SSC CGL 202031)If \(a^2+b^2-c^2=0\), then the value of \(2(a^6+b^6-c^6)\over3a^2b^2c^2\) is :
2
\(a^2+b^2-c^2=0\); \(a^2+b^2=c^2\); {cubing both sides}, \((a^2+b^2)^3=(c^2)^3\); ⇒ \(a^6+b^6+3a^2b^2(a^2+b^2)=c^6\); ⇒ \(a^6+b^6-c^6=-3a^2b^2c^2\);
\({2(a^6+b^6-c^6)\over3a^2b^2c^2}={2\times(-3a^2b^2c^2)\over3a^2b^2c^2}= -2\)
SSC CGL 202032)If p + q = 7 and pq = 5, then the value of \((p^3+q^3)\) is :
238
p + q = 7; cubing both sides \((p^3+q^3)=7^3\); ⇒ \(p^3+q^3+3pq(p+q) =343\); ⇒ \(p^3+q^3=238\)
SSC CGL 202033)If x + 3y + 2 = 0, then value of \(x^3 +27y^3+8-18xy\) is :
0
x + 3y + 2 = 0; ⇒ x + 3y = -2; cubing both sides, \((x+3y)^3=(-2)^3\); ⇒ \(x^3+(3y)^3+3x(3y)(x+3y)=-8\); ⇒ \(x^3+27y^3+9xy(-2)=-8\); ⇒ \(x^3+27y^3+8-18xy=0\)
SSC CGL 202034)If \(a^4+\frac{1}{a^4}=50\), a > 0, then find the value of \((a^3+\frac{1}{a^3})\).
\(\sqrt{2(1+\sqrt{13})}(-1+2\sqrt{13})\)
\(a^4+\frac{1}{a^4}={(a^2+\frac{1}{a^2})}^2-2\times a^2\times\frac{1}{a^2}=50\); \(a^2+\frac{1}{a^2}=2\sqrt{13}\); Similarly \({(a+\frac{1}{a})^2}=2(\sqrt{13}+1)\); \(a+\frac{1}{a}=\sqrt{2(\sqrt{13}+1)}\); Calculate \(a^3+\frac{1}{a^3}=\sqrt{2(\sqrt{13}+1)}(2\sqrt{13}-1)\)
SSC CGL 202035)\(25a^2-9\) is factored as:
(5a + 3)(5a - 3)
\(25a^2-9 =(5a)^2-(3)^2=(5a+3)(5a-3)\)
SSC CGL 202036)Find the product of \((a+b+2c)(a^2+b^2+4c^2-ab-2bc-2ca)\)
\(a^3+b^3+8c^3-6abc\)
\((a+b+2c)(a^2+b^2+4c^2-ab-2bc-2ca) =(a+b+2c)(a^2+b^2+(2c)^2-a\times b-b\times (2c)-(2c)\times a)\); ⇒\(a^3+b^3+(2c)^3-3ab(2c)=a^3 +b^3+8c^3-6abc\)
SSC CGL 202037)If x + y + z = 10 and xy + yz + zx = 15, then find the value of \(x^3+y^3+z^3-3xyz\).
550
Given, x + y + z = 10 and xy + yz + zx = 15; \((x+y+z)^2=x^2+y^2+z^2+2(xy+yz+zx)\); ⇒\(100=x^2+y^2+z^2+2\times15\); ⇒\(x^2+y^2+z^2=70\);
\(x^3+y^3+z^3-3xyz=(x+y+z)(x^2+y^2+z^2-xy-yz-zx)\) = 10(70-15) = 550
SSC CGL 202038)If \(a^2+b^2+c^2=300\) and ab + bc + ca = 50 then what is the value of (a + b + c)? (Given that a, b and c are all positive.)
20
Here, \(a^2+b^2+c^2=300\); ab + bc + ca = 50; \((a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ca)\) = 300 + 2 x 50 = 400; ⇒ So \(a+b+c=\sqrt{400} = 20\)
SSC CGL 202039)Expand \(({x\over3}+{y\over5})^3\) .
\({x^3\over27}+{x^2y\over15}+{xy^2\over25}+{y^3\over125}\)
\(({x\over3}+{y\over5})^3=({x\over3})^3+3({x\over3})^2({y\over5})+3({x\over3})({y\over5})^2+({y\over5})^3\)= \({x^3\over27}+{x^2y\over15}+{xy^2\over25}+{y^3\over125}\) \([\because(a+b)^3=a^3+3a^2b+3ab^2+b^3]\)
SSC CGL 202040)The coefficient of y in the expansion of \((2y-5)^3\), is:
150
\((2y-5)^3=(2y)^3-(5)^3-3(2y)\times5(2y-5)=8y^3-125-60y^2+150y\) ; Co-efficient of y = 150