Solve the following systems of linear in equations:
Question: Solve the following systems of linear in equations: $-2\frac{6-5 x}{4}7$ Solution: $-2\frac{6-5 x}{4}$ and $\frac{6-5 x}{4}7$ $\frac{6-5 x}{4}-2$ Multiplying both the sides by 4 in the above equation $\left(\frac{6-5 x}{4}\right)(4)-2(4)$ $6-5 x-8$ Now subtracting 6 from both the sides $6-5 x-6-8-6$ $-5 x-14$ Multiplying both the sides by -1 in above equation $-5 x(-1)-14(-1)$ $5 x14$ Now, dividing both the sides by 5 in above equation $\frac{5 x}{5}\frac{14}{5}$ $x\frac{14}{5}$ Now wh...
Read More →A box contains 19 balls bearing numbers 1, 2, 3, ..., 19 respectively.
Question: A box contains 19 balls bearing numbers 1, 2, 3, ..., 19 respectively. A ball is drawn at random from the box. Find the probability that the number on the ball is (i) a prime number (ii) an even number (iii) a number divisible by 3. Solution: Total number of possible outcomes $=19$ (i) The prime numbers between 1 and 19 are 2, 3, 5, 7, 11, 13, 17 and 19 . Total number of primes $=8$ $\therefore \mathrm{P}_{(\text {prime number })}=\frac{8}{19}$ (ii) The even numbers between 1 and 19 ar...
Read More →In a survey of 200 ladies, it was found that 82 like coffee while 118 dislike it.
Question: In a survey of 200 ladies, it was found that 82 like coffee while 118 dislike it. From these ladies, one is chosen at random. What is the probability that the chosen lady dislikes coffee? Solution: Total number of ladies surveyed $=200$ Number of ladies who dislike coffee $=118$ If chosen randomly, $P_{(a \text { lady that dislikes coffee })}=\frac{118}{200}=\frac{59}{100}$...
Read More →A die is thrown at random. Find the probability of getting
Question: A die is thrown at random. Find the probability of getting (i) 2 (ii) a number less than 3 (iii) a composite number (iv) a number not less than 4. Solution: The possible outcomes when a dice is thrown at random are $1,2,3,4,5$ and 6 . Total number of outcomes $=6$ (i) $\therefore \mathrm{P}_{\text {(getting 2) }}=\frac{1}{6}$ (ii) The numbers less than 3 are 1 and $2 .$ Number of possible outcomes $=2$ $\therefore \mathbf{P}_{\text {(getting a number }3 \text { ) }}=\frac{2}{6}=\frac{1...
Read More →It is known that a box of 100 electric bulbs contains 8 defective bulbs.
Question: It is known that a box of 100 electric bulbs contains 8 defective bulbs. One bulb is taken out at random from the box. What is the probability that the bulb drawn is (i) defective? (ii) non-defective? Solution: Total number of bulbs in the box $=100$ (i) Number of defective bulbs $=8$ $\therefore \mathrm{P}_{(\text {defective bulb })}=\frac{8}{100}=\frac{2}{25}$ (ii) Number of functioning bulbs $=100-8=92$ $\therefore \mathrm{P}_{(\text {non-defective } \text { bulb })}=\frac{92}{100}=...
Read More →In a lottery, there are 10 prizes and 20 blanks.
Question: In a lottery, there are 10 prizes and 20 blanks. A ticket is chosen at random. What is the probability of getting a prize? Solution: Total number of tickets $=10+20=30$ Number of prize tickets $=10$ $\therefore \mathrm{P}_{\text {(getting a prize) }}=\frac{10}{30}=\frac{1}{3}$...
Read More →A bag contains 5 white, 6 red and 4 green balls. One ball is drawn at random.
Question: A bag contains 5 white, 6 red and 4 green balls. One ball is drawn at random. What is the probability that the ball drawn is (i) green? (ii) white? (iii) non-red? Solution: Total number of balls $=5+6+4=15$ (i) Number of green balls $=4$ $\therefore \mathrm{P}_{(\text {green ball })}=\frac{4}{15}$ (ii) Number of white balls $=5$ $\therefore \mathrm{P}_{(\text {white }}$ ball $)=\frac{5}{15}=\frac{1}{3}$ (iii) Number of balls that are not red (i.e., 5 white and 4 green) $=9$ $\therefore...
Read More →A bag contains 4 white and 5 blue balls. They are mixed thoroughly and one ball is drawn at random.
Question: A bag contains 4 white and 5 blue balls. They are mixed thoroughly and one ball is drawn at random. What is the probability of getting(i) a white ball?(ii) a blue ball? Solution: Total number of balls $=4+5=9$ (i) Number of white balls $=4$ $\therefore \mathrm{P}_{(\text {white ball })}=\frac{4}{9}$ Number of blue balls $=5$ $\therefore \mathrm{P}_{\text {(blue ball) }}=\frac{5}{9}$...
Read More →In a single throw of two coins, find the probability of getting
Question: In a single throw of two coins, find the probability of getting (i) both tails, (ii) at least 1 tail, (iii) at the most 1 tail. Solution: The outcomes when two coins are tossed areHH, HT, THandTT. i.e., total no. of possible outcomes = 4 (i) Getting both tails meansTT. Number of outcomes with two tails = 1 $\therefore \mathrm{P}_{\text {(both tails) }}=\frac{1}{4}$ (ii) Getting at least 1 tail means $H T, T H$ and $T T$. With at least one tail, total number of outcomes $=3$ $\therefore...
Read More →In a single throw of a coin, what is the probability of getting a tail?
Question: In a single throw of a coin, what is the probability of getting a tail? Solution: The possible outcomes in a coin toss are $H$ and $T$. Total number of outcomes $=2$ Number of tails = 1 $\therefore \mathrm{P}_{(\text {tail })}=\frac{1}{2}$...
Read More →A coin is tossed. What are all possible outcomes?
Question: (i) A coin is tossed. What are all possible outcomes? (ii) Two coins are tossed simultaneously. What are all possible outcomes? (iii) A die is thrown. What are all possible outcomes? (iv) From a well-shuffled deck of 52 cards, one card is drawn at random. What is the number of all possible outcomes? Solution: (i) The possible outcomes are head (H) and tail(T).(ii) The possible outcomes areHH, HT, THand TT.(iii) The possible outcomes are 1, 2, 3, 4, 5 and 6.(iv) The total number of poss...
Read More →Solve the following systems of linear in equations:
Question: Solve the following systems of linear in equations: $5 x-7(x+3), 1-\frac{3 x}{2} \geq x-4$ Solution: When, $5 x-7x+3$ Adding 7 to both the sides in the above equation $5 x-7+7x+3+7$ $5 xx+10$ Now, subtracting x from both the sides $5 x-xx+10-x$ $4 x10$ Dividing both the sides by 4 in above equation $\frac{4 x}{4}\frac{10}{4}$ $x\frac{5}{2}$ Now when, $1-\frac{3 x}{2} \geq x-4$ Subtracting 1 from both the sides in the above equation $1-\frac{3 x}{2}-1 \geq x-4-1$ $\frac{-3 x}{2} \geq x-...
Read More →Solve the following systems of linear in equations:
Question: Solve the following systems of linear in equations: $-11 \leq 4 x-3 \leq 13$ Solution: $-11 \leq 4 x-3$ and $4 x-3 \leq 13$ When $-11 \leq 4 x-3$ $4 x-3 \geq-11$ Adding 3 to both the sides $4 x-3+3 \geq-11+3$ $4 x \geq-8$ Divide both the sides by 4 in above equation $\frac{4 x}{4} \geq \frac{-8}{4}$ $x \geq-2$ Now when, $4 x-3 \leq 13$ Adding 3 to both the sides in the above equation $4 x-3+3 \leq 13+3$ $4 x \leq 16$ Dividing both the sides by 4 in the above question $\frac{4 x}{4} \le...
Read More →After 18 years, Swarnim will be 4 times as
Question: After 18 years, Swarnim will be 4 times as old as he is now. His present age is. Solution: 6 years Let Swarnim's present age be $x$ yr. After $18 \mathrm{yr}$, Swarnim's age $=(x+18) \mathrm{yr}$ According to the question, $x+18=4 x$ $\Rightarrow \quad x-4 x=-18 \quad$ [transposing $4 x$ to LHS and 18 to RHS] $\Rightarrow \quad-3 x=-18$ $\Rightarrow \quad \frac{-3 x}{-3}=\frac{-18}{3}$ $\Rightarrow \quad \frac{-3 x}{-3}=\frac{-18}{3} \quad$ [dividing both sides by $-3$ ] $\therefore \q...
Read More →prove that
Question: If $\frac{2}{5} x-2=5-\frac{3}{5} x$, then $x=--$ Solution: 7 Given, $\frac{2}{5} x-2=5-\frac{3}{5} x$ $\Rightarrow$ $\frac{2 x}{5}+\frac{3 x}{5}=5+2\left[\right.$ transposing $-\frac{3 x}{5}$ to LHS and $-2$ to RHS $]$ $\Rightarrow$ $\frac{5 x}{5}=7$ $\Rightarrow$ $x=7$ Hence, the value of $x$ is 7 ....
Read More →Solve the following systems of linear in equations:
Question: Solve the following systems of linear in equations: $\frac{4}{x+1} \leq 3 \leq \frac{6}{x+1}, x0$ Solution: $\frac{4}{x+1} \leq 3$ and $3 \leq \frac{6}{x+1}$ When, $\frac{4}{x+1} \leq 3$ Subtracting 3 from both the sides $\frac{4}{x+1}-3 \leq 3-3$ $\frac{4-3(x+1)}{x+1} \leq 0$ $\frac{4-3 x-3}{x+1} \leq 0$ $\frac{1-3 x}{x+1} \leq 0$ Signs of $1-3 x$ : $1-3 x=0 \rightarrow x=\frac{1}{3}$ (Subtract 1 from both the sides and then divide both sides by -3) $1-3 x0 \rightarrow x\frac{1}{3}$ (...
Read More →When 9 is subtracted from the product
Question: When 9 is subtracted from the product of p and 4, the result is 11. The value of p is-. Solution: 5 Given, 9 is subtracted from the product of $p$ and $4 .$ Then, $4 p-9=11$ $\Rightarrow \quad 4 p=11+9 \quad$ [transposing $-9$ to RHS] $\Rightarrow \quad 4 p=20$ $\Rightarrow \quad \quad \frac{4 p}{4}=\frac{20}{4} \quad \quad$ [dividing both sides by 4$]$ $\therefore \quad p=5$ Hence, the value of $p$ is 5 ....
Read More →When a number is divided by 8,
Question: When a number is divided by 8, the result is -3. The number is. Solution: $-24$ Let the number be $x$. According to the question, $\frac{x}{8}=-3$ $\Rightarrow$ $x=8 \times(-3)$ $\Rightarrow \quad x=-24$[by cross-multiplication] Hence, the required number is $-24$....
Read More →(x/5) + 30 = 18 has the solution as .
Question: (x/5) + 30 = 18 has the solution as . Solution: (x/5) + 30 = 18 has the solution as -60. Given, (x/5) + 30 = 18 Transposing 30 to RHS and it becomes -30. (x/5) = 18 30 (x/5) = -12 X = -12 5 X = -60...
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. $x-41, x \neq 4$ Solution: Given: $x-41, x \neq 4$ Adding 4 to both the sides in above equation $x-4+41+4$ $x5$ Therefore $x \in(5, \infty)$...
Read More →On subtracting 8 from x,
Question: On subtracting 8 from x, the result is 2. The value of x is. Solution: 10 Given, $x-8=2$ $\Rightarrow$ $x=8+2$ [transposing $-8$ to RHS] $\therefore \quad x=10$ Hence, the value of $x$ is 10 ....
Read More →A term of an equation can be transposed
Question: A term of an equation can be transposed to the other side by changing its-. Solution: signe.g. x + a = 0 is a linear equation. .= x = -aHence, the term of an equation can be transposed to the other side by changing its sign....
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. $|x+a|+|x|3, x \in R$ Solution: Given: $|x+a|+|x|3, x \in R$ $|x+a|=-(x+a)$ or $(x+a)$ $|x|=-x$ or $x$ When $|x+a|=-(x+a)$ and $|x|=-x$ Then, $|x+a|+|x|3 \rightarrow-(x+a)+(-x)3$ $-x-a-x3$ $-2 x-a3$ Adding a on both the sides in above equation $-2 x-a+a3+a$ $-2 x3+a$ Dividing both the sides by 2 in above equation $\frac{-2 x}{2}\frac{3+a}{2}$ $-x\frac{3+a}{2}$ Multiplying both the sides by -1 in...
Read More →The share of A when Rs 25 are divided between
Question: The share of A when Rs 25 are divided between A and B, so that A gets Rs 8 more than B, is. Solution: $₹ 16.5$ Let, $B$ 's share be $₹ x$. Then, $A$ 's share $=₹(x+8)$ According to the question, $x+x+8=25$ $\Rightarrow \quad 2 x+8=25$ $\Rightarrow \quad 2 x=25-8$ [transposing 8 to RHS] $\Rightarrow \quad 2 x=17$ $\Rightarrow$ $\frac{2 x}{2}=\frac{17}{2}$ [dividing both sides by 2 ] $\Rightarrow$ $x=8.5$ Hence, $A$ 's share $=8.5+8=₹ 16.5$...
Read More →Three consecutive numbers
Question: Three consecutive numbers whose sum is 12 are _________, _________ and _________. Solution: Three consecutive numbers whose sum is 12 are 3, 4 and 5. 3 + 4 + 5 = 12...
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