights is: Step 1: For similar solids, the ratio of their volumes is equal to the cube of the ratio of their corresponding linear dimensions (like heights, radii, or sides). Let V_1 and V_2 be the volumes, and h_1 and h_2 be the corresponding heights. (V_1)/(V_2) = ((h_1)/(h_2))^3 Step 2: Substitute the given volumes. (125)/(27) = ((h_1)/(h_2))^3 Step 3: Take the cube root of both sides to find the ratio of the heights. 3/(27) = (h_1)/(h_2) (5)/(3) = (h_1)/(h_2) The ratio of their corresponding heights is 5:3. The correct option is B) 5:3. 12) Slope of the line y = 5x + 3 is: Step 1: The equation of a line in slope-intercept form is y = mx + c, where m is the slope and c is the y-intercept. Compare y = 5x + 3 with y = mx + c. The slope m is 5. The correct option is B) 5. 13) An equilateral triangle: An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal to 60^. • can be isosceles: An equilateral triangle is a special case of an isosceles triangle (where at least two sides are equal). • can be right angled: A right-angled triangle has one angle of 90^. An equilateral triangle has all angles 60^, so it cannot be right-angled. • can be obtuse angled: An obtuse-angled triangle has one angle greater than 90^. An equilateral triangle has all angles 60^, so it cannot be obtuse-angled. • has each angle equal to 50^: This is incorrect; each angle is 60^. The most accurate description among the choices is that an equilateral triangle can be isosceles. The correct option is A) can be isosceles. 14) Which data takes only some specific values? • Continuous data: Can take any value within a given range (e.g., height, temperature). • Discrete data: Can only take specific, distinct values (e.g., number of students, shoe size). • Grouped data: Data organized into groups or intervals. • Ungrouped data: Raw data not yet organized. Data that takes only some specific values is discrete data. The correct option is B) Discrete data. 15) While rolling a pair of dice, what will be the probability of double 2? Step 1: When rolling a pair of dice, the total number of possible outcomes is 6 × 6 = 36. Step 2: "Double 2" means both dice show a 2. There is only one outcome for this: (2, 2). Step 3: The probability is the number of favorable outcomes divided by the total number of outcomes. P(double 2) = (1)/(36) The correct option is A) (1)/(36). Drop the next question.