UTME 2022 Mathematics | Objective

Practice these randomly selected questions to test your readiness.

Question 1
If \( 314_{10}-256_{7}=340_{x} \), find \( x \).
A. 10
B. 9
C. 8
D. 7
Question 2
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
D. 7
Question 3
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
D. 50.0
Question 4
In the diagram, PR is a diameter of the circle PQRS. PST and QRT are straight lines. Find QSR.
A. \( 20^{\circ} \)
B. \( 25^{\circ} \)
C. \( 30^{\circ} \)
D. \( 35^{\circ} \)
Question 5
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 6
Factorize \( r^{2}-r(2 p+q)+2 p q \)
A. (r-2q)(2r-p)
B. (r-q)(r+p)
C. (r-q)(r-2p)
D. (2r-q) r+p)
Question 7
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 8
A trader realizes \( 10 x-x^{2} \) naira profit from the sale of x bags of corn. How many bags will give him the maximum profit?
A. 7
B. 6
C. 5
D. 4
Question 9
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 10
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 11
Solve the inequality \( 2-x>x^2 \)
A. \( -2 < x < 1 \)
B. \( x > 2 \) or \( x<-1 \)
C. \( x<-2 \) or \( x>1 \)
D. \( -1< x < 2 \)
Question 12
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 13
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. \( -1 \leq m \leq 7 \)
B. \( -2 < m < 6 \)
C. \( -3 \leq m \leq 9 \)
D. \( -4 \leq m \leq 8 \)
Question 14
If \( \alpha \) and \( \beta \) are the roots of the equation \( 3x^{2} + 5x - 2 = 0 \), find the value of \(\frac{1}{\alpha} + \frac{1}{\beta}. \)
A. \( -5 / 2 \)
B. \( -2 / 3 \)
C. \( 1 / 2 \)
D. \( 5 / 2 \)
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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