IB myp 4-5 MATHEMATICS – Practice Questions- All Topics
Topic :Number-Representing and Solving Inequalities
Topic :Number- Weightage : 21 %
All Questions for Topic : Absolute Values,Representing and Solving Inequalities,Including compound and double inequalities,Irrational numbers,Surds, roots and radicals, including simplifying,Standard form (scientific notation),Laws of exponents, including integer and negative exponents,Number systems notation,Direct and Inverse Proportion,Number Sequence(Prediction ,Description)
Question (4 Marks)
To raise money for their graduation party, senior students organize a cookie and muffin sale. $x$ represents the number of cookies and $y$ represents the number of muffins. The amount of cookies and muffins sold are represented by the shaded region in the diagram below.
Using the information provided in the diagram below:
- Identify the shaded region by completing the inequalities below.
- The first constraint is that they cannot sell more than a total of 500 cookies and muffins. State the other three constraints in the spaces provided.
▶️Answer/Explanation
Ans:
The shaded region can be identified by completing the inequalities as follows:
Inequalities:
1. $x \geq 100$: The number of cookies is more than or equal to 100.
2. $y \geq 50$: The number of muffins is more than or equal to 50.
3. $y \leq x$: The number of muffins is less than or equal to the number of cookies.
Constraints:
1. Total number of cookies and muffins cannot exceed 500: $x + y \leq 500$.
2. The number of cookies is at least 100.
3. The number of muffins is at least 50.
4. The number of muffins cannot exceed the number of cookies.
The completed table is as follows:
The shaded region in the diagram represents the values of $x$ and $y$ that satisfy these inequalities and constraints.
Question (4 Marks)
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Given that $f(a)=11$, find the value of $a$.
▶️Answer/Explanation
Ans:
To find the value of $a$ given that $f(a) = 11$, we need to substitute $f(a)$ into the function $f(x) = (x – 3)^2 + 2$ and solve for $a$.
Substituting $f(a) = 11$ into the function, we have:
$11 = (a – 3)^2 + 2$
Next, we can subtract 2 from both sides of the equation:
$11 – 2 = (a – 3)^2$
Simplifying further:
$9 = (a – 3)^2$
To solve for $a$, we can take the square root of both sides of the equation:
$\sqrt{9} = \sqrt{(a – 3)^2}$
Simplifying:
$3 = |a – 3|$
The equation $3 = |a – 3|$ indicates that the distance between $a$ and 3 is equal to 3. This can be satisfied by two possible values for $a$:
1. $a – 3 = 3$
Solving for $a$:
$a = 3 + 3 = 6$
2. $-(a – 3) = 3$
Solving for $a$:
$-a + 3 = 3$
$-a = 0$
$a = 0$
Therefore, the values of $a$ that satisfy $f(a) = 11$ are $a = 6$ and $a = 0$.
Since, $x \geq 3$ so $a=6$ will be correct answer.