Question 13 Marks
If $a+2 b+3 c=0$, show that $a^3+8 b^3+27 c^3=18 a b c$.
Answer[Hint. $x+y+z=0 \Rightarrow x^3+y^3+z^3=3 x y z$.]
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If $(a+3 b)=6$, show that $a^3+27 b^3+54 a b=216$.
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If $(3 x-2 y)=5$ and $x y=6$, find the value of $\left(27 x^3-8 y^3\right)$.
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If $(4 a+3 b)=10$ and $a b=2$, find the value of $\left(64 a^3+27 b^3\right)$.
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Simplify using suitable identity :
\[(3 x-5 y-4)\left(9 x^2+25 y^2+16+15 x y-20 y+12 x\right)\]
Answer$27 x^3-125 y^3-64-180 x y$
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If $\left(x+\frac{1}{x}\right)=3$, show that $\left(x^3+\frac{1}{x^3}\right)=0$.
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If $\left(a^2+b^2+c^2\right)=89$ and $(a b-b c-c a)=16$, find the value of $(a+b-c)$.
Answer$\pm 11$
[Hint. $(a+b-c)^2=\left(a^2+b^2+c^2\right)+2(a b-b c-c a)$.]
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If $\left(a^2+b^2+c^2\right)=50$ and $(a b+b c+c a)=47$, find the value of $(a+b+c)$.
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If $(a+b+c)=15$ and $(a b+b c+c a)=74$, find the value of $\left(a^2+b^2+c^2\right)$.
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If $(a+b+c)=14$ and $\left(a^2+b^2+c^2\right)=74$, find the value of $(a b+b c+c a)$.
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Using $(a-b)^2=\left(a^2+b^2-2 a b\right)$, evaluate :$(9.98)^2$
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Using $(a-b)^2=\left(a^2+b^2-2 a b\right)$, evaluate :$(992)^2$
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Using $(a-b)^2=\left(a^2+b^2-2 a b\right)$, evaluate : $(97)^2$
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Using $(a+b)^2=\left(a^2+b^2+2 a b\right)$, evaluate : $(11.6)^2$
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Using $(a+b)^2=\left(a^2+b^2+2 a b\right)$, evaluate : $(1008)^2$
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Using $(a+b)^2=\left(a^2+b^2+2 a b\right)$, evaluate : $(137)^2$
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If $(a-b)=0.9$ and $a b=0.36$, find the values of : $\left(a^2-b^2\right)$.
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If $(a-b)=0.9$ and $a b=0.36$, find the values of : $(a+b)$
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If $(a+b)=2$ and $(a-b)=10$, find the values of : $a b$.
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If $(a+b)=2$ and $(a-b)=10$, find the values of : $\left(a^2+b^2\right)$
Answer52
[Hint. $(a+b)^2+(a-b)^2=2\left(a^2+b^2\right)$ and $(a+b)^2-(a-b)^2=4 a b$.]
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If $(3 a+4 b)=16$ and $a b=4$, find the value of $\left(9 a^2+16 b^2\right)$.
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If $x-y=5$ and $x y=24$, find the value of $(x+y)$.
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If $a+b=7$ and $a b=10$, find the value of $(a-b)$.
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Using (a - b)2 = (a2 + b2 - 2ab), evaluate :
(i) (97)2 (ii) (992)2 (iii) (9.98)2
Answer(i) 9409, (ii) 984064, (iii) 99.6004
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Using (a+b)2 = (a2 + b2 + 2ab), evaluate :
(i) (137)2 (ii) (1008)2 (iii) (11.6)2
Answer(i) 18769, (ii) 1016064, (iii) 134.56
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If $a=\frac{1}{(a-5)}$, where $a \neq 5$ and $a \neq 0$, find the values of :
(i) $\left(a-\frac{1}{a}\right)$ (ii) $\left(a+\frac{1}{a}\right)$ (iii) $\left(a^2-\frac{1}{a^2}\right)$ (iv) $\left(a^2+\frac{1}{a^2}\right)$
[Hint. $a=\frac{1}{a-5} \Rightarrow a-5=\frac{1}{a} \Rightarrow\left(a-\frac{1}{a}\right)=5$.]
Answer(i) 5, (ii) $\pm \sqrt{29}$, (iii) $\pm 5 \sqrt{29}$, (iv) $3 \sqrt{3}$
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If $\left(a^2-4 a-1\right)=0$ and $a \neq 0$, find the values of:
(i) $\left(a-\frac{1}{a}\right)$ (ii) $\left(a+\frac{1}{a}\right)$ (iii) $\left(a^2-\frac{1}{a^2}\right)$ (iv) $\left(a^2+\frac{1}{a^2}\right)$
Answer(i) 4, (ii) $\pm 2 \sqrt{5}$, (iii) $\pm 8 \sqrt{5}$, (iv) 18
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If $\left(x^2+\frac{1}{25 x^2}\right)=9 \frac{2}{5}$, find the value of $\left(x-\frac{1}{5 x}\right)$
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If $\left(x^2+\frac{1}{x^2}\right)=7$, find the values of (i) $\left(x+\frac{1}{x}\right)$ (ii) $\left(x-\frac{1}{x}\right)$ (iii) $\left(2 x^2-\frac{2}{x^2}\right)$.
Answer(i) $\pm 3$, (ii) $\pm \sqrt{5}$, (iii) $\pm 6 \sqrt{5}$
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If $\left(x-\frac{1}{x}\right)=8$, find the values of (i) $\left(x+\frac{1}{x}\right)$ (ii) $\left(x^2-\frac{1}{x^2}\right)$.
Answer(i) $\pm 2 \sqrt{17}$, (ii) $\pm 16 \sqrt{17}$
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If $\left(x+\frac{1}{x}\right)=6$, find the values of (i) $\left(x-\frac{1}{x}\right)$ (ii) $\left(x^2-\frac{1}{x^2}\right)$.
Answer(i) $\pm 4 \sqrt{2}$, (ii) $\pm 24 \sqrt{2}$
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If $x-2=\frac{1}{3 x}$, find the values of (i) $\left(x^2+\frac{1}{9 x^2}\right)$ (ii) $\left(x^4+\frac{1}{81 x^4}\right)$.
[Hint. $x-2=\frac{1}{3 x} \Rightarrow\left(x-\frac{1}{3 x}\right)=2$.]
Answer(i) $\frac{14}{3}$, (ii) $\frac{194}{9}$
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If $\left(x-\frac{1}{x}\right)=4$, find the values of (i) $\left(x^2+\frac{1}{x^2}\right)$ (ii) $\left(x^4+\frac{1}{x^4}\right)$.
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If $\left(x+\frac{1}{x}\right)=5$, find the values of (i) $\left(x^2+\frac{1}{x^2}\right)$ (ii) $\left(x^4+\frac{1}{x^4}\right)$.
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If (a - b) = 0.9 and ab = 0.36, find the values of (i) (a + b) (ii) (a2 - b2).
Answer(i) $\pm 1.5$, (ii) $\pm 1.35$
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If (a + b) = 2 and (a - b) = 10, find the values of: (i) (a2 + b2), (ii) ab.
[Hint. $(a+b)^2+(a-b)^2=2\left(a^2+b^2\right)$ and $(a+b)^2-(a-b)^2=4 a b$.]
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If $(3 a+4 b)=16$ and $a b=4$, find the value of $\left(9 a^2+16 b^2\right)$.
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If x - y = 5 and xy = 24, find the value of (x + y).
Question 393 Marks
If a + b = 7 and ab = 10, find the value of (a - b).