Determine the respective values of 𝑝 𝑎𝑛𝑑 𝑞 if (𝑥 − 1) 𝑎𝑛𝑑 (𝑥 + 2) are factors of 2𝑥3 + 𝑝𝑥2 − 𝑥 + 𝑞 = 0
A. −5, 6
B. 6, 5
C. 5, −6
D. 5, 6
2. If 𝛼 𝑎𝑛𝑑 𝛽 are the roots of the quadratic equation 𝑎𝑥2 + 𝑏𝑥 + 𝑐 =0, obtain the equation whose roots are 1/α 𝑎𝑛𝑑 1/β
A. 𝑐𝑥2 + 𝑏𝑥 + 𝑎
B. 𝑏𝑥2 + 𝑐𝑥 + 𝑎
C. 𝑐𝑥2 + 𝑎𝑏𝑥 + 1
D. 𝑥2 + 𝑏𝑥 + 𝑎
3. Solve the inequality 2𝑥2 + 3𝑥 − 2 ≤ 0
A. (2, 1/2)
B. [−2, 1/2]
C. [2, 1/2]
D. (−2, 1/2)
4. The sum of the first 𝑛 terms of a geometric series is 127 and the sum of their reciprocals is 127/64, if the first term is 1, find 𝑛 and the common ratio.
A. 5, 2
B. 7, 2
C. 2, 7
D. 5, 7
5. Solve for tan 𝜃 in the equation 7𝑠𝑒𝑐2𝜃 = 6𝑡𝑎𝑛𝜃 + 8
A. 1, 1/7
B. 1, - 1/7
C. 2, 1/7
D. -2, 1/7
6. Given that 𝐴 = {𝑥 ∶ 𝑥 < 4, 𝑥 𝑖𝑠 𝑎 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟} and 𝐵 = {𝑥 ∶ 𝑥 ≥ 2, 𝑥 𝑖𝑠 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟}. Deduce (𝐴 ∩ 𝐵) × (𝐴 ∖ 𝐵).
A. {(1,2), (1,3)}
B. {(2,3), (2,1)}
C. {(2,1), (3,1)}
D. {(3,2), (1,2)}7. An ellipse is a conic section whose eccentricity (e) is _____
A. e = 1
B. 0 < e < 1
C. e > 1
D. e > 2
8. If 𝑥 = 𝑡2 + 1 and 𝑦 = 𝑡3 + 2, find 𝑦′.
A. 2𝑡
B. 3𝑡
C. 3𝑡/2
D. 2𝑡/3
9. Evaluate ∫1-1𝑥2(𝑥3+1)𝑑𝑥
A. 2/3
B. 1/3
C. -2
D. 2
10. Compute limx→4{𝑥−4√𝑥−2}
A. 2
B. -4
C. 4
D. 0
11. Solve the differential equation:
A. 3𝑦2 - x3 = C
B. 2y2 - 𝑥2 + 𝐶 = 0
C. 3y2 – 2x3 = C
D. 2y3 – 3x2 = C
12. The following are turning points except _____
A. maximum point
B. minimum point
C. inflexion point
D. maximum and minimum
13. If y = 1/𝑥 then the nth derivative of y with respect to x equals ____
14. Integrate the function 𝑥𝑒𝑥
A. 𝑒𝑥 (𝑥 − 1)
B. 𝑒𝑥 (𝑥 + 1)
C. 𝑥𝑒𝑥 (𝑥 + 1)
D. 𝑥 (𝑒𝑥 − 1)
15. The motion of a particle along a straight line is specified by the equation x = 4t4 – 3t3, evaluate the velocity after 3 seconds.
A. 513𝑚𝑠−1
B. 378𝑚𝑠−2
C. 351𝑚𝑠−1
D. 486𝑚𝑠−2
16. Given that 𝑎 = 𝒊 + 𝒋 − 2𝒌, 𝑏 = 𝒊 + 𝒌, 𝑐 = 2𝒊 − 𝒋 + 3𝒌. Evaluate |2𝑎 + 𝑏 + 2𝑐|
A. √85
B.√58
C. ±√58
D. −√85
17. A body of mass 6kg is suspended at a position P by two light inextensible strings AP and BP which are inclined respectively at angle 20° and 60°, and to the upward vertical. If the system is in equilibrium; calculate the total tension in the string.
A. 62.75N
B. 87.71N
C. 32.75N
D. 25.57N
18. A particle moving along the centre defined by 𝑟 = 𝑎 cos 𝜔𝑡 𝒊 + 𝑎 sin 𝜔𝑡 𝒋. Determine its speed at any time 𝑡.
A. – 𝑎𝜔 cos 𝜔𝑡 𝒊 + 𝑎𝜔 sin 𝜔𝑡 𝒋
B. – 𝑎𝜔
C. 𝑎𝜔
D. – 𝜔2𝑟
19. Two forces 10N and 6N act in the direction 060o and 330o respectively. Find the x-component of their resultant.
A. 5√3−3
B. 3−5√3
C. 5−3√3
D. 3√3−5
20. 'Negative Correlation' means _____
A. Linear dependence
B. Direct relationship
C. Inverse relationship
D. No relationship
21. In how many ways can a committee of 5 be formed from a group of 8 people consisting of 3 boys, 3 girls and a brother-sister pair; if the committee must include the brother-sister pair?
A. 40
B. 20
C. 56
D. 36
22. On a final examination in Mathematics, the mean was 72 and the standard deviation was 15. Determine the standard score of students receiving the grade of 60.
A. -0.5
B. 1
C. -0.8
D. 2.5
23. Compute the expected number of males and its standard deviation in a random sample of 100 families.
A. 25, 5
B. 50, 25
C. 50, 5
D. 10, 5
24. The equation is called _____
A. a correlation equation.
B. a prediction equation.
C. a linear regression equation.
D. a non-linear regression equation.
25. An unbiased six sided die has the score 2 engraved on two sides, the score 4 on three sides and score 6 on the remaining one side. The die is thrown once. Compute the probability of obtaining an odd score.
A. 0
B. 1
C. 2/6
D. 5/6
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