POST UTME LAUTECH 2023 Mathematics | Objective

Practice these randomly selected questions to test your readiness.

Question 1
In a right-angled triangle, the length of the hypotenuse is 10 cm and one of the acute angles is 30°. Find the length of the side opposite the 30° angle.
A. 5
B. 5√3
C. 10√3
D. 10√2
Question 2
Find the derivative of the function ( f(x) = \frac{1}{x^2 + 1} ) u\sing the chain rule.
A. ( f'(x) = -\frac{2x}{\( x^2 + 1 \)^2} )
B. ( f'(x) = \frac{2x}{\( x^2 + 1 \)^2} )
C. ( f'(x) = -\frac{1}{\( x^2 + 1 \)^2} )
D. ( f'(x) = \frac{1}{\( x^2 + 1 \)^2} )
Question 3
In a random sample of 100 students, the mean height is 175 cm with a s\tandard deviation of 5 cm. If the population s\tandard deviation is 6 cm, calculate the s\tandard error of the mean.
A. 1.25
B. 1.67
C. 2.00
D. 2.50
Question 4
Find the surface area of the sphere with radius 5 cm.
A. 50π cm²
B. 100π cm²
C. 150π cm²
D. 200π cm²
Question 5
Find the equation of the circle with center \( -2, 3 \ \) ) and radius 4.
A. \( x + 2 \ \)^2 + \( y - 3 \)^2 = 16 )
B. \( x - 2 \ \)^2 + \( y + 3 \)^2 = 16 )
C. \( x + 2 \ \)^2 + \( y + 3 \)^2 = 16 )
D. \( x - 2 \ \)^2 + \( y - 3 \)^2 = 16 )
Question 6
Solve the inequality \( \frac{x^2 - 4}{x + 2} > 0 \) for ( x in mathbb{R} ).
A. \( x < -2 \) or \( x > 2 \)
B. \( x > -2 \) or \( x < 2 \)
C. \( x = -2 \) or \( x = 2 \)
D. \( x = -2 \) and \( x = 2 \)
Question 7
Solve the quadratic equation \( x^2 + 5x + 6 = 0 \).
A. \( x = -2, x = -3 \)
B. \( x = 2, x = 3 \)
C. \( x = -3, x = 2 \)
D. \( x = 2, x = -3 \)
Question 8
Find the equation of the plane pas\sing through the points (1, 2, 3), (4, 5, 6), and (7, 8, 9).
A. x + y + z = 12
B. x - y - z = 0
C. x + 2y - z = 6
D. 2x - 3y + z = 0
Question 9
Find the value of \( \sin 2\theta \) given that \( \sin \theta = \frac{3}{5} \) and \( \cos \theta = \frac{4}{5} \).
A. \( \frac{24}{25} \)
B. \( \frac{16}{25} \)
C. \( \frac{20}{25} \)
D. \( \frac{12}{25} \)
Question 10
Solve for x in the equation \( egin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} egin{bmatrix} x \ 1 \end{bmatrix} = egin{bmatrix} 7 \ 10 \end{bmatrix} \).
A. x = 3
B. x = 4
C. x = 5
D. x = 6
Question 11
In the diagram below, the equation of the circle is given by \( x - 2 \ \)^2 + \( y - 3 \)^2 = 4 ). Find the equation of the \tangent line to the circle at the point ( (5, 7) ).
A. y = -x + 12
B. y = x - 1
C. y = -x + 11
D. y = x + 1
Question 12
Solve the inequality $|x^2 - 4x + 3| geq 2$.
A. \( x leq -1 \) or ( x geq 3 )
B. ( x leq 1 ) or ( x geq 3 )
C. \( x leq -1 \) or ( x geq 2 )
D. ( x leq 1 ) or ( x geq 2 )
Question 13
A random variable ( X ) has a probability distribution given by \( P\( X = x \ \) = egin{cases} 0.2 & \text{if } x = 1 \ 0.3 & \text{if } x = 2 \ 0.5 & \text{if } x = 3 \end{cases} ). Find the probability that ( X ) takes on a value greater than 2.
A. 0.4
B. 0.5
C. 0.6
D. 0.7
Question 14
Find the volume of the solid formed by revolving the region bounded by the curve $y = \frac{1}{2}x^2 + 1$, the $x$-axis, and the line $x = 2$ about the $x$-axis.
A. \( \frac{8pi}{3} \)
B. \( \frac{16pi}{3} \)
C. \( \frac{32pi}{3} \)
D. \( \frac{64pi}{3} \)
Question 15
Solve the inequality \( 2x^2 + 5x - 3 > 0 \).
A. \( x < -\frac{3}{2} \) or \( x > \frac{1}{2} \)
B. \( x < -\frac{3}{2} \) or \( x < \frac{1}{2} \)
C. \( x > -\frac{3}{2} \) or \( x > \frac{1}{2} \)
D. \( x > -\frac{3}{2} \) or \( x < \frac{1}{2} \)

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