Calculate the magnitude of vector (3,4), find the resultant of (2,3) and (4,1), add matrices [[1,2],[3,4]] and [[2,0],[1,5]], and find the determinant of [[1,2],[3,4]].

Here are the solutions to the problems: Vectors 1. Find magnitude of vector (3,4) Step 1: The magnitude of a vector (x,y) is given by the formula sqrt(x^2 + y^2). Step 2: Substitute the values x=3 and y=4 into the formula. Magnitude = sqrt(3^2 + 4^2) Magnitude = sqrt(9 + 16) Magnitude = sqrt(25) Magnitude = 5 The magnitude of the vector is 5. 2. Find resultant of (2,3) and (4,1) Step 1: To find the resultant of two vectors, add their corresponding components. Let v_1 = (2,3) and v_2 = (4,1). Step 2: Add the x-components and the y-components separately. Resultant = (2+4, 3+1) Resultant = (6,4) The resultant vector is (6,4). Matrices 1. Add matrices 1 & 2 \\ 3 & 4 and 2 & 0 \\ 1 & 5 Step 1: To add matrices, add the elements in the corresponding positions. 1 & 2 \\ 3 & 4 + 2 & 0 \\ 1 & 5 = 1+2 & 2+0 \\ 3+1 & 4+5 Step 2: Perform the additions. = 3 & 2 \\ 4 & 9 The sum of the matrices is 3 & 2 \\ 4 & 9 . 2. Find determinant of 2 & 3 \\ 1 & 4 Step 1: For a 2 × 2 matrix a & b \\ c & d , the determinant is calculated as ad - bc. Step 2: Substitute the values a=2, b=3, c=1, and d=4. Determinant = (2 × 4) - (3 × 1) Determinant = 8 - 3 Determinant = 5 The determinant of the matrix is 5. Probability 1. Find probability of getting a head when tossing a coin Step 1: Identify the total number of possible outcomes. When tossing a fair coin, the possible outcomes are Head (H) or Tail (T). So, there are 2 possible outcomes. Step 2: Identify the number of favorable outcomes. Getting a head is 1 favorable outcome. Step 3: Calculate the probability using the formula: Probability = (Number of favorable outcomes) / (Total number of possible outcomes). P(Head) = (1)/(2) The probability of getting a head is (1)/(2). 2. A bag has 3 red and 2 blue balls. Find probability of picking red Step 1: Determine the total number of balls in the bag. Total balls = Number of red balls + Number of blue balls Total balls = 3 + 2 = 5 balls. Step 2: Identify the number of favorable outcomes (picking a red ball). Number of red balls = 3. Step 3: Calculate the probability. P(Picking Red) = Number of red ballsTotal number of balls P(Picking Red) = (3)/(5) The probability of picking a red ball is (3)/(5).