Bacalaureat Național (Romania) · Mathematics

Algebră, Geometrie și Analiză Matematică II

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

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In an arithmetic progression $(a_{n})_{n \geq 1}$ , the first term is $a_{1} = 3$ and the common difference is $r = 4$ . What is the tenth term $a_{10}$ ?
In a geometric progression $(b_{n})_{n \geq 1}$ , $b_{1} = 2$ and the common ratio is $q = 3$ . What is the sum of the first 4 terms $S_{4}$ ?
What are the coordinates of the vertex of the parabola associated with $f (x) = x^{2} - 4 x + 3$ ?
If $x_{1}$ and $x_{2}$ are the roots of $x^{2} - 5 x + 6 = 0$ , what is the value of $\frac{1}{x _{1}} + \frac{1}{x _{2}}$ ?
What is the real solution of the logarithmic equation $lo g_{2} (x^{2} - 1) = 3$ ?
What is the solution of the exponential equation $3^{2 x - 1} = 27$ ?
Calculate the value of the expression $lo g_{3} 18 - lo g_{3} 2$ .
What is the modulus of the complex number $z = 1 - i 3$ ?
What is the imaginary part of the complex number $z = (2 + i) (1 - 2 i)$ ?
How many three-digit numbers with distinct digits can be formed using elements from the set ${1, 2, 3, 4, 5}$ ?
What is the probability that a randomly chosen two-digit natural number is a multiple of 10?
Calculate the number of subsets with 2 elements of a set with 6 elements.
In the Cartesian plane, find the distance between the points $A (1, 2)$ and $B (4, 6)$ .
What are the coordinates of the midpoint $M$ of segment $A B$ , where $A (- 2, 5)$ and $B (4, 1)$ ?
Find the real value of $m$ for which vectors $u = 2 i + 3 j$ and $v = 4 i + m j$ are collinear.
Find the real value of $a$ for which vectors $u = 3 i - 2 j$ and $v = 4 i + a j$ are perpendicular.
What is the slope of a line parallel to the line $d : 2 x - 3 y + 5 = 0$ ?
What is the slope of a line perpendicular to the line $d : y = 2 x - 7$ ?
Determine the area of triangle $A B C$ with vertices $A (0, 0)$ , $B (4, 0)$ and $C (0, 3)$ .
Calculate $sin^{2} 3 0^{\circ} + cos^{2} 6 0^{\circ}$ .
In a triangle $A B C$ , $a = 6$ , $b = 8$ and $sin C = \frac{1}{2}$ . What is the area of triangle $A B C$ ?
If $sin x = \frac{3}{5}$ and $x \in (0, \frac{π}{2})$ , what is $cos x$ ?
What is the exact value of $cos 12 0^{\circ}$ ?
In a triangle $A B C$ , side $a = 5$ and opposite angle $A = 3 0^{\circ}$ . What is the radius $R$ of the circumscribed circle?
What is the domain of definition $D$ for the real function $f (x) = 2 x - 6$ ?
If $f (x) = 2 x + 1$ and $g (x) = x^{2} - 1$ , calculate $(g \circ f) (1)$ .
Find the inverse function $f^{- 1} (x)$ for the bijective function $f : R \to R$ , $f (x) = 3 x - 6$ .
What is the discriminant $Δ$ of the quadratic equation $2 x^{2} - 4 x + 5 = 0$ ?
Calculate the derivative $f^{'} (x)$ of the function $f (x) = x^{4} - 2 x^{2} + 7$ .
At what value of $x$ does the function $f (x) = x^{2} - 6 x + 10$ achieve its minimum?
Calculate the definite integral $\int_{0}^{2} (2 x + 1) d x$ .
Calculate the integral $\int_{1}^{3} x^{2} d x$ .
What is the limit $lim_{x \to 2} \frac{x ^{2} - 4}{x - 2}$ ?
Calculate the derivative of $f (x) = ln (3 x + 1)$ for $x > - 1/3$ .
What is the determinant of the matrix $B = (32 - 1 4)$ ?
Calculate the trace of matrix $A = (5 - 1 23)$ .
If matrix $A = (1001)$ is the identity matrix $I_{2}$ , what is $A^{2026}$ ?
Determine the solution of the trigonometric equation $sin x = 1$ in the interval $[0, 2 π)$ .
What is the value of $tan 4 5^{\circ} + cos 0^{\circ}$ ?
Find the real number $x$ satisfying $2^{x + 3} = 32$ .
What is the sum of roots for the equation $3 x^{2} - 12 x + 7 = 0$ ?
Find the product of roots for the equation $2 x^{2} + 8 x - 10 = 0$ .
Calculate the derivative of $f (x) = e^{2 x}$ .

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