Write the set of values of x satisfying the inequations 5x + 2 < 3x + 8 and
Question: Write the set of values of $x$ satisfying the inequations $5 x+23 x+8$ and $\frac{x+2}{x-1}4$. Solution: We have: $5 x+23 x+8$ and $\frac{x+2}{x-1}4$ $\Rightarrow 2 x6$ and $\frac{x+2}{x-1}-40$ $\Rightarrow x3$ and $\frac{x+2-4 x+4}{x-1}0$ $\Rightarrow x \in(-\infty, 3)$ and $\frac{-3 x+6}{x-1}0$ $\Rightarrow x \in(-\infty, 3)$ and $\frac{-x+2}{x-1}0$ For $\frac{-x+2}{x-1}0$, critical points are 1 and $2 .$ $\Rightarrow x \in(2, \infty) \cup(-\infty, 1)$ $\therefore x \in(-\infty, 1) \...
Read More →Is the following situation possible? If so,
Question: Is the following situation possible? If so, determine their present ages. The sum of the ages of two friends is 20 years ago, the product of their ages in years was 48. Solution: Let the present age of two friends bexyears and (20 x) years respectively. Then, 4 years later, the age of two friends will be (x 4) years and (20 x 4) years respectively Then according to question, $(x-4)(20-x-4)=48$ $(x-4)(16-x)=48$ $16 x-x^{2}-64+4 x=48$ $-x^{2}+20 x-64-48=0$ $x^{2}-20 x+112=0$ LetDbe the d...
Read More →Write the number of integral solutions of
Question: Write the number of integral solutions of $\frac{x+2}{x^{2}+1}\frac{1}{2}$. Solution: We have: $\frac{x+2}{x^{2}+1}\frac{1}{2}$ $\Rightarrow \frac{x+2}{x^{2}+1}-\frac{1}{2}0$ $\Rightarrow \frac{2(x+2)-\left(x^{2}+1\right)}{2\left(x^{2}+1\right)}0$ $\Rightarrow \frac{2 x+4-x^{2}-1}{2\left(x^{2}+1\right)}0$ $\Rightarrow \frac{-x^{2}+2 x+3}{2\left(x^{2}+1\right)}0$ To make the fraction of the left side positive, either the numerator or the denominator should be positive or both should be ...
Read More →Factorise
Question: Factorise $(2 a+1)^{3}+(a-1)^{3}$ Solution: $(2 a+1)^{3}+(a-1)^{3}$ $=(2 a+1+a-1)\left[(2 a+1)^{2}+(a-1)^{2}-(2 a+1)(a-1)\right]$ $=(3 a)\left[4 a^{2}+1+4 a+a^{2}+1-2 a-2 a^{2}+2 a-a+1\right]$ $=3 a\left[3 a^{2}+3 a+3\right]=9 a\left(a^{2}+a+1\right)$...
Read More →The product of Ramu's age (in years) five years ago
Question: The product of Ramu's age (in years) five years ago and his age (in years) nice years later is 15. Determine Ramu's present age. Solution: Let the present age of Ramu be $x$ years Then, 9 years later, age of her $=(x+9)$ years Five years ago, her age $=(x-5)$ years Then according to question, $(x-5)(x+9)=15$ $x^{2}+9 x-5 x-45=15$ $x^{2}+4 x-45-15=0$ $x^{2}+4 x-60=0$ $x^{2}+4 x-60=0$ $x^{2}-6 x+10 x-60=0$ $x(x-6)+10(x-6)=0$ $(x-6)(x+10)=0$ So, either $(x-6)=0$ $x=6$ Or $(x+10)=0$ $x=-10...
Read More →Factorise
Question: Factorise $(x+1)^{3}+(x-1)^{3}$ Solution: $(x+1)^{3}+(x-1)^{3}$ $=(x+1+x-1)\left[(x+1)^{2}+(x-1)^{2}-(x-1)(x+1)\right]$ $=(2 x)\left[(x+1)^{2}+(x-1)^{2}-\left(x^{2}-1\right)\right]$ $=2 x\left(x^{2}+1+2 x+x^{2}+1-2 x-x^{2}+1\right)$ $=2 x\left(x^{2}+3\right)$...
Read More →Write the solution set of the inequation
Question: Write the solution set of the inequation $\left|\frac{1}{x}-2\right|4$ Solution: We have: $\left|\frac{1}{x}-2\right|4$ Here, two cases arise. CASE 1: When $\frac{1}{x}-20$, then $\left|\frac{1}{x}-2\right|=\frac{1}{x}-2$ $\therefore \frac{1}{x}-24$ $\Rightarrow \frac{1}{x}-2-40$ $\Rightarrow \frac{1}{x}6$ $\Rightarrow x \in\left(0, \frac{1}{6}\right)$ ...(i) CASE $2:$ When $\frac{1}{x}-20$, then $\left|\frac{1}{x}-2\right|=-\left(\frac{1}{x}-2\right)$ $\therefore-\frac{1}{x}+24$ $\Rig...
Read More →Factorise
Question: Factorise $x^{3}-3 x^{2}+3 x+7$ Solution: $x^{3}-3 x^{2}+3 x+7$ $=x^{3}-3 x^{2}+3 x+7$ $=x^{3}-3 x^{2}+3 x+8-1=x^{3}-3 x^{2}+3 x-1+8$ $=\left(x^{3}-3 x^{2}+3 x-1\right)+8$ $=(x-1)^{3}+2^{3}$ $=(x-1+2)\left[(x-1)^{2}+4-2(x-1)\right]$ $=(x+1)\left[x^{2}+1-2 x+4-2 x+2\right]$ $=(x+1)\left(x^{2}-4 x+7\right)$...
Read More →The product of Shikha's age five years ago and her age 8 years later is 30,
Question: The product of Shikha's age five years ago and her age 8 years later is 30, her age at both times being given in years. Find her present age. Solution: Let the present age of Shikha be $x$ years Then, 8 years later, age of her $=(x+8)$ years Five years ago, her age $=(x-5)$ years Then according to question, $(x-5)(x+8)=30$ $x^{2}+8 x-5 x-40=30$ $x^{2}+3 x-40-30=0$ $x^{2}+3 x-70=0$ $x^{2}+3 x-70=0$ $x^{2}-7 x+10 x-70=0$ $x(x-7)+10(x-7)=0$ $(x-7)(x+10)=0$ So, either $(x-7)=0$ $x=7$ Or $(...
Read More →Write the set of values of x satisfying
Question: Write the set of values ofxsatisfying |x 1| 3 and |x 1| 1. Solution: We have: $|x-1| \leq 3$ and $|x-1| \geq 1$ We know, $|x-a| \leq r \Rightarrow a-r \leq x \leq a+r$ $A$ nd, $|x-a| \geq r \Rightarrow x \leq a-r$ or $x \geq a+r$ $\therefore 1-3 \leq x \leq 1+3$ and $x \leq 1-1$ or $x \geq 1+1$ $\Rightarrow-2 \leq x \leq 4$ and $\mathrm{x} \leq 0$ or $x \geq 2$ $\Rightarrow x \in[-2,4]$ and $x \in(-\infty, 0] \cup[2, \infty)$ $\Rightarrow x \in[-2,0] \mathrm{U}[2,4]$...
Read More →The sum of ages of a man and his son is 45 years.
Question: The sum of ages of a man and his son is 45 years. Five years ago, the product of their ages was four times the man's age at the time. Find their present ages. Solution: Let the present age of the man be $x$ years Then present age of his son is $=(45-x)$ years Five years ago, man's age $=(x-5)$ years And his son's age $(45-x-5)=(40-x)$ years Then according to question, $(x-5)(40-x)=4(x-5)$ $40 x-x^{2}+5 x-200=4 x-20$ $-x^{2}+45 x-200=4 x-20$ $-x^{2}+45 x-200-4 x+20=0$ $-x^{2}+41 x-180=0...
Read More →Factorise
Question: Factorise $x^{6}-7 x^{3}-8$ Solution: Let $x^{3}=y$ So, the equation becomes $y^{2}-7 y-8=y^{2}-8 y+y-8$ $=y(y-8)+(y-8)$ $=(y-8)(y+1)$ $=\left(x^{3}-8\right)\left(x^{3}+1\right)$ $=(x-2)\left(x^{2}+4+2 x\right)(x+1)\left(x^{2}+1-x\right)$...
Read More →Write the solution set of the equation
Question: Write the solution set of the equation |2 x| =x 2. Solution: We have, $|2-x|=x-2$ Now 2 cases arise. CASE 1 : When $2-x \geq 0$, then $|2-x|=2-x$ $\Rightarrow|2-x|=x-2$ $\Rightarrow 2-x=x-2$ $\Rightarrow 2 x=4$ $\Rightarrow x=2$ So, this condition is satisfied when $x=2$. CASE 2 : When $2-x0$ (i. e. when $x2$ ), then $|2-x|=-(2-x)$ $\Rightarrow|2-x|=x-2$ $\Rightarrow-(2-x)=x-2$ $\Rightarrow-2+x=x-2$ $\Rightarrow-2=-2$ So, this condition is satisfied when $\mathrm{x}2$ Hence, from the g...
Read More →Write the set of values of x satisfying the inequation
Question: Write the set of values ofxsatisfying the inequation (x2 2x+ 1) (x 4) 0. Solution: We have, $\left(x^{2}-2 x+1\right)(x-4)0$ $\Rightarrow(x-1)^{2}(x-4)0$ Equating each one to zero, we obtain $\mathrm{x}=1$ and $\mathrm{x}=4$. Therefore, 1 and 4 are critical points. Drawing the number lines, we get: Therefore, the solution set of the given inequality is $x \in(-\infty, 1) \cup(1,4)$...
Read More →Factorise
Question: Factorise $a^{12}-b^{12}$ Solution: $a^{12}-b^{12}$ $=\left(a^{6}+b^{6}\right)\left(a^{6}-b^{6}\right)$ $=\left[\left(a^{2}\right)^{3}+\left(b^{2}\right)^{3}\right]\left[\left(a^{3}\right)^{2}-\left(b^{3}\right)^{2}\right]$ $=\left[\left(a^{2}+b^{2}\right)\left(a^{4}+b^{4}-a^{2} b^{2}\right)\right]\left[\left(a^{3}-b^{3}\right)\left(a^{3}+b^{3}\right)\right]$ $=\left[\left(a^{2}+b^{2}\right)\left(a^{4}+b^{4}-a^{2} b^{2}\right)\right]\left[(a-b)\left(a^{2}+b^{2}+a b\right)(a+b)\left(a^{...
Read More →Ashu is x years old while his mother Mrs Veena is x2 years old
Question: Ashu is $x$ years old while his mother Mrs Veena is $x^{2}$ years old. Five years hence Mrs Veena will be three times old as Ashu. Find their present ages. Solution: Given that Ashu's present age is $=x$ years and his mother Mrs. Veena is $=x^{2}$ years Then according to question, Five years later, Ashu's is $=(x+5)$ years And his mother Mrs. Veena is $=\left(x^{2}+5\right)$ years Thus $x^{2}+5=3(x+5)$ $x^{2}+5=3 x+15$ $x^{2}+5-3 x-15=0$ $x^{2}-3 x+10=0$ $x^{2}-5 x+2 x+10=0$ $x(x-5)+2(...
Read More →Write the solution set of the inequation
Question: Write the solution set of the inequation $x+\frac{1}{x} \geq 2$. Solution: We have, $x+\frac{1}{x} \geq 2$ $\Rightarrow \frac{x^{2}+1}{x} \geq 2$ $\Rightarrow \frac{x^{2}+1}{x}-2 \geq 0$ $\Rightarrow \frac{x^{2}-2 x+1}{x} \geq 0$ $\Rightarrow \frac{(x-1)^{2}}{x} \geq 0$ $\Rightarrow$ Either $(x-1)^{2} \geq 0$ and $x0$ or $(x-1)^{2}0$ and $x0$ But, $(x-1)^{2}$ is always greater than zero. $\therefore(x-1)^{2} \geq 0$ and $x0$ $\Rightarrow x0$ $\Rightarrow x \in(0, \infty)$...
Read More →Write the solution of the inequation
Question: Write the solution of the inequation $\frac{x^{2}}{x-2}0$. Solution: We have, $\frac{x^{2}}{x-2}0$ Equating both the numerator and the denominator with zero, we obtain $x=0$ and $x=2$ as critical points. Plotting these points on the real line, we see that the real line is divided into three regions. Therefore, the solution set of the given inequality is $x \in(2, \infty)$....
Read More →Factorise
Question: Factorise $a^{6}+b^{6}$ Solution: $a^{6}+b^{6}=\left(a^{2}\right)^{3}+\left(b^{2}\right)^{3}$ $=\left(a^{2}+b^{2}\right)\left[\left(a^{2}\right)^{2}-a^{2} b^{2}+\left(b^{2}\right)^{2}\right]$ $=\left(a^{2}+b^{2}\right)\left(a^{4}-a^{2} b^{2}+b^{4}\right)$...
Read More →An aeroplane left 50 minutes later than its scheduled time,
Question: An aeroplane left 50 minutes later than its scheduled time, and in order to reach the destination, 1250 km away, in time, it had to increase its speed by 250 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 $=(x+250) \mathrm{km} / \mathrm{hr}$ Time taken by the aero plane under usual speed to cover $1250 \mathrm{~km}=\frac{1250}{x} \mathrm{hr}$ Time taken by the aero ...
Read More →Factorize:
Question: Factorize: $2 a^{3}+16 b^{3}-5 a-10 b$ Solution: $2 a^{3}+16 b^{3}-5 a-10 b=2\left[a^{3}+8 b^{3}\right]-5(a+2 b)$ $=2\left[a^{3}+(2 b)^{3}\right]-5(a+2 b)$ $=2(a+2 b)\left[a^{2}-a \times 2 b+(2 b)^{2}\right]-5(a+2 b)$ $=2(a+2 b)\left(a^{2}-2 a b+4 b^{2}\right)-5(a+2 b)$ $=(a+2 b)\left[2\left(a^{2}-2 a b+4 b^{2}\right)-5\right]$...
Read More →The solution set of the inequation
Question: The solution set of the inequation $\frac{|x|+1}{|x|-1}0$ is _______________ Solution: for $\quad \frac{|x|+1}{|x|-1}0$ Since $|x|+10 \quad$ (always) $\Rightarrow \quad|x|-10 \quad\left(\because \frac{|x|+1}{|x|-1}0\right)$ i. e. $|x|1$ i.e. $\quad-1x1$ i.e. $x \in(-1,1)$ i. e. solution set of $\frac{|x|+1}{|x|-1}0$ is $(-1,1)$...
Read More →Solve the following
Question: If $\frac{x-3}{|x-3|} \geq 0$, then $x$ belongs to the interval ________________ Solution: If $\frac{x-3}{|x-3|} \geq 0$ $\Rightarrow \quad|x-3| \geq 0$ i. e. $\quad x-3 \geq 0$ i.e. $\quad x \geq 3$ i.e. $\quad x \in[3, \infty)$...
Read More →Factorize:
Question: Factorize: $a^{3}-\frac{1}{a^{3}}-2 a+\frac{2}{a}$ Solution: $a^{3}-\frac{1}{a^{3}}-2 a+\frac{2}{a}=\left(a^{3}-\frac{1}{a^{3}}\right)-2\left(a-\frac{1}{a}\right)$ $=\left[(a)^{3}-\left(\frac{1}{a}\right)^{3}\right]-2\left(a-\frac{1}{a}\right)$ $=\left(a-\frac{1}{a}\right)\left[a^{2}+a \times \frac{1}{a}+\left(\frac{1}{a}\right)^{2}\right]-2\left(a-\frac{1}{a}\right)$ $=\left(a-\frac{1}{a}\right)\left(a^{2}+1+\frac{1}{a^{2}}\right)-2\left(a-\frac{1}{a}\right)$ $=\left(a-\frac{1}{a}\rig...
Read More →An express train takes 1 hour less than a passenger train to travel 132 km
Question: An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore (without taking into consideration the time they stop at intermediate stations). If the average speed of the express train is 11 km/hr more than that of the passenger train, find the average speed of two trains. Solution: Let the speed of the passenger train be $x \mathrm{~km} / \mathrm{hr}$. Then, Speed of the express train $=(x+11) \mathrm{km} / \mathrm{hr}$ Time taken by the passe...
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