نمونه سوالات تشریحی فصل 5 ریاضی نهم همراه با جواب تشریحی

\( - 5{x^3},\sqrt {13} ,6x{y^3}\)

لازم به ذکر است عبارت x+1، تک جمله ای نیست و چند جمله ای می باشد.

\(\begin{array}{l}1) - 4\sqrt 2 {a^3}{b^2} \Rightarrow a = \sqrt 2 ,b = \sqrt 3 \\\\ = - 4\sqrt 2 {(\sqrt 2 )^3}{(\sqrt 3 )^2} = - 48\\\\2)\frac{1}{{\sqrt 5 }}(a{b^6}) \Rightarrow a = \sqrt {50} ,b = - 1\\\\ = \frac{1}{{\sqrt 5 }}(\sqrt {50} \times 1) = \frac{{\sqrt {50} }}{{\sqrt 5 }} = \sqrt {10} \end{array}\)

\(\begin{array}{l}1)4{y^2} - 5{x^3} - 3{y^2} + 4{x^3} = \\\\ \Rightarrow (4{y^2} - 3{y^2}) + ( - 5{x^3} + 4{x^3}) = {y^2} - {x^3}\\\\2)3x{y^2} - x + 4\sqrt 2 x{y^2} + 5x - y = \\\\ \Rightarrow (3 + 4\sqrt 2 )x{y^2} + ( - x + 5x) - y = \\\\ \Rightarrow (3 + 4\sqrt 2 )x{y^2} + 4x - y\end{array}\)

\(\begin{array}{l}1)5{x^2}y - 10x + 4y - 10x{y^2} + 4{x^2}y - 3 \Rightarrow y\\\\ - 10x{y^2} + 9{x^2}y + 4y - 10x - 3\\\\2)({x^5} + 2)(1 - x) + 2({x^6} + 2)(x - 1) \Rightarrow x\\\\({x^5} + 2 - {x^6} - 2x) + 2({x^7} + 2x - {x^6} - 2) = \\\\2{x^7} - 3{x^6} + {x^5} + 2x - 2\end{array}\)

\(\begin{array}{l}(x + y)({x^2} + {y^2} - 2xy) - (x + y)(x - y) = \\\\ \Rightarrow {x^3} + x{y^2} - 2{x^2}y + {x^2}y + {y^3} - 2x{y^2} - ({x^2} - {y^2}) = \\\\ \Rightarrow {x^3} + {y^3} - {x^2}y - x{y^2} - {x^2} + {y^2}\end{array}\)

تعداد جملات: 6 جمله

درجه نسبت به x: 3

درجه نسبت به y: 3

درجه نسبت به همه متغیر ها: 3

الف)

\( = (x + y) \times (x - y) = {x^2} - {y^2}\) مساحت

ب)

\( = \frac{{\sqrt 3 }}{4}{(a + \sqrt a )^2} = \frac{{\sqrt 3 }}{4}({a^2} + 2a\sqrt a + a) = \) مساحت

\( \Rightarrow \frac{{\sqrt 3 }}{4}{a^2} + \frac{{\sqrt 3 }}{2}a\sqrt a + \frac{{\sqrt 3 }}{4}a\)

\(\begin{array}{l}1)\frac{{{x^2} - {y^2}}}{{{x^2} + 2xy + {y^2}}} = \\\\ \Rightarrow \frac{{(x + y)(x - y)}}{{{{(x + y)}^2}}} = \frac{{x - y}}{{x + y}}\\\\2)\frac{{{x^3} + {y^3}}}{{x + y}} = \\\\ \Rightarrow \frac{{(x + y)({x^2} - xy + {y^2})}}{{x + y}} = {x^2} - xy + {y^2}\\\\3)\frac{{{{(x + y)}^2} + (x + y)(x - y)}}{x} = \\\\ \Rightarrow \frac{{({x^2} + 2xy + {y^2}) + ({x^2} - {y^2})}}{x} = \\\\ \Rightarrow \frac{{2{x^2} + 2xy}}{x} = \frac{{x(2x + 2y)}}{x} = 2x + 2y\end{array}\)

\(\begin{array}{l}1){(2a + b - 2c)^2} = \\\\ \Rightarrow {(2a)^2} + {b^2} + {( - 2c)^2} + 2(2a)(b) + \\\\ \Rightarrow + 2(2a)( - 2c) + 2(b)( - 2c) = \\\\ \Rightarrow 4{a^2} + {b^2} + 4{c^2} + 4ab - 8ac - 4bc\\\\2){(a - b - c)^2} = \\\\ \Rightarrow {a^2} + {b^2} + {c^2} + 2a( - b) + 2a( - 2) + 2( - b)( - 2) = \\\\ \Rightarrow {a^2} + {b^2} + {c^2} - 2ab - 2ac + 2bc\end{array}\)

\(\begin{array}{l}1)(\sqrt 3 + \sqrt 5 )(\sqrt 3 - \sqrt 5 )(5 + 3) = \\\\ \Rightarrow ({(\sqrt 3 )^2} - {(\sqrt 5 )^2})(3 + 5) = (3 - 5)(3 + 5) = \\\\ \Rightarrow 9 - 25 = - 16\\\\2)(x - y)(x + y)({x^2} + {y^2}) = \\\\ \Rightarrow ({x^2} - {y^2})({x^2} + {y^2}) = {x^4} - {y^4}\end{array}\)

\( = (5x + 4y)((x - 2y) + (4x + 3y)) = \)مساحت ذوزنقه

\(\begin{array}{l}\frac{{(5x + 4y)(5x + y)}}{2} = \frac{{5{x^2} + 5xy + 20xy = 4{y^2}}}{2} = \\\\\frac{5}{2}{x^2} + \frac{{25}}{2}xy + 2{y^2}\end{array}\)

\(\begin{array}{l}1)2{x^2} + 14x + 20 = 2({x^2} + 7x + 10) = \\\\2(x + 50)(x + 2)\\\\2){x^2} - 11x + 28 = (x - 4)(x - 7)\\\\3){x^3} + 5x{}^2 + 6x = x({x^2} + 5x + 6) = \\\\x(x + 2)(x + 3)\end{array}\)

\(\begin{array}{l}1){2^{10}} - {2^9} + {2^6} = {({2^5})^2} - 2({2^5})({2^3}) + {({2^3})^2} = \\\\{({2^5} - {2^3})^2}\\\\2){3^8} - 2 \times {6^4} + {2^8} = {({3^4})^2} + 2({3^4})({2^4}) + {({2^4})^2} = \\\\({3^4} + {2^4})\end{array}\)

\(\begin{array}{l}1)104 \times 96 = (100 + 4)(100 - 4) = {100^2} - {4^2} = \\\\1000 - 16 = 9984\\\\2){5^2} - 5 \times {2^3} + {2^4} = {5^2} - 2 \times 5 \times {2^2} + {2^4} = \\\\{(5 - {2^2})^2} = {(5 - 4)^2} = {1^2} = 1\end{array}\)

\(\begin{array}{l}1)9{x^2} - 36{y^2} = {(3x)^2} - {(6y)^2} = (3x + 6y)(3x - 6y)\\\\2){(x - \frac{1}{x})^2} - {(x + \frac{1}{x})^2} = \\\\((x - \frac{1}{x}) + (x + \frac{1}{x}))((x - \frac{1}{x}) - (x + \frac{1}{x})) = \\\\2x \times \frac{{ - 2}}{x} = - 4\\\\3){x^3} + 27 = {x^3} + {3^2}(x + 3)({x^2} - 3x + 9)\\\\4)8 - 216{y^3} = {(2)^3} - {(6y)^3} = \\\\(2 - 6y)(4 + 12y + 36{y^2})\end{array}\)

\(\begin{array}{l}1) - 4 < x < 4 \Rightarrow |x| < 4\\\\2)x \ge - 6\\\\3)(x < - 2) \cup (x \ge 3)\end{array}\)

الف) <

ب) <

پ) <

\(\begin{array}{l}1)\frac{{a{b^2}}}{{{c^3}}} > 0\\\\2)\frac{{ac}}{{ - \sqrt {{b^2}{c^2}} }} > 0\end{array}\)

الف)

\(2x + 5 < - 4\)

ب)

\(|x| \ge x\)

\(\begin{array}{l}1)\frac{{x + 4}}{3} - \frac{{x - 2}}{6} \le 5 \Rightarrow \frac{{x + 4}}{3} - \frac{{x - 2}}{6} - 5 \le 0\\\\ \Rightarrow \frac{{2(x + 4) - (x - 2) - 30}}{6} \le 0\\\\ \Rightarrow 2x + 8 - x + 2 - 30 \le 0 \Rightarrow x - 20 \le 0 \Rightarrow x \le 20\end{array}\)

\(\begin{array}{l}2)4x + 2(x - 5) > 3 \Rightarrow 4x + 2x - 10 - 3 > 0\\\\ \Rightarrow 6x - 13 > 0 \Rightarrow 6x > 13 \Rightarrow \frac{{13}}{6}\end{array}\)

اگر سن شخص را x در نظر بگیریم، داریم:

\(\begin{array}{l}4x + (x - 2) = 4(x + 2) \Rightarrow 4x + (x - 2) = 4x + 8\\\\ \Rightarrow x - 2 = 8 \Rightarrow x = 10\end{array}\)

سن فعلی این شخص، 10 سال می باشد. 5 سال آینده سن این شخص 15 سال می باشد!

متغیر

عددی – حرفی

ضریب عددی

جملات جبری متشابه

چند جمله ای جبری

اتحاد

\((x + 5)(x - 2)\)

مزدوج

مزدوج

قرینه

مرجع دو جمله - \({(x - 5)^2}\)

سه

است

نمی توان

منفی

\({x^2} + 6 \ge 3\)

مثبت

\(x \ge - 1\)

نابرابری خطی

\(\sqrt 3 + \sqrt 7 \)