What do the Grade 11 Mathematics cards cover?

28 cards across 8 chapters of the national textbook: Relations and Functions, Rational expressions and rational, Probability, Determinants and their properties, Vectors, Transformations of the plane, Statistics and 1 more. You can take any chapter one card at a time on the page itself.

Is there a national exam in Grade 11?

No. Ethiopia sets national exams in Grade 6, Grade 8 and Grade 12 only. These cards are for your school's own exams, and for the national exam that comes a few years later.

Where do these cards come from?

They are drafted from Grade 11 Mathematics, the Ministry of Education textbook for this grade. A second pass that cannot see the chapter then re-derives every formula, constant and table row, and anything it cannot confirm is held back instead of published.

Yes. Every card here is free to read and the printable sheet is free to download. Neither needs an account.

When was this last checked?

6 September 2026. Cards arrive chapter by chapter, and the line under each one says when that card was last read through.

Free short notes and revision cards

Grade 11 Mathematics

Grade 11 Mathematics on Temari has 28 revision cards, arranged by the chapters of the Ethiopian national curriculum. Every card says when the rule applies, what each symbol in it stands for, and the mistake students most often make with it. They are free to read and need no account.

28
Cards: 8
Chapters: 13
Formulas: 8
Reference tables: Grade 11 Mathematics
Textbook: Free
Price: 6 September 2026
Last checked

Everything28Formula13Reference table8Rule3Common mistake2Method2

Across the whole subject

Complex number arithmeticFormula

(a+bi)(c+di) = (ac-bd)+(ad+bc)i

(a+bi)(c+di) = (ac-bd)+(ad+bc)i

$i$: The imaginary unit. Its square is minus one, and every sign change on this card comes from that.
$a, c$: The real parts: a belongs to the first number and c to the second.
$b, d$: The imaginary parts, the numbers sitting with i: b in the first number and d in the second.

Operation	Rule	Why it works
Add	$(a+bi)+(c+di) = (a+c)+(b+d)i$	Real parts add together and imaginary parts add together.
Subtract	$(a+bi)-(c+di) = (a-c)+(b-d)i$	Take the second bracket apart term by term, keeping both signs.
Multiply	$(a+bi)(c+di) = (ac-bd)+(ad+bc)i$	The minus sign appears because bd times i squared turns negative.
Multiply by the conjugate	$(a+bi)(a-bi) = a^2+b^2$	The conjugate keeps the real part and flips the sign of the imaginary part, so the product is always real.
Modulus	$\|a+bi\| = \sqrt{a^2+b^2}$	The distance from the origin to the point with coordinates a and b.
Divide	$\dfrac{a+bi}{c+di} = \dfrac{(ac+bd)+(bc-ad)i}{c^2+d^2}$	Multiply top and bottom by the conjugate of the denominator, which must not be zero.
Negative square root	$\sqrt{-a} = i\sqrt{a}$	True when a is zero or positive. Convert before you multiply anything.

When you use it

Reach for this whenever a question hands you numbers of the form $a+bi$ , or the square root of a negative number. It covers adding, subtracting, multiplying and dividing them, along with the conjugate and the modulus you need to finish a division.

Watch out

The sign that $i^2 = -1$ hides in the last term of a product. In $(2+3i)(4+5i)$ that term is $15i^2 = -15$ , so the answer is $-7+22i$ and not $23+22i$ . The same slip shows up twice more. In a division the conjugate you multiply by belongs to the denominator, so reaching for the numerator's conjugate leaves an $i$ still sitting on the bottom. And a negative root has to be written in $i$ form first, because $\sqrt{-4} \cdot \sqrt{-9}$ is $2i \cdot 3i = -6$ , and not $6$ .

Relations and Functions

Domain and Range of a RelationRule

When you use it

Use this to identify the domain and range from a relation given as a set of ordered pairs.

Watch out

Domain contains only the first coordinates and range contains only the second coordinates. Do not write duplicate elements in either set.

Drafted from Grade 11 Mathematics, pages 4-83, then checked twice before it went up

Equality of Ordered PairsCommon mistake

When you use it

Use this when checking if two ordered pairs are equal or when finding unknown coordinates.

Watch out

In an ordered pair (x, y), order matters. The pair (x, y) equals (y, x) only when x equals y.

Drafted from Grade 11 Mathematics, pages 4-83, then checked twice before it went up

Rational expressions and rational

Graph of an Inverse FunctionMethod

When you use it

Use this to sketch the graph of an inverse function from the graph of the original function.

Watch out

Reflect every point across the line y = x by interchanging the x and y coordinates.

Drafted from Grade 11 Mathematics, pages 84-130, then checked twice before it went up

Inverse Function ConditionRule

When you use it

Use this to verify whether two given functions f and g are inverses of each other.

Watch out

You must check both compositions. Showing only one direction is not sufficient.

Drafted from Grade 11 Mathematics, pages 84-130, then checked twice before it went up

Inverse Function NotationCommon mistake

When you use it

Use this when interpreting the notation for the inverse of a function.

Watch out

The inverse function f^{-1}(x) is not equal to -f(x) and is not equal to 1/f(x).

Drafted from Grade 11 Mathematics, pages 84-130, then checked twice before it went up

Rational Expression DefinitionRule

When you use it

Use this to identify whether an algebraic expression is a rational expression.

Watch out

Both the numerator and the denominator must be polynomials, and the denominator cannot be the zero polynomial.

Drafted from Grade 11 Mathematics, pages 84-130, then checked twice before it went up

Probability

Asymptotes of Rational FunctionsReference tabledeg(p) < deg(q) · deg(p) = deg(q) · deg(p) = deg(q) + 1

Degree condition	Asymptote type	Equation
$\text{deg}(p) < \text{deg}(q)$	Horizontal asymptote	$y = 0$
$\text{deg}(p) = \text{deg}(q)$	Horizontal asymptote	$y = \frac{a_n}{b_n}$
$\text{deg}(p) = \text{deg}(q) + 1$	Oblique asymptote	$y = mx + b$ via polynomial division

When you use it

Use this to determine horizontal or oblique asymptotes when graphing rational functions.

Watch out

Oblique asymptotes occur only when the numerator degree is exactly one greater than the denominator degree.

Drafted from Grade 11 Mathematics, pages 131-206, then checked twice before it went up

Operations on Rational ExpressionsReference tableAddition · Subtraction · Multiplication · +1

Operation	Rule	Condition
Addition	$\frac{p(x)}{q(x)} + \frac{r(x)}{s(x)} = \frac{p(x)s(x) + r(x)q(x)}{q(x)s(x)}$	$q(x)$ and $s(x)$ are not zero
Subtraction	$\frac{p(x)}{q(x)} - \frac{r(x)}{s(x)} = \frac{p(x)s(x) - r(x)q(x)}{q(x)s(x)}$	$q(x)$ and $s(x)$ are not zero
Multiplication	$\frac{p(x)}{q(x)} \cdot \frac{r(x)}{s(x)} = \frac{p(x)r(x)}{q(x)s(x)}$	$q(x)$ and $s(x)$ are not zero
Division	$\frac{p(x)}{q(x)} \div \frac{r(x)}{s(x)} = \frac{p(x)s(x)}{q(x)r(x)}$	$q(x)$ , $s(x)$ , and $r(x)$ are not zero

When you use it

Use these rules to combine and simplify sums, differences, products, and quotients of rational expressions.

Watch out

In division, the numerator of the divisor fraction also cannot be zero because it moves to the denominator.

Drafted from Grade 11 Mathematics, pages 131-206, then checked twice before it went up

Work FormulaFormula

W = r t

W = r t

$W$: Work completed $1$
$r$: Combined work rate $\text{h}^{-1}$
$t$: Time taken $\text{h}$

When you use it

Use this when multiple workers or pipes complete a task together at constant rates.

Watch out

Add individual rates of work per unit time, never add completion times directly.

Drafted from Grade 11 Mathematics, pages 131-206, then checked twice before it went up

Determinants and their properties

Elementary Row OperationsReference tableSwapping · Scaling · Pivoting

Operation	Procedure
Swapping	Interchanging two rows of a matrix.
Scaling	Multiplying a row of a matrix by a non-zero constant.
Pivoting	Adding a constant multiple of one row to another row.

When you use it

Use these three valid transformations to row reduce a matrix to Row Echelon Form or Reduced Row Echelon Form.

Drafted from Grade 11 Mathematics, pages 207-250, then checked twice before it went up

Inverse of a Matrix ProductFormula

(AB)^{-1} = B^{-1}A^{-1}

(AB)^{-1} = B^{-1}A^{-1}

$A$: Invertible square matrix A $dimensionless$
$B$: Invertible square matrix B of the same size $dimensionless$

When you use it

Use this rule to compute or expand the inverse of multiplied square matrices.

Watch out

The order of factors reverses in the inverse. Expanding without switching the order is a common mistake.

Drafted from Grade 11 Mathematics, pages 207-250, then checked twice before it went up

Matrix Inversion by Elementary Row OperationsMethod

When you use it

Use this method to find the inverse of a square matrix or to verify whether it is nonsingular.

Watch out

If row reduction of the augmented matrix produces a row of zeros on the left side, the matrix is singular and has no inverse.

Drafted from Grade 11 Mathematics, pages 207-250, then checked twice before it went up

Special Square MatricesReference tableDiagonal · Scalar · Identity · +2

Matrix Type	Defining Condition
Diagonal	Entries outside the main diagonal are all zero.
Scalar	Diagonal matrix where all diagonal entries are equal.
Identity	Diagonal matrix where all diagonal entries are 1.
Upper Triangular	All entries below the main diagonal are zero.
Lower Triangular	All entries above the main diagonal are zero.

When you use it

Use this reference to classify square matrices based on the patterns of their zero and diagonal entries.

Drafted from Grade 11 Mathematics, pages 207-250, then checked twice before it went up

Vectors

Area of a Triangle in xy-planeFormula

\text{Area} = \frac{1}{2} \left| \det \begin{pmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{pmatrix} \right|

\text{Area} = \frac{1}{2} \left| \det \begin{pmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{pmatrix} \right|

$\text{Area}$: Area of the triangle
$x_1, y_1$: Coordinates of first vertex
$x_2, y_2$: Coordinates of second vertex
$x_3, y_3$: Coordinates of third vertex

When you use it

Use this formula to find the area of a triangle given the coordinates of its three vertices.

Watch out

The determinant can evaluate to a negative value. Always take the absolute value to ensure the area is positive.

Drafted from Grade 11 Mathematics, pages 251-312, then checked twice before it went up

Collinear Points TestFormula

\det \begin{pmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{pmatrix} = 0

\det \begin{pmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{pmatrix} = 0

$x_1, y_1$: Coordinates of point A
$x_2, y_2$: Coordinates of point B
$x_3, y_3$: Coordinates of point C

When you use it

Use to test whether three given points lie on the exact same straight line.

Watch out

Do not confuse this with finding area. Here the determinant must strictly equal zero for collinearity.

Drafted from Grade 11 Mathematics, pages 251-312, then checked twice before it went up

Equation of Line Passing Through Two PointsFormula

\det \begin{pmatrix} x & y & 1 \\ x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \end{pmatrix} = 0

\det \begin{pmatrix} x & y & 1 \\ x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \end{pmatrix} = 0

$x, y$: Variables representing any point on the line
$x_1, y_1$: Coordinates of first known point
$x_2, y_2$: Coordinates of second known point

When you use it

Use to find the Cartesian equation of a line passing through two distinct given points.

Watch out

Keep x and y in the first row as variables. Do not substitute numerical values into them.

Drafted from Grade 11 Mathematics, pages 251-312, then checked twice before it went up

Matrix Inverse via AdjointFormula

A^{-1} = \frac{1}{\det(A)} \text{Adj}(A)

A^{-1} = \frac{1}{\det(A)} \text{Adj}(A)

$A^{-1}$: Inverse of square matrix A
$\det(A)$: Determinant of matrix A, provided it is not zero
$\text{Adj}(A)$: Adjoint matrix, transpose of the cofactor matrix

When you use it

Use to compute the inverse of an invertible matrix using cofactors and determinant.

Watch out

The adjoint matrix is the transpose of the cofactor matrix, so rows and columns must be swapped.

Drafted from Grade 11 Mathematics, pages 251-312, then checked twice before it went up

Transformations of the plane

Scalar Product and OrthogonalityFormula

\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}||\mathbf{v}|\cos\theta = u_1 v_1 + u_2 v_2

\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}||\mathbf{v}|\cos\theta = u_1 v_1 + u_2 v_2

$\mathbf{u}, \mathbf{v}$: given vectors $dimensionless$
$|\mathbf{u}|, |\mathbf{v}|$: magnitudes of the vectors $dimensionless$
$\theta$: angle between the two vectors $rad$
$u_1, u_2, v_1, v_2$: scalar components of the vectors $dimensionless$

When you use it

Use to calculate the dot product, find the angle between vectors, or check if two vectors are perpendicular.

Watch out

Two vectors are perpendicular if and only if their scalar product is zero. The dot product is a scalar value, not a vector.

Drafted from Grade 11 Mathematics, pages 313-354, then checked twice before it went up

Tangent Line to a Circle at the OriginFormula

x x_1 + y y_1 = r^2

x x_1 + y y_1 = r^2

$x_1, y_1$: coordinates of the point of tangency $dimensionless$
$x, y$: coordinates of an arbitrary point on the tangent line $dimensionless$
$r$: radius of the circle centered at the origin $dimensionless$

When you use it

Use to find the equation of the tangent line when the circle is centered at the origin (0, 0).

Watch out

This equation only applies when the circle center is at (0, 0). For shifted centers, use the general formula.

Drafted from Grade 11 Mathematics, pages 313-354, then checked twice before it went up

Tangent Line to a General CircleFormula

(x - x_0)(x_1 - x_0) + (y - y_0)(y_1 - y_0) = r^2

(x - x_0)(x_1 - x_0) + (y - y_0)(y_1 - y_0) = r^2

$x_0, y_0$: coordinates of the center of the circle $dimensionless$
$x_1, y_1$: coordinates of the point of tangency $dimensionless$
$x, y$: coordinates of an arbitrary point on the tangent line $dimensionless$
$r$: radius of the circle $dimensionless$

When you use it

Use to find the equation of the tangent line to a circle with center (x_0, y_0) at a given point of tangency (x_1, y_1).

Watch out

Do not confuse the point of tangency with the variable coordinates of the line. The right hand side must be r^2 and not r.

Drafted from Grade 11 Mathematics, pages 313-354, then checked twice before it went up

Statistics

Reflection Across the Line y = mxFormula

\begin{cases} x' = x\cos(2\theta) + y\sin(2\theta) \\ y' = x\sin(2\theta) - y\cos(2\theta) \end{cases}

\begin{cases} x' = x\cos(2\theta) + y\sin(2\theta) \\ y' = x\sin(2\theta) - y\cos(2\theta) \end{cases}

$x'$: x coordinate of the reflected point $dimensionless$
$y'$: y coordinate of the reflected point $dimensionless$
$x$: x coordinate of the original point $dimensionless$
$y$: y coordinate of the original point $dimensionless$
$\theta$: inclination angle of line y = mx where m = tan theta $\text{rad}$

When you use it

Use this formula to find the reflection of any point (x, y) across a line passing through the origin with slope m = tan(theta).

Watch out

The formula uses 2 theta inside sine and cosine, not theta.

Drafted from Grade 11 Mathematics, page 355 onwards, then checked twice before it went up

Rotation About an Arbitrary PointFormula

\begin{cases} x' = a + (x - a)\cos\theta - (y - b)\sin\theta \\ y' = b + (x - a)\sin\theta + (y - b)\cos\theta \end{cases}

\begin{cases} x' = a + (x - a)\cos\theta - (y - b)\sin\theta \\ y' = b + (x - a)\sin\theta + (y - b)\cos\theta \end{cases}

$x'$: x coordinate of the image point $dimensionless$
$y'$: y coordinate of the image point $dimensionless$
$x$: x coordinate of the original point $dimensionless$
$y$: y coordinate of the original point $dimensionless$
$a$: x coordinate of the centre of rotation $dimensionless$
$b$: y coordinate of the centre of rotation $dimensionless$
$\theta$: angle of rotation $\text{rad}$

When you use it

Use this formula to find the image of a point rotated counterclockwise through an angle about any given center point (a, b).

Watch out

Do not forget to subtract the center coordinates (a, b) before applying trigonometric functions, then add them back at the end.

Drafted from Grade 11 Mathematics, page 355 onwards, then checked twice before it went up

Rotations About the OriginReference table90^° · 180^° · 270^° · +1

Angle of Rotation	Image of (x, y)
$90^\circ$	$(-y, x)$
$180^\circ$	$(-x, -y)$
$270^\circ$	$(y, -x)$
$360^\circ$	$(x, y)$

When you use it

Use this table for quick evaluation of counterclockwise rotations about the origin (0, 0) by standard angles.

Watch out

Clockwise rotation by an angle corresponds to counterclockwise rotation by 360 degrees minus that angle.

Drafted from Grade 11 Mathematics, page 355 onwards, then checked twice before it went up

Standard Reflections in the PlaneReference tablex axis (y = 0) · y axis (x = 0) · y = x · +1

Line of Reflection	Image of (x, y)
$x\text{ axis } (y = 0)$	$(x, -y)$
$y\text{ axis } (x = 0)$	$(-x, y)$
$y = x$	$(y, x)$
$y = -x$	$(-y, -x)$

When you use it

Use this reference table when finding the coordinates of a point reflected across common axes and diagonal lines.

Watch out

Reflecting across y = -x swaps the variables and negates both, unlike reflecting across y = x which only swaps them.

Drafted from Grade 11 Mathematics, page 355 onwards, then checked twice before it went up

Unit 8

Absolute Value FunctionFormula

|a| = \begin{cases} a & \text{if } a > 0 \\ 0 & \text{if } a = 0 \\ -a & \text{if } a < 0 \end{cases}

|a| = \begin{cases} a & \text{if } a > 0 \\ 0 & \text{if } a = 0 \\ -a & \text{if } a < 0 \end{cases}

$a$: real number $dimensionless$
$|a|$: absolute value or modulus of a $dimensionless$

When you use it

Use this definition to evaluate the modulus of any real number or to simplify expressions inside absolute value bars.

Watch out

The absolute value of any real number is always nonnegative.

Drafted from Grade 11 Mathematics, page 44 onwards, then checked twice before it went up

Negative Rational Power FunctionsReference tablef(x) = x^-1/n · f(x) = x^-m/n

Function	m	n	Domain	Range
$f(x) = x^{-1/n}$	$1$	even	$(0, \infty)$	$(0, \infty)$
$f(x) = x^{-1/n}$	$1$	odd	$\mathbb{R} \setminus \{0\}$	$\mathbb{R} \setminus \{0\}$
$f(x) = x^{-m/n}$	odd	even	$(0, \infty)$	$(0, \infty)$
$f(x) = x^{-m/n}$	even	odd	$\mathbb{R} \setminus \{0\}$	$(0, \infty)$

When you use it

Use this table to determine the domain and range of power functions with negative rational exponents.

Watch out

Zero is never in the domain of a negative power function because division by zero is undefined.

Drafted from Grade 11 Mathematics, page 44 onwards, then checked twice before it went up

Positive Rational Power FunctionsReference tablef(x) = x^1/n · f(x) = x^m/n

Function	m	n	Domain	Range
$f(x) = x^{1/n}$	$1$	even	$[0, \infty)$	$[0, \infty)$
$f(x) = x^{1/n}$	$1$	odd	$\mathbb{R}$	$\mathbb{R}$
$f(x) = x^{m/n}$	odd	even	$[0, \infty)$	$[0, \infty)$
$f(x) = x^{m/n}$	even	odd	$\mathbb{R}$	$[0, \infty)$

When you use it

Use this table to find the domain and range of power functions with positive rational exponents.

Watch out

When the denominator n is even, negative inputs are not allowed in the real numbers.

Drafted from Grade 11 Mathematics, page 44 onwards, then checked twice before it went up

The same subject in other years

An exam paper keeps asking for what the year below taught. Those cards are here too.

Your other Grade 11 subjects

Everything else in this year that has cards so far.

Questions students ask

What do the Grade 11 Mathematics cards cover?: 28 cards across 8 chapters of the national textbook: Relations and Functions, Rational expressions and rational, Probability, Determinants and their properties, Vectors, Transformations of the plane, Statistics and 1 more. You can take any chapter one card at a time on the page itself.
Is there a national exam in Grade 11?: No. Ethiopia sets national exams in Grade 6, Grade 8 and Grade 12 only. These cards are for your school's own exams, and for the national exam that comes a few years later.
Where do these cards come from?: They are drafted from Grade 11 Mathematics, the Ministry of Education textbook for this grade. A second pass that cannot see the chapter then re-derives every formula, constant and table row, and anything it cannot confirm is held back instead of published.
Is this free?: Yes. Every card here is free to read and the printable sheet is free to download. Neither needs an account.
When was this last checked?: 6 September 2026. Cards arrive chapter by chapter, and the line under each one says when that card was last read through.

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