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Mathematics
UTME / JAMB 1997
Sets, Logic & Venn Diagrams
Solution
If U = (s, p, i, e, n, d, o, u, r), X = (s, p, e, n, d) Y = (s, e, n, o, u), Z = (p, n, o, u, r) find X ∩( Y ∪ Z)
A
(p, o, u, r)
B
(s, p, d, r)
C
(s, p, n, e)
D
(n, d, u)
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
\(Y \cup Z\) = \({s, e, n, o, u, p, r}\)
\(X \cap (Y \cup Z)\) = \({p, e, n}\)
Source: Myschool Past Questions Archive (Record #30797)
Mathematics
UTME / JAMB 1997
Sets, Logic & Venn Diagrams
Solution
A survey of 100 students in an institution shows that 80 students speak Hausa and 20 students speak Igbo, while only 9 students speak both language. How many students speak neither Hausa nor Igbo?
A
0
B
9
C
11
D
20
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
Let the number of students that speak Hausa and Igbo be represented by H and I respectively.
n(H only) = 80 - 9 = 71.
n(I only) = 20 - 9 = 11.
n(neither H nor I) = 100 - (71 + 9 + 11)
= 100 - 91 = 9 students.
Source: Myschool Past Questions Archive (Record #30798)
Mathematics
UTME / JAMB 1997
Plane Geometry, Polygons & Circle Theorems
Solution
If the function f(fx) = x3 + 2x2 + qx - 6 is divisible by x + 1, find q
A
-5
B
-2
C
2
D
5
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
x + 1 = 0, x = -1; f(x) = x3 + 2x2 + qx - 6
0 = -1 + 2 - q - 6
q = -5
Source: Myschool Past Questions Archive (Record #30799)
Mathematics
UTME / JAMB 1997
Differentiation & Applications
Solution
Find the minimum value of X2 - 3x + 2 for all real values of x
A
-\(\frac{1}{4}\)
B
-\(\frac{1}{2}\)
C
\(\frac{1}{4}\)
D
\(\frac{1}{2}\)
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
y = X2 - 3x + 2, \(\frac{dy}{dx}\) = 2x - 3
at turning pt, \(\frac{dy}{dx}\) = 0
∴ 2x - 3 = 0
∴ x = \(\frac{3}{2}\)
\(\frac{d^2y}{dx^2}\) = \(\frac{d}{dx}\)(\(\frac{d}{dx}\))
= 270
∴ ymin = 2\(\frac{3}{2}\) - 3\(\frac{3}{2}\) + 2
= \(\frac{9}{4}\) - \(\frac{9}{2}\) + 2
= -\(\frac{1}{4}\)
Source: Myschool Past Questions Archive (Record #30804)
Mathematics
UTME / JAMB 1997
Algebraic Expressions, Polynomials & Factorisation
Solution
Make F the subject of the formula t = \(\sqrt{\frac{v}{\frac{1}{f} + \frac{1}{g}}}\)
A
\(\frac{gv-t^2}{gt^2}\)
B
\(\frac{gt^2}{gv-t^2}\)
C
\(\frac{v}{\frac{1}{t^2} - \frac{1}{g}}\)
D
\(\frac{gv}{t^2 - g}\)
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
t = \(\sqrt{\frac{v}{\frac{1}{f} + \frac{1}{g}}}\)
t2 = \(\frac{v}{\frac{1}{f} + \frac{1}{g}}\)
= \(\frac{vfg}{ftg}\)
\(\frac{1}{f} + \frac{1}{g}\) = \(\frac{v}{t^2}\)
= (g + f)t2 = vfg
gt2 = vfg - ft2
gt2 = f(vg - t2)
f = \(\frac{gt^2}{gv-t^2}\)
Source: Myschool Past Questions Archive (Record #30807)
Mathematics
UTME / JAMB 1997
Linear, Simultaneous & Quadratic Equations
Solution
What value of g will make the expression 4x2 - 18xy + g a perfect square?
A
9
B
\(\frac{9y^2}{4}\)
C
81y^2
D
\(\frac{18y^2}{4}\)
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
4x2 - 18xy + g = g \(\to\) (\(\frac{18y}{4}\))2
= \(\frac{18y^2}{4}\)
Source: Myschool Past Questions Archive (Record #30808)
Mathematics
UTME / JAMB 1997
Algebraic Expressions, Polynomials & Factorisation
Solution
Find the value of k if \(\frac{5 + 2r}{(r + 1)(r - 2)}\) expressed in partial fraction is \(\frac{k}{r - 2}\) + \(\frac{L}{r + 1}\) where K and L are constants
A
3
B
2
C
1
D
-1
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
5 + 2r = k(r + 1) + L(r - 2)
but r - 2 = 0 and r = 2
9 = 3k
k = 3
Source: Myschool Past Questions Archive (Record #30809)
Mathematics
UTME / JAMB 1997
Inequalities & Linear Programming
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
Inequalities & Linear Programming
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
Sequences & Series (AP & GP)
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
Sequences & Series (AP & GP)
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
Binary Operations, Matrices & Determinants
Solution
Two binary operations \(\ast\) and \(\oplus\) are defines as m \(\ast\) n = mn - n - 1 and m \(\oplus\) n = mn + n - 2 for all real numbers m, n.
Find the value of 3 \(\oplus\) (4 \(\ast\) 5)
A
60
B
57
C
54
D
42
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
m \(\ast\) n = mn - n - 1, m \(\oplus\) n = mn + n - 2
3 \(\oplus\) (4 \(\ast\) 5) = 3 \(\oplus\) (4 x 5 - 5 - 1) = 3 \(\oplus\) 14
3 \(\oplus\) 14 = 3 x 14 + 14 - 2
= 54
Source: Myschool Past Questions Archive (Record #30818)
Mathematics
UTME / JAMB 1997
Binary Operations, Matrices & Determinants
Solution
If X \(\ast\) Y = X + Y - XY, find x when (x \(\ast\) 2) + (x \(\ast\) 3) = 68
A
24
B
22
C
-12
D
-21
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
x \(\ast\) y = x + y - xy
(x \(\ast\) 2) + (x \(\ast\) 3) = 68
= x + 2 - 2x + x + 3 - 3x = 68
-3x = 63
x = -21
Source: Myschool Past Questions Archive (Record #30819)
Mathematics
UTME / JAMB 1997
Plane Geometry, Polygons & Circle Theorems
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
Linear, Simultaneous & Quadratic Equations
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)