(a) A see-saw pivoted at the middle is kept in balance by weights of Richard, John and Philip such that only Richard whose mass is 60 kg sits on one side. If they sit at distances 2 m , 3 m , and 4 m respectively from the pivot and Philip is 15 kg, find the mass of John.
(bi) A body of mass 12 kg rests on a rough plane inclined at an angle of 30ΒΊ to the horizontal. The coefficient of friction between the body and the plane is \(\frac{2}{3}\). A force of magnitude P Newton acts on the body along the inclined plane. Find the value of P, if the body is at the point of moving:
down the plane;
[Take \(g = 10 ms ^{-2}\)]
(bii) A body of mass 12 kg rests on a rough plane inclined at an angle of 30ΒΊ to the horizontal. The coefficient of friction between the body and the plane is \(\frac{2}{3}\). A force of magnitude P Newton acts on the body along the inclined plane. Find the value of P, if the body is at the point of moving:
up the plane;
[Take \(g = 10 ms ^{-2}\)]
(a) A particle of mass 2 kg moves under the action of a constant force, F N , with an initial velocity \((3 i + 2 j ) ms^{ -1}\) and a velocity of \((15 i – 4 j ) ms^{ -1}\) after 4 seconds . Find the:
acceleration of the particle;
(b) A particle of mass 2 kg moves under the action of a constant force, F N , with an initial velocity \((3 i + 2 j ) ms^{ -1}\) and a velocity of \((15 i – 4 j ) ms^{ -1}\) after 4 seconds . Find the:
magnitude of the force F ;
(c) A particle of mass 2 kg moves under the action of a constant force, F N , with an initial velocity \((3 i + 2 j ) ms^{ -1}\) and a velocity of \((15 i – 4 j ) ms^{ -1}\) after 4 seconds . Find the:
magnitude of the velocity of the particle after 8 seconds , correct to three decimal places.
(ai) A bag contains 16 identical balls of which 4 are green. A boy picks a ball at random from the bag and replaces it. If this is repeated 5 times, what is the probability that he:
did not pick a green ball;
(aii) A bag contains 16 identical balls of which 4 are green. A boy picks a ball at random from the bag and replaces it. If this is repeated 5 times, what is the probability that he:
picked a green ball at least three times?
(b) The deviations from a mean of values from a set of data are \(-2, ( m – 1), ( m ^2 + 1), -1, 2, (2 m – 1)\) and \(-2\). Find the possible values of \(m\) .
(a) The table shows the distribution of marks scored by some candidates in an examination.
| Marks | 11 – 20 | 21 – 30 | 31 – 40 | 41 – 50 | 51 – 60 | 61 – 70 | 71 – 80 | 81 – 90 |
91 – 100 |
| Num of candidates | 5 | 39 | 14 | 40 | 57 | 25 | 11 | 8 | 1 |
Construct a cumulative frequency table for the distribution.
(b) The table shows the distribution of marks scored by some candidates in an examination.
| Marks | 11 – 20 | 21 – 30 | 31 – 40 | 41 – 50 | 51 – 60 | 61 – 70 | 71 – 80 | 81 – 90 |
91 – 100 |
| Num of candidates | 5 | 39 | 14 | 40 | 57 | 25 | 11 | 8 | 1 |
Draw a cumulative frequency curve for the distribution.
(ci) Use the curve to estimate the:
number of candidates who scored marks between 24 and 58 ;
(cii) Use the curve to estimate the:
lowest mark for distinction, if 12% of the candidates passed with distinction.
(a) Express \(\frac{8x^2 + 8x + 9}{(x – 1)(2x + 3)^2}\) in partial fractions.
(b) The coordinates of the centre and circumference of a circle are (-2, 5) and 6Ο units respectively. Find the equation of the circle.
(ai) A quadratic polynomial, g (x) has (2x + 1) as a factor. If g (x) is divided by (x – 1) and (x – 2), the remainder are -6 and -5 respectively. Find;
g (x);
(aii) A quadratic polynomial, g (x) has (2x + 1) as a factor. If g (x) is divided by (x – 1) and (x – 2), the remainder are -6 and -5 respectively. Find;
the zeros of g (x).
(b) Find the third term when (\(\frac{x}{2}-1\))\(^8\)is expanded in descending powers of \(x\).
(a) Find the derivative of \(4x-\frac{7}{x^2}\)with respect to \(x\), from first principle.
(b) Given that tan \(P =\frac{3}{x – 1}\) and tan \(Q\) =\frac{2}{x + 1}\), find tan \(( P – Q )\)
P is the mid-point of \(\overline{NO}\) and equidistant from \(\overline{MN}\) and \(\overline{MO}\) . If \(\overline{MN}\) = 8i + 3j and \(\overline{MO}\) = 14i – 5j, find \(\overline{MP}\) .
(a) A bus travels with a velocity of \(6 ms ^{-1}\). It then accelerates uniformly and travels a distance of 70 m. If the final velocity is \(20 ms ^{-1}\), find, correct to one decimal place, the:
acceleration;
(b) A bus travels with a velocity of \(6 ms ^{-1}\). It then accelerates uniformly and travels a distance of 70 m. If the final velocity is \(20 ms ^{-1}\), find, correct to one decimal place, the:
time to travel this distance.
There are 6 boys and 8 girls in a class. If five students are selected from the class, find the probability that more girls than boys are selected
(a)The table shows the distribution of heights ( cm ) of 60 seedlings in a vegetable garden.
| Heights(cm) | 0.1 – 0.3 | 0.4 – 0.6 | 0.7 – 0.9 | 1.0 – 1.4 | 1.5 – 1.9 | 2.0 – 22 | 2.3 – 2.5 |
| Frequency | 6 | 9 | 12 | 15 | 3 | 6 | 9 |
Draw a histogram for the distribution.
(b) The table shows the distribution of heights ( cm ) of 60 seedlings in a vegetable garden.
| Heights(cm) | 0.1 – 0.3 | 0.4 – 0.6 | 0.7 – 0.9 | 1.0 – 1.4 | 1.5 – 1.9 | 2.0 – 2.2 | 2.3 – 2.5 |
| Frequency | 6 | 9 | 12 | 15 | 3 | 6 | 9 |
Use the histogram to estimate the modal height of the seedlings.
(a) The first term of an Arithmetic Progression is -8, the last term is 52 and the sum of terms is 286. Find the:
number of terms in the series;
(b) The first term of an Arithmetic Progression is -8, the last term is 52 and the sum of terms is 286. Find the:
common difference.
(a) The inverse of a function \(f\) is given by \(f^{-1}(x)=\frac{5x – 6}{4 – x},x β 4\).Find the:
function, \(f (x)\)
(b) The inverse of a function \(f\) is given by \(f^{-1}(x)=\frac{5x – 6}{4 – x},x β 4\).Find the:
value of x for which \(f (x) = 5\)
The volume of a cube is increasing at the rate of \(3\frac{1}{2} cm ^3 s^{ -1}\). Find the rate of change of the side of the base when its length is 6 cm .
If \((x – 5)\) is a factor of \(x^3 – 4x^2 – 11x + 30\), find the remaining factors.
- A. \((x + 3) and (x - 2)\)
- B. \((x - 3) and (x + 2)\)
- C. \((x - 3) and (x - 2)\)
- D. \((x + 3) and (x + 2)\)
In how many ways can four Mathematicians be selected from six ?
- A. 90
- B. 60
- C. 15
- D. 360
Find the coefficient of the \(6^{th}term\) in the binomial expansion of \((1 – \frac{2x}{3})10\) in ascending powers of \(x\).
- A. \(-\frac{896x^6}{9}\)
- B. \(-\frac{896x^5}{9}\)
- C. \(-\frac{896x^5}{27}\)
- D. \(-\frac{896x^6}{27}\)
If m and ( m + 4) are the roots of \(4x^2 – 4x – 15 = 0\), find the equation whose roots are 2 m and (2 m + 8).
- A. \(x^2+8x-15=0\)
- B. \(x^2-2x-15=0\)
- C. \(x^2-8x-15=0\)
- D. \(x^2+2x+15=0\)
Given that \(p = \begin{bmatrix} x&4\\3&7\end{bmatrix} Q =\begin{bmatrix} x&3\\1&2x\end{bmatrix}\) and the determinant of \(Q\) is three more than that of \(P\) , find the values of \(x\).
- A. \(-2,\frac{3}{2}\)
- B. \(2,\frac{3}{2}\)
- C. \(-2,-\frac{3}{2}\)
- D. \(2-,\frac{3}{2}\)
The probabilities that Atta and Tunde will hit a target in a shooting contest are \(\frac{1}{6}\) and \({1}{9}\) respectively. Find the probability that only one of them will hit the target.
- A. \(\frac{1}{54}\)
- B. \(\frac{41}{54}\)
- C. \(\frac{20}{27}\)
- D. \(\frac{13}{54}\)