The probability of getting at least one head in a toss of two coins is:

Here are the solutions to the questions: Question 9: To find the probability of getting at least one head in a toss of two coins, list all possible outcomes. Step 1: List all possible outcomes when tossing two coins. The possible outcomes are: (Head, Head), (Head, Tail), (Tail, Head), (Tail, Tail). Total number of outcomes = 4. Step 2: Identify the outcomes with at least one head. The outcomes with at least one head are: (Head, Head), (Head, Tail), (Tail, Head). Number of favorable outcomes = 3. Step 3: Calculate the probability. Probability = Number of favorable outcomesTotal number of outcomes = (3)/(4) The correct option is C. C) (3)/(4) Question 10: Find the missing term in the pattern 2, 3, 7, 10, 15, ..., 32. Let's analyze the differences between consecutive terms: 3 - 2 = 1 7 - 3 = 4 10 - 7 = 3 15 - 10 = 5 The sequence of differences is 1, 4, 3, 5. This pattern suggests an alternating sequence of differences: the odd-positioned differences are 1, 3, 5, (increasing by 2), and the even-positioned differences are 4, . Step 1: Apply the differences to find the terms. a_1 = 2 a_2 = 2 + 1 = 3 a_3 = 3 + 4 = 7 a_4 = 7 + 3 = 10 a_5 = 10 + 5 = 15 Step 2: Find the missing term (a_6) using the next difference in the pattern (which would be 7, following 1, 3, 5). a_6 = 15 + 7 = 22 Step 3: Check if the next term (a_7) matches the given last term (32) using the next difference (which would be 10, following 4, and assuming an increase of 3 for even-positioned differences, i.e., 4, 7, 10). a_7 = 22 + 10 = 32 This pattern perfectly matches the given sequence and the last term. The missing term is 22. The correct option is B. B) 22 Question 11: Find the percentile rank of 6 for the data set: 2, 3, 5, 6, 7, 7, 9, 12. Step 1: Arrange the data in ascending order (it is already arranged). Data set: 2, 3, 5, 6, 7, 7, 9, 12. Step 2: Count the total number of data points (N). N = 8. Step 3: Count the number of data points less than or equal to 6 (C). The data points less than or equal to 6 are 2, 3, 5, 6. C = 4. Step 4: Calculate the percentile rank using the formula: Percentile Rank = Number of values less than or equal to xTotal number of values × 100. Percentile Rank = (4)/(8) × 100\% = (1)/(2) × 100\% = 50\% The correct option is C. C) 50\% Question 12: Convert (3)/(5) radians to degrees. Step 1: Use the conversion factor 1 radian = (180^)/(). (3)/(5) radians = (3)/(5) × (180^)/() Step 2: Simplify the expression. = (3 × 180^)/(5) = 3 × 36^ = 108^ The correct option is A. A) 108^ Question 13: Evaluate the definite integral _1^3 (3x^2 + 2x + 5)\,dx. Step 1: Find the antiderivative of the function. (3x^2 + 2x + 5)\,dx = 3(x^3)/(3) + 2(x^2)/(2) + 5x + C = x^3 + x^2 + 5x + C Step 2: Evaluate the antiderivative at the upper and lower limits and subtract. [x^3 + x^2 + 5x]_1^3 = (3^3 + 3^2 + 5 × 3) - (1^3 + 1^2 + 5 × 1) = (27 + 9 + 15) - (1 + 1 + 5) = 51 - 7 = 44 The correct option is B. B) 44 Question 14: A student has gotten the following grades on his test: 87, 95, 76, and 88. He wants an 85 or better overall. Find the minimum grade he must get on the last test to achieve that average. Step 1: Let x be the grade on the last test. The total number of tests is 5. Step 2: Set up the inequality for the average grade. (87 + 95 + 76 + 88 + x)/(5) 85 Step 3: Sum the known grades. 87 + 95 + 76 + 88 = 346 Step 4: Substitute the sum into the inequality and solve for x. (346 + x)/(5) 85 346 + x 85 × 5 346 + x 425 x 425 - 346 x 79 The minimum grade he must get on the last test is 79. The correct option is D. D) 79 Question 15: Find the exact value of (7-2i)(7+2i). Step 1: Recognize this as a product of complex conjugates, which follows the form (a-bi)(a+bi) = a^2 + b^2. Here, a=7 and b=2. Step 2: Apply the formula. (7-2i)(7+2i) = 7^2 + 2^2 = 49 + 4 = 53 The correct option is A. A) 53 Question 16: Find the argument of the complex number 5 + 2i. The argument of a complex number z = x + yi is given by = (y)/(x). For z = 5 + 2i, = (2)/(5), so = ((2)/(5)). However, the given options are A) 5, B) sqrt(29), C) sqrt(21), D) 2. None of these are angles. It is highly probable that the question intended to ask for the modulus of the complex number, not the argument, given the options. Step 1: Calculate the modulus of the complex number z = x + yi, which is |z| = sqrt(x^2 + y^2). For z = 5 + 2i, x=5 and y=2. |z| = sqrt(5^2 + 2^2) |z| = sqrt(25 + 4) |z| = sqrt(29) Assuming the question meant to ask for the modulus, the correct option is B. B) sqrt(29) Question 17: Given f(x) = 3x^3 - 8x^2 - 5, find f''(1). Step 1: Find the first derivative, f'(x). f'(x) = (d)/(dx)(3x^3 - 8x^2 - 5) = 9x^2 - 16x Step 2: Find the second derivative, f''(x). f''(x) = (d)/(dx)(9x^2 - 16x) = 18x - 16 Step 3: Evaluate f''(x) at x=1. f''(1) = 18(1) - 16 = 18 - 16 = 2 The correct option is C. C) 2 Question 18: Which of the following formulas best describes the formula for standard deviation of a grouped data? The standard deviation for grouped data is typically given by the formula: s = sqrt( f(x - x))^2 f where f is the frequency, x is the midpoint of the class interval, and x is the mean. This formula represents the population standard deviation for grouped data. Let's examine the given options: A) sqrt( (x - x))^2n (This is for ungrouped data, typically sample standard deviation if n-1 is in denominator) B) f(x - x)n (This is not the formula for standard deviation) C) sqrt( f(x - x))^2 f (This is the population standard deviation for grouped data) D) sqrt( f(x - x))^2 f - 1 (This is the sample standard deviation for grouped data) Option C is a standard formula for the standard deviation of grouped data (specifically, the population standard deviation). The correct option is C. C) sqrt( f(x - x))^2 f Question 19: Express 14 × 13 × 12 × 11 as factorial form. Step 1: Recall the definition of factorial: n! = n × (n-1) × × 1. Step 2: To express the given product in factorial form, we need to multiply and divide by the missing terms to complete the factorial. 14 × 13 × 12 × 11 = (14 × 13 × 12 × 11 × (10 × 9 × × 1))/(10 × 9 × × 1) = (14!)/(10!) The correct option is D. D) (14!)/(10!) Question 20: Simplify sqrt(1 - ^2 x). Step 1: Use the Pythagorean trigonometric identity ^2 x + ^2 x = 1. Step 2: Rearrange the identity to solve for ^2 x. ^2 x = 1 - ^2 x Step 3: Substitute this into the given expression. sqrt(1 - ^2 x) = sqrt(^2 x) Step 4: Simplify the square root. sqrt(^2 x) = | x| In multiple-choice questions, often the absolute value is omitted if the context implies x 0 or if it's a general simplification. Given the options, x is the intended answer. The correct option is A. A) x 3 done, 2 left today. You're making progress.