Quyidagi rasmda qirrasining uzunligi \(6\;{\rm{sm}}\) bo‘lgan muntazam tetraedr tasvirlangan:

Agar \(BD = FE = 4\;{\rm{sm}};{\rm{\;\;}}DC = AF = 2\;{\rm{sm}}\) va \(O\) nuqta \(AC\) qirrada ekanligi ma’lum bo‘lsa, \(DO + OF\) yig‘indining eng kichik qiymatini \(\left( {{\rm{sm}}} \right)\) toping.

\(\;\left( {x - 5} \right) \cdot \left( {\frac{1}{5}x + 4} \right) = 0\) bo‘lsa, \(\frac{1}{5}x + 4\) qanday qiymatlar qabul qiladi?

Soddalashtiring: \(\frac{{{{\left( {a + 1} \right)}^3} + 1}}{{{a^3} - 1}}\)

Ishlab chiqarish samaradorligi birinchi yili \(15\% \) ga, ikkinchi yili \(16\% \) ga ortdi. Shu ikki yil ichida samaradorlik necha \(\% \) ga ortgan?

\(\frac{{{{\left( {2\left| x \right| - 3} \right)}^2} - \left| x \right| - 6}}{{4x + 1}} = 0\) tenglamaning haqiqiy ildizlari yig‘indisini toping.

\({\left( {\sqrt {6 + \sqrt {35} } } \right)^x} + {\left( {\sqrt {6 - \sqrt {35} } } \right)^x} = 12\) tenglamaning butun ildizlari ko‘paytmasini toping.

Quyidagi rasmda \(y = f\left( x \right)\) funksiya grafigi tasvirlangan:

Rasmda berilgan ma’lumotlardan foydalanib, \(\left( {x - 3} \right) \cdot f\left( x \right) > 0\) tengsizlikni qanoatlantiradigan butun sonlar yig‘indisini toping.

\({9^{{x^2} - 1}} + {3^{{x^2}}} - 4 = 0\) tenglamaning butun ildizlari yig‘indisini toping.

Quyidagi rasmda tasvirlangan \(ABC\) uchburchakning tomonlarida \(9\) ta nuqta olingan:

Uchlari ko‘rsatilgan nuqtalarda bo‘lgan nechta uchburchak bor?

\(\;\frac{{10!}}{{7!}} \cdot \left( {\frac{{8!}}{{10!}} + \frac{{3!}}{{5!}}} \right)\) ni hisoblang.

\(P\left( x \right) = {\left( {x - 1} \right)^a} \cdot {\left( {x + 2} \right)^b}\) va \(P\left( 2 \right) \cdot P\left( 1 \right) = 64\) bo‘lsa, \(a + b\) ning qiymatini toping.

\(f\left( x \right) = \left| {\left| {x - 2} \right| - 2} \right|\) funksiya grafigini chizing.

\(4{x^2} + \frac{1}{{{x^2} - 4}} = 16 - \frac{1}{{4 - {x^2}}}\) tenglamani yeching.

Teng yonli \(ABCD\) trapetsiyaning ichida \(P\) nuqta shunday olinganki, \(PA = 2;\;\;PB = 3;\;\;PC = 4\) va \(PD = 5\) ga teng. Agar \(AD\) katta asos bo’lsa, \(\frac{{BC}}{{AD}}\) ning qiymatini toping.

\(f\left( {\frac{{3x - 1}}{{x + 2}}} \right) = \frac{{x + 1}}{{x - 1}}\) bo‘lsa, \(f\left( x \right)\) ni toping.

\(\cos 3x - \sqrt 3 \sin 3x < 0\) tengsizlikni yeching.

Muntazam oltiburchakka tashqi chizilgan aylananing radiusi \(4\sqrt 3 \) \({\rm{sm}}\;\)ga teng bo‘lsa, uning kichik diagonalini \(\left( {{\rm{sm}}} \right)\) toping.

Quyidagi chizmada berilgan ma’lumotlardan foydalanib, \(x\;\)ni toping.

\(32;\;\;18\) va \(24\) sonlarining o‘rta geometrigini toping.

Sardor \(420\) \({\rm{m}}\) masofani \(7\) \({\rm{daqiqa}}\)da bosib o‘tadi. Sardorning tezligini \(\left( {{\rm{m}}/{\rm{min}}} \right)\) toping.

\(n - \;\)hadi \({a_n}\) bo‘lgan arifmetik progressiya uchun \({a_n} = 2 + {a_{n - 1}}\) tenglik o‘rinli bo‘lib, dastlabki o‘nta hadining yig‘indisi \(115\) ga teng bo‘lsa, arifmetik progressiyaning birinchi hadini toping.

\({\left( {2{x^2} - \frac{1}{{{x^2}}}} \right)^7} = \ldots + p \cdot {x^6} + \ldots \) bo‘lsa, \(p\) ning qiymatini toping.

Quyidagi rasmda ko‘rsatilgan krandan konus ichiga doimiy bir xil suv oqib tushadi.

Agar kran konusni bo‘yalgan qismini \(4\) minutda to‘ldirsa, butun konusni necha minutda to‘ldiradi?

Aniq integralni hisoblang:
\(\mathop \smallint \nolimits_0^3 \frac{x}{{\sqrt {1 + {x^2}} }}dx + \mathop \smallint \nolimits_0^3 \frac{1}{3}dx\)

\(n + 7\) soni juft son bo‘lsa, quyidagilardan qaysi biri toq son bo‘ladi?

\(f\left( x \right) = f\left( 5 \right) \cdot x - 20\) bo‘lsa, \(f\left( 4 \right)\) ning qiymatini toping.

\(\frac{{1 + 3 + 5 + \ldots + 43}}{{2 + 4 + 6 + \ldots + 42}}\) ni hisoblang.

\({\left( {{4^{\frac{{{{\log }_4}7}}{{{{\log }_7}4}}}} - {7^{{{\log }_4}7}} + {3^{{{\log }_9}16}}} \right)^{\frac{1}{2}}}\) ni hisoblang.

\(\;\left( {e - \pi } \right)x \ge 7\left( {\pi - e} \right)\) tengsizlikni yeching.

\(f\left( x \right) = y\) va \({x^2} - 2x = {e^{x - y}}\) bo‘lsa, \(f'\left( 4 \right)\) ning qiymatini toping.