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Full Grade 11 test

Full Grade 11 mathematics test with mixed topics, dynamically selected questions and a choice of duration. Practise before an assessment or review the whole course at your own pace.

Questions: 36Estimated time: 61 minutes

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Question 1out of 36

Solve 2^x = 32.

Question 2out of 36

Calculate the 2th root of the 4th root of 6561.

A root taken from another root can be written as one root.

Multiply the indices of the outer and inner roots.
Take the root with that new index from the original number.
Check the answer: raising it to the product of the two indices must give the original number.

Rule: the nth root of the mth root of a number is the (n × m)th root of that number.

Question 3out of 36

Calculate the 5th root of 243.

Question 4out of 36

Simplify sin(−60°).

−sin 60°

−cos 60°

cos 60°

sin 60°

Question 5out of 36

Solve (x − (-5))(x − (8)) = 0.

x = 0

x = -40

x = 3

x = -5 arba x = 8

Question 6out of 36

Calculate log₁₀(1).

-1

-4

Question 7out of 36

Solve the inequality: log_1⁄5(x + 5) > log_1⁄5(3 − x).

-5 < x < -1

-1 < x < 3

-5 ≤ x < -1

-5 < x ≤ -1

Determine the domain conditions for both logarithms. Since the base is between 0 and 1, the logarithmic function is decreasing, so reverse the inequality when comparing the arguments. Intersect the solution of the linear inequality with the domain of both arguments.

Question 8out of 36

How does a change inside f(x) differ from a change outside f?

It changes only the domain name

It never changes the graph

It always changes vertically

Changing x inside f changes the graph horizontally

Question 9out of 36

For a geometric progression b₁ = 1, q = 2. Find the sum of the first 5 terms.

1645

299

1684

Question 10out of 36

After squaring an irrational equation, a candidate root is obtained. What should be done?

It must always be negative

Squaring never changes solutions

It may be extraneous and must be checked in the original equation

It is always valid

Question 11out of 36

What does the intersection A ∩ B mean?

Elements in at least one set

Elements belonging to both sets

Elements only in the first set

Elements outside both sets

Question 12out of 36

Solve ∛(x³ + -8) = ∛-35.

-7

-3

-5

Question 13out of 36

An algebraic result makes a logarithm argument negative. What should be done?

Check only the base

Accept every algebraic result

Reject it because every logarithm argument must be positive

Arguments may be zero

Question 14out of 36

Calculate arcsin(√3/2).

30°

60°

90°

0°

Question 15out of 36

Given −2 < x < 2, simplify: √((x − 2)²) + √((x + 2)²).

−2x

Use the identity √(u²) = |u|. If −a < x < a, then x − a < 0 and x + a > 0. Therefore |x − a| = a − x and |x + a| = x + a. Add the expressions to obtain the final numerical answer.

Question 16out of 36

Which formula shifts y = f(x) by 8 units to the right?

y = 8f(x)

y = f(x) − 8

y = f(x + 8)

y = f(x − 8)

Question 17out of 36

How do algebraic and graphical equation solving differ?

The methods are unrelated

Algebra usually gives exact roots, while a graph often gives approximate intersections

Algebra cannot find roots

A graph is always exact

Question 18out of 36

Calculate the value of the expression: |1 − (-11)| + |1 + (-11)|.

First calculate the expressions inside the two absolute-value signs separately. Then apply the definition of absolute value: a positive number remains unchanged, while a negative number is replaced by its positive opposite. Finally add the two non-negative values.

Question 19out of 36

Simplify tan(144° + 180° · 1).

sin 144°

tan 144°

cos 144°

−tan 144°

The period of the tangent function is 180°. Therefore, adding or subtracting any integer multiple of 180° does not change the tangent value.

Rule: tan(α + 180° · k) = tan α, where k is an integer.

Notice that 180° · k has been added to the angle.
This is a whole number of tangent periods, so that part may be removed.
The expression becomes tan α.

Thus, keep the tangent function and the original angle that appears before the added multiple of 180°.

Question 20out of 36

Calculate 5 · log₆(6).

Question 21out of 36

What do solutions of f(x) = a represent on a graph?

Solutions are the x-coordinates of intersections with the horizontal line y = a

The period is the solution

They are all y-coordinates

There is always one solution

Question 22out of 36

Which statement about trigonometric periods is correct?

All periods are 360°

Tangent has no period

All periods are 180°

Sine and cosine have period 360°, tangent has period 180°

Question 23out of 36

How can arithmetic and geometric progressions be distinguished using adjacent terms?

A constant difference indicates arithmetic; a constant ratio indicates geometric

Both use a constant difference

Only the first term matters

Both use a constant ratio

Question 24out of 36

Calculate 10^log₁₀(2).

Question 25out of 36

Solve the equation: √(x − 1) = x − 3.

x = 2

x = 5

x = 3

x = 6

First note that the right-hand side of the equation cannot be negative. Then square both sides and solve the resulting quadratic equation. Check every candidate in the original equation: one of them may be an extraneous root introduced by squaring.

Question 26out of 36

Solve sin x = √3/2 on [0°, 360°].

60° ir 120°

45° ir 315°

60° ir 300°

30° ir 330°

Question 27out of 36

Solve log₂(x) − 1 = 0.

603

496

538

548

Question 28out of 36

Solve 2^{x + 5} = 2¹¹.

-6

Question 29out of 36

Find the maximum value of the function y = 8 sin x + 15 cos x.

The expression a sin x + b cos x can be written as R sin(x + φ), where R = √(a² + b²). Since the maximum value of sine is 1, the maximum value of the whole expression is √(a² + b²).

Question 30out of 36

Set A has 16 elements, B has 8, and their intersection has 7. How many are in A ∪ B?

Question 31out of 36

Calculate arccos(√2/2).

90°

0°

45°

Question 32out of 36

Solve (x − (-5))/(x − (6)) < 0.

-5 < x < 6

x > 6

x < -5 arba x > 6

-5 ≤ x ≤ 6

Question 33out of 36

Solve the equation: log₂(x − 15) + log₂(x + 15) = 6.

x = 15

x = 8

x = 17

x = −17

First determine the domain: both logarithm arguments must be positive. Then use log_au + log_av = log_a(uv) and convert to an equation without logarithms. Solve the resulting quadratic equation and reject any root outside the domain.

Question 34out of 36

What multiplier represents a 15% increase per period?

0.15

1.15

0.85

Question 35out of 36

Solve the system -6 < x < 4 and x ≥ -1.

x < -6 arba x ≥ 4

-6 < x < -1

-1 ≤ x < 4

-6 ≤ x ≤ 4

Question 36out of 36

Solve 4^{x + -4} = 4^-7.

-3

-4

-10

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Full Grade 11 mathematics test

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