Exam Body

Year

Subject

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UTME 2022 Mathematics | Objective

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Question 1

The identity element with respect to the multiplication shown in the table below is: \( \begin{array}{c|cccc} \otimes & p & q & r & s \\ \hline p & r & p & r & p \\ q & p & q & r & s \\ r & r & r & r & r \\ s & q & s & r & q \\ \end{array} \)

A. p

Correct B. q

C. r

D. s

Correct Answer: B

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Question 2

The binary operation * is defined by \( x^{*} y=x y-y -x \) for all real values x and y. If \( x^{*} 3=2 * x \), find x.

A. -1

B. 0

Correct C. 1

D. 5

Correct Answer: C

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Question 3

The determinant of matrix \( \begin{pmatrix} x & 1 & 0 \\ 1-x & 2 & 3 \\ 1 & 1+x & 4 \end{pmatrix}\) in terms of x is

A. \( -3 x^{2}-17 \)

Correct B. \( 3 x^{2}+9 x-1 \)

C. \( 3 x^{2}+17 \)

D. \( 3 x^{2}-9 x+5 \)

Correct Answer: B

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Question 4

Let I = \( \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \), P = \( \begin{pmatrix} 2 & 3 \\ 4 & 5 \end{pmatrix} \), Q = \( \begin{pmatrix} u & 4+u \\ -2v & v \end{pmatrix} \) be 2 x 2 matrices such that PQ = I. Find (u, v).

A. \( \begin{pmatrix}-5 & -1 \\ 2 & 2\end{pmatrix} \)

Correct B. \( \begin{pmatrix} -5 & 3/2 \\ 2 & 2 \end{pmatrix} \)

C. \( \begin{pmatrix} -5/6 & -1 \\ 2 & 2 \end{pmatrix} \)

D. \( \begin{pmatrix} 2/2 & 2 \\ 2 & 3 \end{pmatrix} \)

Correct Answer: B

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Question 5

In the diagram, PR is a diameter of the circle PQRS. PST and QRT are straight lines. Find QSR.

A. \( 20^{\circ} \)

B. \( 25^{\circ} \)

Correct C. \( 30^{\circ} \)

D. \( 35^{\circ} \)

Correct Answer: C

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Question 6

In the diagram, \( \mathrm{PQ} \| \mathrm{ST} \) and \( \mathrm{PQR}=120^{\circ},\mathrm{RST}= 130^{\circ} \), Find the angle marked x.

A. 50

B. 65

Correct C. 70

D. 80

Correct Answer: C

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Question 7

In the figure, PQST is a parallelogram and TSR is a straight line. The area of QRS is

A. \( 35 \mathrm{~cm}^{2} \)

Correct B. \( 65 \mathrm{~cm}^{2} \)

C. \( 70 \mathrm{~cm}^{2} \)

D. \( 140 \mathrm{~cm}^{2} \)

Correct Answer: B

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Question 8

TQ is tangent to circle \( \mathrm{XYTR},\mathrm{YXT}=32^{\circ}, \mathrm{RTQ}= 40^{\circ} \). Find YTR.

Correct A. \( 108^{\circ} \)

B. \( 121^{\circ} \)

C. \( 140^{\circ} \)

D. \( 148^{\circ} \)

Correct Answer: A

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Question 9

A chord of a circle radius 3 cm subtends an angle of 60 on the circumference of the circle. Find the length of the chord

A. \( \sqrt{3} \mathrm{~cm} \)

B. \( 3\sqrt{2} \mathrm{~cm} \)

Correct C. \( 3\sqrt{3} \mathrm{~cm} \)

D. \( 6 \mathrm{~cm} \)

Correct Answer: C

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Question 10

A cylindrical drum of diameter 56 cm contains 123.2 litres of oil when full. Find the height of the drum in centimetres. \( \left(\pi={ }^{22} / 7\right) \)

A. 12.5

B. 25.0

C. 45.0

Correct D. 50.0

Correct Answer: D

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Question 11

In the venn diagram, the shaded region is

A. \( (P \cap Q) \cup R \)

B. \( (P \cap Q) \cap R \)

Correct C. \( (P \cap Q^c) \cap R \)

D. \( (P \cap Q^c) \cup R \)

Correct Answer: C

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Question 12

When the expression \( \mathrm{pm}^{2}+\mathrm{qm}+1 \) is divided by (m-1), it has a remainder 2 and when divided by \( (m+1) \) the remainder is 4 . Find \( p \) and \( q \) respectively.

Correct A. 2,-1

B. -1,2

C. \( 3,-2 \)

D. -2,3

Correct Answer: A

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Question 13

Factorize \( r^{2}-r(2 p+q)+2 p q \)

A. (r-2q)(2r-p)

B. (r-q)(r+p)

Correct C. (r-q)(r-2p)

D. (2r-q) r+p)

Correct Answer: C

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Question 14

Solve the equation \( x - (x-2)^{-1} = 0 \)

Correct A. \( 1 \pm \sqrt{2} \)

B. \( 2 \pm \sqrt{2} \)

C. \( 1 \pm 2\sqrt{2} \)

D. \( 2 \pm 2\sqrt{2} \)

Correct Answer: A

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Question 15

Find the range of values of \( m \) for which the roots of the equation \( 3 x^{2}-3 m x+(m^{2}-m-3)= 0 \) are real.

A. \( -1 \leq m \leq 7 \)

Correct B. \( -2 < m < 6 \)

C. \( -3 \leq m \leq 9 \)

D. \( -4 \leq m \leq 8 \)

Correct Answer: B

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Question 16

If \( 2_{9} \times(Y 3)_{9}=3_{5} \times(Y 3)_{5} \), find the value of Y.

A. 4

B. 3

C. 2

Correct D. 1

Correct Answer: D

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Question 17

Divide 243 by \( 42{ }_{6} \)

Correct A. \( 23{ }_{6} \)

B. \( 35_{6} \)

C. 526

D. \( 55_{6} \)

Correct Answer: A

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Question 18

Simplify \( \sqrt{\frac{0.0023 \times 750}{0.00345 \times 1.25}} \)

A. 15

Correct B. 20

C. 40

D. 75

Correct Answer: B

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Question 19

A trader bought 100 oranges at 5 for N1.20, 20 oranges got spoilt and the remaining were sold at 4 for N1.50. Find the percentage gain or loss.

A. \( 30\% \) gain

Correct B. \( 25\% \) gain

C. \( 30\% \) loss

D. \( 25 \% \) loss.

Correct Answer: B

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Question 20

If \( \log_{8} 10=x \), evaluate \( \log _{8} 5 \) in terms of \( x \).

A. \( \frac{1}{2}x \)

B. \( x - \frac{1}{4} \)

Correct C. \( x - \frac{1}{3} \)

D. \( x - \frac{1}{2} \)

Correct Answer: C

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Question 21

Find the value of \( x \) if \( \frac{\sqrt{2}}{x+\sqrt{2}}=\frac{1}{x-\sqrt{2}} \)

A. \( 3 \sqrt{2}+4 \)

B. \( 3 \sqrt{2}-4 \)

C. \( 3-2 \sqrt{2} \)

Correct D. \( 4+2 \sqrt{2} \)

Correct Answer: D

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Question 22

A group of market women sell at Least one yam, plantain and maize. 12 of them sell maize, 10 sell yam and 14 sell plantain. 5 sell plantain and maize, 4 sell yam and maize, 2 sell yam and plantain only while 3 sell all the three items. How many women are in the group?

Correct A. 25

B. 19

C. 18

D. 17

Correct Answer: A

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Question 23

If \( \frac{(a^{2}b^{-3}c)^{3/4}}{a^{-1}b^{4}c^{5}} = a^{p}b^{q}c^{r} \), what is the value of p + 2q?

A. 5

B. \( \frac{-5}{4} \)

C. \(\frac{25}{4} \)

Correct D. \( -10 \)

Correct Answer: D

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Question 24

If U= \( \{1,2,3,4,5,6\} \) P=\( \{3,4,5\} \), Q=\( \{2,4,6\} \) and R= \( \{1,2,3,4\} \), list the elements of \( (P \cup Q)^{\prime} \cap R \).

A. {1,2,3,4,5,6}

B. {1,2,3,4}

Correct C. {1}

D. {1, 5 }

Correct Answer: C

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Question 25

Three consecutive positive integers \( k, l \) and \( m \) are such that \( l^{2}=3(k+m) \). Find the value of \( m \).

A. 4

B. 5

C. 6

Correct D. 7

Correct Answer: D

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