EmSAT math practice questions are presented along with their answers at the bottom of the page. There are four groups of questions: algebra, geometry, statistics and probabilities, and calculus.
The detailed solutions to the questions are included.
What is the solution set to the equation \( 4 = -(x +2)(x-1) \)?
Find all solutions to the equation \( 9^{-x\left(-x+5\right)}\:= \dfrac{1}{3^{-12}} \).
Write the expression \( \dfrac{1}{3 + \sqrt{-4}} \) as a complex number of the form \( a + i b \).
The graph the complex number \( -2(4 - i) \) is in quadrant
Ahmed rented a car for two days and he paid a total of 261 AED. He was charged a fixed fee of 80 AED per day plus 20 fils for every kilometer traveled.
What is the total distance traveled?
(Give your answer to the nearest kilometer)
Solve for \( x \) and \( y \) the following system of equations.
\[
\begin{cases}
\dfrac{x}{2}+\dfrac{y}{2}=1\\\\
\dfrac{x}{3}+\dfrac{y}{4}=2
\end{cases}
\]
Simplify.
\(
\dfrac{x^2 - 4y^2}{2y - x}
\)
Simplify the expression \( \sqrt{200} + \sqrt{32} \).
We suppose that the point with coordinates \( (a , - 3) \) is on the graph of the curve \( y = x(x-2) \). Find the constant \( a \) if possible.
The solution set of the equation \( \log_4(x) - \log_4(x+10) = -\log_4(x-2)\) is
Find all the solutions of the equation \( 2 - \dfrac{1}{x(x+1)} = \dfrac{3}{x+1} \).
Find \( f^{-1}(-2) \) if \( f(x) = \sqrt{x + 2} - 4 \).
Which of the following is equivalent to the expression \( \sin(4x) \) for all values of \( x \)?
What is the magnitude of vector \( \overrightarrow v \) given by \( \overrightarrow v = 2 \overrightarrow{v_1} - 3 \overrightarrow {v_2} \) where \( \overrightarrow{v_1} = <1,-2> \) and \( \overrightarrow{v_2} = <-2,4> \)?
A factory produced 2000 toys two years ago and 2400 toys this year. If we assume that the production varies linearly with the time, what will be the production of the factory in four years?
Find the solution set of the inequality \( \quad x + 4 \le \dfrac{3}{x+2} \).
Simplify: \( (-x+2)(x-1) - ( x^2 - 2x +1) \)
Factor \( 3x^2 + 4x - 4 \) completely.
Find two positive real numbers such that their difference is \( 2 \) and their product is \( 99 \).
Solve the equation \( \quad x y = \dfrac{y - 1}{x - 1} \) for \( y \).
The sum of three real numbers \( x , y \) and \( z \) is equal to 96. The sum of \( y \) and \( z \) is equal to \( 74 \) and the difference \( z - y \) is equal to \( 12 \). Find the three numbers.
Which of the following equation could be the equation to the third degree polynomial \( P \) whose graph is shown below?
Let \( f(x) = x^2 - 1 \) and \( g(x) = \dfrac{1}{x-2} \); find \( (f_o g)(0) \).
Mansour drank a cup of coffee with 120 milligrams of caffeine at 8:00 am. After five hours, the caffeine in Mansour's body decreased exponentially to half the initial amount. How much caffeine is left in Mansour's body at 6:00 pm?
Write the system of equations
\(
\begin{cases}
-2x + y - z = 1\\\\
5x - y = -3\\\\
-2x + 2z - 4 y = 0
\end{cases}
\)
in matrix form.
The circle with equation \( \quad 2(x - 2)^2 + 2(y + 2 )^2 = 32 \) has
Find \( x \) in the right triangle shown below.
In the figure below AB is parallel to CD, and AD and CB intersect at the point O. Find \( x \) and \( y \).
In the figure below, a square in inscribed in a circle. The shaded (light blue) region has an area of 10 unit2. Find the radius of the circle and round your answer to the nearest tenth.
In the figure below, \( \overline{BC} = 100 \) and \( \overline{AH} = 48 \). Find \( x \) and \( y \), such that \( x \gt y \).
Find all internal angles of the parallelogram below whose area is 300 units2.
Find \( x \) so that the volume of the U-shaped rectangular structure is equal to \( 165 \) cm 3.
The probability that Saef will watch the match on TV is 0.7 and the probability that his team wins is 0.5. What is the probability that Saef will not watch the match and his team wins the match?
A number is selected at random from the set: \( \{ -4,-1,0,2,5,6,7,10\} \). What is the probability that the number is either negative or greater than 6?
The probability that Sultan travels to Spain is 0.5. The probability that he will travel to Spain and then England is 0.3. If Sultan travels to Spain, what is the probability that he will travel to England as well?
Find the first derivative of the function \( f(x) = - 4x^3 + 3x^2 - 2x - 2 \).
Find the first derivative of the function \( f(x) = (x^3 - 2x^2 + x)(2x - 7) \).
Find the first derivative of the function \( f(x) = \sqrt{-3x+3} \).
Find the second derivative of the function \( f(x) = (x^2+1)^5 \).
Find the first derivative of the function \( f(x) = \dfrac{1}{x-1} \).
Find \( f'(2) \) if \( f(x) = \dfrac{x-1}{x+3} \).
Find \( f'(x) \) if \( f(x) = \cos(2x - 2) \).
Find the value of the constant \( k \) if \( f'(1) = 0 \) and \( f(x) = k x^2 + 2x -1 \).
Find all values of \( x \) that make the first derivative of the function \( f(x) = \dfrac{2x^2 + x}{x^2-1} \) equal to zero.
Find the limit \( \lim_{x\to\infty} \dfrac{x^3-2x+4}{-2x^3+x^2-1} \).
Find the limit \( \lim_{x\to + 4} \dfrac{\sqrt{x} - 2}{x - 4} \).
Find the limit \( \lim_{x\to - 3} \dfrac{x^2 + 4x + 3}{x^2 - 9} \).