Which of the following is an expression for the first derivative with respect to $x$ of $$\frac { x ^ { 3 } - 5 x ^ { 2 } } { 2 x \sqrt { x } }$$ A $- \frac { \sqrt { x } } { 2 }$ B $\frac { \sqrt { x } } { 4 }$ C $\frac { 3 x - 5 } { 4 \sqrt { x } }$ D $\frac { 3 \sqrt { x } - 5 } { 4 \sqrt { x } }$ E $\frac { 3 \sqrt { x } - 10 } { 3 \sqrt { x } }$ F $\frac { 3 x ^ { 2 } - 10 x } { 3 \sqrt { x } }$
$(2x+1)$ and $(x-2)$ are factors of $2x^3 + px^2 + q$ What is the value of $2p + q$? A $-10$ B $-\frac{38}{5}$ C $-\frac{22}{3}$ D $\frac{22}{3}$ E $\frac{38}{5}$ F $10$
Find the complete set of values of $x$ for which $$(x+4)(x+3)(1-x) > 0 \text{ and } (x+2)(x-2) < 0$$ A $1 < x < 2$ B $-2 < x < 1$ C $-2 < x < 2$ D $x < -2$ or $x > 1$ E $x < -4$ or $x > 2$ F $x < -4$ or $-3 < x < 1$ G $-4 < x < -2$ or $x > 1$
The $1^{\text{st}}, 2^{\text{nd}}$ and $3^{\text{rd}}$ terms of a geometric progression are also the $1^{\text{st}}, 4^{\text{th}}$ and $6^{\text{th}}$ terms, respectively, of an arithmetic progression. The sum to infinity of the geometric progression is 12. Find the $1^{\text{st}}$ term of the geometric progression. A $1$ B $2$ C $3$ D $4$ E $5$ F $6$
The curve $S$ has equation $$y = px^2 + 6x - q$$ where $p$ and $q$ are constants. $S$ has a line of symmetry at $x = -\frac{1}{4}$ and touches the $x$-axis at exactly one point. What is the value of $p + 8q$? A $6$ B $18$ C $21$ D $25$ E $38$
Find the maximum value of the function $$\mathrm{f}(x) = \frac{1}{5^{2x} - 4(5^x) + 7}$$ A $\frac{1}{7}$ B $\frac{1}{4}$ C $\frac{1}{3}$ D $3$ E $4$ F $7$
Given that $$2^{3x} = 8^{(y+3)}$$ and $$4^{(x+1)} = \frac{16^{(y+1)}}{8^{(y+3)}}$$ what is the value of $x + y$? A $-23$ B $-22$ C $-15$ D $-14$ E $-11$ F $-10$
The function f is defined for all real $x$ as $$\mathrm{f}(x) = (p-x)(x+2)$$ Find the complete set of values of $p$ for which the maximum value of $\mathrm{f}(x)$ is less than 4. A $-2 - 4\sqrt{2} < p < -2 + 4\sqrt{2}$ B $-2 - 2\sqrt{2} < p < -2 + 2\sqrt{2}$ C $-2\sqrt{5} < p < 2\sqrt{5}$ D $-6 < p < 2$ E $-4 < p < 0$ F $-2 < p < 2$
The quadratic expression $x^2 - 14x + 9$ factorises as $(x - \alpha)(x - \beta)$, where $\alpha$ and $\beta$ are positive real numbers. Which quadratic expression can be factorised as $(x - \sqrt{\alpha})(x - \sqrt{\beta})$? A $x^2 - \sqrt{10}x + 3$ B $x^2 - \sqrt{14}x + 3$ C $x^2 - \sqrt{20}x + 3$ D $x^2 - 178x + 81$ E $x^2 - 176x + 81$ F $x^2 + 196x + 81$
The following sequence of transformations is applied to the curve $y = 4x^2$ 1. Translation by $\binom{3}{-5}$ 2. Reflection in the $x$-axis 3. Stretch parallel to the $x$-axis with scale factor 2 What is the equation of the resulting curve? A $y = -x^2 + 12x - 31$ B $y = -x^2 + 12x - 41$ C $y = x^2 + 12x + 31$ D $y = x^2 + 12x + 41$ E $y = -16x^2 + 48x - 31$ F $y = -16x^2 + 48x - 41$ G $y = 16x^2 - 48x + 31$ H $y = 16x^2 - 48x + 41$
The quadratic function shown passes through $(2,0)$ and $(q, 0)$, where $q > 2$. What is the value of $q$ such that the area of region $R$ equals the area of region $S$? A $\sqrt{6}$ B $3$ C $\frac{18}{5}$ D $4$ E $6$ F $\frac{33}{5}$
How many real solutions are there to the equation $$3\cos x = \sqrt{x}$$ where $x$ is in radians? A $0$ B $1$ C $2$ D $3$ E $4$ F $5$ G infinitely many
The area enclosed between the line $y = mx$ and the curve $y = x^3$ is 6. What is the value of $m$? A $2$ B $4$ C $\sqrt{3}$ D $\sqrt{6}$ E $2\sqrt{3}$ F $2\sqrt{6}$
Find the positive difference between the two real values of $x$ for which $$(\log_2 x)^4 + 12(\log_2(\frac{1}{x}))^2 - 2^6 = 0$$ A $4$ B $16$ C $\frac{15}{4}$ D $\frac{17}{4}$ E $\frac{255}{16}$ F $\frac{257}{16}$
The circle $C_1$ has equation $(x+2)^2 + (y-1)^2 = 3$ The circle $C_2$ has equation $(x-4)^2 + (y-1)^2 = 3$ The straight line $l$ is a tangent to both $C_1$ and $C_2$ and has positive gradient. The acute angle between $l$ and the $x$-axis is $\theta$ Find the value of $\tan\theta$ A $\frac{1}{2}$ B $2$ C $\frac{\sqrt{2}}{2}$ D $\sqrt{2}$ E $\frac{\sqrt{6}}{2}$ F $\frac{\sqrt{6}}{3}$ G $\frac{\sqrt{3}}{3}$ H $\sqrt{3}$
Find the complete set of values of $m$ in terms of $c$ such that the graphs of $y = mx + c$ and $y = \sqrt{x}$ have two points of intersection. A $0 < m < \frac{1}{4c}$ B $0 < m < 4c^2$ C $m > \frac{1}{4c}$ D $m < \frac{1}{4c}$ E $m > 4c^2$ F $m < 4c^2$
Find the number of solutions and the sum of the solutions of the equation $$1 - 2\cos^2 x = |\cos x|$$ where $0 \leq x \leq 180^{\circ}$ A Number of solutions $= 2$ Sum of solutions $= 180^{\circ}$ B Number of solutions $= 2$ Sum of solutions $= 240^{\circ}$ C Number of solutions $= 3$ Sum of solutions $= 180^{\circ}$ D Number of solutions $= 3$ Sum of solutions $= 360^{\circ}$ E Number of solutions $= 4$ Sum of solutions $= 240^{\circ}$ F Number of solutions $= 4$ Sum of solutions $= 360^{\circ}$
For how many values of $a$ is the equation $$(x - a)(x^2 - x + a) = 0$$ satisfied by exactly two distinct values of $x$? A $0$ B $1$ C $2$ D $3$ E $4$ F more than 4