Saudi Secondary School Certificate · Advanced Mathematics

Vectors, Calculus, Analytical Geometry and Probability Distributions

42 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.

All 42 questions

What is the cross product $u \times v$ of vectors $u = ⟨ 1, 0, 0 ⟩$ and $v = ⟨ 0, 1, 0 ⟩$ ?
What is the derivative of $f (x) = ln (x^{2} + 1)$ with respect to $x$ ?
What is the value of the definite integral $\int_{0}^{π /2} cos (x) d x$ ?
What are the coordinates of the foci of the ellipse $\frac{x ^{2}}{25} + \frac{y ^{2}}{9} = 1$ ?
If a random variable $X$ follows a normal distribution with mean $μ = 50$ and standard deviation $σ = 5$ , what percentage of data falls between 45 and 55 according to the Empirical Rule?
What is the determinant of the $3 \times 3$ identity matrix $I_{3}$ ?
What is the slope of the tangent line to the curve $y = x^{3} - 2 x + 4$ at the point where $x = 2$ ?
Which of the following represents the vector equation of a line passing through point $P_{0} (1, 2, 3)$ with direction vector $d = ⟨ 4, - 1, 2 ⟩$ ?
What is the horizontal asymptote of the rational function $f (x) = \frac{5 x ^{2} - 3 x + 1}{2 x ^{2} + 7}$ ?
What is the angle between the two orthogonal vectors $a = ⟨ 2, 0 ⟩$ and $b = ⟨ 0, 5 ⟩$ ?
What is the solution of the trigonometric equation $2 sin (x) - 1 = 0$ for $0 \leq x < 2 π$ ?
What is the second derivative $f^{''} (x)$ of $f (x) = x^{4} - 3 x^{3} + 2 x$ ?
What is the magnitude of the vector $v = 3 \hat{i} - 4 \hat{j} + 12 \hat{k}$ ?
What is the limit $lim_{x \to 0} \frac{s i n ( 3 x )}{x}$ ?
What is the area enclosed between the parabola $y = x^{2}$ and the line $y = 4$ ?
Which of the following is a unit vector in the direction of $u = ⟨ 6, 8 ⟩$ ?
What is the sum of the infinite geometric series with first term $a_{1} = 8$ and common ratio $r = - \frac{1}{2}$ ?
What is the radius of the sphere with equation $x^{2} + y^{2} + z^{2} - 4 x + 6 y - 2 z - 11 = 0$ ?
If matrix $M = (3012)$ , what are the eigenvalues of $M$ ?
What is the derivative of $f (x) = e^{3 x} sin (x)$ at $x = 0$ ?
What is the value of $lim_{x \to \infty} \frac{4 x ^{3} - 2 x + 5}{7 x ^{3} + x ^{2} - 1}$ ?
In a binomial experiment with $n = 10$ trials and probability of success $p = 0.4$ , what is the expected value (mean) $μ$ ?
What is 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 length of the major axis of the ellipse defined by $9 x^{2} + 16 y^{2} = 144$ ?
What is the critical point of the function $f (x) = 2 x^{3} - 6 x$ where a local minimum occurs?
What is the angle in radians equivalent to $22 5^{\circ}$ ?
What is the sum of the entries in row 4 of Pascal's triangle ( $1, 4, 6, 4, 1$ )?
What is the solution to the logarithmic equation $lo g_{2} (x^{2} - 1) = 3$ for $x > 1$ ?
What is the distance from the point $P (2, 3, 1)$ to the plane $2 x - y + 2 z - 9 = 0$ ?
What is the value of $arcsin (\frac{3}{2})$ in radians within its principal domain $[- π /2, π /2]$ ?
What is the slope of the line perpendicular to the line $4 x - 6 y = 15$ ?
What is the value of the integral $\int_{1}^{e} \frac{1}{x} d x$ ?
What is the radius of convergence of the power series $\sum_{n = 0}^{\infty} \frac{x ^{n}}{3 ^{n}}$ ?
What is the solution set of $e^{2 x} - 4 e^{x} + 3 = 0$ ?
What is the eccentricity of the hyperbola $\frac{x ^{2}}{16} - \frac{y ^{2}}{9} = 1$ ?
What is the maximum area of a rectangle that can be inscribed under the parabola $y = 12 - x^{2}$ with its base on the x-axis?
What is the volume of the solid generated by revolving the region bounded by $y = x^{2}$ and $y = 4$ about the y-axis?
Evaluate the improper integral $\int_{1}^{\infty} \frac{1}{x ^{3}} d x$ .
What is the equation of the plane passing through $P (1, - 1, 2)$ perpendicular to normal vector $n = ⟨ 2, 3, - 1 ⟩$ ?
If $z = 2 (cos (3 0^{\circ}) + i sin (3 0^{\circ}))$ and $w = 3 (cos (6 0^{\circ}) + i sin (6 0^{\circ}))$ , what is the product $z \cdot w$ in rectangular form?
What is the eccentricity $e$ of the hyperbola $\frac{x ^{2}}{16} - \frac{y ^{2}}{9} = 1$ ?
What is the general solution of the second-order differential equation $y^{''} - 5 y^{'} + 6 y = 0$ ?

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