ESAT Maths 2 practice questions

Simplify ${\displaystyle \frac{\sqrt{12}+3}{\sqrt{3}-1}}$.

Evaluate $2{\log }_{3}6-{\log }_{3}4+{\log }_3\frac{1}{27}$.

Find the set of all real values of $x$ that satisfy both $2x-5<x+1$ and $x^2>3x+10$.

The points $A$ and $B$ have coordinates $(2,-1)$ and $(6,5)$. The line $L$ is the perpendicular bisector of $AB$. What is the $x$-coordinate of the point where $L$ crosses the $x$-axis?

Given that $y={\displaystyle \frac{x^2+8}{\sqrt{x}}}$, find the value of ${\displaystyle \frac{dy}{dx}}$ when $x=4$.

Given that $\cos \theta =-\frac{1}{3}$ and $\pi <\theta <\frac{3\pi }{2}$, what is the value of $\tan \theta$?

What is the sum of all the integers from 1 to 200 inclusive that are not multiples of 3?

Find the set of values of $k$ for which the equation $k{x}^2+4x+k-3=0$ has two distinct real roots.

The polynomial $p(x)=2x^3+a{x}^2+bx-6$, where $a$ and $b$ are constants, has $(x-2)$ as a factor. When $p(x)$ is divided by $(x+1)$ the remainder is $-12$.

What is the remainder when $p(x)$ is divided by $(2x+1)$?

Evaluate ${\displaystyle {\int }_1^9\frac{x-3}{\sqrt{x}}dx}$.

What is the coefficient of $x^3$ in the expansion of $(1+2x)(3-x{)}^5$?

What is the sum of the real solutions of the equation $4^x-2^{x+3}+12=0$?

A geometric series has second term 12 and sum to infinity 64. Every term of the series is less than 20.

What is the third term of the series?

The normal to the curve $y=x^3-3x^2+4$ at the point where $x=3$ crosses the $x$-axis at the point $P$. What is the $x$-coordinate of $P$?

In triangle $ABC$, $\sin A:\sin B:\sin C=4:5:6$. What is the value of $\cos C$?

The curve $y=f(x)$ has a local minimum at the point $(4,-3)$. Which one of the following is true of the curve $y=1-f(2x)$?

An arithmetic sequence has a non-zero common difference. Its 2nd, 5th and 14th terms, in that order, are three consecutive terms of a geometric sequence. What is the common ratio of the geometric sequence?

The line $y=x$ meets the circle $x^2+y^2-4x-6y-12=0$ at the points $P$ and $Q$. What is the length of $PQ$?

The trapezium rule with 4 strips of equal width is used to estimate ${\displaystyle {\int }_0^4{2}^{x}dx}$. Which of the following gives the estimate and correctly says whether it is an overestimate or an underestimate?

Given that ${\displaystyle {\int }_1^{4}f(x)dx=7}$, ${\displaystyle {\int }_1^{6}f(x)dx=3}$ and ${\displaystyle {\int }_4^{6}g(x)dx=5}$, find ${\displaystyle {\int }_4^6[2f(x)-3g(x)+1]dx}$.

A curve has gradient ${\displaystyle \frac{dy}{dx}}=3\sqrt{x}-{\displaystyle \frac{2}{x^2}}$ for $x>0$, and passes through the point $(1,5)$. What is the value of $y$ when $x=4$?

A sector of a circle has radius $r$ cm and angle $\theta$ radians, where $0<\theta <2\pi$. The perimeter of the sector is 20 cm and its area is 16 cm². What is the value of $\theta$?

Find the total area of the regions enclosed between the curve $y=x^3-x^2-2x$ and the $x$-axis.

In triangle $PQR$, $PQ=6$ cm and angle $QPR={30}^{\circ }$. The length of $QR$ is $x$ cm.

What is the complete set of values of $x$ for which there are exactly two different (non-congruent) triangles $PQR$ that fit this information?

How many real solutions does the equation $\mathrm{\mid }{x}^2-5\mathrm{\mid }=2x-2$ have?

Let $f(x)=2x^3-9x^2+12x+c$, where $c$ is a constant. For which values of $c$ does the equation $f(x)=0$ have three distinct real roots?

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