Solve each of the following in equations and represent the solution set on
Question: Solve each of the following in equations and represent the solution set on the number line. $\left|\frac{2 x-1}{x-1}\right|2, x \in \mathbf{R} .$ Solution: Given: $\left|\frac{2 x-1}{x-1}\right|2, x \in R$ $-2\left|\frac{2 x-1}{x-1}\right|2$ $\frac{2 x-1}{x-1}-2$ and $\frac{2 x-1}{x-1}2$ When, $\frac{2 x-1}{x-1}-2$ Adding 2 to both sides in the above equation $\frac{2 x-1}{x-1}+2-2+2$ $\frac{2 x-1+2(x-1)}{x-1}0$ $\frac{2 x-1+2 x-2}{x-1}0$ $\frac{4 x-3}{x-1}0$ Signs of 4x 3: $4 x-3=0 \r...
Read More →The shifting of a number from one side
Question: The shifting of a number from one side of an equation to other is called (a) transposition (b) distributivity (c) commutativity (d) associativity Solution: (a) The shifting of a number from one side of an equation to other side is called transposition. e.g. x +a = 0is the equation, x = -a Here, number a shifts from left hand side to right hand side....
Read More →If 8x – 3 = 25 + 17x, then x is
Question: If 8x 3 = 25 + 17x, then x is (a) a fraction (b) an integer (c) a rational number (d) Cannot be solved Solution: (c) Given, 8x-3 = 25+17x $\Rightarrow \quad 8 x-17 x=25+3 \quad$ [transposing $17 x$ to LHS and $-3$ to RHS] $\Rightarrow \quad-9 x=28$ $\therefore$$x=\frac{-28}{9}$[dividing both sides by $-9$ ] Hence, $x$ is a rational number....
Read More →The solution of the equation ax + b = 0 is
Question: The solution of the equation ax + b = 0 is (a) $x=\frac{a}{b}$ (b) $x=-b$ (c) $x=\frac{-b}{a}$ (d) $x=\frac{b}{a}$ Solution: (c) Given equation is $a x+b=0$ $\Rightarrow$ $a x=-b \quad$ [transposing $b$ to RHS] $\Rightarrow$ $\frac{a x}{a}=\frac{-b}{a}$ [dividing both sides by a] $\therefore$ $x=\frac{-b}{a}$ Hence, the solution of the equation $a x+b=0$ is $x=-\frac{b}{a}$....
Read More →Solve each of the following in equations and represent the solution set on
Question: Solve each of the following in equations and represent the solution set on the number line. $\frac{|x-3|-x}{x}2, x \in \mathbf{R} .$ Solution: Given: $\frac{|x-3|-x}{x}2, x \in R$ Intervals of $|x-3|$ $|x-3|=-(x-3)$ or $(x-3)$ When $|x-3|=x-3$ $x-3 \geq 0$ Therefore, $x \geq 3$ When $|x-3|=-(x-3)$ $(x-3)0$ Therefore, $x3$ Intervals: $x \geq 3$ or $x3$ Domain of $\frac{|x-3|-x}{x}2$ : $\frac{|x-3|-x}{x}$ is not defined for $x=0$ Therefore, x 0 or x 0 Now, combining intervals and domain:...
Read More →The following data shows the agricultural production in India during a certain year.
Question: The following data shows the agricultural production in India during a certain year. Draw a pie chart to represent the above data. Solution: Total production $=(57+76+38+19)=190$ Central angle of each foodgrain $=\left(\frac{\text { production of each foodgrain }}{\text { total production }} \times 360\right)^{\circ}$ Calculation of central angles Construction of pie chart Steps of construction: 1. Draw a circle of any convenient radius. 2. Draw a horizontal radius of the circle. 3. Dr...
Read More →The solution of which of the following
Question: The solution of which of the following equations is neither a fraction nor an integer? (a) -3x + 2=5x + 2 (b)4x-18=2 (c)4x + 7 = x + 2 (d)5x-8 = x +4 Solution: For option (c) Given linear equation is $\quad 3 x+2=5 x+2$ $\Rightarrow \quad 3 x-5 x=2-2 \quad$ [transposing $5 x$ to LHS and 2 to RHS] $\Rightarrow \quad-2 x=0$ $\Rightarrow$$\frac{-2 x}{-2}=\frac{0}{-2}$[dividing both sides by $-2$ ] $\therefore$ $x=0$ Hence, $x=0$ is an integer. For option (b) Given lingar equation is $\qua...
Read More →The following table gives the number of different fruits kept in a hamper.
Question: The following table gives the number of different fruits kept in a hamper. Represent the above data by a pie chart. Solution: Total number of fruits $=(26+30+21+5+8)=90$ Central angle of each fruit $=\left(\frac{\text { number of each type of fruit }}{\text { total number of fruits }} \times 360\right) \circ$ Calculation of central angles Construction of pie chart Steps of construction: 1. Draw a circle of any convenient radius. 2. Draw a horizontal radius of the circle. 3. Draw sector...
Read More →13 and 31 is a strange pair of numbers,
Question: 13 and 31 is a strange pair of numbers, such that their squares 169 and 961 are also mirror of each other. Can yqu find two other such pairs? Solution: First pair is 12 and 21 Squares of numbers, 122= 144 and 212= 441 Second pair is 102 and 201 Squares of numbers, 1022= 10404 and 2012= 40401...
Read More →A three-digit perfect square is such that,
Question: A three-digit perfect square is such that, if it is viewed upside down, the number seen is also a perfect square. What is the number? (Hint: The digits 1, 0 and 8 stay the same when viewed upside down, whereas 9 becomes 6 and 6 becomes 9.) Solution: Three-digit perfect squares are 196 and 961, which looks same when viewed upside down....
Read More →The perimeters of two squares are 40 and 96
Question: The perimeters of two squares are 40 and 96 metres respectively. Find the perimeter of another square equal in area to the sum of the first two squares. Solution: From the question it is given that, Perimeters of first squares = 40m Perimeter of second square = 96m Let us assume side of first square = x1 and assume side of second square = x2 WKT, perimeter of first square = 4 side 40 = 4x1 X1= 40/4 X1= 10 m Then, Perimeter of second square = 4 side 96 = 4x2 X2= 96/4 X2= 24 m As per the...
Read More →Solve each of the following in equations and represent the solution set on
Question: Solve each of the following in equations and represent the solution set on the number line. $\frac{1}{|x|-3} \leq \frac{1}{2}, x \in \mathbf{R}$ Solution: Given $\frac{1}{|x|-3} \leq \frac{1}{2}, x \in R$ Intervals of $|x|$ : $|x|=-x, x0$ $|x|=x, x \geq 0$ Domain of $\frac{1}{|x|-3} \leq \frac{1}{2}$ $|x|+3=0$ $x=-3$ or $x=3$ Therefore, $-3x3$ Now, combining intervals with domain: $x-3,-3x0,0 \leq x3, x3$ For $x-3$ $\frac{1}{|x|-3} \leq \frac{1}{2} \rightarrow \frac{1}{-x-3} \leq \frac...
Read More →Put three different numbers in the circles,
Question: Put three different numbers in the circles, so that when you add the numbers at the end of each line, you always get a perfect square. Solution: $\because \quad 6+19=25$ (perfect square) $19+30=49$ (perfect square) and $30+6=36$ (perfect square)...
Read More →The marks obtained by Sudhir in an examination are given below:
Question: The marks obtained by Sudhir in an examination are given below: Represent the above data by a pie chart. Solution: Total marks obtained $=(105+75+150+120+90)=540$ Central angle of each subject $=\left(\frac{\text { marks obtained in each subject }}{\text { total marks obtained }} \times 360\right)^{\circ}$ Calculation of central angles Construction of pie chart Steps of construction: 1. Draw a circle of any convenient radius. 2. Draw a horizontal radius of this circle. 3. Draw sectors ...
Read More →A perfect square number has four digits,
Question: A perfect square number has four digits, none of which is zero. The- digits from left to right have valfles, that are even, even, odd, even. Find the number. Solution: Suppose abed is a perfect square.where, a = even numberb = even numberc = odd numberd = even numberHence, 8836 is one of the number which satisfies the given condition....
Read More →prove that
Question: Evaluate: $\left\{6^{2}+\left(8^{2}\right)^{1 / 2}\right\}^{3}$ Solution: Given, {(62+ (82)1/2)}3 = {(36 + (64)1/2)}3 = {(36 + (8)}3 = {44}3 = 44 44 44 = 85,184...
Read More →The data on religion wise division of 1080 workers of a factory are given below:
Question: The data on religion wise division of 1080 workers of a factory are given below: Draw a pie chart to represent the above data. Solution: Total number of workers $=1080$ Central angle of each religion $=\left(\frac{\text { number of workers in each religion }}{\text { total number of workers }} \times 360\right)^{\circ}$ Calculation of central angles Construction of pie chart Steps of construction: 1. Draw a circle of any convenient radius. 2. Draw a horizontal radius of this circle. 3....
Read More →prove that
Question: {(52+ (122)1/2)}3 Solution: We have, $\left\{5^{2}+\left(12^{2}\right)^{1 / 2}\right\}^{3}=\{25+12\}^{3}$ $=(37)^{3}=37 \times 37 \times 37=50653$...
Read More →The data given below shows number of hours spent by a school boy on different activities on a working day.
Question: The data given below shows number of hours spent by a school boy on different activities on a working day. Represent the above data by a pie chart. Solution: Total number of hours $=24$ Central angle of each component $=\left(\frac{\text { number of hours spent on each activity }}{\text { total number of hours }} \times 360\right)^{\circ}$ Calculation of central angles Construction of pie chart Steps of construction: 1. Draw a circle of any convenient radius. 2. Draw a horizontal radiu...
Read More →Evaluate: 3√27 + 3√0.008 + 3√0.064
Question: Evaluate:327 +30.008 +30.064 Solution: We have, $\sqrt[3]{27}+\sqrt[3]{0.008}+\sqrt[3]{0.064}$ $=(27)^{1 / 3}+(0.008)^{1 / 3}+(0.064)^{1 / 3}$ $=(3 \times 3 \times 3)^{\frac{1}{3}}+(0.2 \times 0.2 \times 0.2)^{\frac{1}{3}}+(0.4 \times 0.4 \times 0.4)^{\frac{1}{3}}$ $=(3)^{3 \times \frac{1}{3}}+(0.2)^{3 \times \frac{1}{3}}+(0.4)^{3 \times \frac{1}{3}}$ $=3+0.2+0.4=3.6$...
Read More →Various modes of transport used by 1260 students in a given school are given below:
Question: Various modes of transport used by 1260 students in a given school are given below: Represent the above data by a pie chart. Solution: Total number of students $=1260$ Central angle of each component $=\left(\frac{\text { number of students using that mode }}{\text { total number of students }} \times 360\right)^{\circ}$ Calculation of central angles Construction of pie chart Steps of construction: 1. Draw a circle of any convenient radius. 2. Draw a horizontal radius of this circle. 3...
Read More →Three numbers are in the ratio 2 : 3 : 4.
Question: Three numbers are in the ratio 2 : 3 : 4. The sum of their cubes is 0.334125. Find the numbers. Solution: Let the numbers be $2 x, 3 x$ and $4 x$, respectively. $\because$ Sum of their cubes $=0.334125$ [given] According to the question, $(2 x)^{3}+(3 x)^{3}+(4 x)^{3}=0.334125$ $\Rightarrow 8 x^{3}+27 x^{3}+64 x^{3}=0.334125$ $\Rightarrow \quad 99 x^{3}=0.334125$ $\Rightarrow \quad x^{3}=\frac{0.334125}{99}$ $\Rightarrow \quad x^{3}=0.003375$ $\Rightarrow \quad x^{3}=\frac{3375}{100000...
Read More →Solve each of the following in equations and represent the solution set on
Question: Solve each of the following in equations and represent the solution set on the number line. $|4 x-5| \leq \frac{1}{3}, x \in R$ Solution: Given: $|4 x-5| \leq \frac{1}{3}, x \in R$ $4 x-5 \leq \frac{1}{3}$ or $4 x-5 \geq-\frac{1}{3}$ $4 x-5 \leq \frac{1}{3}$ Adding 5 to both the sides in the above equation $4 x-5+5 \leq \frac{1}{3}+5$ $4 x \leq \frac{1+15}{3}$ $4 x \leq \frac{16}{3}$ Now, dividing both the sides by 4 in the above equation $\frac{4 x}{4} \leq \frac{16}{3 \cdot(4)}$ $x \...
Read More →There are 900 creatures in a zoo as per list given below:
Question: There are 900 creatures in a zoo as per list given below: Represent the above data by a pie chart. Solution: Total number of creatures $=900$ Central angle of each component $=\left(\frac{\text { number of creatures in each type }}{\text { total number of creatures }} \times 360\right)^{\circ}$ Calculation of central angles Construction of pie chart Steps of construction: 1. Draw a circle of any convenient radius. 2. Draw a horizontal radius of this circle. 3. Draw sectors whose centra...
Read More →Find the square root of 324
Question: Find the square root of 324 by the method of repeated subtraction. Solution: Given number is $324 .$ Now, we subtract successive odd numbers starting from 1 as follows: Here, $\quad 324-1=323, \quad 323-3=320$ $320-5=315, \quad 315-7=308$ $308-9=299, \quad 299-11=288$ $288-13=275,275-15=260$ $260-17=243,243-19=224$ $224-21=203,203-23=180$ $180-25=155,155-27=128$ $128-29=99, \quad 99-31=68$ $68-33=35, \quad 35-35=0$ We observe that the number 324 reduced to zero $(0)$ after subracting 1...
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