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Advanced Mathematics

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In mathematics, what are the solutions to the quadratic equation $3 x^{2} - 12 x + 9 = 0$ ?
In mathematics, what is the derivative of the rational function $f (x) = \frac{3 x - 1}{x ^{2} + 2}$ at $x = 1$ ?
In mathematics, what is the sum of the first 20 terms of an arithmetic progression with first term $a_{1} = 5$ and common difference $d = 3$ ?
In mathematics, what is the sum of the infinite geometric series $12 + 4 + 4/3 + 4/9 + \dots$ ?
In mathematics, which exact value equals $sin (15 0^{\circ})$ in standard trigonometry?
In mathematics, how many solutions does the equation $2 cos^{2} (x) - 3 cos (x) + 1 = 0$ have in the interval $[0, 2 π)$ ?
In mathematics, what is the real solution to the equation $lo g_{2} (x + 6) - lo g_{2} (x) = 2$ ?
In mathematics, what is the maximal real domain of the function $f (x) = ln (9 - x^{2})$ ?
In mathematics, how many distinct subsets of 3 elements can be chosen from a set of 8 distinct elements?
In mathematics, in how many ways can 5 books be arranged on a shelf if two specific books must always stand next to each other?
In mathematics, what is the Euclidean distance between points $A (- 1, 4)$ and $B (5, - 4)$ in the Cartesian plane?
In mathematics, what is the equation of the line passing through point $P (2, - 1)$ perpendicular to the line $3 x - 4 y + 5 = 0$ ?
In mathematics, what are the coordinates of the centre $S$ and radius $r$ of the circle $x^{2} + y^{2} - 6 x + 8 y = 0$ ?
In mathematics, what is the slope of the tangent line to the parabola $y = 2 x^{2} - 5 x + 3$ at the point where $x = 2$ ?
In mathematics, what is the exact value of the definite integral $\int_{0}^{3} (3 x^{2} - 2 x + 1) d x$ ?
In mathematics, what is the area of the finite region bounded by the curves $y = x^{2}$ and $y = 2 x$ ?
In mathematics, what is the probability of obtaining exactly 2 heads when flipping a fair balanced coin 4 times?
In mathematics, if two fair standard six-sided dice are rolled, what is the probability that their sum is 8 given that the first die shows an even number?
In mathematics, what is the dot product of vectors $u = (3, - 2)$ and $v = (4, 5)$ in two dimensions?
In mathematics, what is the cosine of the angle between vectors $a = (1, 2)$ and $b = (2, - 1)$ ?
In mathematics, what is the solution to the exponential equation $4^{x + 1} = 64$ ?
In mathematics, a radioactive sample decays according to $N (t) = N_{0} \cdot 2^{- t /5}$ . After how many hours does the sample reach $N_{0} /8$ ?
In mathematics, what is the absolute value (modulus) of the complex number $z = 5 - 12 i$ ?
In mathematics, what is the algebraic form of the product $(3 + 2 i) (1 - 4 i)$ ?
In mathematics, what is the solution $(x, y)$ to the linear system $2 x + y = 7$ and $x - y = 2$ ?
In mathematics, what is the determinant of the $2 \times 2$ matrix $M = (43 - 2 5)$ ?
In mathematics, what is the remainder when the polynomial $P (x) = 2 x^{3} - 4 x^{2} + 3 x - 5$ is divided by $x - 2$ ?
In mathematics, at which $x$ -coordinate does the cubic function $f (x) = x^{3} - 6 x^{2} + 9 x + 2$ have an inflection point?
In mathematics, in triangle $A B C$ , if side $a = 6$ , angle $α = 3 0^{\circ}$ , and angle $β = 4 5^{\circ}$ , what is the length of side $b$ ?
In mathematics, in triangle $A B C$ , if side $a = 5$ , side $b = 8$ , and included angle $γ = 6 0^{\circ}$ , what is the exact length of side $c$ ?
In mathematics, what is the volume of a right circular cylinder with radius $r = 3$ and height $h = 7$ ?
In mathematics, what is the volume of a sphere whose surface area equals $36 π$ ?
In mathematics, what is the inverse function $f^{- 1} (x)$ for the linear bijection $f (x) = 4 x - 7$ ?
In mathematics, if $f (x) = 2 x + 3$ and $g (x) = x^{2} - 1$ , what is the expression for the composite function $(g \circ f) (x)$ ?
In mathematics, what is the median of the data set ${14, 7, 23, 19, 5, 12, 31}$ ?
In mathematics, what is the sample variance of the set of three values ${2, 4, 6}$ ?
In mathematics, what is the solution set of the linear inequality $5 - 2 x \geq 1$ ?
In mathematics, what is the solution set of the rational inequality $\frac{x - 3}{x + 1} \leq 0$ ?
In mathematics, what is the limit $lim_{x \to \infty} \frac{5 x ^{3} - 2 x + 1}{2 x ^{3} + 7 x ^{2}}$ ?
In mathematics, what is the limit $lim_{x \to 0} \frac{s i n ( 4 x )}{2 x}$ ?
In mathematics, what are all real solutions to the equation $∣2 x - 3∣ = 7$ ?
In mathematics, what geometric shape represents the locus of points equidistant from a fixed point and a fixed line in the plane?
In mathematics, what is the value of $(2 10) + (3 10)$ using Pascal's identity?
In mathematics, what is the volume of a sphere whose surface area equals $144 π$ ?

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