# UTME CBT Practice Test – Mathematics (Diagnostic Test) – Set 2

UTME CBT Practice Test – Mathematics (Diagnostic Test) – Set 2

Hello and Welcome to UTME CBT Practice Test - Mathematics (Diagnostic Test) - Set 2

1. You are to attempt 40 Objectives Questions ONLY for  30 minutes.
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Geometry/Trigonometry

Find the value of p if the line joining (p, 4) and (6 —2) is perpendicular to the line joining (2, p) and (-1, 3).

A. 0    B. 3    C. 4   D. 6

Number and Numeration

Given that p = $1+\sqrt{2}$   and q = $1-\sqrt{2}$,   evaluate  $\frac{p-q}{2pq}$

A. $-2\left&space;(&space;2+\sqrt{2}&space;\right&space;)$      B. $2\left&space;(&space;2+\sqrt{2}&space;\right&space;)$      C. $-2\sqrt{2}$    D. $2\sqrt{2}$

Algebra
Given the matrix K = $\begin{pmatrix}&space;2&space;&&space;1\\&space;3&space;&&space;4&space;\end{pmatrix}$ , the matrix K2 + K + l where l is the 2 X 2 identity matrix, is _____

A. $\begin{pmatrix}&space;9&space;&&space;8\\&space;22&space;&&space;23&space;\end{pmatrix}$     B. $\begin{pmatrix}&space;10&space;&&space;7\\&space;21&space;&&space;24&space;\end{pmatrix}$    C. $\begin{pmatrix}&space;7&space;&&space;2\\&space;12&space;&&space;21&space;\end{pmatrix}$     D. $\begin{pmatrix}&space;6&space;&&space;3\\&space;13&space;&&space;20&space;\end{pmatrix}$

Calculus

Find the rate of change of the volume v of a sphere with respect to its radius r when r = 1.

A. 4 π   B. 8 π   C. 12 π    D. 24 π.

Number and Numeration

A car dealer bought a second-hand car for N 250,000.00 and spent N 70,000.00 refurbishing it. He then sold the car for N 400,000.00. What is the percentage gain?

A. 20%   B. 25%    C. 32%    D. 60%

Statistics

If 6Pr = 6, find the value of 6Pr+1 .

A. 15    B. 30  C. 33   D. 35

Number and Numeration

Find the principal which amounts to N 5,500 at simple interest in 5 years at 2% per annum

A. N 5,000   B. N 4,900    C. N 4,800    D. N 4,700.

Number and Numeration

Evaluate 21.105347 — 1.6324 x 0.43, to 3 decimal places.

A. 20.351    B. 20.352    C. 20.980    D. 20.981

Number and Numeration

Simplify $\left&space;(\sqrt[3]{64a^{3}}&space;\right&space;)^{-1}$

A. 8a     B. 4a     C. $\frac{1}{4a}$   D. $\frac{1}{8a}$

Statistics

The bar chart above shows different colours of cars passing a particular point of a certain street in two minutes. What fraction of the total number of cars is yellow?

A. $\frac{4}{15}$     B. $\frac{1}{5}$     C. $\frac{3}{25}$    D. $\frac{2}{25}$

Algebra
Evaluate $\begin{vmatrix}&space;-1&space;&&space;-1&space;&&space;-1\\&space;3&space;&&space;1&space;&&space;-1\\&space;1&space;&&space;2&space;&&space;1&space;\end{vmatrix}$

A. 4     B. -2     C. -4   D. -12

Calculus

If the gradient of the curve y = 2kx2 + x + 1 at x = 1 is 9, find k

A. 1    B. 2    C. 3    D. 4

Algebra

Divide a3x - 26a2x + 156ax - 216  by  a2x - 24ax + 108

A. ax — 18.     B. ax — 6     C. ax — 2    D. ax + 2

Statistics

Teams P and Q are involved in a game of football. What is the probability that the game ends in a draw.

A. $\frac{1}{4}$    B. $\frac{1}{3}$    C. $\frac{1}{2}$    D. $\frac{2}{3}$

Number and Numeration

Evaluate  $\frac{\left&space;(&space;0.14&space;\right&space;)\times&space;0.275}{\left&space;(&space;0.02&space;\right&space;)}$  correct to 3 decimal places.

A. 0.033    B. 0.039   C. 0308   D. 0.358

Statistics

The distribution of colours of heath in a bowl is given above. What is the probability that a bead selected at random will be blue or white?

A. $\frac{1}{15}$    B. $\frac{1}{3}$    C. $\frac{2}{5}$   D. $\frac{7}{15}$

The minimum point on the curve y = x2 - 6x + 5 is at _______

A. (1,5) B. (2, 3) C. (3, -4) D. (-3, 4)

Statistics

The histogram above shows the distribution of passengers in taxis of a certain motor park. How many taxis have more than 4 passengers?

A. 14    B. 15   C.16    D. 17

Geometry/Trigonometry

The bearings of P and Q from a common point N are 020° and 300° respectively. If P and Q are also equidistant from N, find the bearing of P from Q.

A. 320°    B. 280°    C. 070°   D. 040°

Arrange $\frac{3}{5}$, $\frac{9}{16}$, $\frac{34}{59}$ and $\frac{71}{97}$ in ascending order of magnitude.

A. $\frac{3}{5}$, $\frac{9}{16}$, $\frac{34}{59}$, $\frac{71}{97}$ B. $\frac{9}{16}$, $\frac{3}{5}$, $\frac{71}{97}$, $\frac{34}{59}$ C. $\frac{9}{16}$, $\frac{34}{59}$, $\frac{3}{5}$, $\frac{71}{97}$ D. $\frac{34}{59}$, $\frac{71}{97}$, $\frac{3}{5}$, $\frac{9}{16}$

Number and Numeration

If  $\frac{y}{2}$  = x, evaluate $\left&space;(&space;\frac{x}{y}+\frac{1}{2}&space;\right&space;)&space;+&space;\left&space;(&space;\frac{1}{2}+\frac{x^{2}}{y^{2}}&space;\right&space;)$

A. $\frac{5}{16}$    B. $\frac{5}{8}$    C. $\frac{5}{4}$    D. $\frac{5}{2}$

Statistics

Find the range of  $\frac{1}{6}$$\frac{1}{3}$$\frac{3}{2}$$\frac{2}{3}$$\frac{8}{9}$  and  $\frac{4}{3}$

A. $\frac{4}{3}$    B. $\frac{7}{6}$   C. $\frac{5}{6}$   D. $\frac{3}{4}$

Algebra

A man saves N 100.00 in his first year of work and each year saves N 20.00 more than in the
preceding year In how many years will he save N 5,800.00?

A. 20 years   B. 29 years   C. 58 years   D. 100 years

Geometry/Trigonometry

A sector of a circle of radius 7.2 cm which subtends an angle of 300° at the centre is used to form a cone. What is the radius of the base of the cone?

A. 6 cm   B. 7 cm   C. 8 cm    D. 9 cm

Statistics

The mean score is _____

A. 11.0    B. 9.5    C. 8.7   D. 7.0

Geometry/Trigonometry

In the figure above, PQR is a straight line segment, PQ = QT. Triangle PQT is an isosceles triangle, ∠SRQ is 75° and ∠QPT is 25°. Calculate the value of ∠RST.

A. 25°    B. 45°    C. 50°    D. 55°

Calculus

Find the dimensions of the rectangle of greatest area which has a fixed perimeter p.

A. Square of sides $\frac{p}{4}$     B. square of sides $\frac{p}{2}$    C. Square of sides p.    D. Square of ides. 2p.

Algebra
If P = $\begin{pmatrix}&space;3&space;&&space;-2&space;&&space;4\\&space;5&space;&&space;0&space;&&space;6\\&space;7&space;&&space;5&space;&&space;-1&space;\end{pmatrix}$, then -2P is ______

A. $\begin{pmatrix}&space;-6&space;&&space;4&space;&&space;-8\\&space;5&space;&&space;0&space;&&space;6\\&space;7&space;&&space;5&space;&&space;-1&space;\end{pmatrix}$     B. $\begin{pmatrix}&space;-6&space;&&space;4&space;&&space;-8\\&space;-10&space;&&space;0&space;&&space;6\\&space;-14&space;&&space;5&space;&&space;-1&space;\end{pmatrix}$     C. $\begin{pmatrix}&space;-6&space;&&space;-4&space;&&space;2\\&space;-10&space;&&space;-2&space;&&space;-12\\&space;-14&space;&&space;-10&space;&&space;2&space;\end{pmatrix}$     D. $\begin{pmatrix}&space;-6&space;&&space;-4&space;&&space;-8\\&space;-10&space;&&space;0&space;&&space;-12\\&space;-14&space;&&space;-10&space;&&space;2&space;\end{pmatrix}$

Calculus

If y = x sin x, find  $\frac{dy}{dx}$  when x = $\frac{\pi&space;}{2}$

A. $\frac{\pi&space;}{2}$    B. 1   C. -1   D. $\frac{\pi&space;}{-2}$

Statistics

Find the variance of 2, 6, 8, 6, 2 and 6

A. $\sqrt{5}$     B. $\sqrt{6}$     C. 5     D. 6

Geometry/Trigonometry

Find the number of sides of a regular polygon whose interior angle is twice the exterior angle

A. 2    B. 3    C. 6   D. 8

Algebra

The identity element with respect to the multiplication shown in the table above is ______

A. k     B. l     C. n    D. o

Calculus

Differentiate (2x + 5)2 (x - 4) with respect to x.

A. (2x + 5)(6x — 11)
B. (2x + 5)(2x — 13)
C. 4(2x + 5)(x — 4)
D. 4(2x + 5)(4x — 3)

Calculus

Find the area bounded by the curves y 4 — x2 and y = 2x + 1

A. $10\frac{1}{3}$ sq. units.     B. $10\frac{2}{3}$ sq. units.    C. $20\frac{1}{3}$ sq. units.     D. $20\frac{2}{3}$ sq. units.

Algebra

An operation is defined on the set of real numbers by a * b = a + b ÷ 1. If the identity element is -1, find the inverse of the element 2 under *.

A. -4    B -2    C. 0    D. 4

Algebra

Find the integral values of x and y satisfying the inequality 3y + 5x ≤ 15 given that y > 0, y < 3 and x > 0.

A. (1x1), (2,1), (1,3)
B. (1,1), (1,2,), (1,3)
C. (1,1), (1,2), (2,1)
D. (1,1), (3,1), (2,2)

Geometry/Trigonometry

A cylindrical tank has a capacity of 3080 m3. What is the depth of the tank if the diameter of its base is 14 m?

A. 20 m   B. 22 m   C. 23 m   D. 25 m

Geometry/Trigonometry

A point P moves such that it is equidistant from points Q and R. Find OR when PR = 8cm and ∠PRO = 30°.

A. 4 cm    B. 43 cm    C. 8 cm    D. $8\sqrt{3}$ cm

Calculus

Evaluate. $\int&space;2\left&space;(&space;2x-3&space;\right&space;)^{\frac{2}{3}}&space;dx$.

A. 2x -3 + K     B. 2(2x - 3) + K     C.  $\frac{6}{5}\left&space;(&space;2x-3&space;\right&space;)^{\frac{5}{3}}+K$     D.  $\frac{3}{5}\left&space;(&space;2x-3&space;\right&space;)^{\frac{5}{3}}+K$

Geometry/Trigonometry

Find the value of θ in the diagram above.

A. 30°   B. 60°   C. 100°   D. 120°

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