Find the cardinality of the set A=x | -3 < x < 33.

Here are the solutions to the questions: Question 4: To find the cardinality of the set A = \x : -3 < x < 33\, we need to count the number of integers between -3 and 33 (exclusive). The integers in the set are -2, -1, 0, 1, , 32. Step 1: Identify the smallest and largest integers in the set. The smallest integer is -2. The largest integer is 32. Step 2: Calculate the cardinality using the formula: (Largest integer - Smallest integer) + 1. Cardinality = 32 - (-2) + 1 Cardinality = 32 + 2 + 1 Cardinality = 35 The correct option is C. The final answer is C. 35. Question 5: Given that = 70^ where is an acute angle. Step 1: Use the trigonometric identity = (90^ - ). So, we can write the given equation as: (90^ - ) = 70^ Step 2: Since is an acute angle, 90^ - must also be an acute angle. Therefore, we can equate the angles. 90^ - = 70^ Step 3: Solve for . = 90^ - 70^ = 20^ The correct option is C. The final answer is C. 20^. Question 6: A fair die is rolled once. We need to find the probability of obtaining a 3 or 4. Step 1: Identify the total possible outcomes when rolling a fair die. The possible outcomes are \1, 2, 3, 4, 5, 6\. Total number of outcomes = 6. Step 2: Identify the favorable outcomes (obtaining a 3 or 4). The favorable outcomes are \3, 4\. Number of favorable outcomes = 2. Step 3: Calculate the probability. P(3 or 4) = Number of favorable outcomesTotal number of outcomes P(3 or 4) = (2)/(6) P(3 or 4) = (1)/(3) The correct option is A. The final answer is A. (1)/(3). Question 7: A bag contains two black, three green, and four red balls. A ball is picked at random. We need to find the probability that it is either black or red. Step 1: Determine the total number of balls in the bag. Number of black balls = 2 Number of green balls = 3 Number of red balls = 4 Total number of balls = 2 + 3 + 4 = 9 Step 2: Determine the number of favorable outcomes (balls that are either black or red). Number of black or red balls = Number of black balls + Number of red balls Number of black or red balls = 2 + 4 = 6 Step 3: Calculate the probability. P(black or red) = Number of black or red ballsTotal number of balls P(black or red) = (6)/(9) P(black or red) = (2)/(3) The correct option is B. The final answer is B. (2)/(3). Question 8: Find the sum to infinity of a Geometric Progression whose first term is 10 and common ratio (2)/(3). Step 1: Recall the formula for the sum to infinity of a Geometric Progression. The sum to infinity S_ is given by S_ = (a)/(1-r), where a is the first term and r is the common ratio, provided |r| < 1. Step 2: Identify the given values. First term a = 10. Common ratio r = (2)/(3). Since |(2)/(3)| < 1, the sum to infinity exists. Step 3: Substitute the values into the formula and calculate. S_ = (10)/(1 - 2)3 S_ = (10)/(3)3 - (2)/(3) S_ = (10)/(1)3 S_ = 10 × 3 S_ = 30 The correct option is C. The final answer is C. 30. Question 9: The length and breadth of a rectangular piece of land are (x+2)m and (x-3)m respectively. The perimeter of the land is 30 m. We need to find the value of x. Step 1: Write down the given dimensions and the perimeter formula for a rectangle. Length L = (x+2) m Breadth B = (x-3) m Perimeter P = 2(L+B) Given P = 30 m. Step 2: Substitute the expressions for length and breadth into the perimeter formula. 30 = 2((x+2) + (x-3)) Step 3: Simplify and solve for x. 30 = 2(x+2+x-3) 30 = 2(2x-1) Divide both sides by 2: (30)/(2) = 2x-1 15 = 2x-1 Add 1 to both sides: 15 + 1 = 2x 16 = 2x Divide both sides by 2: x = (16)/(2) x = 8 The correct option is A. The final answer is A. 8 m. 3 done, 2 left today. You're making progress.