ALGEBRAIC FORMULA APPLICATION | MATH CHALLENGER | বীজগাণিতিক সূত্রের প্রয়োগ

---

01. 75×75 - 25×25 =?

Show Answer

সমাধান : $75×75 - 25×25 $

$= (75)^2 - (25)^2 $

[ সূত্র : $a^2 -b^2 = (a+b)(a-b)$ ]

$ = (75+25)(75-25)$

$= 100 \times 50$

$= 5000$

02. 3.75×3.75 - 2.75×2.75 =?

Show Answer

সমাধান : $3.75×3.75 - 2.75×2.75 $

$= (3.75)^2 - (2.75)^2 $

[ সূত্র : $a^2 -b^2 = (a+b)(a-b)$ ]

$ = (3.75+2.75)(3.75-2.75)$

$= 6.50 \times 1.00$

$= 6.5 $

03. $\frac{(1.25×1.25 - 0.23×0.23)}{1.48} = ? $

Show Answer

সমাধান : $\frac{(1.25×1.25 - 0.23×0.23)}{1.48} $

$= \frac{(1.25)^2 - (0.23)^2}{1.48} $

$= \frac{(1.25+0.23) (1.25-0.23)^2}{1.48} $

[ সূত্র : $a^2 -b^2 = (a+b)(a-b)$ ]

$= \frac{1.48 \times 1.02}{1.48} $

$= 1.02 $

04. $\frac{(256×256 - 144×144)}{112} = ?$

Show Answer

সমাধান : $\frac{(256×256 - 144×144)}{112} $

$= \frac{256)^2 - (144)^2}{112} $

$= \frac{(256+144) (256-144)^2}{112} $

[ সূত্র : $a^2 -b^2 = (a+b)(a-b)$ ]

$= \frac{400 \times 112}{112} $

$= 400 $

05. 72×72 + 28×28 + 2×72×28 = ?

Show Answer

সমাধান : $72×72 + 28×28 + 2×72×28 $

$= (72)^2 + 2×72×28 + (28)^2 $

[ সূত্র : $ a^2 + 2ab + b^2 = (a+b)^2 $ ]

$= (72+28)^2 $

$= (100)^2 $

$= 10, 000 $

06. 3.75×3.75 + 4.25×4.25 + 2×3.75×4.25 = ?

Show Answer

সমাধান : $ 3.75×3.75 + 4.25×4.25 + 2×3.75×4.25 $

$= (3.75)^2 + 2×3.75×4.25 + (4.25)^2 $

[ সূত্র : $ a^2 + 2ab + b^2 = (a+b)^2 $ ]

$= (3.75+4.25)^2 $

$= (8.00)^2 $

$= 64 $

07. $(2x-3y)^2 + 2(2x-3y)(x+3y)+ (x+3y)^2 = ?$

Show Answer

সমাধান : $(2x-3y)^2 + 2(2x-3y)(x+3y)+ (x+3y)^2 $

$= \{(2x-3y) + (x+3y)\}^2 $

[ সূত্র : $ a^2 + 2ab + b^2 = (a+b)^2 $ ]

$= \{2x-3y + x+3y \}^2 $

$= \{2x + x \}^2 $

$= \{ 3x \}^2 $

$ = 9x^2 $

08. 61×61 +21×21 - 2×61×21 = ?

Show Answer

সমাধান : $61×61 +21×21 - 2×61×21 $

$= (61)^2 - 2×61×21 +(21)^2 $

$= (61-21)^2 $

[ সূত্র : $ a^2 - 2ab + b^2 = (a-b)^2 $ ]

$= (40)^2 $

$= 1600 $

09. 2.73×2.73 + 1.23×1.23 - 2×2.73×1.23 = ?

Show Answer

সমাধান : নিজে নিজে করো।

[ সূত্র সংকেত : $ a^2 - 2ab + b^2 = (a-b)^2 $ ]

10. $(2p+3q)^2 - 2(2p+3q)(3p+3q)+ (3p+3q)^2 = ?$

Show Answer

সমাধান : নিজে নিজে করো।

[ সূত্র সংকেত : $ a^2 - 2ab + b^2 = (a-b)^2 $ ]

11. 53×53×53 + 3×53×53×27 + 3×53×27×27 + 27×27×27 = ?

Show Answer

সমাধান : $ 53×53×53 + 3×53×53×27 + 3×53×27×27 $ $+ 27×27×27 $

$= (53)^3 + 3×(53)^2×27 + 3×53×(27)^2 + (27)^3 $

$ = (53+27)^3$

[ সূত্র : $ a^3+3a^2b+3ab^2 + b^3 = (a+b)^3 $ ]

$ = (80)^3$

$ = 512000 $

12. $(5a-b)^3 + 3(5a-b)^2(a+b)$ $+ 3(5a-b)(a+b)^2 + (a+b)^3 = ?$

Show Answer

সমাধান : $(5a-b)^3 + 3(5a-b)^2(a+b)$ $+ 3(5a-b)(a+b)^2 + (a+b)^3 $

$ = \{(5a-b)+(a+b)\}^3 $

$ = \{ 5a-b+a+b \}^3 $

$ = \{ 5a+a \}^3 $

$ = \{ 6a \}^3 $

$ = 216a^3 $

13. 2.73×2.73×2.73 + 3.27×3.27×3.27 + 18×2.73×3.27 = ?

Show Answer

সমাধান : $2.73×2.73×2.73 + 3.27×3.27×3.27$ $+ \underline {18} ×2.73×3.27 $

$ = (2.73)^3 + (3.27)^3 + \underline {3×6} ×2.73×3.27 $

$ = (2.73)^3 + (3.27)^3 + \underline {3× (2.73+3.27)}$ $×2.73×3.27 $

$ = (2.73)^3 + (3.27)^3 + 3 ×2.73×3.27 (2.73+3.27) $

[ সূত্র : $ a^3 + b^3 + 3ab(a+b) = (a+b)^3 $ ]

$= (2.73+3.27)^3 $

$= (6.00)^3 $

$= 216 $

14. 37×37×37 - 3×37×37×33 + 3×37×33×33 - 33×33×33 = ?

Show Answer

সমাধান : 37×37×37 - 3×37×37×33 + 3×37×33×33 - 33×33×33

$ = (37)^3 - 3×(37)^2×33 + 3×37×(33)^2 - (33)^3 $

$ = (37-33)^3$

[ সূত্র : $ a^3-3a^2b+3ab^2 - b^3 = (a-b)^3 $ ]

$ = (4)^3$ $ = 64 $

15. $(5a-6b)^3 - 3(5a-6b)^2(7a-6b)$ $+ 3(5a-6b)(7a-6b)^2$ $- (7a-6b)^3 = ?$

Show Answer

সমাধান : নিজে নিজে করো।

[ সূত্র সংকেত : $ a^3-3a^2b+3ab^2 - b^3 = (a-b)^3 $ ]

16. 6.19×6.19×6.19 - 4.69×4.69×4.69 - 4.5×6.19×4.69 = ?

Show Answer

সমাধান : $6.19×6.19×6.19 - 4.69×4.69×4.69$ $- \underline {4.5}×6.19×4.69 $

$ = (6.19)^3 - (4.69)^3 - \underline {3×1.5} ×6.19×4.69 $

$ = (6.19)^3 - (4.69)^3 - \underline {3× (6.19-4.69)}$ $×6.19×4.69 $

$ = (6.19)^3 - (4.69)^3 - 3 ×6.19×4.69 (6.19-4.69) $

[ সূত্র : $ a^3 - b^3 - 3ab(a-b) = (a-b)^3 $ ]

$= (6.19-4.69)^3 $

$= (1.5)^3 $

$= 3.375 $

17. 1.2×1.2 + 1.3×1.3 + 1.5×1.5 + 2×1.2×1.3 + 2×1.3×1.5 + 2×1.2×1.5 = ?

Show Answer

সমাধান : 1.2×1.2 + 1.3×1.3 + 1.5×1.5 + 2×1.2×1.3 + 2×1.3×1.5 + 2×1.2×1.5

$ = (1.2)^2 + (1.3)^2 + (1.5)^2 + 2×1.2×1.3$ $+ 2×1.3×1.5 + 2×1.2×1.5 $

$ = (1.2+1.3+1.5)^2 $

[ সূত্র : $ a^2 + b^2 + c^2 + 2ab + 2bc + 2ca = (a+b+c)^2 $ ]

$= (4.0)^2$

$ = 16 $

18. 8.3×8.3 + 1.2×1.2 + 3.5×3.5 + 2×8.3×1.2 - 2×1.2×3.5 - 2×8.3×3.5 = ?

Show Answer

সমাধান : 8.3×8.3 + 1.2×1.2 + 3.5×3.5 + 2×8.3×1.2 - 2×1.2×3.5 - 2×8.3×3.5

$ = (8.3)^2 + (1.2)^2 + (-3.5)^2 + 2×8.3×1.2$ $+ 2×1.2×(-3.5) + 2×8.3×(-3.5) $

$ = \{8.3+1.2+(-3.5) \}^2 $

[ সূত্র : $ a^2 + b^2 + c^2 + 2ab + 2bc + 2ca = (a+b+c)^2 $ ]

$ = \{ 8.3+1.2-3.5 \}^2 $

$ = \{ 9.5 -3.5 \}^2 $

$ = \{ 6.0 \}^2 $

$ = 36 $

19. $ (\sqrt[3]{1.2} +$ $ \sqrt[3]{4.8})\{(\sqrt[3]{1.2})^2$ $- \sqrt[3]{5.76} +$ $(\sqrt[3]{4.8})^2 \} = ?$

Show Answer

সমাধান : $ (\sqrt[3]{1.2} + \sqrt[3]{4.8})\{(\sqrt[3]{1.2})^2 - \sqrt[3]{5.76} + (\sqrt[3]{4.8})^2 \} $

$ = (\sqrt[3]{1.2} + \sqrt[3]{4.8})\{(\sqrt[3]{1.2})^2$ $- \sqrt[3]{1.2} \times \sqrt[3]{4.8} + (\sqrt[3]{4.8})^2 \} $

$ = (\sqrt[3]{1.2})^3 + (\sqrt[3]{4.8})^3 $

[ সূত্র : $ (a+b)(a^2 -ab+b^2) = a^3 +b^3 $ ]

$ = 1.2 + 4.8 $

$ = 6 $

20. $(\sqrt[3]{7.3} - \sqrt[3]{2.8})\{(\sqrt[3]{7.3})^2$ $- \sqrt[3]{20.44} +$ $(\sqrt[3]{2.8})^2 \} = ?$

Show Answer

সমাধান : $(\sqrt[3]{7.3} - \sqrt[3]{2.8})\{(\sqrt[3]{7.3})^2 - \sqrt[3]{20.44} + (\sqrt[3]{2.8})^2 \} $

$ = (\sqrt[3]{7.3} - \sqrt[3]{2.8})\{(\sqrt[3]{7.3})^2$ $- \sqrt[3]{7.3} \times \sqrt[3]{2.8} + (\sqrt[3]{2.8})^2 \} $

$ = (\sqrt[3]{7.3})^3 - (\sqrt[3]{2.8})^3 $

[ সূত্র : $ (a-b)(a^2 + ab + b^2) = a^3 -b^3 $ ]

$ = 7.3 - 2.8 $

$ =4.5 $

21. $\frac{(2.75×2.75×2.75 - 2.25×2.25×2.25)}{2.75×2.75 + 2.75×2.25 + 2.25×2.25)} = ?$

Show Answer

সমাধান : $\frac{(2.75×2.75×2.75 - 2.25×2.25×2.25)}{2.75×2.75 + 2.75×2.25 + 2.25×2.25)}$

$= \frac{(2.75)^3 - (2.25)^3}{(2.75)^2 + 2.75×2.25 + (2.25)^2}$

$= \frac{(2.75-2.25)\{(2.75)^2 + 2.75×2.25 + (2.25)^2\}}{\{(2.75)^2 + 2.75×2.25 + (2.25)^2\}}$

$ = (2.75-2.25) $

$ = 0.50 $

$ = \frac{50}{100} = \frac{1}{2} $

22. $\frac{(0.05×0.05×0.05-0.04×0.04×0.04)}{(0.05×0.05+0.002+0.04×0.04)} = ?$

Show Answer

সমাধান : $\frac{(0.05×0.05×0.05-0.04×0.04×0.04)}{(0.05×0.05+\underline{0.002}+0.04×0.04)} $

$= \frac{(0.05)^3-(0.04)^3}{(0.05)^2+ \underline{0.05 \times 0.04} +(0.04)^2} $

$= \frac{(0.05-0.04)\{(0.05)^2+ 0.05 \times 0.04 +(0.04)^2\}}{\{(0.05)^2+ \underline{0.05 \times 0.04} +(0.04)^2\}} $

$ = (0.05-0.04) $

$ = 0.01 $

23. $ \frac{(0.008+0.125)}{(0.04-0.1+0.25)} = ?$

Show Answer

সমাধান : $ \frac{(0.008+0.125)}{(0.04- \underline{0.1}+0.25)} $

$= \frac{(0.2)^3 +(0.5)^3}{(0.2)^2- \underline{0.2 \times 0.5} + (0.5)^2} $

$= \frac{(0.2 +0.5)\{(0.2)^2-{0.2 \times 0.5} + (0.5)^2\}}{\{(0.2)^2- \underline{0.2 \times 0.5} + (0.5)^2\}} $

$ = (0.2 +0.5) $

$ = 0.7 $

24. $ \frac{\{(3.2)^3-0.008\}}{\{(3.2)^2+0.64+0.04\}} = ?$

Show Answer

সমাধান : $ \frac{\{(3.2)^3-0.008\}}{\{(3.2)^2+\underline{0.64}+0.04\}} $

$ = \frac{\{(3.2)^3-(0.2)^3\}}{\{(3.2)^2+ \underline{3.2 \times 0.2}+(0.2)^2\}} $

$ = \frac{(3.2-0.2)\{(3.2)^2+{3.2 \times 0.2}+(0.2)^2\}}{\{(3.2)^2+ \underline{3.2 \times 0.2}+(0.2)^2\}} $

$ = (3.2-0.2) $

$ = 3 $

25. $(2.3×2.3×2.3 - 1)\over(2.3×2.3+2.3+1)$$ = ?$

Show Answer

সমাধান : $ \frac{(2.3×2.3×2.3 - 1)}{(2.3×2.3+2.3+1)} $

$ = \frac{(2.3)^3 - 1^3}{(2.3)^2+2.3 \times 1 +1^2} $

$ = \frac{(2.3 - 1)\{(2.3)^2+2.3 \times 1 +1^2\}}{\{(2.3)^2+2.3 \times 1 +1^2\}} $

$ = (2.3 - 1) $

$ = 1.3 $

26. $(2.52×2.52×2.52 + 1)\over(2.52×2.52 - 2.52+1)$$ = ?$

Show Answer

সমাধান : নিজে নিজে করো ।

27. $\{(2.5)^3 + (5.2)^3 + (4.3)^3 - 3(2.5)(5.2)(4.3)\}\over\{(2.5)^2 + (5.2)^2 + (4.3)^2 - (2.5)(5.2) - (5.2)(4.3) - (2.5)(4.3)\}$$ = ?$

Show Answer

সমাধান : $\frac{(2.5)^3 + (5.2)^3 + (4.3)^3 - 3(2.5)(5.2)(4.3)}{(2.5)^2 + (5.2)^2 + (4.3)^2 - (2.5)(5.2) - (5.2)(4.3) - (2.5)(4.3)}$

= $\frac{(2.5+5.2+4.3)\{(2.5)^2 + (5.2)^2 + (4.3)^2 - (2.5)(5.2) - (5.2)(4.3) - (2.5)(4.3)\}}{ \{(2.5)^2 + (5.2)^2 + (4.3)^2 - (2.5)(5.2) - (5.2)(4.3) - (2.5)(4.3)\}}$

[ সূত্র : $ a^3 + b^3 +c^3 - 3abc =$ $(a+b+c)(a^2+b^2+c^2 -ab-bc-ca)$ ]

$ = (2.5+5.2+4.3) $

$ = 12 $

28. $ \frac{(0.5×0.5×0.5-0.2×0.2×0.2+0.3×0.3×0.3+3×0.5×0.2×0.3)}{(0.5×0.5+0.2×0.2+0.3×0.3+0.5×0.2+0.3×0.2-0.5×0.3)} = ?$

Show Answer

সমাধান : নিজে নিজে করো ।

[ সূত্র সংকেত : $ a^3 + b^3 +c^3 - 3abc =$ $(a+b+c)(a^2+b^2+c^2 -ab-bc-ca)$ ]

29. $x + y = 5$ এবং $xy = 6$ হলে, $x - y = ?$

Show Answer

সমাধান : $(x-y)^2 = (x+y)^2 -4xy $

[ সূত্র : $ (a-b)^2 = (a+b)^2 - 4ab$ ]

$ = 5^2 - 4 \times 6 $      [ $x + y = 5$ এবং $xy = 6$]

$ = 25-24 $

$ = 1 $

∴ $(x-y) = \sqrt{1} = \pm 1 $

30. $a + b = 5$ এবং $ab = 5$ হলে, $a^2 + b^2 = ?$

Show Answer

সমাধান : $(a + b)^2 = 5^2$

or, $a^2 +2ab+b^2 = 25 $

or, $a^2 +2 \times 5 +b^2 = 25 $

[$ab = 5$]

or, $a^2 +b^2 +10 = 25 $

or, $a^2 +b^2 = 25 -10 = 15 $

31. $a - b = 3$ এবং $ab = 4$ হলে, $a + b = ?$

Show Answer

সমাধান : $(a+b)^2 = (a-b)^2 + 4ab $

[ সূত্র : $ (a+b)^2 = (a-b)^2 + 4ab $ ] $ = 3^2 + 4 \times 4 $

[ $a - b = 3$ এবং $ab = 4$ ]

$ = 9+16 $

$ = 25 $

∴ $(a+b) = \sqrt{25} = \pm 5 $

32. $a + b = 5$ এবং $a^2 + b^2 = 26$ হলে, $ab = ?$

Show Answer

সমাধান : $a^2 + b^2 = 26$

[ সূত্র : $ a^2 + b^2 = (a+b)^2 -2ab $ ]

or, $ (a+b)^2 -2ab = 26$ or, $ (5)^2 -2ab = 26$

[$a + b = 5$]

or, $ 25 -2ab = 26$

or, $ -2ab = 26 - 25 = 1 $

or, $ ab = \frac{1}{-2} = - \frac{1}{2} $

33. $5a + \frac{1}{3a} = 5 $ হলে $9a^2 + \frac{1}{25a^2} = ?$

Show Answer

সমাধান : $5a + \frac{1}{3a} = 5 $

or, $5a \times \frac{3}{5} + \frac{1}{3a} \times \frac{3}{5} = 5 \times \frac{3}{5} $

or, $3a + \frac{1}{5a} = 3 $

or, $(3a + \frac{1}{5a})^2 = 3^2 $

or, $(3a)^2 + 2 \times 3a \times \frac{1}{5a} + (\frac{1}{5a})^2 = 9 $

or, $9a^2 + \frac{6}{5} + \frac{1}{25a^2} = 9 $

or, $9a^2 + \frac{1}{25a^2} = 9 - \frac{6}{5}$

or, $9a^2 + \frac{1}{25a^2} =\frac{45-6}{5} =\frac{39}{5} $

34. $2x - \frac{1}{2x} = 6$ হলে $x^2 + \frac{1}{16x^2} = ?$

Show Answer

সমাধান : $2x - \frac{1}{2x} = 6$

or, $(2x \times \frac{1}{2}) - (\frac{1}{2x} \times \frac{1}{2} ) = 6 \times \frac{1}{2} $

or, $x - \frac{1}{4x} = 3$

or, $(x - \frac{1}{4x})^2 = 3^2$

or, $x^2 - 2 \times x \times \frac{1}{4x} + (\frac{1}{4x})^2 = 9 $

or, $x^2 - \frac{1}{2} +\frac{1}{16x^2} = 9 $

or, $x^2 + \frac{1}{16x^2} = 9 + \frac{1}{2} =\frac{18+1}{2} =\frac{19}{2} $

35. $6x^2 - 1 = 4x$ হলে, $36x^2 + \frac{1}{x^2} = ?$

Show Answer

সমাধান : $6x^2 - 1 = 4x$

or, $\frac{6x^2 - 1}{x} = \frac{4x}{x}$

or, $\frac{6x^2}{x} - \frac{1}{x} = 4 $

or, $ 6x - \frac{1}{x} = 4 $

or, $ (6x - \frac{1}{x})^2 = 4^2 $

or, $(6x)^2 - 2 \times 6x \times \frac{1}{x} + (\frac{1}{x})^2 = 16 $

or, $ 36x^2 - 12 + \frac{1}{x^2} = 16 $

or, $ 36x^2 + \frac{1}{x^2} = 16 + 12 = 28 $

You May Read Also :