The solution set of the inequation
Question: The solution set of the inequation|x+ 2| 5 is Solution: for $|x+2|5$ We get, $x+25$ or $x+2-5$ i.e. $\quad x3 \quad($ By adding $-2$ to both sides of $x+25)$ and $x-7 \quad($ By adding $-2$ to both sides of $x+2-5)$ Hence, solution set of $|x+2|5$ is $(-\infty,-7) \cup(3, \infty)$...
Read More →Factorize:
Question: Factorize: $a^{3}+3 a^{2} b+3 a b^{2}+b^{3}-8$ Solution: $a^{3}+3 a^{2} b+3 a b^{2}+b^{3}-8=\left(a^{3}+b^{3}+3 a^{2} b+3 a b^{2}\right)-8$ $=\left[a^{3}+b^{3}+3 a b(a+b)\right]-8$ $=(a+b)^{3}-2^{3}$ $=(a+b-2)\left[(a+b)^{2}+2(a+b)+2^{2}\right]$ $=(a+b-2)\left[(a+b)^{2}+2(a+b)+4\right]$...
Read More →The solution set of the inequation
Question: The solution set of the inequation |x+ 1| 3 is __________. Solution: $|x+1|3$ i.e. $\quad-3x+13$ i.e. $-3x+13$ By adding $-1$, throughout the inequality We get, $-3-1x+1-13-1$ i.e. $\quad-4x2$ i.e. $\quad x \in(-4,2)$ $\therefore$ Solution set of $|x+1|3$ is $(-4,2)$...
Read More →Factorize:
Question: Factorize: $8 a^{3}-b^{3}-4 a x+2 b x$ Solution: $8 a^{3}-b^{3}-4 a x+2 b x=\left[(2 a)^{3}-(b)^{3}\right]-2 x(2 a-b)$ $=(2 a-b)\left[(2 a)^{2}+2 a b+b^{2}\right]-2 x(2 a-b)$ $=(2 a-b)\left(4 a^{2}+2 a b+b^{2}\right)-2 x(2 a-b)$ $=(2 a-b)\left(4 a^{2}+2 a b+b^{2}-2 x\right)$...
Read More →If x > y and z < 0, then
Question: Ifxyandz 0, then xz_______ yz. Solution: If $xy$ and $z0$ Now, multiplying by $z$, we get $x zy z$ Now, multiplying inequality by $-1$ We get $-x z \geq-y z$...
Read More →A train travels 360 km at a uniform speed.
Question: A train travels 360 km at a uniform speed. If the speed had been 5 km/hr more, it would have taken 1 hour less for the same journey. Find the speed of the train. Solution: Let the original speed of train be $x \mathrm{~km} / \mathrm{hr}$. Then, Increased speed of the train $=(x+5) \mathrm{km} / \mathrm{hr}$ Time taken by the train under usual speed to cover $360 \mathrm{~km}=\frac{360}{x} \mathrm{hr}$ Time taken by the train under increased speed to cover $360 \mathrm{~km}=\frac{360}{(...
Read More →Solve the following
Question: If $-\frac{3 x}{4} \leq-3$, then $x$ ______________ 4. Solution: Given $-\frac{3 x}{4} \leq-3$ i.e. $\quad-3 x \leq-12$ Now, multiplying both sides by $-\frac{1}{3}$, We get, $\left(-\frac{1}{3}\right)(-3 x) \geq\left(-\frac{1}{3}\right)(-12)$ i. e. $\quad x \geq 4$...
Read More →Factorize:
Question: Factorize: $(a+b)^{3}-(a-b)^{3}$ Solution: $(a+b)^{3}-(a-b)^{3}=[(a+b)-(a-b)]\left[(a+b)^{2}+(a+b)(a-b)+(a-b)^{2}\right]$ $=(a+b-a+b)\left[a^{2}+2 a b+b^{2}+a^{2}-b^{2}+a^{2}-2 a b+b^{2}\right]$ $=2 b\left(3 a^{2}+b^{2}\right)$...
Read More →Factorise
Question: Factorise $x^{9}-y^{9}$ Solution: $x^{9}-y^{9}=\left(x^{3}\right)^{3}-\left(y^{3}\right)^{3}$ we know $a^{3}-b^{3}=(a-b)\left(a^{2}+b^{2}+a b\right)$ $a=x^{3}, b=y^{3}$ So, $x^{9}-y^{9}=\left(x^{3}\right)^{3}-\left(y^{3}\right)^{3}=\left(x^{3}-y^{3}\right)\left(x^{6}+y^{6}+x^{3} y^{3}\right)$ $=(x-y)\left(x^{2}+y^{2}+x y\right)\left(x^{6}+y^{6}+x^{3} y^{3}\right)$...
Read More →A train covers a distance of 90 km at a uniform speed.
Question: A train covers a distance of 90 km at a uniform speed. Had the speed been 15 km/hour more, it would have taken 30 minutes less for a journey. Find the original speed of the train. Solution: Let the original speed of train be $x \mathrm{~km} / \mathrm{hr}$. Then, Increased speed of the train $=(x+15) \mathrm{km} / \mathrm{hr}$ Time taken by the train under usual speed to cover $90 \mathrm{~km}=\frac{90}{x} \mathrm{hr}$ Time taken by the train under increased speed to cover $90 \mathrm{~...
Read More →Factorize:
Question: Factorize: $x^{6}-729$ Solution: $x^{6}-729=\left(x^{2}\right)^{3}-(9)^{3}$ $=\left[x^{2}-9\right]\left[\left(x^{2}\right)^{2}+x^{2} \times 9+9^{2}\right]$ $=\left[x^{2}-3^{2}\right]\left(x^{4}+9 x^{2}+81\right)$ $=(x+3)(x-3)\left(x^{4}+18 x^{2}+81-9 x^{2}\right)$ $=(x+3)(x-3)\left[\left(x^{2}\right)^{2}+2 \times x^{2} \times 9+9^{2}-9 x^{2}\right]$ $=(x+3)(x-3)\left[\left(x^{2}+9\right)^{2}-(3 x)^{2}\right]$ $=(x+3)(x-3)\left(x^{2}+9+3 x\right)\left(x^{2}+9-3 x\right)$ $=(x+3)(x-3)\le...
Read More →If – 4x ≥ 2, then x
Question: If 4x 2, thenx_______ 3. Solution: Since, 4x 2 i. e. $-x \geq \frac{2}{4}$ (dividing both sides by 4 ) i. e. $-x \geq \frac{1}{2}$ Now, multiply both sides by 1, We get, $x \leq \frac{-1}{2}$...
Read More →An aeroplane take 1 hour less for a journey of 1200 km
Question: An aeroplane take 1 hour less for a journey of 1200 km if its speed is increased by 100 km/hr from its usual speed. Find its usual speed. Solution: Let the usual speed of aero plane be $x \mathrm{~km} / \mathrm{hr}$. Then, Increased speed of the aero plane $=(x+100) \mathrm{km} / \mathrm{hr}$ Time taken by the aero plane under usual speed to cover $1200 \mathrm{~km}=\frac{1200}{x} \mathrm{hr}$ Time taken by the aero plane under increased speed to cover $1200 \mathrm{~km}=\frac{1200}{(x...
Read More →Factorise
Question: Factorise $1029-3 x^{3}$ Solution: $1029-3 x^{3}$ $=3\left(343-x^{3}\right)$ $=3\left(7^{3}-x^{3}\right)$ $=3(7-x)\left(49+x^{2}+7 x\right)$...
Read More →If |3x – 7| > 2, then x
Question: If $|3 x-7|2$, then $x$ _______________$\frac{5}{3}$ or, $x$ __________________$3 .$ Solution: |3x 7| 2 By defination of modulus inequality; 3x 7 2 or 3x 7 2 If 3x 7 2 By adding 7 to both sides, We get, 3x 9 i.e.x 9 If 3x 7 2 Now, by adding 7 to both sides, We get, $3 x-7+7-2+7$ i.e. 3x 5 i.e. $x\frac{5}{3}$...
Read More →If |x – 1| ≤ 2 then
Question: If |x 1| 2 then 1 _______x 3. Solution: If |x 1| 2 i.e. 2 x 1 2 (By defination of modulus inequality) Now, By adding 1 throught the inequality, We get, 2 + 1 x 1 + 1 2 + 1 i.e 1 x 3...
Read More →Factorise
Question: Factorise $8 x^{2} y^{3}-x^{5}$ Solution: $8 x^{2} y^{3}-x^{5}=x^{2}\left(8 y^{3}-x^{3}\right)$ $=x^{2}(2 y-x)\left(4 y^{2}+x^{2}+2 x y\right)$...
Read More →Solve the following
Question: If $\frac{1}{x-2}0$, then $x$________________ $2 .$ Solution: If, $\frac{1}{x-2}0$ $\Rightarrow x-20 \quad(\because+10)$ $\Rightarrow x2 \quad$ (adding 2 on both sides)...
Read More →Factorise
Question: Factorise $x^{4} y^{4}-x y$ Solution: Using the identity $a^{3}-b^{3}=(a-b)\left(a^{2}+b^{2}+a b\right)$ $x^{4} y^{4}-x y=x y\left(x^{3} y^{3}-1\right)$ $=x y(x y-1)\left(x^{2} y^{2}+1+x y\right)$...
Read More →Solve the following
Question: If $\frac{1}{x-2}0$, then $x$ $2 .$ Solution: If, $\frac{1}{x-2}0$ $\Rightarrow x-20 \quad(\because+10)$ $\Rightarrow x2 \quad$ (adding 2 on both sides)...
Read More →Solve the following
Question: If $\frac{1}{x-2}0$, then $x$ $2 .$ Solution: If, $\frac{1}{x-2}0$ $\Rightarrow x-20 \quad(\because+10)$ $\Rightarrow x2 \quad$ (adding 2 on both sides)...
Read More →If –x ≤ –4,
Question: If x 4, then 2x_______ 8. Solution: Ifx 4 i.e.x4(multiplyingby1) i.e.2x8(multiplyingby2)...
Read More →If x ≥ –3,
Question: Ifx 3, thenx+ 5 _______ 2. Solution: Ifx 3 Byadding5tobothsides, weget x+ 5 3 + 5 i.e.x+ 5 2...
Read More →Factorize:
Question: Factorize: $3 a^{7} b-81 a^{4} b^{4}$ Solution: $3 a^{7} b-81 a^{4} b^{4}=3 a^{4} b\left(a^{3}-27 b^{3}\right)$ $=3 a^{4} b\left[a^{3}-(3 b)^{3}\right]$ $=3 a^{4} b(a-3 b)\left[a^{2}+a \times 3 b+(3 b)^{2}\right]$ $=3 a^{4} b(a-3 b)\left(a^{2}+3 a b+9 b^{2}\right)$...
Read More →The shaded part of the number line in the given figure can also be described as
Question: The shaded part of the number line in the given figure can also be described as (a) (, 1) (2, ) (b) (, 1] [2, ) (c) (1, 2) (d) [1, 2] Solution: Sinceshadedregiondoesnotinclude1and2butincludeseveryvaluebefore1andeveryvalueafter2. Hence,shadedregionisgivenby(, 1) (2, ) Hence, the correct answer is option A....
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