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

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

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

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

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

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

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

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

\(n - \;\)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.

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

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

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

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

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

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?

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

\(P\left( x \right) = {\left( {x - 1} \right)^a} \cdot {\left( {x + 2} \right)^b}\) 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.

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

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

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

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

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

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.

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

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

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