A company produces three products every day.
Question: A company produces three products every day. Their production on a certain day is 45 tons. It is found that the production of third product exceeds the production of first product by 8 tons while the total production of first and third product is twice the production of second product. Determine the production level of each product using matrix method. Solution: Let $x, y$ and $z$ be the production level of the first, second and third product, respectively. According to the question, $...
Read More →A bag contains 15 white and some black balls.
Question: A bag contains 15 white and some black balls. If the probability of drawing a black ball from the bag is thrice that of drawing a white ball find the number of black balls in the bag. Solution: The number of white balls in a bag = 15.Let the number of black balls in that bag bex.Then, the total number of balls in bag = 15+x. Now, $\mathrm{P}($ black ball $)=\frac{x}{15+x}$ and $\mathrm{P}($ white ball $)=\frac{15}{15+x}$. According to question, P(black ball) = 3 P(white ball) $\Rightar...
Read More →A rectangular piece is 20 m long and 15 m wide.
Question: A rectangular piece is 20 m long and 15 m wide. From its four corners, quadrants of radii 3.5 m have been cut. Find the area of the remaining part. Solution: It is given that the length of the rectangular piece is $20 \mathrm{~m}$ and its width is $15 \mathrm{~m}$. And, from each corner a quadrant each of radius $3.5 \mathrm{~m}$ has been cut out. A rough figure for this is given below : $\therefore$ Area of the remaining part $=$ Area of the rectangular piece $-(4 \times$ Area of a qu...
Read More →A rectangular piece is 20 m long and 15 m wide.
Question: A rectangular piece is 20 m long and 15 m wide. From its four corners, quadrants of radii 3.5 m have been cut. Find the area of the remaining part. Solution: It is given that the length of the rectangular piece is $20 \mathrm{~m}$ and its width is $15 \mathrm{~m}$. And, from each corner a quadrant each of radius $3.5 \mathrm{~m}$ has been cut out. A rough figure for this is given below : $\therefore$ Area of the remaining part $=$ Area of the rectangular piece $-(4 \times$ Area of a qu...
Read More →Susan invested certain amount of money
Question: Susan invested certain amount of money in two schemes A and B, which offer interest at the rate of 8% per annum and 9% per annum, respectively. She received ₹ 1860 as annual interest. However, had she interchanged the amount of investments in the two schemes, she would have received ₹ 20 more as annual interest. How much money did she invest in each scheme? Solution: Let the amount of investments in schemes A and 6 be ₹ x and ₹ y, respectively. Case I Interest at the rate of 8% per ann...
Read More →A playground has the shape of a rectangle,
Question: A playground has the shape of a rectangle, with two semi-circles on its smaller sides as diameters, added to its outside. If the sides of the rectangle are 36 m and 24.5 m, find the area of the playground. (Take= 22/7). Solution: It is given that the playground is in the shape of a rectangle with two semicircles on its smaller sides. Length of the rectangular portion is $36 \mathrm{~m}$ and its width is $24.5 \mathrm{~m}$ as shown in the figure below. Thus, the area of the playground w...
Read More →An amount of Rs 10,000 is put into three investments at the rate of 10,
Question: An amount of Rs 10,000 is put into three investments at the rate of 10, 12 and 15% per annum. The combined income is Rs 1310 and the combined income of first and second investment is Rs 190 short of the income from the third. Find the investment in each using matrix method. Solution: Let $x, y$ and $z$ be the investments at the rates of interest of $10 \%, 12 \%$ and $15 \%$ per annum respectively. Total investment $=$ Rs 10,000 $\Rightarrow x+y+z=10,000$ Income from the first investme...
Read More →A shopkeeper sells a saree at 8% profit
Question: A shopkeeper sells a saree at 8% profit and a sweater at 10% discount, thereby, getting a sum ₹ 1008. If she had sold the saree at 10% profit and the sweater at 8% discount, she would have got ₹ 1028 then find the cost of the saree and the list price (price before discount) Of the sweater. Solution: Let the cost price of the saree and the list price of the sweater be ₹ x and ₹ y, respectively. Case I Sells a saree at $8 \%$ profit + Sells a sweater at $10 \%$ discount $=₹ 1008$ $\Right...
Read More →A plot is in the form of a rectangle ABCD having semi-circle on BC as shown in Fig. 20.23.
Question: A plot is in the form of a rectangleABCDhaving semi-circle onBCas shown in Fig. 20.23. IfAB= 60 m andBC= 28 m, find the area of the plot. Solution: The given figure has a rectangle with a semicircle on one of its sides. Total area of the plot $=$ Area of rectangle $\mathrm{ABCD}+$ Area of semicircle with radius $\left(\mathrm{r}=\frac{28}{2}=14 \mathrm{~m}\right)$ $\therefore$ Area of the rectangular plot with sides $60 \mathrm{~m}$ and $28 \mathrm{~m}=60 \times 28=1680 \mathrm{~m}^{2}...
Read More →A bag contains 18 balls out of which x balls are red.
Question: A bag contains 18 balls out of whichxballs are red.(i) If the ball is drawn at random from the bag, what is the probability that it is not red? (ii) If two more red balls are put in the bag, the probability of drawing a red ball will be $\frac{9}{8}$ times the probability of drawing a red ball in the first case. Find the value of $x$. Solution: Total number of balls = 18.Number of red balls =x.(i) Number of balls which are not red = 18x $\therefore \mathrm{P}($ getting a ball which is ...
Read More →A bag contains 18 balls out of which x balls are red.
Question: A bag contains 18 balls out of whichxballs are red.(i) If the ball is drawn at random from the bag, what is the probability that it is not red? (ii) If two more red balls are put in the bag, the probability of drawing a red ball will be $\frac{9}{8}$ times the probability of drawing a red ball in the first case. Find the value of $x$. Solution: Total number of balls = 18.Number of red balls =x.(i) Number of balls which are not red = 18x $\therefore \mathrm{P}($ getting a ball which is ...
Read More →A flooring tile has the shape of a parallelogram whose base is 24 cm and the corresponding height is 10 cm.
Question: A flooring tile has the shape of a parallelogram whose base is 24 cm and the corresponding height is 10 cm. How many such tiles are required to cover a floor of area 1080 m2? Solution: Given: Base of a flooring tile that is in the shape of a parallelogram $=\mathrm{b}=24 \mathrm{~cm}$ Corresponding height $=\mathrm{h}=10 \mathrm{~cm}$ Now, in a parallelogram : Area $(\mathrm{A})=$ Base $(\mathrm{b}) \times$ Height $(\mathrm{h})$ $\therefore$ Area of a tile $=24 \mathrm{~cm} \times 10 \...
Read More →The probability of selecting a red ball at random from the jar that contains only red,
Question: The probability of selecting a red ball at random from the jar that contains only red, blue and orange balls is $\frac{1}{4}$. The probability of selecting a blue ball at random from the same jar is $\frac{1}{3}$. If the jar contains 10 orange balls, find the total number of balls in the jar. Solution: It is given that, $P($ getting a red ball $)=\frac{1}{4}$ and $P($ getting a blue ball $)=\frac{1}{3}$ Let P(getting an orange ball) bex.Since, there are only 3 types of balls in the jar...
Read More →A railway half ticket cost half the full fare but the reservation
Question: A railway half ticket cost half the full fare but the reservation charges are the same on a half ticket as on a full ticket. One reserved first class ticket from the stations A to B costs ₹ 2530. Also, one reserved first class ticket and one reserved first class half ticket from stations A to B costs ₹ 3810. Find the full first class fare from stations A to B and also the reservation charges for a ticket. Solution: Let the cost of full and half first class fare be $₹ \frac{X}{2}$ and $...
Read More →A jar contains 54 marbles, each of which some are blue, some are green and some are white.
Question: A jar contains 54 marbles, each of which some are blue, some are green and some are white. The probability of selecting a blue marble at random is $\frac{1}{3}$ and the probability of selecting a green marble at random is $\frac{4}{9}$. How many white marbles does the jar contain? Solution: Total number of marbles = 54. It is given that, $P$ (getting a blue marble) $=\frac{1}{3}$ and $P$ (getting a green marble) $=\frac{4}{9}$ Let P(getting a white marble) bex.Since, there are only 3 t...
Read More →The sum of three numbers is 2.
Question: The sum of three numbers is 2. If twice the second number is added to the sum of first and third, the sum is 1. By adding second and third number to five times the first number, we get 6. Find the three numbers by using matrices. Solution: Let the three numbers bex,yandz. According to the question, $x+y+2$ $x+2 y+z=1$ $5 x+y+z=6$ The given system of equations can be written in matrix form as follows: $\left[\begin{array}{lll}1 1 1 \\ 1 2 1 \\ 5 1 1\end{array}\right]\left[\begin{array}{...
Read More →A jar contains 54 marbles, each of which some are blue, some are green and some are white.
Question: A jar contains 54 marbles, each of which some are blue, some are green and some are white. The probability of selecting a blue marble at random is $\frac{1}{3}$ and the probability of selecting a green marble at random is $\frac{4}{9}$. How many white marbles does the jar contain? Solution: Total number of marbles = 54. It is given that, $P$ (getting a blue marble) $=\frac{1}{3}$ and $P$ (getting a green marble) $=\frac{4}{9}$ Let P(getting a white marble) bex.Since, there are only 3 t...
Read More →The sum of three numbers is 2.
Question: The sum of three numbers is 2. If twice the second number is added to the sum of first and third, the sum is 1. By adding second and third number to five times the first number, we get 6. Find the three numbers by using matrices. Solution: Let the three numbers bex,yandz. According to the question, $x+y+2$ $x+2 y+z=1$ $5 x+y+z=6$ The given system of equations can be written in matrix form as follows: $\left[\begin{array}{lll}1 1 1 \\ 1 2 1 \\ 5 1 1\end{array}\right]\left[\begin{array}{...
Read More →A jar contains 24 marbles. Some of these are green and others are blue.
Question: A jar contains 24 marbles. Some of these are green and others are blue. If a marble is drawn at random from the jar, the probability that it is green is $\frac{2}{3}$. Find the number of blue marbles in the jar. Solution: Total number of marbles = 24.Let the number of blue marbles bex.Then, the number of green marbles = 24 x $\therefore \mathrm{P}$ (getting a green marble) $=\frac{\text { Number of favourable outcomes }}{\text { Number of all possible outcomes }}$ $=\frac{24-x}{24}$ Bu...
Read More →A bag contains 5 white balls, 7 red balls, 4 black balls and 2 blue balls. A ball is drawn at random from the bag.
Question: A bag contains 5 white balls, 7 red balls, 4 black balls and 2 blue balls. A ball is drawn at random from the bag. Find the probability that the drawn ball is (i) white or blue, (ii) neither white nor black. Solution: Number of white balls in the bag = 5Number of red balls in the bag = 7Number of black balls in the bag = 4Number of blue balls in the bag = 2Total number of balls in the bag = 5 + 7 + 4 + 2 = 18 Total number of outcomes = 18(i) There are 7 balls (5 white and 2 blue) in th...
Read More →A bag contains 4 white balls, 5 red balls, 2 black balls and 4 green balls.
Question: A bag contains 4 white balls, 5 red balls, 2 black balls and 4 green balls. A ball is drawn at random from the bag. Find the probability that it is(i) black,(ii) not green,(iii) red or white,(iv) neither red nor green. Solution: Total number of balls = 15(i) Number of black balls = 2 $\therefore P($ getting a black ball $)=\frac{\text { Number of favourable outcomes }}{\text { Number of all possible outcomes }}$ $=\frac{2}{15}$ Thus, the probability of getting a black ball is $\frac{2}...
Read More →Verify Euler's formula for each of the following polyhedrons:
Question: Verify Euler's formula for each of the following polyhedrons: Solution: (i) In the given polyhedron: Edges $\mathrm{E}=15$ Faces $\mathrm{F}=7$ Vertices $\mathrm{V}=10$ Now, putting these values in Euler's formula: LHS : F+V $=7+10$ $=17$ LHS : E+2 $=15+2$ $=17$ LHS = RHS Hence, the Euler's formula is satisfied. (ii) In the given polyhedron: Edges $\mathrm{E}=16$ Faces $\mathrm{F}=9$ Vertices $\mathrm{V}=9$ Now, putting these values in Euler's formula: RHS : F+V $=9+9$ $=18$ LHS : E+2 ...
Read More →Solve this
Question: (i) If $A=\left[\begin{array}{rrr}1 -2 0 \\ 2 1 3 \\ 0 -2 1\end{array}\right]$, find $A^{-1} .$ Using $A^{-1}$, solve the system of linear equations $x-2 y=10,2 x+y+3 z=8,-2 y+z=7$ (ii) $A=\left[\begin{array}{rrr}3 -4 2 \\ 2 3 5 \\ 1 0 1\end{array}\right]$, find $A^{-1}$ and hence solve the following system of equations: $3 x-4 y+2 z=-1,2 x+3 y+5 z=7, x+z=2$ (iii) $A=\left[\begin{array}{ccc}1 -2 0 \\ 2 1 3 \\ 0 -2 1\end{array}\right]$ and $B=\left[\begin{array}{ccc}7 2 -6 \\ -2 1 -3 \\...
Read More →17 cards numbered 1, 2, 3, 4, .... ,17 are put in a box and mixed thoroughly.
Question: 17 cards numbered 1, 2, 3, 4, .... ,17 are put in a box and mixed thoroughly. A card is drawn at random from the box. Find the probability that the card drawn bears (i) an odd number (ii) a number divisible by 5. Solution: Total number of cards = 17(i)Let E1be the event of choosing an odd number.These numbers are 1, 3, 5, 7, 9, 11, 13, 15 and 17. $\therefore P($ getting an odd number $)=P\left(E_{1}\right)=\frac{\text { Number of outcomes favourable to } E_{1}}{\text { Number of all po...
Read More →A game of chance consists of spinning an arrow which is equally likely to come to rest pointing to one of the numbers 1, 2, 3,..., 12 as shown in the figure.
Question: A game of chance consists of spinning an arrow which is equally likely to come to rest pointing to one of the numbers 1, 2, 3,..., 12 as shown in the figure. What is the probability that it will point to (i) 6?(ii) an even number(iii) a prime number?(iv) a number which is a multiple of 5?(v) a number which is a factor of 8? Solution: The arrow can come to rest at any one of the numbers 1, 2, 3, 4,..., 12. Total number of outcomes = 12(i) There is only one 6 in the figure. So, there is ...
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