Question 14 Marks
If $\left(x-\frac{2}{x}\right)=6$, find the value of $\left(x^3-\frac{8}{x^3}\right)$.
View full question & answer→Question 24 Marks
If $\left(x-\frac{1}{x}\right)=5$, find the value of $\left(x^3-\frac{1}{x^3}\right)$.
View full question & answer→Question 34 Marks
If $\left(x+\frac{1}{x}\right)=3$, find the value of $\left(x^3+\frac{1}{x^3}\right)$.
View full question & answer→Question 44 Marks
If $\frac{a}{b}=\frac{b}{c}$, prove that $(a+b+c)(a-b+c)=\left(a^2+b^2+c^2\right)$.
Answer[Hint. Let $\frac{a}{b}=\frac{b}{c}=k$. Then, $b=c k$ and $a=b k$.$\therefore b=c k$ and $a=c k^2$.
Put these values in required identity and show that L.H.S. $=$ R.H.S.]
View full question & answer→Question 54 Marks
If $\left(x^2+\frac{1}{25 x^2}\right)=9 \frac{2}{5}$, find the value of $\left(x-\frac{1}{5 x}\right)$.
View full question & answer→Question 64 Marks
If $\left(x^2+\frac{1}{x^2}\right)=7$, find the values of $\left(2 x^2-\frac{2}{x^2}\right)$.
View full question & answer→Question 74 Marks
If $\left(x^2+\frac{1}{x^2}\right)=7$, find the values of $\left(x-\frac{1}{x}\right)$
View full question & answer→Question 84 Marks
If $\left(x^2+\frac{1}{x^2}\right)=7$, find the values of $\left(x+\frac{1}{x}\right)$
View full question & answer→Question 94 Marks
If $\left(x-\frac{1}{x}\right)=8$, find the values of $\left(x^2-\frac{1}{x^2}\right)$.
View full question & answer→Question 104 Marks
If $\left(x-\frac{1}{x}\right)=8$, find the values of $\left(x+\frac{1}{x}\right)$
View full question & answer→Question 114 Marks
If $\left(x+\frac{1}{x}\right)=6$, find the values of $\left(x^2-\frac{1}{x^2}\right)$.
View full question & answer→Question 124 Marks
If $\left(x+\frac{1}{x}\right)=6$, find the values of $\left(x-\frac{1}{x}\right)$
View full question & answer→Question 134 Marks
If $x-2=\frac{1}{3 x}$, find the values of $\left(x^4+\frac{1}{81 x^4}\right)$.
Answer$\frac{194}{9}$
[Hint. $x-2=\frac{1}{3 x} \Rightarrow\left(x-\frac{1}{3 x}\right)=2$.]
View full question & answer→Question 144 Marks
If $x-2=\frac{1}{3 x}$, find the values of $\left(x^2+\frac{1}{9 x^2}\right)$
Answer$\frac{14}{3}$
[Hint. $x-2=\frac{1}{3 x} \Rightarrow\left(x-\frac{1}{3 x}\right)=2$.]
View full question & answer→Question 154 Marks
If $\left(x-\frac{1}{x}\right)=4$, find the values of $\left(x^4+\frac{1}{x^4}\right)$.
View full question & answer→Question 164 Marks
If $\left(x-\frac{1}{x}\right)=4$, find the values of $\left(x^2+\frac{1}{x^2}\right)$
View full question & answer→Question 174 Marks
If $\left(x+\frac{1}{x}\right)=5$, find the values of $\left(x^4+\frac{1}{x^4}\right)$.
View full question & answer→Question 184 Marks
If $\left(x+\frac{1}{x}\right)=5$, find the values of $\left(x^2+\frac{1}{x^2}\right)$
View full question & answer→Question 194 Marks
Simplify:
$\left(x+\frac{1}{x}\right)^2-\left(x-\frac{1}{x}\right)^2$
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Simplify:
$\left(x+\frac{1}{x}\right)^2+\left(x-\frac{1}{x}\right)^2$
Answer$2\left(x^2+\frac{1}{x^2}\right)$
View full question & answer→Question 214 Marks
Simplify:
$\left(\frac{a}{2 b}+\frac{2 b}{a}\right)^2-\left(\frac{2 b}{a}-\frac{a}{2 b}\right)^2$
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Simplify:
$\left(3 x-\frac{1}{3 x}\right)^2-\left(3 x+\frac{1}{3 x}\right)\left(3 x-\frac{1}{3 x}\right)$
Answer$2\left(\frac{1}{9 x^2}-1\right)$
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Simplify:
$(a+b)^2-(a-b)^2$
View full question & answer→Question 244 Marks
Simplify:
$(a+b)^2+(a-b)^2$
Answer$2\left(a^2+b^2\right)$
View full question & answer→Question 254 Marks
Simplify:
$(5 a+3 b)^2-(5 a-3 b)^2-60 a b$
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Simplify:
$(3 x+1)^2-(3 x+2)(3 x-1)$
View full question & answer→Question 274 Marks
Simplify using suitable identity:
(3x - 5y - 4) (9x2 + 25y2 + 16 +15xy - 20y + 12x)
Answer$27 x^3-125 y^3-64-180 x y$
View full question & answer→Question 284 Marks
If $\frac{a}{b}=\frac{b}{c}$, prove that $(a+b+c)(a-b+c)=\left(a^2+b^2+c^2\right)$.
(Hint. Let $\frac{a}{b}=\frac{b}{c}=k$. Then, $b=c k$ and $a=b k$.
$\therefore b=c k$ and $a=c k^2$.
Put these values in required identity and show that L.H.S. = R.H.S.]
View full question & answer→Question 294 Marks
If $\left(x+\frac{1}{x}\right)^2=3$, show that $\left(x^3+\frac{1}{x^3}\right)=0$.
View full question & answer→Question 304 Marks
If $\left(x^2+\frac{1}{25 x^2}\right)=8 \frac{3}{5}$, find the values of : (i) $\left(x+\frac{1}{5 x}\right)$ (ii) $\left(x^3+\frac{1}{125 x^3}\right)$.
Answer(i) $\pm 3$, (ii) $\pm 25 \frac{1}{5}$
View full question & answer→Question 314 Marks
If $\left(a^2+\frac{1}{a^2}\right)=27$, find the values of: (i) $\left(a-\frac{1}{a}\right)$ (ii) $\left(a^3-\frac{1}{a^3}\right)$.
Answer(i) $\pm 5$, (ii) $\pm 140$
View full question & answer→Question 324 Marks
If $\left(a-\frac{1}{a}\right)=\sqrt{5}$, find the values of : (i) $\left(a+\frac{1}{a}\right)$ (ii) $\left(a^3+\frac{1}{a^3}\right)$.
Answer(i) $\pm 3$, (ii) $\pm 18$
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If $\left(a^2+\frac{1}{a^2}\right)=23$, find the values of: (i) $\left(a+\frac{1}{a}\right)$ (ii) $\left(a^3+\frac{1}{a^3}\right)$.
Answer(i) $\pm 5$, (ii) $\pm 110$
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If $\left(x+\frac{1}{x}\right)=4$, find the values of: (i) $\left(x^3+\frac{1}{x^3}\right)$ (ii) $\left(x-\frac{1}{x}\right)$ (iii) $\left(x^3-\frac{1}{x^3}\right)$.
Answer(i) 52, (ii) $\pm 2 \sqrt{3}$, (iii) $\pm 30 \sqrt{3}$
View full question & answer→Question 354 Marks
If $\left(x-\frac{2}{x}\right)=6$, find the value of $\left(x^3-\frac{8}{x^3}\right)$.
View full question & answer→Question 364 Marks
If $\left(x-\frac{1}{x}\right)=5$, find the value of $\left(x^3-\frac{1}{x^3}\right)$.
View full question & answer→Question 374 Marks
If $\left(x+\frac{1}{x}\right)=3$, find the value of $\left(x^3+\frac{1}{x^3}\right)$.
View full question & answer→Question 384 Marks
If $a+2 b+3 c=0$, show that $a^3+8 b^3+27 c^3=18 a b c.$
[Hint. $x+y+z=0 \quad \Rightarrow x^3+y^3+z^3=3 x y z$.]
View full question & answer→Question 394 Marks
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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If (a2 +b2 +c2) = 89 and (ab - bc - ca) = 16, find the value of (a + b - c).
[Hint. (a + b - c)2 = (a2 + b2 + c2) + 2 (ab - bc - ca).]
Question 434 Marks
If (a2 + b2 + c2) = 50 and (ab + bc + ca) = 47, find the value of (a + b + c).
Question 444 Marks
If (a + b + c) = 15 and (ab + bc + ca) = 74, find the value of (a2 +b2 +c2).
Question 454 Marks
If (a + b + c) = 14 and (a2 + b2 +c2) = 74, find the value of (ab + bc + ca).