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. $\frac{x-3}{x+4}0, x \in R$ Solution: Given: $\frac{x-3}{x+4}0, x \in R$ Signs of $x-3$ $x-3=0 \rightarrow x=3$ (Adding both the sides by 3 ) $x-30 \rightarrow x3$ (Adding both the sides by 3 ) $x-30 \rightarrow x3$ (Adding both the sides by 3 ) Signs of x + 4 $x+4=0 \rightarrow x=-4$ (Subtracting both the sides by 4 ) $x+40 \rightarrow x-4$ (Subtracting both the sides by 4 ) $x+40 \rightarrow x...
Read More →What is the least number that should
Question: What is the least number that should be added to 6200 to make it a perfect square? Solution: First, find the square root of 1385 by long division method. Hence, the least number is 16 , which should be subtracted from 1385 to get a perfect square and the required perfect square number $=1385-16=1369$ $\therefore$ Square root of $1369=\sqrt{37 \times 37}=37$...
Read More →A man started his journey on his car from location a and came back.
Question: A man started his journey on his car from location a and came back. The graph given below shows his position at different times during the whole journey. Study the above graph carefully and answer the questions given below:(i) At what time did he start and end his journey?(ii) What was the total duration of the journey?(iii) Which journey, onward or return, was of longer duration?(iv) For how many hours did he not move?(v) At what time did he have the fastest speed? Solution: (i) He st...
Read More →A courier-person cycles from a town to a neighbouring suburban area to deliver a parcel to a merchant.
Question: A courier-person cycles from a town to a neighbouring suburban area to deliver a parcel to a merchant. His distances from the town at different times are shown by the given graph. Study the above graph carefully and answer the questions given below:(i) What is the scale taken for the time-axis?(ii) How much time did the person take for the travel?(iii) How far is the place of the merchant from the town?(iv) Did the person stop on his way? Explain.(v) During which period did the ride fa...
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. $\frac{x-3}{x+1}0, x \in R$ Solution: Given: $\frac{x-3}{x+1}0, x \in R$ Signs of $x-3$ $x-3=0 \rightarrow x=0$ (Adding both the sides by 3 ) $x-30 \rightarrow x3$ (Adding both the sides by 3 ) $x-30 \rightarrow x3$ (Adding both the sides by 3 ) Signs of x + 1 $x+1=0 \rightarrow x=-1$ (Subtracting both the sides by 1 ) $x+10 \rightarrow x-1$ (Subtracting both the sides by 1 ) $x+10 \rightarrow x...
Read More →A car is travelling from city P to city Q, which are 350 km apart.
Question: A car is travelling from cityPto cityQ, which are 350 km apart. The line graph given below describes the distances of the car from the cityPat different times. Study the above graph and answer the questions given below:(i) What information is given on the two axes?(ii) From where and when did the car begin its journey?(iii) How far dis the car go in the first hour?(iv) How far did the car go during (a) the 2nd hour and (b) the 3rd hour?(v) Was the speed same during first three hours? H...
Read More →Find the square root of the following
Question: Find the square root of the following by long division method. (a) 1369 (b) 5625 Solution: (a) We have, 1369 $\therefore \quad \sqrt{1369}=37$ (b) We have, 5625 $\therefore \quad \sqrt{5625}=75$...
Read More →The following table shows the percentage of students who dropped out of school after completing high school
Question: The following table shows the percentage of students who dropped out of school after completing high school Now, use the graph to answer the following question: (i) In which year both boys and the girls achieve their maximum height? (ii) Who grows faster at puberty (14 years to 16 years of age) Solution: (i) Both boys and girls achieve their maximum height in 18th years. (ii) Boys grows faster than girls in puberty....
Read More →By what smallest number should 3600 be multiplied,
Question: By what smallest number should 3600 be multiplied, so that the quotient is a perfect cube. Also, find the cube root of the quotient. Solution: Prime factors of 3600 = 2x2x2x2x3x3x5x5 Grouping the factors into triplets of equal factors, we get $3600=\underline{2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 5}$ We know that, if a number is to be a perfect cube, then each of its prime factors must occur thrice. We find that 2 occurs once 3 and 5 occurs twice only. Hence, t...
Read More →The following table shows the percentage of students who dropped out of school after completing high school.
Question: The following table shows the percentage of students who dropped out of school after completing high school. Study the above table carefully and draw a line graph to depict it. Solution:...
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. $\frac{(2 x-1)}{3} \geq \frac{(3 x-2)}{4}-\frac{(2-x)}{5}$ where $x \in \mathbf{R}$. Solution: Given: $\frac{(2 x-1)}{3} \geq \frac{(3 x-2)}{4}-\frac{(2-x)}{5}$, where $x \in R$ Multiplying by 60 on both the sides in the above equation. $(60) \frac{(2 x-1)}{3} \geq(60) \frac{(3 x-2)}{4}-(60) \frac{(2-x)}{5}$ $20(2 x-1) \geq 15(3 x-2)-12(2-x)$ $40 x-20 \geq 45 x-30-24+12 x$ $40 x-20 \geq 57 x-54$...
Read More →Ajeeta starts off from home at 7 a.m. with her father on a scooter that goes at a uniform speed of 30 km/hr.
Question: Ajeeta starts off from home at 7 a.m. with her father on a scooter that goes at a uniform speed of 30 km/hr. Her father drops her at her school after half an hour. She stays in the school till 1.30 p.m. and takes an autorickshaw to return home. The autorickshaw has a uniform speed of 10 km/hr. Draw the line graph for the given situation and also determine the distance of Ajeets's school from her home. Solution: distance of Ajeets's school from her home is given by speed $=\frac{\text {...
Read More →By what smallest number should 216 be divided,
Question: By what smallest number should 216 be divided, so that the quotient is a perfect square? Also, find the square root of the quotient. Solution: Prime factors of $216=2 \times 2 \times 2 \times 3 \times 3 \times 3$ Grouping the factors into pairs of equal factors, we get $216=2 \times 2 \times 2 \times 3 \times 3 \times 3$ We find that there is no prime factor to form a pair with 2 and $3 .$ Therefore, we must divide the number by 6 , so that the quotient becomes a perfect square. If we ...
Read More →The table given below depicts the annual gross profit of a company for a period of 5 years.
Question: The table given below depicts the annual gross profit of a company for a period of 5 years. Study the table and draw a line graph for the same. Solution:...
Read More →Write two Pythagorean triplets,
Question: Write two Pythagorean triplets, each having one of the numbers as 5. Solution: As, $5^{2}=3^{2}+4^{2}$ and $13^{2}=12^{2}+5^{2}$ Hence, $3,4,5$ and $12,5,13$ are the two pythagorean triplets....
Read More →Consider the following input/output table. Draw a line graph for it.
Question: Consider the following input/output table. Draw a line graph for it. Now, use the graph drawn to predict the outputs for the inputs of 3 and 8. Solution: From the graph, we can see that outout for the inputs of 3 and 8 are 8 and 23 respectively....
Read More →The following table depicts the maximum temperature on the seven days of a particular week.
Question: The following table depicts the maximum temperature on the seven days of a particular week. Study the table and draw a line graph for the same. Solution:...
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. $\frac{x}{4}\frac{(5 x-2)}{3}-\frac{(7 x-3)}{5}$ where $x \in \mathbf{R}$. Solution: Given: $\frac{x}{4}\frac{(5 x-2)}{3}-\frac{(7 x-3)}{5}$, where $x \in R$ Multiplying 60 on both the sides in the above equation, $\frac{x}{4}(60)\frac{(5 x-2)}{3}(60)-\frac{(7 x-3)}{5}(60)$ $15 x20(5 x-2)-12(7 x-3)$ $15 x100 x-40-84 x+36$ $15 x16 x-4$ Now, subtracting 16x from both sides in above equation $15 x-...
Read More →Is 9720 a perfect cube?
Question: Is 9720 a perfect cube? If not, find the smallest number by which it should be divided to get a perfect cube. Solution: First we have to find out the factors by using prime factorisation method. So, prime factors of 9720 = 2 2 2 3 3 3 3 3 5 Now, grouping the prime factors = (2 2 2) (3 3 3) 3 3 5 = 23 33 3 3 5 Factors 3 and 4 has no pair. 9720 is not a perfect cube. The smallest number it should be divided to get a perfect cube is 3 3 5 = 45. Then, = 9720 45 = 216 Factors of 216 = (2 2 ...
Read More →Is 176 a perfect square?
Question: Is 176 a perfect square? If not, find the smallest number by which it should be multiplied to get a perfect square. Solution: First we have to find out the factors by using prime factorisation method. So, prime factors of 176 = 2 2 2 2 11 Now, grouping the prime factors = (2 2) (2 2) (11) = 22 22 11 Factor 11 has no pair. 176 is not a perfect square. The smallest number it should be multiplied to get a perfect square is 11. Then, 176 11 = 1936 Factors of 1936 = 2 2 2 2 11 11 = 1936 = (...
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. $\frac{2 x-1}{12}-\frac{x-1}{3}\frac{3 x+1}{4}$ where $x \in \mathbf{R}$. Solution: Given: $\frac{2 x-1}{12}-\frac{x-1}{3}\frac{3 x+1}{4}$, where $x \in R$ Multiply by 12 on both sides in the above equation $12\left(\frac{2 x-1}{12}\right)-12\left(\frac{x-1}{3}\right)12\left(\frac{3 x+1}{4}\right)$ $(2 x-1)-4(x-1)3(3 x+1)$ $2 x-1-4 x+49 x+3$ $3-2 x9 x+3$ Now, subtracting 3 on both sides in the a...
Read More →Using prime factorisation,
Question: Using prime factorisation, find the square roots of (a) 11025 (b) 4761 Solution: (a) We have, 11025 Now, $\quad 11025=3 \times 3 \times 5 \times 5 \times 7 \times 7$ $\therefore$ Square root of $11025=\sqrt{11025}=3 \times 5 \times 7=105$ (b) We have, 4761 Now, $4761=3 \times 3 \times 23 \times 23$ $\therefore$ Square root of $4761=\sqrt{4761}=3 \times 23=69$...
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. $\frac{1}{2}\left(\frac{2}{3} x+1\right) \geq \frac{1}{3}(x-2)$ where $x \in \mathbf{R}$. Solution: Given: $\frac{1}{2}\left(\frac{2}{3} x+1\right) \geq \frac{1}{3}(x-2)$, where $x \in R$ $\frac{1}{2}\left(\frac{2 x}{3}\right)+\frac{1}{2}(1) \geq \frac{1}{3}(x)-\frac{1}{3}(2)$ $\frac{x}{3}+\frac{1}{2} \geq \frac{x}{3}-\frac{2}{3}$ Now, subtracting $\frac{1}{2}$ from both the sides in the above e...
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. $\frac{5 x}{4}-\frac{4 x-1}{3}1$, where $x \in \mathbf{R}$ Solution: Given: $\frac{5 x}{4}-\frac{4 x-1}{3}1$, where $x \in R$ $\frac{3(5 x)-4(4 x-1)}{12}1$ $\frac{15 x-16 x+4}{12}1$ $\frac{-x+4}{12}1$ Now, multiplying by 12 on both the sides in the above equation, $\left(\frac{-x+4}{12}\right) \cdot(12)1$. (12) $-x+412$ Now, subtracting 4 from both the sides in above equation $-x+4-412-4$ $-x8$ ...
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. $\frac{5 x-8}{3} \geq \frac{4 x-7}{2}$, where $x \in \mathbf{R}$ Solution: Given: $\frac{5 x-8}{3} \geq \frac{4 x-7}{2}$, where $x \in R$ $(5 x-8) \cdot(2) \geq(4 x-7) \cdot(3)$ $10 x-16 \geq 12 x-21$ Now, adding 16 to both the sides $10 x-16+16 \geq 12 x-21+16$ $10 x \geq 12 x-5$ Now, subtracting 12x from both the sides of the above equation $10 x-12 x \geq 12 x-5-12 x$ $-2 x \geq-5$ Now, multi...
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