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Mathematics
WAEC / WASSCE 1997
Algebra
Solution
Solve the equations: 4x -y = 11; 5x + 2y = 4.
A
x = 6, y = 13
B
x = -2, y = -3
C
x = -2, y = 3
D
x = 2, y = -3
E
x = 2, y = 3.
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
4x - y = 11....(1) x 2 8x - 2y = 22....(3)
5x + 12y = 4....(2) x 1 5x + 2y = 4....(4)
add eqn (3) and (4)
13x = 26
x = 2
4(2) - y = 11 \(\implies\) y = 8 - 11
y = -3.
Source: Myschool Past Questions Archive (Record #23861)
Mathematics
WAEC / WASSCE 1997
Algebra
Solution
Solve the inequality 3x - 8 ≥ 5x
A
x ≤ 4
B
x ≥ 1
C
x ≤ -4
D
x ≤ -1
E
x < 2
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
3x - 8 ≥ 5x
3x - 5x ≥ 8
-2x ≥ 8
x ≤ -4
Source: Myschool Past Questions Archive (Record #23878)
Mathematics
UTME / JAMB 1997
Algebra
Solution
Let f(x) = 2x + 4 and g(x) = 6x + 7 here g(x) > 0. Solve the inequality \(\frac{f(x)}{g(x)}\) < 1
A
x < - \(\frac{3}{4}\)
B
x > - \(\frac{4}{3}\)
C
x > - \(\frac{3}{4}\)
D
x > - 12
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
\(\frac{f(x)}{g(x)}\) 2x + 4
= 6x - 2x > 4 - 7
= 4x > -3
∴ x > -\(\frac{3}{4}\)
Source: Myschool Past Questions Archive (Record #30814)
Mathematics
UTME / JAMB 1997
Algebra
Solution
Find the range of values of x which satisfies the inequality 12x2 < x + 1
A
-\(\frac{1}{4}\) < x < \(\frac{1}{3}\)
B
\(\frac{1}{4}\) < x < -\(\frac{1}{3}\)
C
-\(\frac{1}{3}\) < x < \(\frac{1}{4}\)
D
-\(\frac{1}{4}\) < x < - \(\frac{1}{3}\)
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
12x2 2 - 4x + 3x - 1 0, 3x - 1 -\(\frac{1}{4}\)x 0
-\(\frac{1}{4}\) < x < - \(\frac{1}{3}\)
Source: Myschool Past Questions Archive (Record #30815)
Mathematics
UTME / JAMB 1997
Algebra
Solution
Sn is the sum of the first n terms of a series given by Sn = n\(^2\) - 1. Find the nth term
A
4n + 1
B
4n - 1
C
2n + 1
D
2n - 1
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
\(S_{n} = n^{2} - 1\)
\(T_{n} = S_{n} - S_{n - 1}\)
\(S_{n - 1} = (n - 1)^{2} - 1\)
= \(n^{2} - 2n + 1 - 1\)
= \(n^{2} - 2n\)
\(S_{n} - S_{n - 1} = (n^{2} - 1) - (n^{2} - 2n)\)
= \(2n - 1\)
Source: Myschool Past Questions Archive (Record #30816)
Mathematics
UTME / JAMB 1997
Algebra
Solution
The nth term of a sequence is given \(3^{1 - n}\), find the sum of the first three terms of the sequence.
A
\(\frac{13}{9}\)
B
1
C
\(\frac{1}{3}\)
D
\(\frac{1}{9}\)
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
Tn = \(3^{1 - n}\)
where n = 1, 2, 3.
\(S_3 = 3^{1 - 1} + 3^{1 - 2} + 3^{1 - 3}\)
= 1 + \(\frac{1}{3}\) + \(\frac{1}{9}\)
= \(\frac{13}{9}\)
Source: Myschool Past Questions Archive (Record #30817)
Mathematics
UTME / JAMB 1997
Algebra
Solution
Determine x + y if \(\begin{pmatrix} 2 & -3 \\ -1 & 4 \end{pmatrix}\) \(\begin{pmatrix} x \\ y \end{pmatrix}\) = \(\begin{pmatrix}-1 \\ 8 \end{pmatrix}\)
A
3
B
4
C
7
D
12
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
\(\begin{pmatrix} 2 & -3 \\ -1 & 4 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -1 \\ 8 \end{pmatrix}\)
\(\begin{pmatrix} 2x - 3y \\ -x + 4y \end{pmatrix} = \begin{pmatrix} -1 \\ 8 \end{pmatrix}\)
\(2x - 3y = -1 ... (i)\)
\(-x + 4y = 8 ... (ii)\)
From (ii), x = 4y - 8.
\(2(4y - 8) - 3y = -1 \implies 8y - 16 - 3y = -1\)
\(5y = -1 + 16 = 15 \implies y = 3\)
\(x = 4(3) - 8 = 12 - 8 = 4\)
\(\therefore x + y = 3 + 4 = 7\)
Source: Myschool Past Questions Archive (Record #30820)
Mathematics
UTME / JAMB 1997
Algebra
Solution
Find the non-zero positive value of x which satisfies the equation \(\begin{vmatrix} x & 1 & 0 \\ 1 & x & 1 \\ 0 & 1 & x\end{vmatrix}\) = 0
A
2
B
4
C
√3
D
√2
E
1
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
x(x\(^2\) - 1) - x = 0
= x\(^3\) - 2x = 0
x(x\(^2\) - 2) = 0
x\(^2\) - 2 = 0
x\(^2\) = 2
x = \(\sqrt{2}\)
Source: Myschool Past Questions Archive (Record #30821)
Mathematics
UTME / JAMB 1997
Algebra
Solution
The angle between the positive horizontal axis and a given line is 135o. Find the equation of the line if it passes through the point (2,3)
A
x - y = 1
B
x + y = 1
C
x + y = 5
D
x - y = 5
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
m = tan 135o = -tan 45o = -1
\(\frac{y - y_1}{x - x_1}\) = m
\(\frac{y - 3}{x - 2}\) = -1
= y - 3 = -(x - 2)
= -x + 2
x + y = 5
Source: Myschool Past Questions Archive (Record #30833)
Mathematics
WAEC / WASSCE 1997
Algebra
Solution
(a) Given that \(\frac{5y - x}{8y + 3x} = \frac{1}{5}\), find the value of \(\frac{x}{y}\) to two decimal places.
(b) If 3 is a root of the quadratic equation \(x^{2} + bx - 15 = 0\), determine the value of b. Find the other root.
medium Difficulty
Verified Correct Answer: Option
Step-by-Step Mathematical Solution
(a) \(\frac{5y - x}{8y + 3x} = \frac{1}{5}\)
\(5(5y - x) = 8y + 3x \implies 25y - 5x = 8y + 3x\)
\(25y - 8y = 3x + 5x \implies 17y = 8x\)
\(\therefore \frac{x}{y} = \frac{17}{8} = 2.125 \approxeq 2.13\)
(b) \(x^{2} + bx - 15 = 0\)
Since x = 3 is a root of the equation, f(3) = 0.
\(3^{2} + 3b - 15 = 0 \implies 3b = 6\)
\(b = 2\)
\(\therefore\) The equation is \(x^{2} + 2x - 15 = 0\)
\(x^{2} - 3x + 5x - 15 = 0 \implies x(x - 3) + 5(x - 3) = 0\)
\((x - 3)(x + 5) = 0\)
The second root of the equation is x = -5.
Source: Myschool Past Questions Archive (Record #54476)
Mathematics
WAEC / WASSCE 1997
Algebra
Solution
(a) Solve the simultaneous equation : \(\log_{10} x + \log_{10} y = 4\)
\(\log_{10} x + 2\log_{10} y = 3\)
(b) The time, t, taken to buy fuel at a petrol station varies directly as the number of vehicles V on queue and jointly varies inversely as the number of pumps P available in the station. In a station with 5 pumps, it took 10 minutes to fuel 20 vehicles. Find :
(i) the relationship between t, P and V ; (ii) the time it will take to fuel 50 vehicles in the station with 2 pumps ; (iii) the number of pumps required to fuel 40 vehicles in 20 minutes.
medium Difficulty
Verified Correct Answer: Option
Step-by-Step Mathematical Solution
(a) \(\log_{10} x + \log_{10} y = 4 ... (1)\)
\(\log_{10} x + 2 \log_{10} y = 3 ...(2)\)
\((1) \implies \log_{10} (xy) = 4 \therefore xy = 10^{4} ... (3)\)
\((2) \implies \log_{10} (xy^{2}) = 3 \therefore xy^{2} = 10^{3} ... (4)\)
Divide (4) by (3) :
\(y = 10^{-1} = 0.1\)
Put y in (3), we have
\(\frac{x}{10} = 10^{4} \implies x = 10^{5}\)
\((x, y) = (10^{5}, 10^{-1})\)
(b) (i) \(t \propto \frac{V}{P}\)
\(t = \frac{kV}{P}\)
when t = 10, V = 20 and P = 5.
\(10 = \frac{20k}{5} \implies 50 = 20k\)
\(k = \frac{5}{2}\)
\(\therefore t = \frac{5V}{2P}\)
(ii) When V = 50, P = 2, we have
\(t = \frac{5(50)}{2(2)} = \frac{250}{4}\)
= \(62.5 minutes\)
(iii) when V = 40, t = 20 minutes
\(20 = \frac{5(40)}{2P} \implies 40P = 200\)
\(P = \frac{200}{40} = 5 pumps\)
Source: Myschool Past Questions Archive (Record #54486)
Mathematics
WAEC / WASSCE 1997
Algebra
Solution
Above is the graph of the quadratic function \(y = ax^{2} + bx + c\) where a, b and c are constants. Using the graph, find :
(a)(i) the scales on both axes ; (ii) the equation of the line of symmetry of the curve ; (iii) the roots of the quadratic equation \(ax^{2} + bx + c = 0\)
(b) Use the coordinates of D, E and G to find the values of the constants a, b and c hence write down the quadratic function illustrated in the graph.
(c) Find the greatest value of y within the range \(-3 \leq x \leq 5\).
medium Difficulty
Verified Correct Answer: Option
Step-by-Step Mathematical Solution
(a)(i) Scale : On x- axis, 2cm = 1 unit
On y- axis, 2cm = 5 units.
(ii) Equation of line of symmetry is x = 1.25.
(iii) Roots of the equation \(ax^{2} + bx + c = 0\) are x = 0.25 and x = 2.25.
(b) Coordinates are D(0, 1), E(1, -2) and G(3, 4).
Substituting for y and x in \(ax^{2} + bx + c = y\)
D(0, 1) : \(1 = a(0^{2}) + b(0) + c \implies c = 1\)
E(1, -2) : \(-2 = a(1^{2}) + b(1) + c \implies -2 = a + b + c\)
\(a + b = -2 - 1 = -3 ... (1)\)
G(3, 4) : \(4 = a(3^{2}) + b(3) + c \implies 4 = 9a + 3b + c\)
\(9a + 3b = 4 - 1 = 3 ... (2)\)
\(\implies 3a + b = 1 ... (2a)\)
(2a) - (1) : \(2a = 4 \implies a = 2\)
\(a + b = -3 \implies 2 + b = -3\)
\(b = -3 - 2 = -5\)
\(\therefore\) The equation is \(y = 2x^{2} - 5x + 1\)
(c) The greatest value of y = 33.5.
Source: Myschool Past Questions Archive (Record #54492)
Mathematics
WAEC / WASSCE 1996
Algebra
Solution
Convert the decimal number 89 to a binary number
A
101101
B
111001
C
1001001
D
1001101
E
1011001
medium Difficulty
Verified Correct Answer: Option E
Step-by-Step Mathematical Solution
2
89
2
44 r 1
2
22 r 0
2
11 r 0
2
5 r 1
2
2 r 1
2
1 r 0
0 r 1
89\(_{10}\) = 101100\(1_2\)
Source: Myschool Past Questions Archive (Record #23792)
Mathematics
WAEC / WASSCE 1996
Algebra
Solution
Find the 9th term of the arithmetic progression 18, 12, 6, 0, -6 .........
A
-54
B
-30
C
30
D
42
E
66
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
18, 12, 6, 0, -6, 12, -18, 24, -30
9th term = -30
Source: Myschool Past Questions Archive (Record #23797)
Mathematics
WAEC / WASSCE 1996
Algebra
Diagram
Solution
If x is a real number which of the following is more illustrated on the number line?
Attached Figure / Diagram
A
x < 4
B
x > -2
C
-2 < x ≤ 4
D
-3 ≤ x < 4
E
-2 ≤ x < 4
medium Difficulty
Verified Correct Answer: Option E
Step-by-Step Mathematical Solution
The number line illustrates the inequality: -2 \le x < 4
Source: Myschool Past Questions Archive (Record #23802)