\(n - \;\)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.

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.

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

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

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

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?

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.

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

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

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

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.

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