TMUA 2016 Paper 2

Find the value of $${\int }_1^2{\left(x^2-\frac{4}{x^2}\right)}^{2}dx$$

$$f(x)=\frac{(x^2+5)(2x)}{\sqrt[4]{x^3}},x>0$$Which one of the following is equal to $f^{\prime }(x)$?

What is the value, in radians, of the largest angle $x$ in the range $0\le x\le 2\pi$ that satisfies the equation $8{\sin }^{2}x+4{\cos }^{2}x=7$?

Five sealed urns, labelled P, Q, R, S, and T, each contain the same (non-zero) number of balls. The following statements are attached to the urns.

Urn P: This urn contains one or four balls. Urn Q: This urn contains two or four balls. Urn R: This urn contains more than two balls and fewer than five balls. Urn S: This urn contains one or two balls. Urn T: This urn contains fewer than three balls.

Exactly one of the urns has a true statement attached to it.

Which urn is it?

Consider the statement:

$(\ast )$ A whole number $n$ is prime if it is 1 less or 5 less than a multiple of 6.

How many counterexamples to $(\ast )$ are there in the range $0<n<50$?

The sequence of functions $f_1(x),f_2(x),f_3(x),\dots$ is defined as follows: $$f_1(x)=x^{10}$$ $$f_{n+1}(x)=x{f}_n^{\prime }(x)\text{for }n\ge 1$$ where $f_n^{\prime }(x)=\frac{d{f}_n(x)}{dx}$.

Find the value of $$\sum _{n=1}^{20}{f}_n(x)$$

The four real numbers $a,b,c,$ and $d$ are all greater than 1.

Suppose that they satisfy the equation ${\log }_{c}d=({\log }_{a}b{)}^2$.

Use some of the lines given to construct a proof that, in this case, it follows that $$(\ast ){\log }_{b}d=({\log }_{a}b)({\log }_{a}c).$$The available lines are:
- (1) Let $x={\log }_{a}b$ and $y={\log }_{a}c$
- (2) $d=(c^x{)}^2$
- (3) $d=c^{(x^2)}$
- (4) $d=b^{xy}$
- (5) $d=(a^y{)}^{(x^2)}$
- (6) $d=((a^y{)}^x{)}^2$
- (7) $d=(a^x{)}^{xy}$
- (8) $d=a^{(y^{2x})}$
- (9) $d=a^{(x^{2}y)}$

Which sequence of lines gives a correct proof?

A region is defined by the inequalities $x+y>6$ and $x-y>-4$.

Consider the three statements:
1. $x>1$
2. $y>5$
3. $(x+y)(x-y)>-24$

Which of the above statements is/are true for every point in the region?

Triangles $ABC$ and $XYZ$ have the same area.

Which of these extra conditions, taken independently, would imply that they are congruent?

(1) $AB=XY$ and $BC=YZ$
(2) $AB=XY$ and $\angle ABC=\angle XYZ$
(3) $\angle ABC=\angle XYZ$ and $\angle BCA=\angle YZX$

In this question $x$ and $y$ are non-zero real numbers.

Which one of the following is sufficient to conclude that $x<y$?

$f(x)$ is a polynomial with real coefficients.

The equation $f(x)=0$ has exactly two real roots, $x=-p$ and $x=p$, where $p>0$.

Consider the following three statements:
1. $f^{\prime }(x)=0$ for exactly one value of $x$ between $-p$ and $p$
2. The area between the curve $y=f(x)$, the $x$-axis and the lines $x=-p$ and $x=p$ is given by $2{\int }_0^{p}f(x)dx$
3. The graph of $y=-f(-x)$ intersects the $x$-axis at the points $x=-p$ and $x=p$ only

Which of the above statements must be true?

The first term of an arithmetic sequence is $a$ and the common difference is $d$.

The sum of the first $n$ terms is denoted by $S_n$.

If $S_8>3S_6$, what can be deduced about the sign of $a$ and the sign of $d$?

In this question, $a$, $b$, and $c$ are positive integers.

The following is an attempted proof of the false statement:

"If $a$ divides $bc$, then $a$ divides $b$ or $a$ divides $c$."

[$a$ divides $bc$ means $a$ is a factor of $bc$.]

Which line contains the error in this proof?

1. The statement is equivalent to: if $a$ does not divide $b$ and $a$ does not divide $c$, then $a$ does not divide $bc$.
2. Suppose $a$ does not divide $b$ and $a$ does not divide $c$. Then the remainder when dividing $b$ by $a$ is $r$, where $0<r<a$, and the remainder when dividing $c$ by $a$ is $s$, where $0<s<a$.
3. So $b=ax+r$ and $c=ay+s$ for some integers $x$ and $y$.
4. Thus $bc=a(axy+xs+yr)+rs$.
5. So the remainder when dividing $bc$ by $a$ is $rs$.
6. Since $r>0$ and $s>0$, it follows that $rs>0$.
7. Hence $a$ does not divide $bc$.

$f(x)=a{x}^4+b{x}^3+c{x}^2+dx+e$, where $a,b,c,d,$ and $e$ are real numbers.

Suppose $f(x)=1$ has $p$ distinct real solutions, $f(x)=2$ has $q$ distinct real solutions, $f(x)=3$ has $r$ distinct real solutions, and $f(x)=4$ has $s$ distinct real solutions.

Which one of the following is not possible?

Consider the quadratic $f(x)=x^2-2px+q$ and the statement:

$(\ast )$ $f(x)=0$ has two real roots whose difference is greater than 2 and less than 4.

Which one of the following statements is true if and only if $(\ast )$ is true?

In the figure, $PQRS$ is a trapezium with $PQ$ parallel to $SR$.

The diagonals of the trapezium meet at $X$.

$U$ lies on $SP$ and $T$ lies on $RQ$ such that $UT$ is a line segment through $X$ parallel to $PQ$.

The length of $PQ$ is 12 cm and the length of $SR$ is 3 cm.

What, in centimetres, is the length of $UT$?

Consider these simultaneous equations, where $c$ is a constant: $$y=3\sin x+2$$ $$y=x+c$$Which of the following statements is/are true?

1. For some value of $c$: there is exactly one solution with $0\le x\le \pi$ and there is at least one solution with $-\pi <x<0$.
2. For some value of $c$: there is exactly one solution with $0\le x\le \pi$ and there are no solutions with $-\pi <x<0$.
3. For some value of $c$: there is exactly one solution with $0\le x\le \pi$ and there are no solutions with $x>\pi$.

Consider this statement about a function $f(x)$:

$(\ast )$ If $(f(x){)}^2\le 1$ for all $-1\le x\le 1$, then ${\int }_{-1}^1(f(x){)}^{2}dx\le {\int }_{-1}^{1}f(x)dx$.

Which one of the following functions provides a counterexample to $(\ast )$?

Some identical unit cubes are used to construct a three-dimensional object by gluing them together face to face.

Sketches of this object are made by looking at it from the right-hand side, from the front and from above. These sketches are called the side elevation, the front elevation, and the plan view respectively.

The side elevation shows a 2-wide by 3-tall grid (top-right cell occupied, then 2 wide rows below). The front elevation shows a 3-wide by 2-tall grid (top-right cell occupied, then a full 3-wide bottom row). The plan view shows an L-shaped 3-wide by 2-tall grid with the top-right cell absent (2 cells in top row, 3 cells in bottom row).

How many cubes were used to construct the object?

Each interior angle of a regular polygon with $n$ sides is $\frac{3}{4}$ of each interior angle of a second regular polygon with $m$ sides.

How many pairs of positive integers $n$ and $m$ are there for which this statement is true?