Section A: Calculus (25 marks) Question 1: Differentiate the following functions

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Section A: Calculus (25 marks)
Question 1: Differentiate the following functions

Section A: Calculus (25 marks)
Question 1: Differentiate the following functions:�=�3+4�2−2�+5y=x3+4×2−2x+5
�=��ln⁡(�)y=exln(x)
Question 2: Integrate the following functions:∫(3�2−2�+4) ��∫(3×2−2x+4)dx
∫1�2 ��∫x21dx
Question 3: Find the limit:lim⁡�→2�2−4�−2limx→2x−2×2−4
Question 4: Solve the differential equation:����=2�2+3�dxdy=2×2+3x with �(0)=1y(0)=1
Section B: Algebra (20 marks)
Question 5: Solve the quadratic equation �2−5�+6=0x2−5x+6=0.
Question 6: Find the inverse of the matrix:
�=[1234]A=[1324]
Question 7: Simplify the expression:
3�3+4�2−2�+5�2−3�+2×2−3x+23×3+4×2−2x+5
Question 8: Solve the complex equation �2+2�+2=0z2+2z+2=0.
Section C: Geometry and Trigonometry (30 marks)
Question 9: Find the area of a triangle with vertices at (1, 2), (4, 6), and (7, 3).
Question 10: Determine the equation of the circle with center (2, -3) and radius 5.
Question 11: Find the angle between the lines �=3�+4y=3x+4 and �=−2�+1y=−2x+1.
Question 12: Solve the trigonometric equation sin⁡2(�)+cos⁡(�)=0sin2(x)+cos(x)=0.
Section D: Probability and Statistics (25 marks)
Question 13: A die is rolled. Find the probability of getting a prime number.
Question 14: Calculate the mean and standard deviation of the following data set: 10, 12, 15, 18, 20.
Question 15: Perform a hypothesis test to determine if the mean weight of a sample is significantly different from 150 pounds, given a sample size of 25 and a sample mean of 155 pounds.
Ensure that each question is clearly presented and includes the necessary instructions. You can adjust the difficulty level of questions according to your preference or the intended audience. Additionally, provide adequate space for students to write their answers and show their work.

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