GCE Advanced Level (Sri Lanka) · Combined Mathematics

Pure Maths: Polynomials, Algebra and Sequences

75 practice questions, each with the answer and an explanation.

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All 75 questions

For what positive value of $k$ does $x^{2} + k x + 16 = 0$ have equal roots?
Find the coefficient of $x^{3}$ in the expansion of $(1 + 2 x)^{8}$ .
Evaluate $r = 1 \sum 8 (r + 5)$ .
The first term of an arithmetic progression is $9$ and the common difference is $7$ . Find the sum of the first $15$ terms.
Find the sum of all integers $x$ satisfying $x^{2} - 4 x - 5 < 0$ .
Solve $lo g_{3} (x + 1) = 3$ .
If $α$ and $β$ are the roots of $x^{2} - 8 x + 7 = 0$ , find $α^{2} + β^{2}$ .
The roots of $x^{2} + 2 x - 24 = 0$ are $α$ and $β$ . Which equation has roots $α^{2}$ and $β^{2}$ ?
For what positive value of $k$ does $x^{2} + k x + 4 = 0$ have equal roots?
Find the term independent of $x$ in the expansion of $(x + \frac{2}{x})^{6}$ .
Evaluate $r = 1 \sum 8 (4 r - 3)$ .
In a geometric progression the first term is $2$ and the common ratio is $3$ . Find the term number $5$ .
Find the sum of all integers $x$ satisfying $x^{2} - 5 x - 6 < 0$ .
Solve $2^{x} \cdot 2^{2} = 128$ .
If $α$ and $β$ are the roots of $x^{2} + 3 x - 10 = 0$ , find $\frac{1}{α} + \frac{1}{β}$ .
For what positive value of $k$ does $x^{2} + k x + 36 = 0$ have equal roots?
Find the remainder when $3 x^{3} + 2 x^{2} - 3 x + 8$ is divided by $(x - 3)$ .
Find the term independent of $x$ in the expansion of $(x + \frac{3}{x})^{4}$ .
Evaluate $r = 1 \sum 16 (3 r + 1)$ .
The first term of an arithmetic progression is $8$ and the common difference is $3$ . Find the sum of the first $13$ terms.
Find the sum of all integers $x$ satisfying $x^{2} - 4 < 0$ .
Solve $2^{x} \cdot 2^{3} = 512$ .
The roots of $x^{2} - 11 x + 30 = 0$ are $α$ and $β$ . Which equation has roots $α^{2}$ and $β^{2}$ ?
Find the remainder when $x^{3} + 5 x^{2} + 6 x - 1$ is divided by $(x - 2)$ .
Evaluate $r = 1 \sum 17 (3 r + 5)$ .
A geometric progression has first term $7$ and common ratio $\frac{3}{4}$ . Find its sum to infinity.
How many integers $x$ satisfy $∣ x + 4∣ < 3$ ?
Solve $lo g_{2} (x + 5) = 5$ .
The roots of $x^{2} - 4 x - 5 = 0$ are $α$ and $β$ . Which equation has roots $α^{2}$ and $β^{2}$ ?
Find the remainder when $2 x^{3} + 2 x^{2} - 5 x + 6$ is divided by $(x + 4)$ .
Find the term independent of $x$ in the expansion of $(x + \frac{3}{x})^{8}$ .
Evaluate $r = 1 \sum 5 r^{2}$ .
In a geometric progression the first term is $2$ and the common ratio is $2$ . Find the term number $6$ .
Evaluate $lo g_{5} 3125 + lo g_{5} 125 - lo g_{5} 5$ .
If $α$ and $β$ are the roots of $x^{2} + 5 x - 6 = 0$ , find $α^{3} + β^{3}$ .
For what positive value of $k$ does $x^{2} + k x + 9 = 0$ have equal roots?
Find the remainder when $2 x^{3} + 3 x^{2} - 5 x + 9$ is divided by $(x + 3)$ .
The first term of an arithmetic progression is $3$ and the common difference is $7$ . Find the sum of the first $12$ terms.
Find the sum of all integers $x$ satisfying $x^{2} - 3 x - 4 < 0$ .
Evaluate $lo g_{3} 27 + lo g_{3} 81 - lo g_{3} 3$ .
If $α$ and $β$ are the roots of $x^{2} + 9 x + 20 = 0$ , find $\frac{1}{α} + \frac{1}{β}$ .
For what positive value of $k$ does $x^{2} + k x + 81 = 0$ have equal roots?
Evaluate $r = 1 \sum 6 r^{2}$ .
The first term of an arithmetic progression is $8$ and the common difference is $4$ . Find the sum of the first $11$ terms.
Find the sum of all integers $x$ satisfying $x^{2} - 2 x - 15 < 0$ .
Evaluate $lo g_{3} 27 + lo g_{3} 9 - lo g_{3} 3$ .
If $α$ and $β$ are the roots of $x^{2} + x - 2 = 0$ , find $α^{3} + β^{3}$ .
The roots of $x^{2} - 6 x + 5 = 0$ are $α$ and $β$ . Which equation has roots $α^{2}$ and $β^{2}$ ?
Find the coefficient of $x^{3}$ in the expansion of $(1 + 2 x)^{5}$ .
Find $r = 1 \sum 15 \frac{1}{r ( r + 1 )}$ .
The first term of an arithmetic progression is $8$ and the common difference is $4$ . Find the sum of the first $29$ terms.
Solve $lo g_{5} (x + 8) = 4$ .
If $α$ and $β$ are the roots of $x^{2} + 4 x + 3 = 0$ , find $α^{3} + β^{3}$ .
The roots of $x^{2} - 10 x + 24 = 0$ are $α$ and $β$ . Which equation has roots $α^{2}$ and $β^{2}$ ?
Find the remainder when $x^{3} - 4 x^{2} + 6 x + 4$ is divided by $(x + 4)$ .
Find the coefficient of $x^{3}$ in the expansion of $(1 + 3 x)^{5}$ .
Find $r = 1 \sum 6 \frac{1}{r ( r + 1 )}$ .
The first term of an arithmetic progression is $8$ and the common difference is $2$ . Find the sum of the first $22$ terms.
If $α$ and $β$ are the roots of $x^{2} + 6 x + 8 = 0$ , find $α^{3} + β^{3}$ .
The roots of $x^{2} - x - 30 = 0$ are $α$ and $β$ . Which equation has roots $α^{2}$ and $β^{2}$ ?
Find the remainder when $3 x^{3} - 3 x^{2} + 5 x - 6$ is divided by $(x + 1)$ .
Find the coefficient of $x^{3}$ in the expansion of $(1 + 2 x)^{7}$ .
Find $r = 1 \sum 10 \frac{1}{r ( r + 1 )}$ .
In a geometric progression the first term is $2$ and the common ratio is $2$ . Find the term number $7$ .
Solve $lo g_{2} (x + 5) = 3$ .
The roots of $x^{2} - 4 x - 12 = 0$ are $α$ and $β$ . Which equation has roots $α^{2}$ and $β^{2}$ ?
Evaluate $r = 1 \sum 18 r^{2}$ .
The first term of an arithmetic progression is $3$ and the common difference is $2$ . Find the sum of the first $29$ terms.
Find the sum of all integers $x$ satisfying $x^{2} - 9 < 0$ .
Solve $lo g_{3} (x + 3) = 4$ .
If $α$ and $β$ are the roots of $x^{2} + 8 x + 15 = 0$ , find $α^{3} + β^{3}$ .
The roots of $x^{2} - 2 x - 8 = 0$ are $α$ and $β$ . Which equation has roots $α^{2}$ and $β^{2}$ ?
Find the remainder when $x^{3} + 4 x^{2} - 3 x - 2$ is divided by $(x - 4)$ .
In a geometric progression the first term is $4$ and the common ratio is $2$ . Find the term number $6$ .
Find the sum of all integers $x$ satisfying $x^{2} - 2 x - 8 < 0$ .

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