Leaving Cert Ireland · Mathematics

Algebra, Functions, Calculus and Statistics

36 practice questions, each with the answer and an explanation.

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All 36 questions

Solve the equation $3 x - 7 = 11$ for $x$ .
What is $\frac{d}{d x} (x^{4})$ ?
What is the sum of the interior angles of a hexagon?
Find the mean of the numbers 4, 8, 9, 11 and 13.
Find the distance between the points $(1, 2)$ and $(4, 6)$ .
Simplify $(2 + 3 i) (2 - 3 i)$ , where $i^{2} = - 1$ .
What are the roots of $x^{2} - 5 x + 6 = 0$ ?
What is the slope of the line $2 x + 3 y = 12$ ?
An arithmetic sequence has first term 5 and common difference 3. What is the 20th term?
Find the sum to infinity of the geometric series $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots$
Evaluate $\int (6 x^{2} + 2) d x$ .
Two fair dice are rolled. What is the probability that the sum is 7?
In a triangle two sides have lengths 5 and 7 and the angle between them is $6 0^{\circ}$ . Find the third side.
Write the equation of the circle with centre $(2, - 3)$ and radius 4.
For $f (x) = x^{3} - 3 x$ , what is the nature of the stationary point at $x = 1$ ?
Simplify $lo g_{10} 4 + lo g_{10} 25$ .
€1000 is invested at 5% per year compound interest. What is its value after 2 years?
In how many different orders can 5 different books be arranged on a shelf?
A data set has mean 12 and standard deviation 3. Ten is added to every value. What happens?
Find the modulus of the complex number $3 + 4 i$ .
In how many ways can a committee of 3 be chosen from 8 people?
Find the area of a sector of a circle with radius 6 and angle $\frac{π}{3}$ radians.
Find the equation of the line through $(1, 2)$ with slope 3.
Differentiate $y = x sin x$ with respect to $x$ .
Differentiate $y = (2 x + 1)^{5}$ with respect to $x$ .
Find the volume of a cone with base radius 3 and height 4.
Let $z = 1 + i$ . Find $z^{8}$ .
The first term of an arithmetic series is 2 and the sum of its first 10 terms is 155. Find the common difference.
For which values of $k$ does $x^{2} + k x + 9 = 0$ have equal roots?
A rectangle has a perimeter of 40 cm. What is the largest area it can have?
Find the area enclosed between the curve $y = x^{2}$ and the line $y = 2 x$ .
What is the $y$ -intercept of the tangent to $y = x^{2}$ at the point where $x = 3$ ?
In a proof by induction that $1 + 2 + \dots + n = \frac{n ( n + 1 )}{2}$ , the inductive step assumes the result for $n = k$ . Which expression must be shown to equal $\frac{( k + 1 ) ( k + 2 )}{2}$ ?
A bag holds 5 red and 3 blue counters. Two are drawn one after the other without replacement. What is the probability that both are red?
Solve $2 sin x = 1$ for $0 \leq x \leq 2 π$ .
What is the value of $50 - 18$ in simplest surd form?

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Algebra, Functions, Calculus and Statistics

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