[
    {
        "question_number": 1,
        "question": "Să se calculeze  $I = \\int_{-2}^{2} \\frac{1}{(x^{2}+4)(3^{x}+1)} dx$.",
        "type": "single-choice",
        "options": [
            "A. $I = \\frac{\\pi}{3}$",
            "B. $I = 0$",
            "C. $I = \\frac{\\pi}{20}$",
            "D. $I = \\frac{\\pi}{4}$",
            "E. $I = \\frac{\\pi}{10}$",
            "F. $I = \\frac{\\pi}{8}$"
        ],
        "year": "2024",
        "source": "Admission, UPB",
        "subject": "math",
        "correct_answer": "F",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 2,
        "question": "Fie matricea $A = \\begin{pmatrix} 1 & 3 & -3 \\\\ 3 & 1 & 0 \\\\ 3 & 0 & 1 \\end{pmatrix} \\in M_3(\\mathbb{R})$. Atunci suma modulelor elementelor de pe diagonala principală a matricei $A^{59}$ este:",
        "type": "single-choice",
        "options": [
            "A. 30799",
            "B. 30789",
            "C. 30790",
            "D. 30800",
            "E. 30795",
            "F. 30788"
        ],
        "year": "2024",
        "source": "Admission, UPB",
        "subject": "math",
        "correct_answer": "A",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 3,
        "question": "Rezolvați ecuația $2^{3x-1} = 4$.",
        "type": "single-choice",
        "options": [
            "A. $x = -2$",
            "B. $x = -1$",
            "C. $x=2$",
            "D. $x=1$",
            "E. $x=4$",
            "F. $x = 0$"
        ],
        "year": "2024",
        "source": "Admission, UPB",
        "subject": "math",
        "correct_answer": "D",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 4,
        "question": "Soluția ecuației $\\sqrt{2x+3}=3$ este:",
        "type": "single-choice",
        "options": [
            "A. $x=2$",
            "B. $x=0$",
            "C. $x=-1$",
            "D. $x=1$",
            "E. $x=3$",
            "F. $x = -3$"
        ],
        "year": "2024",
        "source": "Admission, UPB",
        "subject": "math",
        "correct_answer": "E",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 5,
        "question": "Aflați valorile lui $m \\in \\mathbb{R}$ pentru care ecuația $1-2x-2x^2 = me^{2x}$ admite trei soluții reale distincte.",
        "type": "single-choice",
        "options": [
            "A. $m \\in (-\\frac{3}{e^2},0)$",
            "B. $m \\in (-\\infty,-\\frac{3}{e^2})$",
            "C. $m \\in (\\frac{1}{e},1)$",
            "D. $m \\in (e,\\infty)$",
            "E. $m \\in (1,e)$",
            "F. $m \\in (0,\\frac{1}{e})$"
        ],
        "year": "2024",
        "source": "Admission, UPB",
        "subject": "math",
        "correct_answer": "A",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 6,
        "question": "Mulțimea soluțiilor reale ale inecuației $2x+1 \\leq x+7$ este:",
        "type": "single-choice",
        "options": [
            "A. $(13,\\infty)$",
            "B. $[11,13]$",
            "C. $(-\\infty,6]$",
            "D. $(7,9)$",
            "E. $(6,7)$",
            "F. $(9,11)$"
        ],
        "year": "2024",
        "source": "Admission, UPB",
        "subject": "math",
        "correct_answer": "C",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 7,
        "question": "Fie $a \\in (0,1) \\cup (1,\\infty)$ și $f : (0,\\infty) \\rightarrow (0,\\infty), f(x) = x^x + a^x+x^a$. Determinați valoarea parametrului $a$\npentru care $f'(1)=1$.",
        "type": "single-choice",
        "options": [
            "A. $a = \\frac{1}{e}$",
            "B. $a = \\frac{1}{2}$",
            "C. $a = \\frac{1}{e^2}$",
            "D. $a = e^2$",
            "E. $a = 2$",
            "F. $a=e$"
        ],
        "year": "2024",
        "source": "Admission, UPB",
        "subject": "math",
        "correct_answer": "A",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 8,
        "question": "Fie polinomul $f = (X+1)^{2024} +3X+5$. Să se determine restul împărțirii polinomului $f$ la polinomul $g = X^2 +3X +3$.",
        "type": "single-choice",
        "options": [
            "A. $X+3$",
            "B. $3X+5$",
            "C. $3X+3$",
            "D. $2X-3$",
            "E. $X+1$",
            "F. $2X+3$"
        ],
        "year": "2024",
        "source": "Admission, UPB",
        "subject": "math",
        "correct_answer": "F",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 9,
        "question": "Fie $(a_n)_{n \\geq 1}$ o progresie aritmetică, de rație $r = 2$ și cu primul termen $a_1 = 3$. Calculați $a_5$.",
        "type": "single-choice",
        "options": [
            "A. 13",
            "B. 8",
            "C. 11",
            "D. 9",
            "E. 10",
            "F. 12"
        ],
        "year": "2024",
        "source": "Admission, UPB",
        "subject": "math",
        "correct_answer": "C",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 10,
        "question": "Fie $x_1$ și $x_2$ soluțiile ecuației $\\begin{vmatrix} x+7 & 3 \\\\ x-1 & x \\end{vmatrix} = 0$. Calculați $x_1^2 + x_2^2$.",
        "type": "single-choice",
        "options": [
            "A. 7",
            "B. 8",
            "C. 12",
            "D. 9",
            "E. 10",
            "F. 11"
        ],
        "year": "2024",
        "source": "Admission, UPB",
        "subject": "math",
        "correct_answer": "E",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 1,
        "question": "Să se rezolve sistemul $\\begin{cases} x+y = 5 \\\\ x-y=1 \\end{cases}$",
        "type": "single-choice",
        "options": [
            "A. $x = 0; y = 3$",
            "B. $x = 3; y = 2$",
            "C. $x = 0; y = 1$",
            "D. $x = 1; y = 1$",
            "E. $x = 1; y = -1$",
            "F. $x = 3; y = 4$"
        ],
        "year": "2023",
        "source": "Admission, UPB",
        "subject": "math",
        "correct_answer": "B",
        "tags": [
            "variant:E",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 2,
        "question": "Să se determine mulțimea soluțiilor reale ale ecuației $\\sqrt{1-5x} + x = 1$.",
        "type": "single-choice",
        "options": [
            "A. $\\{1; 3\\}$",
            "B. $\\{-2; 1\\}$",
            "C. $\\{-1; 0\\}$",
            "D. $\\{-3; 0\\}$",
            "E. $\\{-1; 1\\}$",
            "F. $\\{3; 4\\}$"
        ],
        "year": "2023",
        "source": "Admission, UPB",
        "subject": "math",
        "correct_answer": "D",
        "tags": [
            "variant:E",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 3,
        "question": "Fie $f: \\mathbb{R} \\rightarrow \\mathbb{R}, f(x) = x^4 + 3x^2$. Să se calculeze $f'(1)$.",
        "type": "single-choice",
        "options": [
            "A. 6",
            "B. 8",
            "C. 11",
            "D. 7",
            "E. 9",
            "F. 10"
        ],
        "year": "2023",
        "source": "Admission, UPB",
        "subject": "math",
        "correct_answer": "F",
        "tags": [
            "variant:E",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 4,
        "question": "Să se calculeze $l = \\lim_{x \\to \\infty} \\int_{0}^{a} \\frac{2x+1}{x^4+2x^3 +3x^2+2x+2} dx$.",
        "type": "single-choice",
        "options": [
            "A. $l = arctg \\space 2$",
            "B. $l = arctg \\space 3$",
            "C. $l = \\frac{\\pi}{2}$",
            "D. $l = arctg \\space \\frac{1}{3}$",
            "E. $l = \\frac{\\pi}{3}$",
            "F. $l = \\frac{\\pi}{4}$"
        ],
        "year": "2023",
        "source": "Admission, UPB",
        "subject": "math",
        "correct_answer": "F",
        "tags": [
            "variant:E",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 5,
        "question": "Soluția ecuației $9^{x+1} = 81$ este:",
        "type": "single-choice",
        "options": [
            "A. $x = 0$",
            "B. $x = 2$",
            "C. $x = -3$",
            "D. $x = 1$",
            "E. $x = -1$",
            "F. $x = -2$"
        ],
        "year": "2023",
        "source": "Admission, UPB",
        "subject": "math",
        "correct_answer": "D",
        "tags": [
            "variant:E",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 6,
        "question": "Fie $f: \\mathbb{R} \\rightarrow \\mathbb{R}, f(x) = \\frac{x^2 + ax + b}{\\sqrt{x^2+1}}$ unde $a, b$ sunt numere reale. Presupunem că funcția $f$ admite trei puncte de extrem local și are asimptota $y = x + 2$. Atunci",
        "type": "single-choice",
        "options": [
            "A. $ab = 6$",
            "B. $ab \\in (6,7)$",
            "C. $ab = \\frac{1}{4}$",
            "D. $a+b > 7$",
            "E. $a+b \\in (5,6)$",
            "F. $a+b \\in (6,7)$"
        ],
        "year": "2023",
        "source": "Admission, UPB",
        "subject": "math",
        "correct_answer": "D",
        "tags": [
            "variant:E",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 7,
        "question": "Mulțimea soluțiilor reale ale ecuației $x^2 -7x +10 = 0$ este:",
        "type": "single-choice",
        "options": [
            "A. $\\{2; 5\\}$",
            "B. $\\{5; 6\\}$",
            "C. $\\{3; 5\\}$",
            "D. $\\{1; 4\\}$",
            "E. $\\{1; 2\\}$",
            "F. $\\{4; 5\\}$"
        ],
        "year": "2023",
        "source": "Admission, UPB",
        "subject": "math",
        "correct_answer": "A",
        "tags": [
            "variant:E",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 8,
        "question": "Pentru ce valori ale lui $x \\in \\mathbb{R}$, numerele 4, $2x+3$ și 10 (în această ordine) formează o progresie aritmetică?",
        "type": "single-choice",
        "options": [
            "A. $x = -2$",
            "B. $x = -4$",
            "C. $x = 3$",
            "D. $x = 4$",
            "E. $x = 1$",
            "F. $x = 2$"
        ],
        "year": "2023",
        "source": "Admission, UPB",
        "subject": "math",
        "correct_answer": "F",
        "tags": [
            "variant:E",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 9,
        "question": "Fie polinomul $P \\in \\mathbb{R}[X], P = aX^{2024} +bX^{2023} +2X^3+cX^2+7X-3$. Dacă $P$ este divizibil prin $X^2+1$ și restul împărțirii lui $P$ la $X+1$ este 3, să se calculeze $P(1)$.",
        "type": "single-choice",
        "options": [
            "A. 36",
            "B. 31",
            "C. -14",
            "D. 21",
            "E. 27",
            "F. 15"
        ],
        "year": "2023",
        "source": "Admission, UPB",
        "subject": "math",
        "correct_answer": "B",
        "tags": [
            "variant:E",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 10,
        "question": "Pe mulțimea numerelor reale se definește legea de compoziție: $x * y = 2xy-10x-10y+55$. Să se determine suma soluțiilor reale ale ecuației $\\underbrace{x*x*...*x}_{de \\space 2024 \\space ori} = \\frac{11}{2}$.",
        "type": "single-choice",
        "options": [
            "A. 9",
            "B. 14",
            "C. 12",
            "D. 13",
            "E. 11",
            "F. 10"
        ],
        "year": "2023",
        "source": "Admission, UPB",
        "subject": "math",
        "correct_answer": "F",
        "tags": [
            "variant:E",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 1,
        "question": "Fie sistemul $\\begin{cases} mx+y-z = 1 \\\\ x+y-z=2 \\\\ -x + y + z = 0 \\end{cases}$ unde m este un parametru real. Pentru câte valori $m\\in \\mathbb{Z}$ sistemul are soluție unică $(x_0, y_0, z_0)$, cu componentele numere întregi?",
        "type": "single-choice",
        "options": [
            "A. 4",
            "B. 3",
            "C. 1",
            "D. o infinitate",
            "E. 2",
            "F. 5"
        ],
        "source": "Admission, UPB",
        "year": "2022",
        "subject": "math",
        "correct_answer": "E",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 2,
        "question": "Fie $f : \\mathbb{R} \\rightarrow \\mathbb{R}, f(x) = x^3 + x^2$. Să se calculeze $f'(1)$.",
        "type": "single-choice",
        "options": [
            "A. 4",
            "B. 3",
            "C. 0",
            "D. 2",
            "E. 5",
            "F. 7"
        ],
        "source": "Admission, UPB",
        "year": "2022",
        "subject": "math",
        "correct_answer": "E",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 3,
        "question": "Ecuația $2^{2x+1} = 8$ are soluția:",
        "type": "single-choice",
        "options": [
            "A. $x = -1$",
            "B. $x = 2$",
            "C. $x = 1$",
            "D. $x = 0$",
            "E. $x = 3$",
            "F. $x = -2$"
        ],
        "source": "Admission, UPB",
        "year": "2022",
        "subject": "math",
        "correct_answer": "C",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 4,
        "question": "Determinantul matricei $A = \\begin{pmatrix} 2 & 1 \\ 1 & 2 \\\\ \\end{pmatrix}$ este:",
        "type": "single-choice",
        "options": [
            "A. 3",
            "B. 6",
            "C. 1",
            "D. 5",
            "E. 4",
            "F. 0"
        ],
        "source": "Admission, UPB",
        "year": "2022",
        "subject": "math",
        "correct_answer": "A",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 5,
        "question": "Fie $f : \\mathbb{R} \\rightarrow \\mathbb{R}, f(x) = \\int_{0}^{1} |x - t|dt$. Să se calculeze $I = \\int_{-1}^{2} f(x)dx$.",
        "type": "single-choice",
        "options": [
            "A. $I = \\frac{11}{2}$",
            "B. $I = \\frac{8}{5}$",
            "C. $I = \\frac{4}{3}$",
            "D. $I = \\frac{1}{2}$",
            "E. $I = \\frac{1}{5}$",
            "F. $I = \\frac{7}{3}$"
        ],
        "source": "Admission, UPB",
        "year": "2022",
        "subject": "math",
        "correct_answer": "F",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 6,
        "question": "Fie $(a_n)_{n \\geq 1}$ o progresie aritmetică astfel ca $a_2 = 3$ și $a_3 = 5$. Să se calculeze $a_4$.",
        "type": "single-choice",
        "options": [
            "A. 8",
            "B. 11",
            "C. 9",
            "D. 6",
            "E. 7",
            "F. 10"
        ],
        "source": "Admission, UPB",
        "year": "2022",
        "subject": "math",
        "correct_answer": "E",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 7,
        "question": "Să se afle valorile parametrului real $m$ astfel încât ecuația $x^2 +1 = me^{-\\frac{1}{x}}$ să aibă trei soluții reale distincte.",
        "type": "single-choice",
        "options": [
            "A. $m > 2e$",
            "B. $m \\in (1, e)$",
            "C. $m \\in (1, e^2)$",
            "D. $m \\in (e, 2e)$",
            "E. $m < 2e$",
            "F. $m \\in (0,1)$"
        ],
        "source": "Admission, UPB",
        "year": "2022",
        "subject": "math",
        "correct_answer": "A",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 8,
        "question": "Să se rezolve ecuația $\\sqrt{x + 1} + x = 5$.",
        "type": "single-choice",
        "options": [
            "A. $x = 0$",
            "B. $x = 5$",
            "C. $x = -1$",
            "D. $x = 4$",
            "E. $x = 7$",
            "F. $x = 3$"
        ],
        "source": "Admission, UPB",
        "year": "2022",
        "subject": "math",
        "correct_answer": "F",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 9,
        "question": "Mulțimea soluțiilor reale ale ecuației $x^2 -11x + 18 = 0$ este:",
        "type": "single-choice",
        "options": [
            "A. $\\{1,4\\}$",
            "B. $\\{3,6\\}$",
            "C. $\\{2,9\\}$",
            "D. $\\{1,3\\}$",
            "E. $\\{0,1\\}$",
            "F. $\\{2, 7\\}$"
        ],
        "source": "Admission, UPB",
        "year": "2022",
        "subject": "math",
        "correct_answer": "C",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 10,
        "question": "Fie $f : \\mathbb{N}^* \\rightarrow \\mathbb{R}, f(n) = n + [\\frac{2022}{n}]$, unde prin $[x]$ notăm partea întreagă a numărului real $x$. Pentru câte valori $n \\in \\mathbb{N}^*$, funcția $f$ își atinge cea mai mică valoare?",
        "type": "single-choice",
        "options": [
            "A. 6",
            "B. 2",
            "C. 4",
            "D. 1",
            "E. 3",
            "F. 5"
        ],
        "source": "Admission, UPB",
        "year": "2022",
        "subject": "math",
        "correct_answer": "E",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 1,
        "question": "Fie funcția $f : [1,\\infty) \\rightarrow \\mathbb{R}, f(x) = \\int_{1}^{x} t(1-\\ln^2 t)dt$. Aflați abscisa punctului de maxim local.",
        "type": "single-choice",
        "options": [
            "A. $e$",
            "B. $2\\sqrt{e}$",
            "C. $\\sqrt{e^2}$",
            "D. $\\frac{1}{e}$",
            "E. $\\sqrt{e}$",
            "F. $e^2$"
        ],
        "source": "Admission, UPB",
        "year": "2021",
        "subject": "math",
        "correct_answer": "A",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 2,
        "question": "Să se determine numărul real $m$ astfel încât $\\begin{vmatrix} m & 6 \\\\ 1 & 2 \\end{vmatrix} = 0$.",
        "type": "single-choice",
        "options": [
            "A. $m=1$",
            "B. $m=3$",
            "C. $m=2$",
            "D. $m=5$",
            "E. $m=0$",
            "F. $m=4$"
        ],
        "source": "Admission, UPB",
        "year": "2021",
        "subject": "math",
        "correct_answer": "B",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 3,
        "question": "Să se determine numărul natural $n$ astfel încât $4, \\frac{n+8}{2}$ și 8 să fie trei termeni consecutivi ai unei progresii aritmetice.",
        "type": "single-choice",
        "options": [
            "A. $n=4$",
            "B. $n=1$",
            "C. $n=2$",
            "D. $n=3$",
            "E. $n=0$",
            "F. $n=6$"
        ],
        "source": "Admission, UPB",
        "year": "2021",
        "subject": "math",
        "correct_answer": "A",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 4,
        "question": "Fie funcția $f : \\mathbb{R} \\rightarrow \\mathbb{R}, f(x)=e^x + x^2$. Atunci $f'(0)$ este:",
        "type": "single-choice",
        "options": [
            "A. 4",
            "B. 1",
            "C. -1",
            "D. 3",
            "E. 0",
            "F. 2"
        ],
        "source": "Admission, UPB",
        "year": "2021",
        "subject": "math",
        "correct_answer": "B",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 5,
        "question": "Să se rezolve în $\\mathbb{R}$ inecuația $3x-1 > x+3$.",
        "type": "single-choice",
        "options": [
            "A. $x<1$",
            "B. $x<-2$",
            "C. $x>2$",
            "D. $x>3$",
            "E. $x<2$",
            "F. $x<3$"
        ],
        "source": "Admission, UPB",
        "year": "2021",
        "subject": "math",
        "correct_answer": "C",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 6,
        "question": "Mulțimea soluțiilor reale ale ecuației $x^2 -6x + 8 = 0$ este:",
        "type": "single-choice",
        "options": [
            "A. $\\{1,3\\}$",
            "B. $\\{1,5\\}$",
            "C. $\\emptyset$",
            "D. $\\{1\\}$",
            "E. $\\{-4, -2\\}$",
            "F. $\\{2,4\\}$"
        ],
        "source": "Admission, UPB",
        "year": "2021",
        "subject": "math",
        "correct_answer": "F",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 7,
        "question": "Pe mulțimea $\\mathbb{Z}$ a numerelor întregi se definește legea de compoziție $x \\circ y = xy - 5x - 5y + 30$. Atunci suma elementelor simetrizabile în raport cu legea de compoziție $\\circ$ este:",
        "type": "single-choice",
        "options": [
            "A. 10",
            "B. 9",
            "C. 6",
            "D. 0",
            "E. 5",
            "F. 8"
        ],
        "source": "Admission, UPB",
        "year": "2021",
        "subject": "math",
        "correct_answer": "A",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 8,
        "question": "Să se rezolve în mulțimea numerelor reale ecuația $\\sqrt{x+3} - 1 = x$.",
        "type": "single-choice",
        "options": [
            "A. $x=3$",
            "B. $x \\in \\emptyset$",
            "C. $x=-5$",
            "D. $x \\in \\{-1, 2\\}$",
            "E. $x=0$",
            "F. $x=1$"
        ],
        "source": "Admission, UPB",
        "year": "2021",
        "subject": "math",
        "correct_answer": "F",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 9,
        "question": "Fie $M = \\{1, 2, 3, ..., 999\\}$. Să se determine numărul elementelor mulțimii $M$ care conțin cifra 9 cel puțin o dată:",
        "type": "single-choice",
        "options": [
            "A. 271",
            "B. 243",
            "C. 270",
            "D. 274",
            "E. 275",
            "F. 272"
        ],
        "source": "Admission, UPB",
        "year": "2021",
        "subject": "math",
        "correct_answer": "A",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 10,
        "question": "Se consideră ecuația $3^{x^2+1} = 9$. Atunci soluțiile acesteia sunt:",
        "type": "single-choice",
        "options": [
            "A. $-1$ și 1",
            "B. 2 și 3",
            "C. -2 și 2",
            "D. $-\\sqrt{2}$ și $\\sqrt{2}$",
            "E. 0",
            "F. 0 și 5"
        ],
        "source": "Admission, UPB",
        "year": "2021",
        "subject": "math",
        "correct_answer": "A",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 1,
        "question": "Să se determine mulțimea valorilor lui $a \\in \\mathbb{R}$ astfel încât ecuația $\\ln(1+ 2x) - x^2 = a$ să aibă o singură soluție strict negativă.",
        "type": "single-choice",
        "options": [
            "A. $a \\in (-e, e)$",
            "B. $a \\in (0, \\ln 2)$",
            "C. $a \\in (-1, \\ln 2)$",
            "D. $a \\in (-\\infty, 0)$",
            "E. $a \\in (0, \\ln 2 - \\frac{1}{4})$",
            "F. $a \\in (\\frac{1}{2}, \\ln 3)$"
        ],
        "source": "Admission, UPB",
        "year": "2019",
        "subject": "math",
        "correct_answer": "D",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 2,
        "question": "Valoarea determinantului $\\begin{vmatrix} 2 & 0 & 1 \\\\ 1 & -1 & 0 \\\\ 1 & 1 & 1 \\end{vmatrix}$ este:",
        "type": "single-choice",
        "options": [
            "A. -1",
            "B. 0",
            "C. 5",
            "D. 1",
            "E. -2",
            "F. 2"
        ],
        "source": "Admission, UPB",
        "year": "2019",
        "subject": "math",
        "correct_answer": "B",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 3,
        "question": "Pentru $r > 0$, fie mulțimea $M = \\{z \\in \\mathbb{C} ; |z| = 1 \\text{ și } |z - 3i| = r\\}$. Fie $A = \\{r > 0; M \\text{ are un singur element}\\}$. Să se determine suma $S$ a elementelor mulțimii $A$.",
        "type": "single-choice",
        "options": [
            "A. $S = 6$",
            "B. $S = 5$",
            "C. $S = 4$",
            "D. $S = 2$",
            "E. $S = 8$",
            "F. $S = 12$"
        ],
        "source": "Admission, UPB",
        "year": "2019",
        "subject": "math",
        "correct_answer": "A",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 4,
        "question": "Știind că numerele $x, x + 1, x + 3$ sunt în progresie geometrică (în această ordine), atunci:",
        "type": "single-choice",
        "options": [
            "A. $x = 3$",
            "B. $x = -1$",
            "C. $x = 1$",
            "D. $x = -2$",
            "E. $x = 4$",
            "F. $x = 2$"
        ],
        "source": "Admission, UPB",
        "year": "2019",
        "subject": "math",
        "correct_answer": "C",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 5,
        "question": "Fie matricele $A = \\begin{pmatrix} 1 & 0 \\\\ 2 & -2 \\end{pmatrix}$ și $B = \\begin{pmatrix} 1 & 1 \\\\ 1 & -1 \\end{pmatrix}$. Dacă $X = A + 2B$, să se calculeze determinantul matricei $X$.",
        "type": "single-choice",
        "options": [
            "A. -10",
            "B. 14",
            "C. -14",
            "D. 10",
            "E. 20",
            "F. -20"
        ],
        "source": "Admission, UPB",
        "year": "2019",
        "subject": "math",
        "correct_answer": "D",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 6,
        "question": "Fie $P$ un polinom cu coeficienți reali astfel încât $P(1) + P(2) + ... + P(n) = n^5$, pentru orice număr natural $n \\ge 1$. Să se calculeze $P(\\frac{3}{2})$.",
        "type": "single-choice",
        "options": [
            "A. $\\frac{225}{49}$",
            "B. $\\frac{121}{16}$",
            "C. $\\frac{114}{31}$",
            "D. $\\frac{47}{15}$",
            "E. $\\frac{91}{17}$",
            "F. $\\frac{169}{25}$"
        ],
        "source": "Admission, UPB",
        "year": "2019",
        "subject": "math",
        "correct_answer": "B",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 7,
        "question": "Dacă a, b și c sunt determinate astfel încât să aibă loc egalitatea $\\lim_{x \\to 0} \\frac{1}{x^5} \\int_{0}^{x} (a+b\\cos t+c\\cos 2t)dt = \\frac{1}{5}$, să se calculeze $S = |a| + |b| + |c|$.",
        "type": "single-choice",
        "options": [
            "A. $S = 16$",
            "B. $S = 18$",
            "C. $S = 14$",
            "D. $S = 24$",
            "E. $S = 20$",
            "F. $S = 22$"
        ],
        "source": "Admission, UPB",
        "year": "2019",
        "subject": "math",
        "correct_answer": "A",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 8,
        "question": "Produsul soluțiilor ecuației $\\sqrt{1 - x} + \\sqrt{x} = 1$ este:",
        "type": "single-choice",
        "options": [
            "A. 0",
            "B. 2",
            "C. -1",
            "D. 1",
            "E. $\\frac{1}{3}$",
            "F. $\\frac{1}{2}$"
        ],
        "source": "Admission, UPB",
        "year": "2019",
        "subject": "math",
        "correct_answer": "A",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 9,
        "question": "Fie ecuația $x^3 + x^2 - 2x = 0$. Suma $S$ a soluțiilor reale este:",
        "type": "single-choice",
        "options": [
            "A. $S = 0$",
            "B. $S = 1$",
            "C. $S = -2$",
            "D. $S = 2$",
            "E. $S = -1$",
            "F. $S = 3$"
        ],
        "source": "Admission, UPB",
        "year": "2019",
        "subject": "math",
        "correct_answer": "E",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 10,
        "question": "Soluția ecuației $4^{x-1} = 16$ este:",
        "type": "single-choice",
        "options": [
            "A. $x = -2$",
            "B. $x = 4$",
            "C. $x = 5$",
            "D. $x = 2$",
            "E. $x = 0$",
            "F. $x = 3$"
        ],
        "source": "Admission, UPB",
        "year": "2019",
        "subject": "math",
        "correct_answer": "F",
        "tags": [
            "variant:A",
            "subcategory:Algebra and Analysis"
        ]
    },
    {
        "question_number": 1,
        "question": "Se consideră dreptele de ecuații $d_1: 2x+y-3=0$ și $d_2: x+2y-3 = 0$. Dacă $y = a_1x + b_1$ și $y = a_2x+b_2$, cu $a_1, b_1, a_2, b_2 \\in \\mathbb{R}$ sunt ecuațiile celor două drepte bisectoare ale unghiurilor rezultate din intersecția dreptelor $d_1$ și $d_2$, atunci suma $S = b_1 + b_2$ este:",
        "type": "single-choice",
        "options": [
            "A. 3",
            "B. 2",
            "C. 0",
            "D. 1",
            "E. 5",
            "F. -3"
        ],
        "source": "Admission, UPB",
        "year": "2023",
        "subject": "math",
        "correct_answer": "B",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 2,
        "question": "Valoarea parametrului real $m$ pentru care punctul $P(0, m)$ aparține dreptei de ecuație $d : 2x + y = 1$ este:",
        "type": "single-choice",
        "options": [
            "A. 0",
            "B. -$\\frac{1}{2}$",
            "C. -1",
            "D. 1",
            "E. 2",
            "F. $\\frac{1}{2}$"
        ],
        "source": "Admission, UPB",
        "year": "2023",
        "subject": "math",
        "correct_answer": "D",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 3,
        "question": "Dacă $tg \\alpha = 1$, atunci valoarea expresiei $E = cos \\alpha - sin \\alpha$ este:",
        "type": "single-choice",
        "options": [
            "A. $\\frac{\\sqrt{3}-\\sqrt{2}}{2}$",
            "B. $\\frac{\\sqrt{3}}{2}$",
            "C. 0",
            "D. 1",
            "E. $\\frac{1}{2}$",
            "F. $\\sqrt{3}$"
        ],
        "source": "Admission, UPB",
        "year": "2023",
        "subject": "math",
        "correct_answer": "C",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 4,
        "question": "Valoarea expresiei $E = sin \\alpha \\cdot cos(3\\alpha)$ pentru $\\alpha = 30^\\circ$ este:",
        "type": "single-choice",
        "options": [
            "A. 1",
            "B. -1",
            "C. $\\frac{\\sqrt{2}}{2}$",
            "D. $\\frac{\\sqrt{3}}{2}$",
            "E. $\\frac{1}{2}$",
            "F. 0"
        ],
        "source": "Admission, UPB",
        "year": "2023",
        "subject": "math",
        "correct_answer": "F",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 5,
        "question": "În reperul cartezian $xOy$, punctele $A(0,0)$ și $B(6,8)$ reprezintă vârfuri ale triunghiului echilateral $ABC$. Dacă vârful $C$ este situat în al doilea cadran, atunci ordonata acestuia este:",
        "type": "single-choice",
        "options": [
            "A. $1 + \\sqrt{10}$",
            "B. $4 + 3\\sqrt{3}$",
            "C. $\\sqrt{10}$",
            "D. 5",
            "E. $4 - 3\\sqrt{3}$",
            "F. 10"
        ],
        "source": "Admission, UPB",
        "year": "2023",
        "subject": "math",
        "correct_answer": "B",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 6,
        "question": "Lungimea laturii unui pătrat cu diagonala $d = 2\\sqrt{2}$ este:",
        "type": "single-choice",
        "options": [
            "A. $\\frac{\\sqrt{2}}{2}$",
            "B. $2\\sqrt{2}$",
            "C. 2",
            "D. $\\sqrt{2}$",
            "E. 1",
            "F. $\\sqrt{3}$"
        ],
        "source": "Admission, UPB",
        "year": "2023",
        "subject": "math",
        "correct_answer": "C",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 7,
        "question": "În reperul \\{O,$\\vec{i}$,$\\vec{j}$\\} se consideră vectorii $\\vec{u} = \\vec{i} - 3\\vec{j}$ și $\\vec{v} = 2\\vec{i} + \\vec{j}$. Atunci vectorul $\\vec{w} = \\vec{u} + 2\\vec{v}$ este:",
        "type": "single-choice",
        "options": [
            "A. $5\\vec{i} - \\vec{j}$",
            "B. $3\\vec{i} - 2\\vec{j}$",
            "C. $-\\vec{i} - 4\\vec{j}$",
            "D. $\\vec{i} - \\vec{j}$",
            "E. $\\vec{i}$",
            "F. $4\\vec{i} - 5\\vec{j}$"
        ],
        "source": "Admission, UPB",
        "year": "2023",
        "subject": "math",
        "correct_answer": "A",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 8,
        "question": "Aria triunghiului dreptunghic $ABC$ cu $m(\\widehat{BAC}) = 90^\\circ$, $m(\\widehat{ABC}) = 60^\\circ$ și $AB = 1$ este:",
        "type": "single-choice",
        "options": [
            "A. $\\sqrt{3}$",
            "B. $\\frac{\\sqrt{3}}{4}$",
            "C. $\\frac{\\sqrt{3}}{2}$",
            "D. 1",
            "E. 2",
            "F. $\\frac{1}{2}$"
        ],
        "source": "Admission, UPB",
        "year": "2023",
        "subject": "math",
        "correct_answer": "C",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 9,
        "question": "În reperul \\{O,$\\vec{i}$,$\\vec{j}$\\} fie vectorii $\\overrightarrow{OA} = -2\\vec{i} + 2\\vec{j}$, $\\overrightarrow{OB} = 4\\vec{i} +3\\vec{j}$, $\\overrightarrow{OC} = m\\vec{i} - \\vec{j}$ și $\\overrightarrow{OD} = -3\\vec{i} - 2\\vec{j}$. Valoarea parametrului real m pentru care $ABCD$ este paralelogram este:",
        "type": "single-choice",
        "options": [
            "A. 0",
            "B. -3",
            "C. -2",
            "D. 2",
            "E. 3",
            "F. 1"
        ],
        "source": "Admission, UPB",
        "year": "2023",
        "subject": "math",
        "correct_answer": "E",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 10,
        "question": "În reperul cartezian $xOy$ se consideră punctele $A(1, -2)$ și $B(5,1)$. Lungimea segmentului $[AB]$ este:",
        "type": "single-choice",
        "options": [
            "A. 5",
            "B. $\\sqrt{7}$",
            "C. $\\sqrt{5}$",
            "D. 25",
            "E. $\\sqrt{3}$",
            "F. 3"
        ],
        "source": "Admission, UPB",
        "year": "2023",
        "subject": "math",
        "correct_answer": "A",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 1,
        "question": "Un pătrat și un romb au același perimetru. Dacă aria pătratului este 16, iar unul dintre unghiurile rombului are 30°, atunci suma pătratelor diagonalelor rombului este:",
        "type": "single-choice",
        "options": [
            "A. 10",
            "B. 32",
            "C. 16",
            "D. 40",
            "E. 36",
            "F. 64"
        ],
        "source": "Admission, UPB",
        "year": "2022",
        "subject": "math",
        "correct_answer": "F",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 2,
        "question": "Valoarea numărului $P = \\sin 60^\\circ \\cdot \\text{tg } 30^\\circ \\cdot \\cos 90^\\circ$ este",
        "type": "single-choice",
        "options": [
            "A. 1",
            "B. $\\frac{\\sqrt{2}}{2}$",
            "C. $\\frac{\\sqrt{3}}{2}$",
            "D. $\\frac{1}{4}$",
            "E. 0",
            "F. $\\frac{\\sqrt{3}}{4}$"
        ],
        "source": "Admission, UPB",
        "year": "2022",
        "subject": "math",
        "correct_answer": "E",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 3,
        "question": "În triunghiul $ABC$ se cunosc $AB = 2, BC = \\sqrt{7}$ și $m(\\widehat{A}) = \\frac{\\pi}{3}$. Lungimea laturii $AC$ este:",
        "type": "single-choice",
        "options": [
            "A. $\\sqrt{7}$",
            "B. 4",
            "C. 2",
            "D. 7",
            "E. $\\sqrt{3}$",
            "F. 3"
        ],
        "source": "Admission, UPB",
        "year": "2022",
        "subject": "math",
        "correct_answer": "F",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 4,
        "question": "Într-un triunghi $ABC$ în care $AB = 6$ are loc relația $2(\\cos A + \\cos B) = 3 + 2 \\cos(A + B)$. Atunci aria triunghiului este:",
        "type": "single-choice",
        "options": [
            "A. $6\\sqrt{3}$",
            "B. 18",
            "C. $9\\sqrt{3}$",
            "D. $3\\sqrt{3}$",
            "E. 12",
            "F. 36"
        ],
        "source": "Admission, UPB",
        "year": "2022",
        "subject": "math",
        "correct_answer": "C",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 5,
        "question": "Fie $x \\in (0, \\frac{\\pi}{2})$. Dacă $\\sin x = \\frac{3}{5}$, atunci $\\cos x$ este:",
        "type": "single-choice",
        "options": [
            "A. 1",
            "B. $\\frac{3}{5}$",
            "C. $\\frac{4}{5}$",
            "D. $\\frac{\\sqrt{3}}{5}$",
            "E. -$\\frac{3}{5}$",
            "F. 0"
        ],
        "source": "Admission, UPB",
        "year": "2022",
        "subject": "math",
        "correct_answer": "C",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 6,
        "question": "Valoarea parametrului $m\\in \\mathbb{R}$ pentru care vectorii $\\vec{u} = m\\vec{i} + \\vec{j}$ și $\\vec{v} = -\\vec{i} + 3\\vec{j}$ sunt ortogonali, este:",
        "type": "single-choice",
        "options": [
            "A. -3",
            "B. 1",
            "C. -1",
            "D. 3",
            "E. $\\frac{1}{3}$",
            "F. 0"
        ],
        "source": "Admission, UPB",
        "year": "2022",
        "subject": "math",
        "correct_answer": "D",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 7,
        "question": "Centrul de greutate al triunghiului $ABC$ de vârfuri $A(0,3)$, $B(-1,0)$ și $C(1,0)$ este:",
        "type": "single-choice",
        "options": [
            "A. $G(0, -1)$",
            "B. $G(0,1)$",
            "C. $G(1,1)$",
            "D. $G(-1,0)$",
            "E. $G(2,0)$",
            "F. $G(0,0)$"
        ],
        "source": "Admission, UPB",
        "year": "2022",
        "subject": "math",
        "correct_answer": "B",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 8,
        "question": "Se consideră dreptele de ecuații $d_1: mx + y = 2$ și $d_2 : x + 2y = -2$. Valoarea parametrului real $m$ pentru care dreptele sunt paralele, este:",
        "type": "single-choice",
        "options": [
            "A. -1",
            "B. 2",
            "C. $\\frac{1}{2}$",
            "D. -$\\frac{1}{2}$",
            "E. 1",
            "F. 0"
        ],
        "source": "Admission, UPB",
        "year": "2022",
        "subject": "math",
        "correct_answer": "C",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 9,
        "question": "Se consideră vectorii $\\vec{u} = \\vec{i} + \\vec{j}$, $\\vec{v} = \\vec{i} - \\vec{j}$ și $\\vec{w} = 2\\vec{i} + \\vec{j}$. Atunci vectorul sumă $\\vec{u} + \\vec{v} + \\vec{w}$ este:",
        "type": "single-choice",
        "options": [
            "A. $\\vec{j}$",
            "B. $\\vec{i}$",
            "C. $-2\\vec{j}$",
            "D. $2\\vec{i} + \\vec{j}$",
            "E. $4\\vec{i} + \\vec{j}$",
            "F. $4\\vec{i} - \\vec{j}$"
        ],
        "source": "Admission, UPB",
        "year": "2022",
        "subject": "math",
        "correct_answer": "E",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 10,
        "question": "În planul $xOy$ se consideră paralelogramul $ABCD$. Fie $M$ punctul de intersecție al diagonalelor și $P$ mijlocul segmentului $[CD]$. Dacă $\\overrightarrow{AB} = \\vec{i} + \\vec{j}$ și $\\overrightarrow{AM} = 2\\vec{i} - \\vec{j}$, atunci vectorul $\\overrightarrow{AP}$ este:",
        "type": "single-choice",
        "options": [
            "A. $\\frac{1}{2}\\vec{i} + \\frac{1}{2}\\vec{j}$",
            "B. $-\\vec{i} + \\vec{j}$",
            "C. $\\frac{1}{2}\\vec{i} - \\frac{5}{2}\\vec{j}$",
            "D. $2\\vec{i} + \\vec{j}$",
            "E. $\\frac{7}{2}\\vec{i} - \\frac{5}{2}\\vec{j}$",
            "F. $\\frac{7}{2}\\vec{i} + \\vec{j}$"
        ],
        "source": "Admission, UPB",
        "year": "2022",
        "subject": "math",
        "correct_answer": "E",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 1,
        "question": "Într-un triunghi dreptunghic $ABC$ avem $m(\\widehat{A}) = 90^\\circ$, $BC = 5$ și $AB = 4$. Atunci aria triunghiului $ABC$ este:",
        "type": "single-choice",
        "options": [
            "A. 6",
            "B. 12",
            "C. 3",
            "D. 10",
            "E. 2",
            "F. 5"
        ],
        "source": "Admission, UPB",
        "year": "2021",
        "subject": "math",
        "correct_answer": "A",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 2,
        "question": "Știind că $sin x = \\frac{\\sqrt{3}}{2}$, atunci $cos^2 x$ este:",
        "type": "single-choice",
        "options": [
            "A. $\\frac{1}{\\sqrt{2}}$",
            "B. $\\frac{1}{2}$",
            "C. $\\frac{3}{4}$",
            "D. $\\frac{1}{4}$",
            "E. 1",
            "F. 0"
        ],
        "source": "Admission, UPB",
        "year": "2021",
        "subject": "math",
        "correct_answer": "D",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 3,
        "question": "Soluția ecuației $sin^3 x = cos^3 x$ din intervalul $[0, \\pi]$ este:",
        "type": "single-choice",
        "options": [
            "A. $x = \\frac{\\pi}{3}$",
            "B. $x = \\frac{\\pi}{5}$",
            "C. $x = \\frac{5\\pi}{6}$",
            "D. $x = \\frac{\\pi}{4}$",
            "E. $x = \\frac{2\\pi}{3}$",
            "F. $x = \\frac{3\\pi}{4}$"
        ],
        "source": "Admission, UPB",
        "year": "2021",
        "subject": "math",
        "correct_answer": "D",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 4,
        "question": "Distanța de la punctul $M(-1,2)$ la dreapta de ecuație $d : 3x + 4y - 3 = 0$ este:",
        "type": "single-choice",
        "options": [
            "A. $\\frac{2}{5}$",
            "B. 1",
            "C. 5",
            "D. $\\frac{1}{5}$",
            "E. 2",
            "F. $\\frac{5}{2}$"
        ],
        "source": "Admission, UPB",
        "year": "2021",
        "subject": "math",
        "correct_answer": "A",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 5,
        "question": "Fie $M$ mulțimea valorilor parametrului $m\\in \\mathbb{R}$ pentru care dreptele de ecuații $d_1 : mx + y = 2$ și $d_2: x + my = 1$ sunt paralele. Atunci:",
        "type": "single-choice",
        "options": [
            "A. $M = \\{1\\}$",
            "B. $M = \\{-1\\}$",
            "C. $M = \\emptyset$",
            "D. $M = \\{0\\}$",
            "E. $M = \\{-1,0,1\\}$",
            "F. $M = \\{-1,1\\}$"
        ],
        "source": "Admission, UPB",
        "year": "2021",
        "subject": "math",
        "correct_answer": "F",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 6,
        "question": "Se consideră triunghiul $ABC$ de vârfuri $A(0, 2), B(2, 0)$ și $C(4,0)$. Centrul cercului circumscris triunghiului $ABC$ are coordonatele:",
        "type": "single-choice",
        "options": [
            "A. $(\\frac{3}{2},3)$",
            "B. $(0,3)$",
            "C. $(3,0)$",
            "D. $(\\frac{3}{2},\\frac{3}{2})$",
            "E. $(3,3)$",
            "F. $(0,\\frac{3}{2})$"
        ],
        "source": "Admission, UPB",
        "year": "2021",
        "subject": "math",
        "correct_answer": "E",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 7,
        "question": "Să se determine valoarea parametrului $m\\in \\mathbb{R}$ pentru care vectorii $\\vec{u} = (2m + 1)\\vec{i} + 3\\vec{j}$ și $\\vec{v} = -\\vec{i} + \\vec{j}$ sunt ortogonali.",
        "type": "single-choice",
        "options": [
            "A. $m = 0$",
            "B. $m = 1$",
            "C. $m = -\\frac{1}{2}$",
            "D. $m = \\frac{1}{2}$",
            "E. $m = -1$",
            "F. $m = -2$"
        ],
        "source": "Admission, UPB",
        "year": "2021",
        "subject": "math",
        "correct_answer": "B",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 8,
        "question": "Valoarea expresiei $E = 2 \\cos 60^\\circ \\cdot \\text{ctg }45^\\circ \\cdot \\text{tg }30^\\circ \\cdot \\sin 90^\\circ$ este:",
        "type": "single-choice",
        "options": [
            "A. $E = -\\frac{\\sqrt{3}}{3}$",
            "B. $E = \\frac{\\sqrt{3}}{3}$",
            "C. $E = 0$",
            "D. $E = \\frac{\\sqrt{3}}{6}$",
            "E. $E = \\frac{\\sqrt{2}}{2}$",
            "F. $E = 1$"
        ],
        "source": "Admission, UPB",
        "year": "2021",
        "subject": "math",
        "correct_answer": "B",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 9,
        "question": "Se dau vectorii $\\vec{u} = \\sqrt{3}\\vec{i} - \\vec{j}$ și $\\vec{v} = -\\sqrt{3}\\vec{i} + 2\\vec{j}$. Calculați $||\\vec{u} + \\vec{v}||$.",
        "type": "single-choice",
        "options": [
            "A. 0",
            "B. 2",
            "C. 1",
            "D. 4",
            "E. 3",
            "F. $\\sqrt{3}$"
        ],
        "source": "Admission, UPB",
        "year": "2021",
        "subject": "math",
        "correct_answer": "C",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 10,
        "question": "Fie $n$ numărul soluțiilor ecuației $\\sin x + \\cos x = \\sqrt{2}$ care aparțin intervalului $[\\frac{\\pi}{4}, \\frac{17\\pi}{4}]$. Atunci:",
        "type": "single-choice",
        "options": [
            "A. $n = 3$",
            "B. $n = 5$",
            "C. $n = 0$",
            "D. $n = 2$",
            "E. $n = 4$",
            "F. $n = 1$"
        ],
        "source": "Admission, UPB",
        "year": "2021",
        "subject": "math",
        "correct_answer": "A",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 1,
        "question": "Știind $\\cos x = \\frac{\\sqrt{3}}{2}$, atunci $\\sin^2 x$ este:",
        "type": "single-choice",
        "options": [
            "A. $\\frac{1}{\\sqrt{2}}$",
            "B. $\\frac{1}{4}$",
            "C. $\\frac{1}{2}$",
            "D. $\\frac{\\sqrt{3}}{2}$",
            "E. $\\frac{1}{8}$",
            "F. $\\frac{\\sqrt{3}}{3}$"
        ],
        "source": "Admission, UPB",
        "year": "2019",
        "subject": "math",
        "correct_answer": "B",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 2,
        "question": "Valoarea expresiei $E = \\frac{\\text{ctg }30^\\circ \\cos 90^\\circ}{\\sin 15^\\circ}$ este:",
        "type": "single-choice",
        "options": [
            "A. 1",
            "B. $\\frac{\\sqrt{3}}{3}$",
            "C. $\\frac{\\sqrt{2}}{2}$",
            "D. $\\frac{1}{2}$",
            "E. 0",
            "F. $\\frac{1}{4}$"
        ],
        "source": "Admission, UPB",
        "year": "2019",
        "subject": "math",
        "correct_answer": "E",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 3,
        "question": "Să se determine valoarea parametrului $m \\in \\mathbb{R}$, știind că punctul $A(m, 2)$ aparține dreptei de ecuație $d: 2x + y = 3$.",
        "type": "single-choice",
        "options": [
            "A. $\\frac{1}{3}$",
            "B. 1",
            "C. 0",
            "D. $\\frac{1}{2}$",
            "E. 2",
            "F. 3"
        ],
        "source": "Admission, UPB",
        "year": "2019",
        "subject": "math",
        "correct_answer": "D",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 4,
        "question": "Se consideră triunghiul $ABC$ în care $AB = 1, BC = \\sqrt{2}, \\widehat{B} = \\frac{\\pi}{4}$. Atunci $AC$ este:",
        "type": "single-choice",
        "options": [
            "A. $\\sqrt{2}$",
            "B. $\\frac{1}{4}$",
            "C. $\\frac{3}{2}$",
            "D. 1",
            "E. $\\frac{1}{2}$",
            "F. 2"
        ],
        "source": "Admission, UPB",
        "year": "2019",
        "subject": "math",
        "correct_answer": "D",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 5,
        "question": "În triunghiul $ABC$ are loc relația $\\cos \\widehat{B} + \\cos \\widehat{C} = \\sin \\widehat{B} + \\sin \\widehat{C}$. Atunci $\\sin \\widehat{A}$ este:",
        "type": "single-choice",
        "options": [
            "A. $\\frac{\\sqrt{2}}{2}$",
            "B. -1",
            "C. $-\\frac{1}{2}$",
            "D. $\\frac{1}{2}$",
            "E. 1",
            "F. $\\frac{\\sqrt{3}}{2}$"
        ],
        "source": "Admission, UPB",
        "year": "2019",
        "subject": "math",
        "correct_answer": "E",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 6,
        "question": "Aria triunghiului de vârfuri $A(0,0), B(2,0), C(1,1)$ este:",
        "type": "single-choice",
        "options": [
            "A. 1",
            "B. $\\frac{1}{4}$",
            "C. 2",
            "D. $\\frac{\\sqrt{2}}{2}$",
            "E. 4",
            "F. $\\frac{1}{2}$"
        ],
        "source": "Admission, UPB",
        "year": "2019",
        "subject": "math",
        "correct_answer": "A",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 7,
        "question": "Să se calculeze $\\sin 105^\\circ$.",
        "type": "single-choice",
        "options": [
            "A. $\\frac{\\sqrt{6}-\\sqrt{3}}{2}$",
            "B. $\\frac{\\sqrt{6}+\\sqrt{2}}{2}$",
            "C. $\\frac{\\sqrt{6}}{2}$",
            "D. $\\frac{\\sqrt{6}+\\sqrt{2}}{4}$",
            "E. $\\frac{\\sqrt{2}}{2}$",
            "F. $\\frac{\\sqrt{6}-\\sqrt{2}}{4}$"
        ],
        "source": "Admission, UPB",
        "year": "2019",
        "subject": "math",
        "correct_answer": "D",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 8,
        "question": "Aflați valoarea parametrului $m \\in \\mathbb{R} \\setminus \\{0\\}$ astfel încât unghiul format de vectorii $\\vec{u} = \\sqrt{3}\\vec{i} - \\vec{j}$ și $\\vec{v} = \\vec{i} + m\\vec{j}$ să fie $\\frac{\\pi}{6}$.",
        "type": "single-choice",
        "options": [
            "A. 1",
            "B. $\\sqrt{5}$",
            "C. $2\\sqrt{3}$",
            "D. 3",
            "E. $\\sqrt{2}$",
            "F. $-\\sqrt{3}$"
        ],
        "source": "Admission, UPB",
        "year": "2019",
        "subject": "math",
        "correct_answer": "F",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 9,
        "question": "Dreapta ce trece prin punctele $A(0,1)$ și $B(1,0)$ are ecuația:",
        "type": "single-choice",
        "options": [
            "A. $x + y = 1$",
            "B. $x - y = 1$",
            "C. $x + y = 0$",
            "D. $x - y = -1$",
            "E. $x - y = 0$",
            "F. $x + y = -1$"
        ],
        "source": "Admission, UPB",
        "year": "2019",
        "subject": "math",
        "correct_answer": "A",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 10,
        "question": "Distanța de la punctul $A(2, -1)$ la dreapta de ecuație $x - y + 1 = 0$ este:",
        "type": "single-choice",
        "options": [
            "A. 1",
            "B. $\\sqrt{2}$",
            "C. 2",
            "D. $\\frac{\\sqrt{2}}{2}$",
            "E. $2\\sqrt{2}$",
            "F. 4"
        ],
        "source": "Admission, UPB",
        "year": "2019",
        "subject": "math",
        "correct_answer": "C",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 11,
        "question": "Mulțimea soluțiilor ecuației $\\cos 2x + \\sin x = 1$ din intervalul $[0, \\frac{\\pi}{2}]$ este:",
        "type": "single-choice",
        "options": [
            "A. $\\{\\frac{\\pi}{4}, \\frac{\\pi}{3}\\}$",
            "B. $\\{0, \\frac{\\pi}{2}\\}$",
            "C. $\\{\\frac{\\pi}{6}, \\frac{\\pi}{4}\\}$",
            "D. $\\{\\frac{\\pi}{6}, \\frac{\\pi}{3}\\}$",
            "E. $\\{0, \\frac{\\pi}{6}\\}$",
            "F. $\\{0, \\frac{\\pi}{3}\\}$"
        ],
        "source": "Admission, UPB",
        "year": "2019",
        "subject": "math",
        "correct_answer": "E",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 12,
        "question": "Determinați valoarea parametrului $m \\in \\mathbb{R}$ astfel încât dreptele $d_1: mx + y - 2 = 0$ și $d_2 : x - y + 2m = 0$ să fie paralele.",
        "type": "single-choice",
        "options": [
            "A. 0",
            "B. $\\sqrt{2}$",
            "C. -1",
            "D. $\\sqrt{3}$",
            "E. 2",
            "F. 3"
        ],
        "source": "Admission, UPB",
        "year": "2019",
        "subject": "math",
        "correct_answer": "C",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 13,
        "question": "Valoarea parametrului $m \\in \\mathbb{R}$ pentru care vectorii $\\vec{u} = m\\vec{i} + \\vec{j}$ și $\\vec{v} = -\\vec{i} + 4\\vec{j}$ sunt perpendiculari este:",
        "type": "single-choice",
        "options": [
            "A. -1",
            "B. 2",
            "C. 1",
            "D. 0",
            "E. 4",
            "F. -2"
        ],
        "source": "Admission, UPB",
        "year": "2019",
        "subject": "math",
        "correct_answer": "E",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 14,
        "question": "Lungimea razei cercului circumscris unui triunghi echilateral de latură $2\\sqrt{3}$ este:",
        "type": "single-choice",
        "options": [
            "A. 1",
            "B. $\\frac{1}{2}$",
            "C. 3",
            "D. 2",
            "E. $\\frac{1}{\\sqrt{3}}$",
            "F. $\\sqrt{3}$"
        ],
        "source": "Admission, UPB",
        "year": "2019",
        "subject": "math",
        "correct_answer": "D",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 15,
        "question": "Se dau vectorii $\\vec{u} = 2\\vec{i} + 3\\vec{j}$ și $\\vec{v} = -2\\vec{i} + 3\\vec{j}$. Atunci vectorul $2\\vec{u} - 3\\vec{v}$ este:",
        "type": "single-choice",
        "options": [
            "A. $3\\vec{j}$",
            "B. $2\\vec{j}$",
            "C. $\\vec{i} + \\vec{j}$",
            "D. $10\\vec{i} - 3\\vec{j}$",
            "E. $8\\vec{i}$",
            "F. $4\\vec{i} + 6\\vec{j}$"
        ],
        "source": "Admission, UPB",
        "year": "2019",
        "subject": "math",
        "correct_answer": "D",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 1,
        "question": "Soluția ecuației $\\sqrt{2}\\sin x = 1$, unde $x \\in [0, \\frac{\\pi}{2}]$ este:",
        "type": "single-choice",
        "options": [
            "A. $\\frac{\\pi}{6}$",
            "B. $\\frac{\\pi}{4}$",
            "C. 0",
            "D. $\\frac{\\pi}{2}$",
            "E. $\\frac{\\pi}{8}$",
            "F. $\\frac{\\pi}{3}$"
        ],
        "source": "Admission, UPB",
        "year": "2018",
        "subject": "math",
        "correct_answer": "B",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 2,
        "question": "Fie triunghiul ascuțitunghic $ABC$ cu aria $3\\sqrt{2}, AB = 3$ și $AC = 4$. Atunci măsura unghiului $\\hat{A}$ este:",
        "type": "single-choice",
        "options": [
            "A. $30^\\circ$",
            "B. $90^\\circ$",
            "C. $75^\\circ$",
            "D. $45^\\circ$",
            "E. $120^\\circ$",
            "F. $60^\\circ$"
        ],
        "source": "Admission, UPB",
        "year": "2018",
        "subject": "math",
        "correct_answer": "D",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 3,
        "question": "Știind că $2 \\cos x = 1$, să se calculeze $\\sin^2 x$.",
        "type": "single-choice",
        "options": [
            "A. $\\frac{1}{5}$",
            "B. $\\frac{1}{2}$",
            "C. $\\frac{2}{3}$",
            "D. $\\frac{3}{4}$",
            "E. 1",
            "F. 0"
        ],
        "source": "Admission, UPB",
        "year": "2018",
        "subject": "math",
        "correct_answer": "D",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 4,
        "question": "Ecuația dreptei care trece prin punctele $M(1,5)$ și $N(2, 1)$ este:",
        "type": "single-choice",
        "options": [
            "A. $x + 2y = 3$",
            "B. $4x - 3y = 1$",
            "C. $4x + 3y = 0$",
            "D. $4x + y = 9$",
            "E. $x - y = 1$",
            "F. $x + y = 5$"
        ],
        "source": "Admission, UPB",
        "year": "2018",
        "subject": "math",
        "correct_answer": "D",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    },
    {
        "question_number": 5,
        "question": "Aria triunghiului $ABC$, unde $A(4,6), B(10,6), C(10,0)$ este:",
        "type": "single-choice",
        "options": [
            "A. 18",
            "B. 11",
            "C. 8",
            "D. 7",
            "E. 10",
            "F. 12"
        ],
        "source": "Admission, UPB",
        "year": "2018",
        "subject": "math",
        "correct_answer": "A",
        "tags": [
            "variant:A",
            "subcategory:Geometry and Trigonometry"
        ]
    }
]