It is given that the expansion of $( a x + b ) ^ { 3 }$ is $8 x ^ { 3 } - p x ^ { 2 } + 18 x - 3 \sqrt { 3 }$, where $a , b$ and $p$ are real constants. What is the value of $p$ ? A $- 12 \sqrt { 3 }$ B $- 6 \sqrt { 3 }$ C $- 4 \sqrt { 3 }$ D $- \sqrt { 3 }$ E $\sqrt { 3 }$ F $4 \sqrt { 3 }$ G $6 \sqrt { 3 }$ H $12 \sqrt { 3 }$
The expression $3 x ^ { 3 } + 13 x ^ { 2 } + 8 x + a$, where $a$ is a constant, has ( $x + 2$ ) as a factor. Which one of the following is a complete factorisation of the expression? A $( x + 2 ) ( x - 1 ) ( 3 x - 2 )$ B $( x + 2 ) ( x + 1 ) ( 3 x - 2 )$ C $( x + 2 ) ( x + 1 ) ( 3 x + 2 )$ D $( x + 2 ) ( x - 3 ) ( 3 x + 2 )$ E $( x + 2 ) ( x + 3 ) ( 3 x - 2 )$ F $( x + 2 ) ( x + 3 ) ( 3 x + 2 )$
A line is drawn normal to the curve $y = \frac { 2 } { x ^ { 2 } }$ at the point on the curve where $x = 1$. This line cuts the $x$-axis at $P$ and the $y$-axis at $Q$. The length of $P Q$ is A $\frac { 3 \sqrt { 5 } } { 2 }$ B $\frac { 3 \sqrt { 17 } } { 4 }$ C $\frac { 7 \sqrt { 17 } } { 4 }$ D $\frac { 35 } { 4 }$ E $\frac { 35 \sqrt { 5 } } { 2 }$ F $\frac { 3 \sqrt { 17 } } { 2 }$
The sequence $a _ { n }$ is defined by the rule: $$a _ { n } = ( - 1 ) ^ { n } - ( - 1 ) ^ { n - 1 } + ( - 1 ) ^ { n + 2 } \text { for } n \geq 1$$ Find the value of $$\sum _ { n = 1 } ^ { 39 } a _ { n }$$ A - 39 B - 3 C - 1 D 0 E 1 F 3 G 39
What is the total area enclosed between the curve $y = x ^ { 2 } - 1$, the $x$-axis and the lines $x = - 2$ and $x = 2$ ? A $\frac { 4 } { 3 }$ B $\frac { 8 } { 3 }$ C 4 D $\frac { 16 } { 3 }$ E 12 F 16
P, Q, and R are each mixtures of red and white paint. The percentage by volume of red paint in P is $30 \%$. The percentage by volume of red paint in Q is 20\%. The mixtures P, Q, and R are combined in the proportion $12 : 5 : 3$ respectively. If the resulting mixture contains $25 \%$ by volume of red paint, what percentage by volume of mixture $R$ is red paint? A $25 \%$ B 23\% C $13 \frac { 1 } { 3 } \%$ D $19 \frac { 1 } { 2 } \%$ E $9 \frac { 3 } { 4 } \%$ F It is impossible to achieve this result.
60\% of a sports club's members are women and the remainder are men. This sports club offers the opportunity to play tennis or cricket. Every member plays exactly one of the two sports. $\frac { 2 } { 5 }$ of the male members of the club play cricket; $\frac { 2 } { 3 }$ of the cricketing members of the club are women. What is the probability that a member of the club, chosen at random, is a woman who plays tennis? A $\frac { 1 } { 5 }$ B $\frac { 7 } { 25 }$ C $\frac { 1 } { 3 }$ D $\frac { 11 } { 25 }$ E $\frac { 3 } { 5 }$
The line segment joining the points $( 3,3 )$ and ( 7,5 ) is a diameter of a circle. This circle is translated by 3 units in the negative $x$-direction, then reflected in the $x$-axis, and then enlarged by a scale factor of 4 about the centre of the resulting circle. The equation of the final circle is A $( x - 2 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 320$ B $( x - 2 ) ^ { 2 } + ( y + 4 ) ^ { 2 } = 320$ C $( x - 2 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 80$ D $( x - 2 ) ^ { 2 } + ( y + 4 ) ^ { 2 } = 80$ E $( x - 2 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 20$ F $( x - 2 ) ^ { 2 } + ( y + 4 ) ^ { 2 } = 20$
A right circular cylinder is contained within a sphere of radius 5 cm in such a way that the whole of the circumferences of both ends of the cylinder are in contact with the sphere. The diagram shows a planar cross section through the centre of the sphere and cylinder. Find, in cubic centimetres, the maximum possible volume of the cylinder. A $250 \pi$ B $500 \pi$ C $1000 \pi$ D $\frac { 250 \sqrt { 3 } } { 3 } \pi$ E $\frac { 500 \sqrt { 3 } } { 9 } \pi$ F $\frac { 1000 \sqrt { 3 } } { 9 } \pi$
The terms of an infinite series $S$ are formed by adding together the corresponding terms in two infinite geometric series, T and U . The first term of T and the first term of U are each 4. In order, the first three terms of the combined series $S$ are 8,3 , and $\frac { 5 } { 4 }$ What is the sum to infinity of $S$ ? A $\frac { 32 } { 5 }$ B $\frac { 20 } { 3 }$ C $\frac { 64 } { 5 }$ D $\frac { 40 } { 3 }$ E 16 F 32
The least possible value of the gradient of the curve $y = ( 2 x + a ) ( x - 2 a ) ^ { 2 }$ at the point where $x = 1$, as $a$ varies, is A $- \frac { 49 } { 4 }$ B - 8 C $- \frac { 25 } { 4 }$ D $\frac { 7 } { 4 }$ E $\frac { 47 } { 16 }$
The function $\frac { 1 - x } { \sqrt [ 3 ] { x ^ { 2 } } }$ is defined for all $x \neq 0$. The complete set of values of $x$ for which the function is decreasing is A $x \leq - 2 , x > 0$ B $- 2 \leq x < 0$ C $x \leq 1 , x \neq 0$ D $x \geq 1$ E $- 2 \leq x \leq 1 , \quad x \neq 0$ F $x \leq - 2 , x \geq 1$
The coefficient of $x ^ { 3 }$ in the expansion of $\left( 1 + 2 x + 3 x ^ { 2 } \right) ^ { 6 }$ is equal to twice the coefficient of $x ^ { 4 }$ in the expansion of $\left( 1 - a x ^ { 2 } \right) ^ { 5 }$. Find all possible values of the constant $a$. A $\pm 2 \sqrt { 2 }$ B $\pm \sqrt { 17 }$ C $\pm \sqrt { 34 }$ D $\pm 2 \sqrt { 17 }$ E There are no possible values of $a$.
The diagram shows a square-based pyramid with base $P Q R S$ and vertex $O$. All the edges of the pyramid are of length 20 metres. Find the shortest distance, in metres, along the outer surface of the pyramid from $P$ to the midpoint of $O R$. A $10 \sqrt { 5 - 2 \sqrt { 3 } }$ B $10 \sqrt { 3 }$ C $10 \sqrt { 5 }$ D $10 \sqrt { 7 }$ E $10 \sqrt { 5 + 2 \sqrt { 3 } }$