SSC CGL 202011)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 202012)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 202013)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 202014)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 202015)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
SSC CGL 202016)If \(1-64x^3-12x+px^2=(1-4x)^3\), then the value of p is:
48
\(1-64x^3-12x+px^2=(1-4x)^3\) ⇒ \(1-64x^3-12x+px^2=(1)^3-(4x)^3-3\times1\times4x(1-4x)\) ⇒ \(1-64x^3-12x+px^2=1-64x^3-12x+48x^2\); Therefore, p = 48
SSC CGL 202017)If b + c = ax, c + a = by, a + b = cz then the value of \({1\over9}[{1\over x+1}+{1\over y+1}+{1\over z+1}]\) is:
\(1\over9\)
Put, a = b = c 1 . then x = 2 ; y = 2 ; z = 2. \({1\over9}[{1\over x+1}+{1\over y+1}+{1\over z+1}]= {1\over9}[{1\over3}+{1\over3}+{1\over3}]={1\over9}\)
SSC CGL 202018)If \(a^3+{1\over a^3}=52\) then the value of \(2(a+{1\over a})\) is :
8
\(a^3+{1\over a^3}=a^3+{1\over a^3}+3(a+{1\over a})\) ; ⇒\((a+{1\over a})^3-3(a+{1\over a}) =52\); From options, If, \(a+{1\over a}=4\), then \((4)^3-3\times4 = 52\); \(2(a+{1\over a}) = 2\times4 = 8\)
SSC CGL 202019)If \(x^2-4x+4=0\), then the value of \(16({x^4}-{1\over x^4})\) is :
255
\(x^2-4x+4=0\); ⇒ \((x-2)^2=0\); ⇒ x = 2; \(\therefore 16({x^4}-{1\over x^4}) =16({2^4}-{1\over 2^4})= 255\)
SSC CGL 202020)The coefficient of x in \((x-3y)^3\) is :
\(27y^2\)
\((x-3y)^3= x^3-(3y)^3-3x\times 3y(x-3y)=x^3-27y^3-9x^2y+27xy^2\); Co-efficient of x = \(27y^2\)