UTME CBT Practice Test – Mathematics (Diagnostic Test) – Set 7

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1. You are to attempt 40 Objectives Questions ONLY for  30 minutes.
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Number and Numeration

If X = {n2+1 : is a positive integer and 1 ≤ n ≤ 5} and Y = {5n : n is a positive integer and 1 ≤ n ≤ 5} find X ∩ Y.

A. {5, 10}     B. {5, 10, 15}     C. {2, 5, 10}     D. {5, 10, 15, 20}

The table above shows the distribution of marks of students in a test. Find the probability of passing the test if the pass mark is 5.

A. $\frac{3}{5}$     B. $\frac{4}{9}$     C. $\frac{7}{20}$     D. $\frac{1}{5}$

Geometry/Trigonometry

What is the value of p if the gradient of the line joining (-1, p) and (p, 4) is $\frac{2}{3}$ ?

A. -2    B. -1    C. 1    D. 2

Geometry/Trigonometry

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°, find the width of the river.

A. 100 m    B. $75\sqrt{3}$ m    C. $100\sqrt{3}$ m    D. $200\sqrt{3}$ m

The probability of a student passing any examination is  $\frac{2}{3}$ . If the student takes three examinations, what is the probability that he will not pass any of them?

A. $\frac{2}{3}$     B. $\frac{4}{9}$     C. $\frac{8}{27}$     D. $\frac{1}{27}$

Algebra

Find to infinity, the sum of the sequence 1, $\frac{9}{10}$, $\left&space;(\frac{9}{10}&space;\right&space;)^{2}$, $\left&space;(\frac{9}{10}&space;\right&space;)^{3}$,.....

A. 10    B. 9    C. $\frac{10}{9}$    D. $\frac{9}{10}$

Calculus

Find the value of x for which the function 3x3 — 9x2 is minimum.

A. 0    B. 2    C. 3    D. 5

In how many ways can 9 people be seated if 3 chairs are available?

A. 720    B. 504     C. 336     D. 210

The cumulative frequency curve above shows the distribution of the scores of 50 students in an examination. Find the 36th percentile score.

A. 18%     B. 28%    C. 36%    D. 50%

Algebra

A binary operation ⊗ defined on the set of integers is such that m ⊗ n = m + n + mn for all integers m and n. find the inverse of -5 under this operation, if the identity element is 0.

A. $-\frac{5}{4}$     B. $-\frac{5}{6}$     C. 0     D. 5

Number and Numeration

If $P&space;=&space;\sqrt{\frac{rs^{3}}{t}}$ , express r in terms of P, s and t.

A. $\frac{P^{2}t}{s^{3}}$     B. $\frac{P^{3}t}{s^{3}}$     C. $\frac{P^{3}t}{s^{2}}$     D. $\frac{Pt}{s^{3}}$

Calculus

If y = 3 cos 4x, $\frac{dy}{dx}$ equals _____

A. 6 sin 8x     B. -24 sin 4x     C. 12 sin 4x     D. -12 sin 4x

Statistics

The pie chart above represents 400 fruits on display in a grocery store. How many apples are in the store?

A. 45    B. 50    C. 60    D. 75

5, 8, 6 and k occur with frequencies 3, 2, 4, and 1 respectively and have a mean of 5.7.

Find the value of k.

A. 4    B. 3    C. 2    D. 1

Calculus

What value of will make the function x(4 - x) a maximum?

A. 4     B. 3     C. 2     D. 1

Statistics

The distribution above shows the number of days a group of 260 students were absent from school in a particular term. How many students were absent for at least four days in the term?

A. 180     B. 120     C. 110     D. 40

Algebra

Which of the following equations represents the graph above?

A. y = 2 + 7x + 4x2     B. y = 2 - 7x + 4x2     C. y = 2 + 7x - 4x2     D. y = 2 - 7x - 4x2

Calculus

If  $\frac{dy}{dx}$  =  x + cos x, find y.

A. x2 — sin x + c     B. x2 + sin x + c     C.  $\frac{x^{2}}{2}$  -  sin x + c     D.  $\frac{x^{2}}{2}$  +  sin x + c

Calculus

If s = (2 + 3t)(5t - 4), find $\frac{ds}{dt}$ when t = $\frac{4}{5}$ secs

A. 0 unit per sec     B. 15 units per sec.     C. 22 units per sec     D. 26 units per sec.

Number and Numeration

A student spent  $\frac{1}{5}$  of his allowances on books,  $\frac{1}{3}$  of the reminder on food and kept the rest for contingencies. What fraction was kept?

A. $\frac{7}{15}$    B. $\frac{8}{15}$    C. $\frac{2}{3}$    D. $\frac{4}{5}$

Number and Numeration

If 55x + 52x = 7710, find x

A. 5    B. 6    C. 7    D. 10

Algebra

Find the range of values of x for which 3x - 7 ≤ 0 and x + 5 > 0

A. -5 < x < $\frac{7}{3}$     B. -5 ≤ x ≤ $\frac{7}{3}$     C. 5 < x ≤ $\frac{7}{3}$     D. -5 ≤ x < $\frac{7}{3}$

Algebra

W is directly proportional to U. If W = 5, when U = 3, find U when W = $-\frac{2}{7}$

A. $-\frac{6}{35}$    B. $-\frac{10}{21}$    C. $-\frac{21}{10}$    D. $-\frac{35}{6}$

Geometry/Trigonometry

What is the value of r if the distance between the points (4, 2) and (1, r) is 3 units?

A. 1    B. 2    C. 3    D. 4

Number and Numeration

Solve $5^{2x+1}\times&space;5^{x+1}$ = 0.04

A. $\frac{1}{3}$    B. $\frac{1}{4}$    C. $-\frac{1}{5}$    D. $-\frac{1}{3}$

Calculus

The distance travelled by a particle from a fixed point is given as s = (t3 - t2 - t + 5) cm. Find the minimum distance that the particle can cover from the fixed point.

A. 2.3 cm    B. 4.0 cm    C. 5.2 cm    D. 6.0 cm

Calculus

Differentiate (cos θ — sin θ)2 with respect to θ

A. -2 cos 2θ    B. -2 sin 2θ     C. 1 - 2 cos 2θ    D. 1 - 2 sin 2θ

Geometry/Trigonometry

Find the value of sin 45° – cos 30°.

A. $\frac{2+\sqrt{6}}{4}$     B. $\frac{\sqrt{2}+\sqrt{3}}{4}$      C. $\frac{\sqrt{2}+\sqrt{3}}{2}$      D. $\frac{\sqrt{2}-\sqrt{3}}{2}$

Algebra

A polynomial in x whose roots are  $\frac{4}{3}$  and  $-\frac{3}{5}$  is

A. 15x2 – 11x – 12     B. 15x2 + 11x – 12     C. 12x2 – x – 12     D. 12x2 + 11x – 15

Algebra

The sum of the first n terms of the arithmetic progression 5, 11, 17, 23, 29, 35,… is

A. n(3n – 0.5)     B. 2(3n + 2)     C. n(3n + 2.5)     D. n(3n + 5)

Simplify  $\frac{5+\sqrt{7}}{3+\sqrt{7}}$

A. $17-\sqrt{7}$     B. $4-\sqrt{7}$     C. $15+\sqrt{7}$     D. $7-\sqrt{7}$

Geometry/Trigonometry

In the figure above, TS//XY and XY = TY, ∠STZ = 34°, ∠TXY = 47°, find the angle marked n .

A. 47°     B. 52°     C. 56°     D. 99°

Geometry/Trigonometry

A regular polygon has 150° as the size of each interior angle. How many sides does it have?

A. 12     B. 10     C. 9     D. 8

Geometry/Trigonometry

If the hypotenuse of a right-angled isosceles triangle is 2cm, what is the area of the triangle ?

A. $\frac{1}{\sqrt{2}}$ cm2     B. 1 cm2     C. $\sqrt{2}$ cm2     D. $2\sqrt{2}$ cm2

Geometry/Trigonometry

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

Statistics

The histogram above represents the number of candidates that sat for Mathematics examination in a school. How many candidates scored more than 50 marks?

A. 80    B. 95    C. 100    D. 115

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