Thanawiya Amma (Egypt) · Pure Mathematics

Spatial Geometry and Integral Calculus

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

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Evaluate the definite integral $\int_{0}^{1} x e^{x} d x$ .
Find the volume of the solid generated by revolving the region bounded by $y = x$ , $y = 0$ , and $x = 4$ about the x-axis.
What is the angle between the two vectors $u = (1, 0, 1)$ and $v = (0, 1, 1)$ ?
Evaluate $\int \frac{2 x + 3}{x ^{2} + 3 x + 5} d x$ .
Find the center and radius of the sphere with equation $x^{2} + y^{2} + z^{2} - 4 x + 6 y - 2 z - 11 = 0$ .
Evaluate $\int_{0}^{π /2} sin^{3} x cos x d x$ .
What is the equation of the plane passing through the point $(1, 2, 3)$ with normal vector $n = (2, - 1, 4)$ ?
Evaluate $\int \frac{1}{x l n x} d x$ .
If $A = (2, - 1, 3)$ and $B = (1, 4, - 2)$ , find the cross product $A \times B$ .
Find the area bounded by the curve $y = x^{2}$ and the line $y = 4$ .
What is the distance from the point $(1, 2, 3)$ to the plane $2 x + 2 y - z - 9 = 0$ ?
Evaluate $\int e^{2 x} cos (3 x) d x$ using integration by parts.
What are the direction cosines of the vector $v = (2, - 2, 1)$ ?
Evaluate $\int_{1}^{e} \frac{l n x}{x} d x$ .
Find the vector equation of the straight line passing through $(1, - 2, 3)$ and parallel to vector $d = (2, 1, - 1)$ .
Evaluate $\int \frac{d x}{x ^{2} + 9}$ .
If two planes $2 x - y + 3 z = 5$ and $x + 2 y + k z = 7$ are perpendicular, what is the value of $k$ ?
Evaluate $\int_{0}^{1} \frac{x}{1 + x ^{2}} d x$ .
What is the angle between the two lines with direction vectors $d_{1} = (1, 2, 2)$ and $d_{2} = (2, - 1, 2)$ ?
Find the area of the region bounded by $y = cos x$ and the x-axis from $x = 0$ to $x = π$ .
What is the length of the perpendicular drawn from the origin to the plane $x + 2 y - 2 z + 6 = 0$ ?
Evaluate $\int \frac{x ^{2} - 1}{x - 1} d x$ for $x \neq = 1$ .
Which condition ensures that three vectors $u, v, w$ in $R^{3}$ are coplanar?
Evaluate $\int_{0}^{2} ∣ x - 1∣ d x$ .
Find the point of intersection of the line $\frac{x - 1}{2} = \frac{y + 1}{- 1} = \frac{z}{1}$ with the plane $x + y + z = 2$ .
Evaluate $\int \frac{1}{x ^{2} - 4} d x$ .
What is the area of the region enclosed by the parabola $y = x^{2}$ and the line $y = x + 2$ ?
The region between $y = x$ and $y = x^{2}$ (for $0 \leq x \leq 1$ ) is rotated about the $x$ -axis. What is the volume of the solid generated?
The points $A (1, 0, 0)$ , $B (0, 2, 0)$ and $C (0, 0, 3)$ are the vertices of a triangle. What is its area?
Evaluate $\int_{0}^{1} x e^{2 x} d x$ .
For which value of $m$ are $u = (m, 3, - 1)$ and $v = (1, 4, 16)$ perpendicular?
Evaluate $\int_{0}^{1} e^{2 x} d x$ .
In triangle $A B C$ , $b = 3$ , $c = 5$ and $∠ A = 12 0^{\circ}$ . Find $a$ .
What is the volume of a sphere of radius 9 cm, in terms of $π$ ?
Find the magnitude of $v = (7, 6, 6)$ .
What is the centre of the circle $x^{2} + y^{2} - 6 x - 2 y - 15 = 0$ ?
What is the centre of the circle $x^{2} + y^{2} + 4 x + 10 y + 25 = 0$ ?
The region under $y = x$ from $x = 0$ to $x = 4$ is rotated about the $x$ -axis. What is the volume generated?
What is the total surface area of a cuboid measuring 10 cm by 7 cm by 4 cm?
In how many ways can a committee of 4 be chosen from 12 people?
Evaluate $\int_{0}^{1} e^{3 x} d x$ .
Find $u \cdot v$ for $u = (- 4, 0, - 4)$ and $v = (- 1, 4, 3)$ .
What is the volume of a sphere of radius 6 cm, in terms of $π$ ?
Find the distance from $M (- 3, 1, 1)$ to the plane $2 x + 6 y + 9 z + 6 = 0$ .
If $θ$ is acute and $sin θ = \frac{7}{25}$ , what is $sin 2 θ$ ?
Find $u \times v$ for $u = (1, - 2, - 1)$ and $v = (4, 1, 1)$ .
The region under $y = x$ from $x = 0$ to $x = 3$ is rotated about the $x$ -axis. What is the volume generated?
Evaluate $\int_{0}^{1} e^{4 x} d x$ .
Find the distance from $M (- 3, 0, 3)$ to the plane $1 x + 4 y + 8 z - 5 = 0$ .
If $θ$ is acute and $sin θ = \frac{5}{13}$ , what is $tan θ$ ?
Two fair dice are rolled. What is the probability that the sum is 4?
Find the distance from $M (3, - 3, 0)$ to the plane $4 x + 4 y + 7 z + 2 = 0$ .
What is the total surface area of a cuboid measuring 5 cm by 5 cm by 7 cm?
Find the distance from $M (3, 2, 3)$ to the plane $1 x + 2 y + 2 z - 3 = 0$ .
If $θ$ is acute and $sin θ = \frac{20}{29}$ , what is $tan θ$ ?
Find the area enclosed between the line $y = 2 x$ and the parabola $y = x^{2}$ .
Evaluate $sin 3 0^{\circ} + cos^{2} 3 0^{\circ} - sin^{2} 4 5^{\circ}$ .
Two fair dice are rolled. What is the probability that the sum is 8?
Find the magnitude of $v = (11, - 10, - 2)$ .
What is the radius of the circle $x^{2} + y^{2} + 4 x - 4 y - 56 = 0$ ?
A team of 2 men and 2 women is chosen from 4 men and 3 women. How many teams are possible?
What is the radius of the circle $x^{2} + y^{2} - 4 x + 8 y - 61 = 0$ ?
What is the centre of the circle $x^{2} + y^{2} + 8 x + 10 y + 37 = 0$ ?
Find the distance from $M (- 1, 2, 2)$ to the plane $1 x + 2 y + 2 z + 5 = 0$ .
What is the volume of a cone of base radius 14 cm and height 18 cm? (Use $π = \frac{22}{7}$ .)

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