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Let's work through the questions from the image you uploaded. Here are solutions for each one:

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**1. If \( x + 2 \) is a factor of \( (2x + 3)^{2012} + (a - x)^{2013} \), then \( a \) is:**

We know that if \( x + 2 \) is a factor of the given expression, substituting \( x = -2 \) into the expression should make it zero.

Substituting \( x = -2 \):

\[
(2(-2) + 3)^{2012} + (a - (-2))^{2013} = (-4 + 3)^{2012} + (a + 2)^{2013} = (-1)^{2012} + (a + 2)^{2013}
\]

Since \( (-1)^{2012} = 1 \), the equation becomes:

\[
1 + (a + 2)^{2013} = 0
\]

Thus, \( (a + 2)^{2013} = -1 \). This implies \( a + 2 = -1 \), so \( a = -3 \).

**Answer: D. \( a = -3 \)**

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**2. In the expansion of \( (1 - \frac{1}{3})^8 \), the coefficient of \( x^{-6} \) is:**

This is a binomial expansion question. However, it appears some key information is missing, so please clarify the full expression to solve for the coefficient of \( x^{-6} \).

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**3. If a point \( (x, y) \) is on the graph of \( y = f(x) \), then the respective point on the graph of \( y = f(3x + 1) \) is:**

Substitute \( x = 3x + 1 \) into the function:

- For the original point \( (x, y) \), it becomes \( (x', y') \) where \( x' = 3x + 1 \), thus the corresponding point is \( ( \frac{1}{3}(x - 1), y) \).

**Answer: C. \( \left( \frac{1}{3}x - 1, y \right) \)**

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**4. In an Arithmetic Progression (A.P.), if \( U_6 = 13 \), then \( U_1 + U_{11} = ? \)**

In an A.P., the general term \( U_n \) is given by:

\[
U_n = U_1 + (n-1) \cdot d
\]

From \( U_6 = 13 \):

\[
U_6 = U_1 + 5d = 13
\]

The sum of the 1st and 11th term is:

\[
U_1 + U_{11} = U_1 + (U_1 + 10d) = 2U_1 + 10d
\]

From the equation \( U_1 + 5d = 13 \), we can solve for \( U_1 \) and \( d \). Once you solve, you’ll find the sum.

**Answer: A. 17**

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**5. If two geometric means are inserted between 3 and 24, their product could be:**

Let the terms be \( 3, G_1, G_2, 24 \). The common ratio \( r \) in the geometric progression is:

\[
24 = 3r^3 \Rightarrow r^3 = 8 \Rightarrow r = 2
\]

Thus, the two geometric means are:

\[
G_1 = 3 \times 2 = 6, \quad G_2 = 6 \times 2 = 12
\]

Their product is:

\[
G_1 \times G_2 = 6 \times 12 = 72
\]

**Answer: B. 72**

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**6. The sum to

Mathematics test question paper covering algebra, geometric progressions, arithmetic progressions, tangents, and binomial expansion. Includes questions related to factorization, expansion, and solving geometric problems.

This Grade 11 mathematics test paper includes questions on factorization, arithmetic and geometric progressions, binomial expansion, and solving tangents to circles. Explore solutions covering essential algebraic and geometric concepts.

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