Section 1

Section 1: Calculus 1. Compute the derivative of � ( � )=3 � 4−2 � 2+7 f ( x )=3 x 4 −2 x 2 +7 . Answer: � ′( � )=12 � 3−4 � f ′ ( x )=12 x 3 −4 x . 2. Evaluate the integral ∫02(4 � 3−6 � ) ��∫ 02 (4 x 3 −6 x ) dx . Answer: ∫02(4 � 3−6 � ) �� =8 ∫ 02 (4 x 3 −6 x ) dx =8 . Section 2: Linear Algebra 3. Solve the system of equations: 2 � +3 � =114 � − � =52 x +3 y 4 x − y =11=5 Answer: � =2, � =3 x =2, y =3 . 4. Find the determinant of the matrix: [35−24] [ 3−254 ] Answer: Determinant = 2222 . Section 3: Probability and Statistics 5. If P(A) = 0.6 and P(B) = 0.4, what is P(A or B) if A and B are independent events? Answer: � ( �  or  � )=0.76 P ( A  or  B )=0.76 . 6. Calculate the mean of the following data set: 5, 7, 9, 12, 15. Answer: Mean = 9.69.6 . Section 4: Discrete Mathematics 7. Find the number of ways to arrange the letters in the word "MATHEMATICS." Answer: Number of arrangements = 11!/2!2!11!/2! 2! . 8. If there are 15 people in a room, how many handshakes occur if each person shakes hands with every other person once? Answer: Number of handshakes = 105105 . Section 5: Differential Equations 9. Solve the differential equation ���� =3 � 2 dxdy =3 x 2 . Answer: � = � 3+ � y = x 3 + C , where � C is the constant of integration. 10. Solve the initial value problem: ���� =2 � dxdy =2 y , � (0)=4 y (0)=4 . Answer: � =4 � 2 � y =4 e 2 x . Section 6: Geometry