UTME CBT Practice Test – Mathematics (Diagnostic Test) – Set 1

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Number and Numeration

lf 31410 - 2567 = 340x, find x .

A. 7   B. 8    C. 9     D. 10

Number and Numeration

Let P = {1, 2, u, v, w, x},  Q = {2, 3, u, v, w, 5, 6, y}, and R = {2, 3, 4, v, x, y}.

Determine (P - Q) ∩ R.

A. {1, x}     B. {x, y}    C. {x}    D. 4

Geometry/Trigonometry

In the diagram above, if ∠RPS = 50o, ∠RPQ = 30o and PQ = QR, find the value of ∠PRS.

A. 80o   B. 70o   C. 60o  D. 50o

Calculus
Find the value of $\int_{0}^{\pi&space;}\frac{cos^{2}\theta&space;-1}{sin^{2}\theta&space;}d\theta$

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

Geometry/Trigonometry

A frustum pyramid with square base has its upper and lower sections as squares of sizes 2 cm and 5 cm respectively and the distance between them is 6 cm. Find the height of the pyramid from which the frustum was obtained.

A. 8.0 m   B. 8.4 m   C. 9.0 m   D. 10.0 m

Number and Numeration

Audu bought an article for N50,000 and sold it Femi at a loss of x%. Femi later sold the article to Oche at a profit of 40%. If Femi made a profit N10,000, find the value of x.

A. 60    B. 50    C. 40    D. 20

Geometry/Trigonometry

An equilateral triangle of 3 sides is inscribed inside a circle. Find the radius of the circle.

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

Calculus

A trader realises 10x - x2 naira profit from the sale of x bags of corn. How many bags will give him the maximum profit?

A. 4    B. 5    C. 6    D. 7

Multiply (x + 3y +5) by (2x2 + 5y + 2)

A. 2x3 + 3yx2 + 10xy + 15y2 + 13y + 10x2 + 2x + 10
B. 2x3 + 6y2 + 5xy + 15y2 + 31y + 10x2 + 2x + 10
C. 2x3 + 3yx2 + 5xy + 10y2 + 13y + 5x2 + 2x + 10
D. 2x3 + 6y2 + 5xy + 15y2 + 13y + 10x2 + 2x + 10

Calculus

Find the minimum value of the function $f\left&space;(&space;\theta&space;\right&space;)=\frac{2}{3-cos\:&space;\theta&space;}$ for 0 ≤ θ ≤ 2π

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

Algebra

A binary operation * is defined by a*b ab. If a*2 = 2 - a, find the possible value of a.

A. 1,-1    B. 1, 2    C. 2,-2    D. 1,-2

The variance of x, 2x, 3x, 4x and 5x is ______

A. $x\sqrt{2}$      B. 2x2    C. x2     D. 3x.

Geometry/Trigonometry

In a regular polygon, each interior angle doubles its corresponding exterior angle. Find the number of sides of the polygon.

A. 8    B. 6    C. 4    D. 3

Statistics

The table above shows the frequency distribution of the ages (in years) of pupils in a certain secondary school. What percentage of the total number of pupils is. over 15 years but less than 21 years?

A. 35%    B. 45%    C. 50%     D. 60%.

Number and Numeration

If the population of a town was 240 000 in January 1998 and it increased by 2% each year, what would be the population of the town in January 2000?

A. 480 000      B. 149 696       C. 249 696      D. 244 800

Number and Numeration

In a youth club of 94 members, 60 like modern music and 50 like traditional music. The number of members who like both traditional and modern music is three times those who do not like any type of music. How many members like only one type of music?

A. 8    B. 24    C. 62    D. 86

Calculus

The expression ax2 + bx + c equals 5 at x = 1, if its derivative is 2x + 1, what are the values of a, b, c, respectively?

A. 1, 3, 1,     B. 1, 2, 1     C. 2, 1, 1    D. 1, 1, 3.

If the mean of the numbers 0, x + 2,  3x +6 and 4x + 8 is 4, find their mean deviations

A. 0    B. 2    C.3    D. 4

Geometry/Trigonometry

In the diagram above, EFGH is a circle center O. FH is a diameter and GH is a chord which meets FH at right angle at the point N. If NH = 8 cm, and EG = 24 cm, calculate FH.

A. 16 cm   B. 20 cm   C. 26 cm   D. 32 cm

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

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

Algebra

The 3rd term of an AP is 4x - 2y and the 9th term is 10x - 8y. Find the common difference.

A. 19x-17y     B. 8x-4y    C. x-y     D. 2x

Algebra

Find the value of t, for which the determinant of the matrix $\begin{pmatrix}&space;t-4&space;&&space;0&space;&&space;0\\&space;-1&space;&&space;t+1&space;&&space;1\\&space;3&space;&&space;4&space;&&space;t-2&space;\end{pmatrix}$ is zero.

A. 0,2,3    B. -4, 2, 3    C. -4, -2, 3, -3    D. 4, -2, 3

Statistics

A dice is rolled 240 times and the result depicted in the table above. If a pie chart is constructed to represent the data, the angle corresponding to 4 is ______

A. 10°    B. 16°    C. 40°    D. 60°

Geometry/Trigonometry

P is a point on a straight line UV and P moves in the same direction as UV. If the straight line is on the locus of P and ∠VUS = 50o, find ∠UST.

A. 300o      B. 130o      C. 80o      D. 50o

Calculus

A function f(x) passes through the origin and its first derivative is 3x + 2. What is f(x)?

A. y = $\frac{3}{2}$ x2 + 2x       B. y = $\frac{3}{2}$ x2 + x    C. y = 3x2 + $\frac{x}{2}$     D. y = 3x2+2x

Calculus

If the volume of a hemisphere is increasing at a steady rate of 18 π m3s-1, at what rate is its radius changing when it is 6m?

A. 2.50 ms-1    B. 2.00 ms-1    C. 0.25 ms-1    D. 0.20 ms-1.

Algebra

Simplify $\frac{3\left&space;(&space;2^{n+1}&space;\right&space;)-4\left&space;(&space;2^{n-1}&space;\right&space;)}{2^{n+1}-2^{n}}$

A. $2^{n+1}$     B. $2^{n+1}$     C. 4    D. $\frac{1}{4}$

If  $\frac{2\sqrt{3}-\sqrt{2}}{\sqrt{3}+2\sqrt{2}}$  =  $m+n\sqrt{6}$   find the values of m and n respectively.

A. 1,  -2    B. -2,  1    C. - $\frac{2}{5}$,  1    D. 2,  $\frac{3}{5}$

Algebra

A matrix $P\begin{pmatrix}&space;a&space;&&space;b\\&space;c&space;&&space;d&space;\end{pmatrix}$ is such that PT = -P where PT is the transpose of P. If b = 1, then P is _____

A. $\begin{pmatrix}&space;0&space;&&space;1\\&space;1&space;&&space;0&space;\end{pmatrix}$     B. $\begin{pmatrix}&space;0&space;&&space;1\\&space;-1&space;&&space;0&space;\end{pmatrix}$     C. $\begin{pmatrix}&space;0&space;&&space;1\\&space;-1&space;&&space;1&space;\end{pmatrix}$     D. $\begin{pmatrix}&space;1&space;&&space;1\\&space;-1&space;&&space;0&space;\end{pmatrix}$

Calculus

If y =  2x cos x- sin 2x,  find $\frac{dy}{dx}$ when x = $\frac{\pi&space;}{4}$

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

Statistics

If U = {x : x is an integer and 1 ≤ x ≤ 20} E1 = {x : x is a multiple of 3} E2 = {x : x is a multiple of 4} and an integer is picked at random from U, the probability that it is not in E2.

A. $\frac{3}{4}$     B. $\frac{3}{10}$     C. $\frac{1}{4}$     D. $\frac{1}{20}$

Geometry/Trigonometry

A predator moves in a circle of radius $\sqrt{2}$ , centre (0, 0), while a prey moves along the line y = x. If 0 ≤ x ≤ 2, at which point(s) will they meet?

A. (1, 1) only     B. (1,) and (1, 2)      C. (0, 0) and (1, 1)    D. ($\sqrt{2}$, $\sqrt{2}$) only.

Algebra

If (x - 1), (x + 1) and (x - 2) are factors of the polynomial ax3 + bx2 + cx - 1, find a, b, c respectively.

A. - $\frac{1}{2}$,  1,  $\frac{1}{2}$        B.  $\frac{1}{2}$,  1,  $\frac{1}{2}$      C.  $\frac{1}{2}$,  1,  - $\frac{1}{2}$      D.  $\frac{1}{2}$,  -1,  $\frac{1}{2}$

Statistics

In how many ways can a delegation of 3 be chosen from among 5 men and 3 women, if at least one man and at least one woman must be included?

A. 15    B. 28    C. 30    D. 45.

Find the sum of the range and the mode of the set of numbers 10 5, 10, 9, 8, 7, 7, 10, 8, 10 8, 4, 6, 9, 10, 9, 10, 9, 7, 10 6, 5.

A. 16    B. 14    C. 12    D. 10.

Algebra

If α and β are the roots of the equation 3x2 - 5x - 2 = 0, find the value of $\frac{1}{\alpha&space;}+\frac{1}{\beta&space;}$

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

Statistics

In how many ways can word MATHEMATICS be arraigned?

A. $\frac{11!}{9!2!}$     B. $\frac{11!}{9!2!2!}$     C. $\frac{11!}{2!2!2!}$     D. $\frac{11!}{2!2!}$

Geometry/Trigonometry

3y = 4x - 1 and Ky = x + 3 are equations of two straight lines. If the two lines are perpendicular to each other, find K.

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

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