UTME 2020 Mathematics | Objective

Practice these randomly selected questions to test your readiness.

Question 1
A cliff on the bank of a river is 300 metres 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.
A. 100 m
B. 75 \( \sqrt{ } \) 3 m
C. 100 \( \sqrt{ } \) 3 m
D. 200 \( \sqrt{ } \) 3 m
Question 2
TQ is tangent to circle XYTR, DYXT=\( 32^\circ \), RTQ= \( 40^\circ \). Find DYTR.
A. \( 108^\circ \)
B. \( 121^\circ \)
C. \( 140^\circ \)
D. \( 148^\circ \)
Question 3
What is the locus of the mid-point of all chords of length 6 cm with a circle of radius 5 cm and with centre O?
A. A circle of radius 4 cm and with centre O
B. The perpendicular bisector of the chords
C. A straight line passing through centre O.
D. A circle of radius 6 cm and with centre O.
Question 4
The locus of all points at a distance 8 cm from a point N passes through points T and S. If S is equidistant from T and N, find the area of triangle STN.
A. \( 4 \sqrt{3} cm^{2} \)
B. \( 16 \sqrt{3} cm^{2} \)
C. \( 32 cm^{2} \)
D. \( 64 cm^{2} \)
Question 5
A binary operation * is defined by a*b=ab+a+b for any real number a and b. If the identity element is zero, find the inverse of 2 under this operation.
A. \( \frac{2}{3} \)
B. \( \frac{1}{2} \)
C. -\( \frac{1}{2} \)
D. -\( \frac{2}{3} \)
Question 6
Evaluate \( 64.764^{2} -35. 236^{2} \) correct to 3 significant figures.
A. 2960
B. 2950
C. 2860
D. 2850
Question 7
Factorize completely \( 9y^{2} \)-\( 16x^{2} \).
A. (3y-2x)(3y+4x)
B. (3y+4x)(3y+4x)
C. (3y+2x)(3y-4x)
D. (3y+4x)(3y-4x)
Question 8
Find the locus of a particle which moves in the first quadrant so that it is equidistant from the lines x=0 and y=0 (where k is a constant).
A. x+y=0
B. x-y=0
C. x+y+k=0
D. x-y-k=0
Question 9
Simplify \( \frac{x^2 -1}{ x^2 + 2x^2 - x - 2} \)
A. \( \frac{1}{x-1} \)
B. \( \frac{x-1}{x+1} \)
C. \( \frac{x-1}{x+2} \)
D. \( \frac{1}{x-2} \)
Question 10
Given that Q= \( \begin{pmatrix}6 & 0 \\ 4 & 5\end{pmatrix} \) and Q+P= \( \begin{pmatrix}7 & -2 \\ 6 & 8\end{pmatrix} \) evaluate |Q+2P|
A. 90
B. 96
C. 102
D. 120
Question 11
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)
Question 12
Resolve \( \frac{3}{ x^{2} + x-2} \) into partial fractions.
A. \( \frac{1}{x-1} \)-\( \frac{1}{x+2} \)
B. \( \frac{1}{x+2} \)+\( \frac{1}{x-1} \)
C. \( \frac{1}{x+1} \)-\( \frac{1}{x-2} \)
D. \( \frac{1}{x-2} \)+\( \frac{1}{x+1} \)
Question 13
Simplify \( \log_2 96 - 2\log_2 6 \)
A. \( 2 - \log_2 3 \)
B. \( 3 - \log_2 3 \)
C. \( \log_2 3 - 3 \)
D. \( \log_2 3 - 2 \)
Question 14
Find the value of \( (0.006)^{3} + (0.0004)^{3} \) in standard form
A. \( 2.8 \times 10^{-9} \)
B. \( 2.8 \times 10^{-7} \)
C. \( 2.8 \times 10^{-5} \)
D. \( 2.8 \times 10^{-6} \)
Question 15
Solve the following inequality (x-3)(x-4) ≤ 0
A. 3 ≤ x ≤ 4
B. 3
C. 3 ≤ x<4
D. 3

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