(a) A boy runs in a line and his displacement at time t seconds after leaving the start point O is X metres, where 20X = 4t\(^2\) + t\(^3\). Find the:
(i) velocity of the body when t = 15 seconds (ii) value of t for which the acceleration of the body is 8 times his initial acceleration
(b) A body of mass 6 kg moves with a velocity of 7 ms\(^{-1}\). It collides with a second body moving in the opposite direction with a velocity of 5 ms\(^{-1}\). After collision, the two bodies move together with a velocity of 4 ms\(^{-1}\). Find the mass of the second body.
PART II
A particle of weight 12 N lying on a horizontal ground is acted by forces F\(_1\) = (10 N, 090º), F\(_2\) = (16 N, 180º), F\(_3\) = (7 N, 300º) and F\(_4\) = (12N, 030º)
(a) Express all the forces acting on the particle as column vectors
(b) Find, correct to two decimal places, the magnitude of the:
(i) resultant forces;
(ii) acceleration with which the particle starts to move.[Take g = 10 ms\(^{-2}\)]
In an examination, 70% of the candidates passed. If 12 candidates are selected at random, find the probability that:
(a) at least two of them failed;
(b) exactly half of them passed;
(c) not more than one – six of them failed.
The data shows the ordered marks scored by students in a test: 11, 12, (2x + y), (x + 2y), 14, and ((y\(^2\) – 2x). Given that the median is 13\(\frac{1}{2}\) and y is greater than x by 1, find:
(a) the values of x and y
(b) correct to three significant figures, the standard deviation of the distribution.
(a) Express \(\frac{9x}{(2x + 1)(x^2 + 1)}\) in partial fraction
(b) If \(^{2m}P_2\) – 10 = \(^m P_2\), find the positive value of m.
Three linear transformations, P, Q, and R in the oxy plane are defined by
P: (x, y) → (-4x – y, 2x)
Q: (x, y) → (y, 6x – 9y)
R: (x, y) → (x – 2y, 3x + 5y)
(a) write down the matrices of P, Q, and R
(b) Find:
(i) 2P – 3R + Q;
(ii) QR;
(iii) the inverse of the matrix R.
SECTION B (PART 1)
A curve is given by y = 8x + \(\frac{27}{2x^2}\),
FIND:
(a) an expression for \(\frac{dy}{dx}\),
(b) the coordinates of the stationary point on the curve and the nature of the stationary point;
(c) the equation of the normal to the curve at (2, 2).
A body of mass 40 kg is placed on a rough inclined plane which makes an angle of 30\(^0\) with the horizontal. If a force of 420 N is applied upwards parallel to the plane. find the:
(a) maximum friction force that will keep the body in equilibrium;
(b) coefficient of friction.[Take g = 10ms\(^{-1}\)
The magnitude of two vectors u and v are 10N and 12N respectively. If the magnitude of their resultant is 15N, calculate the angle between them.
The table shows the distribution of distance (in km) of 60 villages from a state capital.
| Distance (in km) | 0 – 19 | 20 – 29 | 30 – 39 | 40 – 49 | 50 – 59 | 70 – 99 | 100 – 149 |
| Number of villages | 12 | 7 | 6 | 8 | 5 | 9 | 10 |
Two events M and N are such that P(M) = \(\frac{1}{2}\), P(N) = \(\frac{9}{20}\) and P(M ∩ N) = \(\frac{11}{50}\)
(a) P(M ∩ N’): (b) P(M’ ∩ N)
Find the equation of the normal to the curve y = 7x – 5x\(^2\) at x = 2
Using the trapezium rule with seven ordinates, evaluate, correct to three decimal places, \(\int_{2.4}^{3.6} \frac{1}{\sqrt{x^2 – 2}}\)dx
Find the sum of all natural numbers between 403 and 603 which are divisible by 7
If tan x = \(\frac{1}{3}\), where 180º < x < 270º
evaluate \(\frac{sin2 x – cos x}{2 tan x + sin 2x}\), leaving the answer in surd form (radicals)
Find the sum of the first 20 terms of the sequences, -7, – 3, 1, . . . . .
- A. 620
- B. 660
- C. 690
- D. 1240
In a truth table, if p is true and q is false, which of the following notations is false
- A. [(p ⇒ q) ^ q] ⇒ q
- B. [(p ⇒ q) ^ q] ⇔ q
- C. [(p ⇒ q) ^ q] ^ q
- D. [(p ⇒ q) ^ q] v ~ q
The parents of 7 out of every 10 students in a class are farmers. If 12 students were selected at random, find the probability that the parents of 8 of them will be farmers.
- A. 0.0078
- B. 0.0231
- C. 0.0780
- D. 0.2311
The line x + y + 4 = 0 makes an angle of \(\theta\) with the x-axis. Find the value of \(\theta\)
- A. 315\(^0\)
- B. 225\(^0\)
- C. 135\(^0\)
- D. 45\(^0\)
The scores of some students in a class test are 4, 6, 1, 8, 9, 5, and 2. Calculate, correct to one decimal place, the mean deviation of their scores.
- A. 2.3
- B. 2.7
- C. 5.0
- D. 5.2
Given that y = 2x – 1 and Δx = 0.1, find Δ y
- A. 0.20
- B. 0.15
- C. 0.10
- D. 0.05