TMUA Paper 1 practice: Applications of Mathematical Knowledge

Simplify ${\displaystyle \frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}}-{\displaystyle \frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}}$.

How many integers $n$ satisfy both $n^2-4n<21$ and $n^2+n>12$?

The points $A(2,7)$ and $B(6,-1)$ are given. The perpendicular bisector of $AB$ meets the $x$-axis at $P$ and the $y$-axis at $Q$. If $O$ is the origin, what is the area of triangle $OPQ$?

In triangle $ABC$, $AB=5$, $AC=8$ and angle $BAC={60}^{\circ }$. The point $D$ lies on $BC$ and $AD$ is perpendicular to $BC$. What is the length of $AD$?

The third term of an arithmetic sequence is 7, and the sum of its first 11 terms is 33. What is the sum of its first 20 terms?

The equation $x^2+y^2-6x+4y=k$ represents a circle that does not meet either of the coordinate axes. Which of the following describes all the possible values of $k$?

A bag contains $n$ red counters and 3 blue counters, where $n\ge 1$. Two counters are taken out at random, without replacement. The probability that the two counters are the same colour is $\frac{1}{2}$. Which of the following gives all the possible values of $n$?

The tangent to the curve $y=x^3-3x+16$ at the point $P$ passes through the origin. What is the gradient of this tangent?

How many solutions does the equation $2{\sin }^{2}x+3\cos x=0$ have in the interval $-\pi \le x\le \frac{5\pi }{2}$?

In the expansion of $(1+ax{)}^5(1-2x)$, where $a$ is a non-zero constant, the coefficient of $x^2$ is zero. What is the coefficient of $x^3$?

The curve $y=f(x)$ has exactly one stationary point, which is a maximum at $(2,5)$. Which of the following describes the stationary point of the curve $y=1-2f(3-x)$?

When the polynomial $p(x)$ is divided by $(x-1)$ the remainder is 3, and when it is divided by $(x+2)$ the remainder is $-6$. What is the remainder when $p(x)$ is divided by $x^2+x-2$?

What is the sum of all the real solutions of the equation $4^x-5\times 2^x+3=0$?

A convergent geometric series has sum to infinity 12. The series formed by squaring each of its terms has sum to infinity 48. What is the common ratio of the original series?

A sequence is defined by $u_1=2$ and $u_{n+1}={\displaystyle \frac{1}{1-u_n}}$ for $n\ge 1$. What is the value of $u_1+u_2+\cdots +u_{100}$?

What is the greatest value of $3{\sin }^2\theta +2\cos \theta$ as $\theta$ varies over all real numbers?

For how many integer values of $k$ does the equation $x^3-12x+k=0$ have three distinct real solutions?

Let $p=2^{100}$, $q=3^{60}$ and $r={10}^{30}$. Which of the following is true?

The line $y=kx$, where $0<k<4$, and the curve $y=4x-x^2$ enclose a finite region of area $\frac{9}{2}$. What is the value of $k$?

As $k$ varies over the real numbers, what is the least possible value of ${\int }_0^2(x^2-k{)}^{2}dx$?