UNI Admission Exam (Peru) · Mathematics

Algebra, calculus and geometry

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

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All 88 questions

Find the $x$ -coordinate of the local minimum of $f (x) = x^{3} - 3 x$ .
What are the coordinates of the vertex of the parabola $y = - (x + 1)^{2} - 5$ ?
The function $f$ is defined by $f (x) = 4 x - 5$ . Find $f^{- 1} (- 3)$ .
How many real roots does the equation $4 x^{2} + 2 x + 0 = 0$ have?
Find the gradient of the tangent to $y = x^{2} - 3 x$ at the point where $x = - 1$ .
Solve $3^{x + 2} = 81$ for $x$ .
What is the area enclosed by $y = x^{2}$ , the $x$ -axis and the line $x = 2$ (in square units)?
Differentiate $y = e^{4 x}$ with respect to $x$ .
Let $f (x) = x^{2} (4 x + 1)$ . Use the product rule to find $f^{'} (1)$ .
A rectangle has a fixed perimeter of 20 units. What is the greatest possible area of the rectangle?
A radioactive substance has a half-life of 2 years. How much of a 320 g sample remains after 8 years?
For events $A$ and $B$ , $P (B) = 0.4$ and $P (A \cap B) = 0.2$ . What is $P (A ∣ B)$ ?
If $f (x) = 3 x^{3} + 4 x^{2} + 4$ , what is $f^{'} (4)$ ?
What is the value of $lo g_{3} (8 1^{4})$ ?
What is the sum to infinity of the geometric series with first term 6 and common ratio $\frac{1}{3}$ ?
Find the determinant of the matrix with rows $(5, 6)$ and $(4, 5)$ .
Find the dot product of the vectors $(2, - 3, 3)$ and $(5, 3, 5)$ .
What is the distance between the points $(1, 2)$ and $(9, - 13)$ ?
What is the centre of the circle $x^{2} + y^{2} - 4 x - 2 y - 31 = 0$ ?
In how many ways can 4 books be chosen from 6 different books?
What is the modulus of the complex number $5 - 12 i$ ?
If $α$ and $β$ are the roots of $x^{2} - 5 x + 4 = 0$ , what is $α^{2} + β^{2}$ ?
What is the value of $sin 15 0^{\circ}$ ?
What is the sum of the first 22 terms of an arithmetic progression with first term 7 and common difference 4?
Find $lim_{x \to 2} \frac{x ^{2} - 4}{x - 2}$ .
A line is given by $y = 3 x + 1$ . What is the slope of a line perpendicular to it?
If $f (x) = (4 x - 4)^{3}$ , find $f^{'} (2)$ .
What are the coordinates of the vertex of the parabola $y = (x - 5)^{2} - 1$ ?
The function $f$ is defined by $f (x) = 4 x - 3$ . Find $f^{- 1} (8)$ .
How many real roots does the equation $1 x^{2} + 1 x + 2 = 0$ have?
Find the gradient of the tangent to $y = 2 x^{2} + 3 x$ at the point where $x = 1$ .
Solve $2^{x + 2} = 8$ for $x$ .
What is the sum of the first 7 terms of the geometric sequence with first term 5 and common ratio 3?
What is the area enclosed by $y = 2 x^{2}$ , the $x$ -axis and the line $x = 2$ (in square units)?
Differentiate $y = e^{2 x}$ with respect to $x$ .
Let $f (x) = x^{2} (3 x + 3)$ . Use the product rule to find $f^{'} (1)$ .
A rectangle has a fixed perimeter of 40 units. What is the greatest possible area of the rectangle?
A radioactive substance has a half-life of 3 years. How much of a 800 g sample remains after 9 years?
For events $A$ and $B$ , $P (B) = 0.5$ and $P (A \cap B) = 0.25$ . What is $P (A ∣ B)$ ?
If $f (x) = x^{3} + 5 x^{2}$ , what is $f^{'} (1)$ ?
What is the value of $lo g_{5} (312 5^{2})$ ?
What is the sum to infinity of the geometric series with first term 8 and common ratio $\frac{2}{3}$ ?
Find the determinant of the matrix with rows $(- 4, 1)$ and $(0, 3)$ .
Find the dot product of the vectors $(- 3, 5, 2)$ and $(3, 2, - 3)$ .
What is the distance between the points $(0, - 3)$ and $(5, - 15)$ ?
What is the centre of the circle $x^{2} + y^{2} + 4 x - 2 y - 44 = 0$ ?
In how many ways can 4 books be chosen from 7 different books?
What is the modulus of the complex number $3 - 4 i$ ?
If $α$ and $β$ are the roots of $x^{2} - 8 x + 12 = 0$ , what is $α^{2} + β^{2}$ ?
What is the value of $cos 21 0^{\circ}$ ?
What is the sum of the first 19 terms of an arithmetic progression with first term 4 and common difference 7?
Find $lim_{x \to 5} \frac{x ^{2} - 25}{x - 5}$ .
If $f (x) = (4 x + 4)^{2}$ , find $f^{'} (2)$ .
Find the $x$ -coordinate of the local minimum of $f (x) = x^{3} - 12 x$ .
The function $f$ is defined by $f (x) = 2 x + 5$ . Find $f^{- 1} (0)$ .
How many real roots does the equation $4 x^{2} - 8 x + 0 = 0$ have?
Find the gradient of the tangent to $y = 4 x^{2} + 5 x$ at the point where $x = 1$ .
What is the sum of the first 5 terms of the geometric sequence with first term 2 and common ratio 3?
An amount of 2000 dollars is invested at 20% per annum compounded annually. What is its value after 2 years, in dollars?
What is the area enclosed by $y = 3 x^{2}$ , the $x$ -axis and the line $x = 4$ (in square units)?
Let $f (x) = x^{2} (2 x + 3)$ . Use the product rule to find $f^{'} (1)$ .
A radioactive substance has a half-life of 3 years. How much of a 320 g sample remains after 12 years?
For events $A$ and $B$ , $P (B) = 0.4$ and $P (A \cap B) = 0.1$ . What is $P (A ∣ B)$ ?
If $f (x) = 3 x^{3} + 4 x^{2} - 5$ , what is $f^{'} (1)$ ?
What is the value of $lo g_{2} (8^{3})$ ?
What is the sum to infinity of the geometric series with first term 8 and common ratio $\frac{1}{3}$ ?
Find the determinant of the matrix with rows $(- 3, 5)$ and $(1, - 3)$ .
Find the dot product of the vectors $(2, 5, 5)$ and $(- 1, 5, - 1)$ .
What is the distance between the points $(0, 2)$ and $(8, - 13)$ ?
What is the centre of the circle $x^{2} + y^{2} - 4 x + 8 y + 4 = 0$ ?
In how many ways can 3 books be chosen from 10 different books?
What is the modulus of the complex number $8 + 6 i$ ?
If $α$ and $β$ are the roots of $x^{2} + 1 x - 6 = 0$ , what is $α^{2} + β^{2}$ ?
What is the value of $sin 6 0^{\circ}$ ?
What is the sum of the first 13 terms of an arithmetic progression with first term 9 and common difference 4?
Find $lim_{x \to 7} \frac{x ^{2} - 49}{x - 7}$ .
A line is given by $y = - 3 x - 1$ . What is the slope of a line perpendicular to it?
If $f (x) = (5 x + 1)^{5}$ , find $f^{'} (1)$ .
Find the $x$ -coordinate of the local minimum of $f (x) = x^{3} - 27 x$ .
What are the coordinates of the vertex of the parabola $y = - (x + 2)^{2} + 6$ ?
The function $f$ is defined by $f (x) = 4 x + 6$ . Find $f^{- 1} (2)$ .
How many real roots does the equation $2 x^{2} + 4 x - 5 = 0$ have?
Find the gradient of the tangent to $y = 4 x^{2} + 1 x$ at the point where $x = 2$ .
Solve $3^{x + 2} = 27$ for $x$ .
What is the sum of the first 4 terms of the geometric sequence with first term 5 and common ratio 3?
Let $f (x) = x^{2} (1 x + 4)$ . Use the product rule to find $f^{'} (1)$ .
A radioactive substance has a half-life of 5 years. How much of a 640 g sample remains after 15 years?
For events $A$ and $B$ , $P (B) = 0.5$ and $P (A \cap B) = 0.125$ . What is $P (A ∣ B)$ ?

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