JAMB UTME - Mathematics - 2009
Swali 1 Ripoti
If \(P = \begin{bmatrix} x+3 & x+2 \\ x+1 & x-1 \end{bmatrix}\) evaluate x if \(|P| = -10\)
Maelezo ya Majibu
(x+3)(x-1) - {(x+1)(x+2)} = -10
x2 - x + 3x - 3 - {x2 + 2x + x + 2} = -10
x2 + 2x - 3 - {x2 + 3x + 2} = -10
-x - 5 = -10
-5 + 10 = x
5 = x
∴x = 5
Swali 2 Ripoti
Maelezo ya Majibu
= -5 - (3+2)
= -5-5
= -10
Swali 3 Ripoti
Maelezo ya Majibu
52x - 1 x 5x + 1 = 125
53x - 1 = 5-2
3x - 1= -2
3x = -1
therefore, x = -13
Swali 4 Ripoti
Maelezo ya Majibu
x2 + 9 = 25
x2 = 25 – 9
x2 = 16
x = √ 16
= 4 cm
∴ The locus is a circle of radius 4 cm with the center 0
Swali 5 Ripoti
Maelezo ya Majibu
Swali 6 Ripoti
Find to infinity, the sum of the sequence \(1, \frac{9}{10}, \left(\frac{9}{10}\right)^2, \left(\frac{9}{10}\right)^3, \ldots\)
Maelezo ya Majibu
Swali 7 Ripoti
A student spent \( \frac{1}{5} \) of his allowances on books, \( \frac{1}{3} \) of the remainder on food and kept the rest for contingences. What fraction was kept?
Maelezo ya Majibu
spent 13 of 45 on food
45 x 13 = 415
therefore, 45 - 415
= 12−415
= 815
Swali 8 Ripoti
Maelezo ya Majibu
Swali 9 Ripoti
Maelezo ya Majibu
Swali 10 Ripoti
Maelezo ya Majibu
Swali 11 Ripoti
Evaluate \(\frac{81.81+99.44}{20.09+36.16}\) correct to 3 significant figures.
Maelezo ya Majibu
Swali 12 Ripoti
The histogram above represents the number of candidates that sat for Mathematics examination in a school. How many candidate scored more than 50 marks?
Maelezo ya Majibu
Swali 13 Ripoti
| Marks | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| No. of students | 3 | 4 | 1 | 0 | 4 | 5 | 2 | 1 |
The table above shows the distribution of marks of students in a test. Find the probability of passing the test if the pass mark is 5.
Maelezo ya Majibu
= 35
Swali 14 Ripoti
Maelezo ya Majibu
Swali 15 Ripoti
I.S∩T∩W=S II. S∪T∪W=S
III. T∩W=S
If S⊂T⊂W, which of the above statements are true?
Maelezo ya Majibu
S∪T∪W = W (imply W is a universal set for S and T) T∩W = S imply all the elements of S are in T and W)
S ⊂ T ⊂ W
∴ I, II, and III
Swali 16 Ripoti
Maelezo ya Majibu
x2 + 25 = 169
x2 = 144
x = √ 144
= 12
Length of the chord AB = 2x
2*12= 24cm
Swali 17 Ripoti
A cliff on the bank of a river is 300 meter high. if the angle of depression of a point on the opposite side of the river is \(60^\circ\), find the width of the river.
Maelezo ya Majibu
We can solve this problem using trigonometry. Let's draw a diagram of the situation:
A /| / | / | h = 300 m / | ----------- x B
Where point A is the top of the cliff, point B is the unknown point on the opposite side of the river, and x is the width of the river.
We know that the angle of depression from A to B is 60 degrees. This means that the angle of elevation from B to A is also 60 degrees.
Using trigonometry, we can set up the following equation:
tan(60) = h / x
where h is the height of the cliff and x is the width of the river. We can solve for x:
x = h / tan(60) x = 300 / √3 x = 100√3 meters
Therefore, the width of the river is 100√3 meters. Answer is correct.
Swali 18 Ripoti
Evaluate \( \int \sec^2 \theta \, d\theta \)
Swali 19 Ripoti
Find the value of \( \sin 45^\circ - \cos 30^\circ \)
Maelezo ya Majibu
Swali 20 Ripoti
A binary operation \(plus\) defined on the set of integers is such that m \(plus\) n = n + mn for all integers m and n. Find the inverse of -5 under this operation, if the identity element is 0.
Maelezo ya Majibu
Swali 21 Ripoti
Maelezo ya Majibu
Swali 22 Ripoti
Maelezo ya Majibu
Swali 23 Ripoti
Find the value of x in the diagram.
Maelezo ya Majibu
x = 100 - 30
= 70o
Swali 24 Ripoti
Maelezo ya Majibu
∫(cos)-2 dθ,
let u = cos θ
∴∫u-2 = 1/u + c
∫cos θ = sin θ + c
∫sec-2θ = 1/u sin θ + c
= (sinθ / cosθ) + c
= Tan θ + c
Swali 25 Ripoti
| No. of days | 1 | 2 | 3 | 4 | 5 | 6 |
| No. of students | 20 | x | 50 | 40 | 2x | 60 |
The distribution above shows the number of days a group of 260 students were absents from school in a particular term. How many students were absent for at least four days in the term
Maelezo ya Majibu
3X + 170 = 260
3X = 260 - 170
3x = 90
x = 30
Absent for at least 4 days
| 4 | 5 | 6 |
| 40 | 2x | 60 |
i.e 40 + 2x + 60 = 40 + (2 x 30) + 60
= 40 + 60 + 60
= 160
Swali 26 Ripoti
Maelezo ya Majibu
Swali 27 Ripoti
Maelezo ya Majibu
Imply x = 4/3 and - 3/5
3x = 4 and 5x = -3
∴3x-4 = 0 and 5x+3 = 0 are factors
(3x-4)(5x+3) = 0 product of the factors
15x2 + 9x – 20x – 12 = 0 By expansion
15x2 - 11x – 12 = 0
Swali 28 Ripoti
A regular polygon has \(150^\circ\) as the size of each interior angle. How many sides does it have?
Maelezo ya Majibu
Swali 29 Ripoti
Find the range of values of \(x\) for which \(3x - 7 \le 0\) and \(x + 5 > 0\)
Maelezo ya Majibu
Swali 30 Ripoti
In the figure above, TS//XY and XY = TY,
Maelezo ya Majibu
< TYZ = 94∘
n∘ = 180 - (94∘ + 34∘ )
= 180∘ - 128∘
= 52∘
Swali 31 Ripoti
Maelezo ya Majibu
Swali 32 Ripoti
Find the infinity, the sum of the sequence \(1, \frac{9}{10}, \left(\frac{9}{10}\right)^2, \left(\frac{9}{10}\right)^3, ...\)
Maelezo ya Majibu
Swali 33 Ripoti
Simplify \(4\frac{3}{4} - 6\frac{1}{4}\)
Maelezo ya Majibu
194 - 254 ............(A)
215 - 54 .............(B)
Now work out the value of A and the value of B and then find the value AB
A = 194 - 254
= ?64
B = 215 x 520
= 10520
= 214
But then AB = -64
214 = -64 ÷ 214
= -64 x 421
= -2484
= -27
Swali 34 Ripoti
If s = (2 + 3t)(5t - 4), find \( \frac{dy}{dx} \) when t = \( \frac{4}{5} \) sec
Maelezo ya Majibu
= 10t - 8 + 15t2 - 12t
dsdt = 15t2 - 2t - 8
= 30t - 2 ; t = 45
= 30t - 2
= 30 x 45 - 2 → 24 - 2
= 22 sec
Swali 35 Ripoti
Maelezo ya Majibu
Swali 36 Ripoti
| No. of days | 1 | 2 | 3 | 4 | 5 | 6 |
| No. of students | 20 | \(2x\) | 60 | 40 | \(x\) | 50 |
The distribution above shows the number of days a group of 260 students were absent from school in a particular term. How many students were absent for at least four days in the term
Maelezo ya Majibu
3x = 260 - 170
3x = 90 → x = 30
therefore, 20 + 2x + 60 + 40
= 180
Swali 37 Ripoti
The probability of a student passing any examination is \( \frac{2}{3} \). If the student takes three examinations, what is the probability that he will not pass any of them?
Maelezo ya Majibu
Swali 38 Ripoti
Maelezo ya Majibu
Let u = 2+3t ∴du/dt = 3
and v = 5t-4 ∴dv/dt = 5
dx/dt = Vdu/dt + Udv/dt
= (5t-4)3 + (2+3t)5
= 15t - 12 + 10 + 15t
= 30t - 2
= 30x4/5 - 2
= 24 - 2
= 22
Swali 39 Ripoti
Maelezo ya Majibu
Swali 40 Ripoti
Maelezo ya Majibu
Swali 41 Ripoti
Find the acute angle between the straight lines \(y = x\) and \(y = \sqrt{3x}\)
Maelezo ya Majibu
x2 = 3x
3x = 90o
x = 90o3 = 30o
Swali 42 Ripoti
Simplify \( \frac{5+\sqrt{7}}{3+\sqrt{7}} \)
Maelezo ya Majibu
Swali 43 Ripoti
If \(m * n = n-(m + 2)\) for any real numbers m and n, find the value of \(3 * (-5)\)
Maelezo ya Majibu
= n - (m + 2)
3 ∗ (-5) = -5 - (3 + 2)
= -5 - 5
= -10
Swali 44 Ripoti
Maelezo ya Majibu
Swali 45 Ripoti
If \( p = \begin{pmatrix} x + 3 & x + 2 \\ x + 1 & x - 1 \end{pmatrix} \) evaluate x if \(|p| = -10\)
Maelezo ya Majibu
|P| = (x + 3)(x - 1) - (x + 2)(x + 1)
x2 + 2x - 3 - (x2 + 3x + 2)
x2 + 2x - 3 - x2 - 3x - 2
-x - 5 = -10
-x = -10 + 5
x = 5
Swali 46 Ripoti
If \(y = 3 \cos 4x\), \(\frac{dy}{dx}\) equals
Maelezo ya Majibu
Swali 47 Ripoti
Maelezo ya Majibu
Swali 48 Ripoti
i. \(S \cap T \cap W = S\)
ii. \(S \cup T \cup W = W\)
ii. \(T \cap W = S\)
If \(S \subset T \subset W\). Which of the above statements are true?
Maelezo ya Majibu
Swali 49 Ripoti
The cumulative frequency curve above shoes the distribution of the scores of 50 students in an examination. Find the 36th percentile scores
Maelezo ya Majibu
Swali 50 Ripoti
Maelezo ya Majibu
