UTME 2022 Mathematics | Objective

Practice these randomly selected questions to test your readiness.

Question 1
A binary operation * is defined by \( a^{*} b=a^b \). If \( a^{*} 2=2-a \). find the possible values of \( a \).
A. \( 1,-2 \)
B. 1,2
C. \( 2,-2 \)
D. 1,-1
Question 2
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. \( -3 x^{2}-17 \)
B. \( 3 x^{2}+9 x-1 \)
C. \( 3 x^{2}+17 \)
D. \( 3 x^{2}-9 x+5 \)
Question 3
The first term of a geometrical progression is twice its common ratio. Find the sum of the first two terms of the progression if its sum to infinity is 8.
A. \( \frac{8}{5} \)
B. \( \frac{8}{3} \)
C. \( \frac{72}{25} \)
D. \( \frac{56}{9} \)
Question 4
Simplify \( \sqrt{\frac{0.0023 \times 750}{0.00345 \times 1.25}} \)
A. 15
B. 20
C. 40
D. 75
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
In the venn diagram, the shaded region is
A. \( (P \cap Q) \cup R \)
B. \( (P \cap Q) \cap R \)
C. \( (P \cap Q^c) \cap R \)
D. \( (P \cap Q^c) \cup R \)
Question 7
TQ is tangent to circle \( \mathrm{XYTR},\mathrm{YXT}=32^{\circ}, \mathrm{RTQ}= 40^{\circ} \). Find YTR.
A. \( 108^{\circ} \)
B. \( 121^{\circ} \)
C. \( 140^{\circ} \)
D. \( 148^{\circ} \)
Question 8
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.
A. 2,-1
B. -1,2
C. \( 3,-2 \)
D. -2,3
Question 9
Find the matrix T if ST = I where S = \( \begin{pmatrix} -1 & 1 \\ 0 & 1 \end{pmatrix} \) and I is the identity matrix.
A. \( \begin{pmatrix}-2 & 1 \\ 1 & 1\end{pmatrix} \)
B. \( \begin{pmatrix}-2 & -1 \\ -1 & -1\end{pmatrix} \)
C. \( \begin{pmatrix}-1 & -1 \\ 0 & -1\end{pmatrix} \)
D. \( \begin{pmatrix}-1 & 1 \\ -1 & 0\end{pmatrix} \)
Question 10
If m * n = \( \frac{n}{m} - \frac{m}{n} \text{ for } m, n \in \mathbb{R}, \text{ evaluate } -3 * 4 \)
A. \( \frac{-25}{12} \)
B. \( \frac{-7}{12} \)
C. \( \frac{2}{12} \)
D. \( \frac{25}{12} \)
Question 11
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?
A. 25
B. 19
C. 18
D. 17
Question 12
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. \( \begin{pmatrix}-5 & -1 \\ 2 & 2\end{pmatrix} \)
B. \( \begin{pmatrix} -5 & 3/2 \\ 2 & 2 \end{pmatrix} \)
C. \( \begin{pmatrix} -5/6 & -1 \\ 2 & 2 \end{pmatrix} \)
D. \( \begin{pmatrix} 2/2 & 2 \\ 2 & 3 \end{pmatrix} \)
Question 13
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. \( 3-2 \sqrt{2} \)
D. \( 4+2 \sqrt{2} \)
Question 14
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
B. q
C. r
D. s
Question 15
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}
C. {1}
D. {1, 5 }

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