TMUA 2020 Paper 1

Which of the following is an expression for the first derivative with respect to $x$ of $$\frac{x^3-5x^2}{2x\sqrt{x}}$$

$(2x+1)$ and $(x-2)$ are factors of $2x^3+p{x}^2+q$. What is the value of $2p+q$?

Find the complete set of values of $x$ for which $$(x+4)(x+3)(1-x)>0\text{and}(x+2)(x-2)<0$$

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.

The curve $S$ has equation $y=p{x}^2+6x-q$ 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 $p+8q$?

Find the maximum value of the function $$\mathrm{f}(x)=\frac{1}{5^{2x}-4(5^x)+7}$$

Given that $$2^{3x}=8^{(y+3)}$$ and $$4^{(x+1)}=\frac{16^{(y+1)}}{8^{(y+3)}}$$ what is the value of $x+y$?

The function $\mathrm{f}$ 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.

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 $(x-\sqrt{\alpha })(x-\sqrt{\beta })$?

The following sequence of transformations is applied to the curve $y=4x^2$: (1) Translation by $\left(\begin{array}{c}3\\ -5\end{array}\right)$, (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?

The quadratic function shown passes through $(2,0)$ and $(q,0)$, where $q>2$. Region $R$ is bounded by the curve and the $y$-axis (above the $x$-axis), and region $S$ is bounded by the curve and the $x$-axis between $x=2$ and $x=q$ (below the $x$-axis). What is the value of $q$ such that the area of region $R$ equals the area of region $S$?

How many real solutions are there to the equation $$3\cos x=\sqrt{x}$$ where $x$ is in radians?

Find the coefficient of $x^2{y}^4$ in the expansion of $(1+x+y^2{)}^7$.

The area enclosed between the line $y=mx$ and the curve $y=x^3$ is 6. What is the value of $m$?

Find the positive difference between the two real values of $x$ for which $$({\log }_{2}x{)}^4+12{\left({\log }_2\left(\frac{1}{x}\right)\right)}^2-2^6=0$$

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 $\tan \theta$.

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.

Find the number of solutions and the sum of the solutions of the equation $$1-2{\cos }^{2}x=|\cos x|$$ where $0\le x\le 180^{\circ }$.

Find the lowest positive integer for which $x^2-52x-52$ is positive.

For how many values of $a$ is the equation $$(x-a)(x^2-x+a)=0$$ satisfied by exactly two distinct values of $x$?