Sure, let's break down the solutions step by step for each question.
### Question 34
Which of the following properties is not true for a parallelogram?
Given options:
A. Opposite angles are equal.
B. Opposite sides are parallel and equal.
C. Diagonals are equal.
D. Diagonals bisect each other.
To answer this, let's consider the properties of a parallelogram:
- Opposite angles are indeed equal (this is a true property).
- Opposite sides are parallel and equal (this is a true property).
- Diagonals of a parallelogram bisect each other (this is a true property).
- However, the diagonals are not necessarily equal. In special cases like a rectangle or square (which are specific types of parallelograms), the diagonals are equal, but this is not true for a general parallelogram.
Thus, the property that is not true for a general parallelogram is:
Answer: C. Diagonals are equal.
### Question 35
Factorize [tex]\(9 - x^2\)[/tex].
Given options:
A. [tex]\((9 - x)(9 + x)\)[/tex]
B. [tex]\((3 - x)(3 + x)\)[/tex]
C. [tex]\((3 + x)(3 + x)\)[/tex]
D. [tex]\((3 - x)(3 - x)\)[/tex]
This is a classic example of a difference of squares, which follows the formula:
[tex]\[ a^2 - b^2 = (a - b)(a + b) \][/tex]
Here, [tex]\(9\)[/tex] can be written as [tex]\(3^2\)[/tex] and so:
[tex]\[ 9 - x^2 = 3^2 - x^2 \][/tex]
Applying the difference of squares formula:
[tex]\[ 9 - x^2 = (3 - x)(3 + x) \][/tex]
Thus, the correct factorization is:
Answer: B. [tex]\((3 - x)(3 + x)\)[/tex]
### Question 36
Simplify [tex]\(\left(\frac{1}{4}\right)^{\frac{-3}{2}}\)[/tex].
Given options:
A. [tex]\(\frac{1}{8}\)[/tex]
B. [tex]\(\frac{1}{4}\)[/tex]
C. 4
D. 8
The given expression can be simplified using the properties of exponents. Specifically:
[tex]\[ \left( \frac{1}{4} \right)^{-\frac{3}{2}} \][/tex]
First, recognize that a negative exponent indicates a reciprocal:
[tex]\[ \left( \frac{1}{4} \right)^{-\frac{3}{2}} = \left( 4 \right)^{\frac{3}{2}} \][/tex]
Next, convert the exponent [tex]\(\frac{3}{2}\)[/tex]:
[tex]\[ 4 = 2^2 \][/tex]
So:
[tex]\[ \left( 4 \right)^{\frac{3}{2}} = \left( 2^2 \right)^{\frac{3}{2}} = 2^{2 \cdot \frac{3}{2}} = 2^3 = 8 \][/tex]
Thus, the simplified value is:
Answer: D. 8
