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

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

\(n - \;\)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.

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

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

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

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

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.

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

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

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