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Mathematics
UTME / JAMB 1995
Plane Geometry, Polygons & Circle Theorems
Solution
Finds a positive value of p if the expression 2x2 - px + p leaves a remainder 6 when divided by x - p and q
A
1
B
2
C
3
D
4
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
x - p,x = p
2p2 - p2 + p = 6
= p2 + p - 6
= 0
p = 3, 2
Source: Myschool Past Questions Archive (Record #30894)
Mathematics
UTME / JAMB 1995
Plane Geometry, Polygons & Circle Theorems
Solution
Find T in terms of K, Q, and S if S = 2r\(\sqrt{\pi(QT + K)}\)
A
\(\frac{S^2}{2 \pi r^2Q} - \frac{k}{Q}\)
B
\(\frac{S^2}{2 \pi r^2Q}\) - k
C
\(\frac{S^2}{4 \pi r^2Q} - \frac{k}{Q}\)
D
\(\frac{s^2}{4 \pi r^2Q}\)
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
\(\frac{s^2}{4r^2}\) = \(\pi\)(QT + K)
\(\frac{s^2}{4r^2}\) - k\(\pi\) = QT\(\pi\)
T = \(\frac{s^2}{4Q\pi r^2}\) - \(\frac{k}{Q}\)
Source: Myschool Past Questions Archive (Record #30903)
Mathematics
UTME / JAMB 1995
Plane Geometry, Polygons & Circle Theorems
Solution
The graph of f(x) = x2 - 5x + 6 crosses the x-axis at the points
A
(-6, 0), (-1, 0)
B
(-3, 0), (-2,0)
C
(-6, 0),(1, 0)
D
(2, 0), (3, 0)
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
When X = 3, Y = 0(3, 0),
When x = 2, y = 0(2, 0)
Source: Myschool Past Questions Archive (Record #30904)
Mathematics
UTME / JAMB 1995
Algebraic Expressions, Polynomials & Factorisation
Solution
Factorize completely the expression \(abx^2 + 6y - 3ax - 2byx\)
A
(ax - 2y)(bx - 3)
B
(bx + 3)(2y - ax)
C
(bx + 3)(ax - 2y)
D
(ax - 2y)(ax - b)
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
\(abx^{2} + 6y - 3ax - 2byx\)
Collecting like terms, we have
\(abx^{2} - 3ax + 6y - 2byx\)
= \(ax(bx - 3) + 2y(3 - bx)\)
= \(ax(bx - 3) - 2y(bx - 3)\)
= \((ax - 2y)(bx - 3)\)
Source: Myschool Past Questions Archive (Record #30905)
Mathematics
UTME / JAMB 1995
Inequalities & Linear Programming
Solution
Solve the inequality (x - 3)(x - 4) \(\leq\) 0
A
3 \(\leq\) x \(\leq\) 4
B
3 < x < 4
C
3 \(\leq\) x < 4
D
3 < x \(\leq\) 4
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
(x - 3)(x - 4) \(\leq\) 0
Case 1 (+, -) = x - 3 \(\geq\) 0, X - 4 \(\geq\) 0
= X \(\leq\) 3, x \(\geq\) 4
= 3 < x \(\geq\) 4 (solution)
Case 2 = (-, +) = x - 3 \(\leq\) 0, x - 4 \(\geq\) 0
= x \(\leq\) 3, x \(\geq\) 4
therefore = 3 \(\leq\) x \(\leq\) 4
Source: Myschool Past Questions Archive (Record #30906)
Mathematics
UTME / JAMB 1995
Sequences & Series (AP & GP)
Solution
The 4th term of an A.P. is 13 while the 10th term is 31. Find the 21st term.
A
175
B
85
C
64
D
45
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
a + 3d = 13
a + 9d = 31
6d = 18
= d = 3
a = 13 - 9
= 4
a + 20d = 4 + (20 x 3)
= 64
Source: Myschool Past Questions Archive (Record #30910)
Mathematics
UTME / JAMB 1995
Linear, Simultaneous & Quadratic Equations
Solution
Simplify \(\frac{x^2 - 1}{x^3 + 2x^2 - x - 2}\)
A
\(\frac{1}{x + 2}\)
B
\(\frac{x - 1}{x + 1}\)
C
\(\frac{x - 1}{x + 2}\)
D
\(\frac{1}{x - 2}\)
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
\(\frac{x^2 - 1}{(x - 1)(x + 2)(x + 1)}\) = \(\frac{(x -
1)(x + 1)}{(x - 1)(x + 2)(x + 1)}\)
= \(\frac{1}{x + 2}\)
Source: Myschool Past Questions Archive (Record #30912)
Mathematics
UTME / JAMB 1995
Algebraic Expressions, Polynomials & Factorisation
Solution
Express \(\frac{5x - 12}{(x - 2)(x - 3)}\) in partial fractions
A
\(\frac{2}{x + 2} - \frac{3}{x - 3}\)
B
\(\frac{2}{x - 2} + \frac{3}{x - 3}\)
C
\(\frac{2}{x - 3} - \frac{3}{x - 2}\)
D
\(\frac{5}{x - 3} - \frac{4}{x - 2}\)
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
\(\frac{5x - 12}{(x - 2)(x - 3)} = \frac{A}{x - 2} + \frac{B}{x - 3}\)
= \(\frac{A(x - 3) + B(x - 2)}{(x - 2)(x - 3)}\)
\(\implies 5x - 12 = Ax - 3A + Bx - 2B\)
\(A + B = 5 ... (i)\)
\(-(3A + 2B) = -12 \implies 3A + 2B = 12 ... (ii)\)
From (i), \(A = 5 - B\)
\(3(5 - B) + 2B = 12\)
\(15 - 3B + 2B = 12 \implies B = 3\)
\(A + 3 = 5 \implies A = 2\)
\(\frac{5x - 12}{(x - 2)(x - 3)} = \frac{2}{x - 2} + \frac{3}{x - 3}\)
Source: Myschool Past Questions Archive (Record #30913)
Mathematics
UTME / JAMB 1995
Binary Operations, Matrices & Determinants
Solution
Which of the following binary operations is cumulative on the set of integers?
A
a \(\ast\) b = a + 2b
B
a \(\ast\) b = a + b - ab
C
a \(\ast\) b = a2 + b
D
a \(\ast\) b = \(\frac{a(b + 1)}{2}\)
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
\(a \ast b = a + b - ab\)
\(b \ast a = b + a - ba\)
On the set of integers, the two above are cumulative as multiplication and addition are cumulative on the set of integers.
Source: Myschool Past Questions Archive (Record #30916)
Mathematics
UTME / JAMB 1995
Binary Operations, Matrices & Determinants
Solution
If a \(\ast\) b = + \(\sqrt{ab}\), evaluate 2 \(\ast\)(12 \(\ast\) 27)
A
12
B
9
C
6
D
2
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
\(2 \ast (12 \ast 27)\)
\(12 \ast 27 = + \sqrt{12 \times 27}\)
= \(+ \sqrt{324} = 18\)
\(2 \ast 18 = + \sqrt{2 \times 18}\)
= \(+ \sqrt{36} = 6\)
Source: Myschool Past Questions Archive (Record #30918)
Mathematics
UTME / JAMB 1995
Sequences & Series (AP & GP)
Solution
Find the sum to infinity of the following sequence \(1, \frac{9}{10}, (\frac{9}{10})^{2}, (\frac{9}{10})^{3}\)
A
\(\frac{1}{10}\)
B
\(\frac{9}{10}\)
C
\(\frac{10}{9}\)
D
10
medium Difficulty
Verified Correct Answer: Option D
Step-by-Step Mathematical Solution
\(S_{\infty} = \frac{a}{1 - r}\) (Sum to infinity of a G.P)
\(a = 1; r = \frac{9}{10}\)
\(S_{\infty} = \frac{1}{1 - \frac{9}{10}}\)
= \(\frac{1}{\frac{1}{10}} = 10\)
Source: Myschool Past Questions Archive (Record #30919)
Mathematics
UTME / JAMB 1995
Plane Geometry, Polygons & Circle Theorems
Solution
\(\begin{vmatrix} -2 & 1 & 1 \\ 2 & 1 & k \\1 & 3 & -1 \end{vmatrix}\) = 23
A
1
B
2
C
3
D
4
medium Difficulty
Verified Correct Answer: Option B
Step-by-Step Mathematical Solution
-2(-1 - 3k) - 1(-2 -k) + 1(6 - 1) = 23
7k = 14
k = 2
Source: Myschool Past Questions Archive (Record #30921)
Mathematics
UTME / JAMB 1995
Plane Geometry, Polygons & Circle Theorems
Solution
If x = \(\begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix}\) and y = \(\begin{pmatrix} 2 & 1 \\ 4 & 3 \end{pmatrix}\). Find xy.
A
\(\begin{pmatrix} 10 & 7 \\ 12 & 9 \end{pmatrix}\)
B
\(\begin{pmatrix} 2 & 7 \\ 4 & 17 \end{pmatrix}\)
C
\(\begin{pmatrix} 10 & 4 \\ 4 & 6 \end{pmatrix}\)
D
\(\begin{pmatrix} 4 & 3 \\ 10 & 9 \end{pmatrix}\)
medium Difficulty
Verified Correct Answer: Option A
Step-by-Step Mathematical Solution
\(\begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix}\); y = \(\begin{pmatrix} 2 & 1 \\ 4 & 3 \end{pmatrix}\).
= \(\begin{pmatrix} 2 + 8 & 1 + 6 \\ 0 + 12 & 0 + 9 \end{pmatrix}\)
= \(\begin{pmatrix} 10 & 7 \\ 12 & 9 \end{pmatrix}\)
Source: Myschool Past Questions Archive (Record #30923)
Mathematics
UTME / JAMB 1995
Trigonometry, Elevation & Bearings
In a triangle XYZ, < YXZ = 44^\circ and < XYZ = 112^\circ. Calculate the acute angle between the internal bisectors of < XYZ and < XZY.
A
12o
B
56o
C
68o
D
78o
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
No expanded working has been attached for this question yet. The verified answer is Option C.
Source: Myschool Past Questions Archive (Record #30925)
Mathematics
UTME / JAMB 1995
Plane Geometry, Polygons & Circle Theorems
Solution
Find the distance between two towns P(45^\circN, 30^\circW) and Q(15^\circS, 30^\circW) if the radius of the earth is 7000km. [\(\pi = \frac{22}{7}\)]
A
\(\frac{11000}{3}\)km
B
\(\frac{2200}{3}\)km
C
\(\frac{22000}{3}\)km
D
\(\frac{1100}{3}\)km
medium Difficulty
Verified Correct Answer: Option C
Step-by-Step Mathematical Solution
Angular difference = 45^\circ + 15^\circ = 60^\circ.
\(D = \frac{60^\circ}{360^\circ} \times 2 \times \frac{22}{7} \times 7000\)
= \(\frac{22000}{3} km\)
Source: Myschool Past Questions Archive (Record #30930)