Thanawiya Amma (Egypt) · Pure Mathematics

Algebra and Differential Calculus

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

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Random questions from this topic, one at a time. You see the answer and the explanation straight after each one.

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In the binomial expansion of $(x + \frac{1}{x})^{10}$ , what is the term independent of $x$ ?
For $y = ln (\frac{1}{c o s x} + tan x)$ , determine the first derivative $\frac{d y}{d x}$ with respect to $x$ for all $x$ in $(- \frac{π}{2}, \frac{π}{2})$ :
If $z = 1 + i 3$ , what is the polar trigonometric form of $z$ ?
What is the limit $lim_{x \to 0} \frac{e ^{3 x} - 1}{2 x}$ ?
If a continuous curve has inflection point at $x = c$ , which statement must be true regarding $f^{''} (c)$ if it exists?
Find the slope of the tangent to the curve $x^{2} + y^{2} - 4 x + 6 y = 0$ at the origin $(0, 0)$ .
Solve for $n$ if $(2 n) = 45$ .
What is the maximum value of $f (x) = 2 x^{3} - 9 x^{2} + 12 x + 5$ on the closed interval $[0, 3]$ ?
If $ω$ is a complex cube root of unity, what is the value of $(1 - ω + ω^{2}) (1 + ω - ω^{2})$ ?
Find $\frac{d y}{d x}$ if $x = 2 t^{2} - 1$ and $y = 4 t + 3$ .
What is the number of arrangements of 5 distinct books on a shelf if 2 particular books must always be adjacent?
Find the rate of change of the area of a circle with respect to its radius when $r = 7 cm$ .
If $x 3 2 x = 10$ , what are the possible real values of $x$ ?
What is the radius of curvature or behavior of $f (x) = x^{3} - 3 x^{2} + 4$ at its point of inflection?
In how many ways can a committee of 3 men and 2 women be selected from 6 men and 5 women?
Find the equation of the normal to the curve $y = x^{2} - 4 x + 3$ at the point $(1, 0)$ .
What is the rank of matrix $A = 123246369$ ?
If $y = e^{x^{2}}$ , find $\frac{d ^{2} y}{d x ^{2}}$ .
Find the value of $k$ such that the matrix $(24 k 8)$ is singular.
What is the sum of the coefficients in the expansion of $(2 x - 3 y)^{5}$ ?
If $f (x) = 25 - x^{2}$ , on which interval is $f (x)$ strictly decreasing?
How many distinct 4-digit numbers can be formed using the digits ${1, 2, 3, 4, 5}$ without repetition?
Evaluate $lim_{x \to \infty} (1 + \frac{2}{x})^{x}$ .
Find the value of $\frac{d}{d x} [sin^{2} (3 x)]$ .
What is the modulus of the complex number $z = \frac{3 + 4 i}{1 - 2 i}$ ?
A ladder 10 m long leans against a vertical wall. If the bottom of the ladder slides away from the wall at $2 m/s$ , how fast is the top sliding down when the bottom is $6 m$ from the wal…
What is the minimum value of $f (x) = x^{3} - 12 x + 5$ on the closed interval $[0, 3]$ ?
Water runs into a conical tank, vertex downward, with height $10$ m and top radius $5$ m, at $3 m^{3} / min$ . How fast is the water level rising when the depth is $4$ m?
For the curve $x^{2} + x y + y^{2} = 7$ , what is $\frac{d y}{d x}$ at the point $(2, 1)$ ?
The roots of $x^{2} - 5 x + 3 = 0$ are $α$ and $β$ . What is the value of $α^{3} + β^{3}$ ?
Which pair gives the square roots of $- 3 + 4 i$ ?
Evaluate $lim_{x \to - 1} \frac{x ^{2} + 3 x + 2}{x + 1}$ .
Find the local minimum value of $f (x) = x^{3} - 12 x + 5$ .
A farmer has 160 m of fencing to enclose a rectangular field. What is the largest possible area in square metres?
Find the local minimum value of $f (x) = x^{3} - 12 x - 1$ .
What is the term independent of $x$ in the expansion of $(x + \frac{1}{x})^{6}$ ?
What is the remainder when $x^{3} + 4 x^{2} - 3 x + 1$ is divided by $x - 4$ ?
Evaluate $lim_{x \to 2} \frac{x ^{2} - 6 x + 8}{x - 2}$ .
In an arithmetic sequence $a_{3} = 21$ and $a_{11} = 53$ . What is $a_{15}$ ?
What is the equation of the tangent to $y = x^{2} - 6 x + 1$ at $x = - 1$ ?
A farmer has 60 m of fencing to enclose a rectangular field. What is the largest possible area in square metres?
If $f (x) = (x^{2} + 3) (x - 3)$ , find $f^{'} (0)$ .
What is the remainder when $3 x^{3} - 3 x^{2} - 3 x - 8$ is divided by $x - 3$ ?
What is the 18th term of the arithmetic sequence $- 6, - 11, - 16, \dots$ ?
If $f (x) = 4 x$ and $g (x) = 2 x + 6$ , what is $f (g (- 2))$ ?
What is the solution set of $x^{2} - 3 x - 18 > 0$ ?
What is the term independent of $x$ in the expansion of $(x + \frac{1}{x})^{10}$ ?
A geometric sequence has first term 4 and common ratio 3. What is its 5th term?
If $f (x) = \frac{3 x + 1}{x + 6}$ , find $f^{'} (2)$ .
If $f (x) = (x^{2} + 3) (x - 4)$ , find $f^{'} (0)$ .
What is the domain of $f (x) = - 3 x - 6$ ?
Evaluate $i^{431}$ .
What is the equation of the tangent to $y = x^{2} + x + 7$ at $x = 4$ ?
Evaluate $lim_{x \to \infty} \frac{2 x ^{2} + 3 x}{5 x ^{2} - 5}$ .
A farmer has 200 m of fencing to enclose a rectangular field. What is the largest possible area in square metres?
Find the local minimum value of $f (x) = x^{3} - 27 x + 3$ .
What is the remainder when $3 x^{3} - 2 x^{2} + 6 x + 1$ is divided by $x - 3$ ?
What is the solution set of $x^{2} - 4 x - 32 < 0$ ?
Find the local minimum value of $f (x) = x^{3} - 27 x + 8$ .
What is the coefficient of $x^{4}$ in the expansion of $(1 + 2 x)^{7}$ ?
Find the determinant of $(- 2 - 1 - 1 - 5)$ .
If $A$ is a $2 \times 2$ matrix with $det A = 2$ , what is $det (3 A)$ ?
What are the solutions of $x^{2} - 5 x + 4 = 0$ ?
A spherical balloon is inflated so that its radius increases at 2 cm/s. How fast is its volume increasing, in cm³/s, when the radius is 6 cm? Give the answer as a multiple of $π$ .
What is the solution set of $x^{2} - 8 x + 12 < 0$ ?
A geometric sequence has first term 5 and common ratio 2. What is its 7th term?
How many terms are there in the expansion of $(x + 2)^{10}$ ?
What are the solutions of $x^{2} + 7 x + 12 = 0$ ?
What is the remainder when $3 x^{3} + 4 x^{2} + 6 x - 2$ is divided by $x + 2$ ?
If $A$ is a $2 \times 2$ matrix with $det A = 5$ , what is $det (2 A)$ ?

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