A wooden cylindrical pole is 7 m high and its base radius is 10 cm.
Question: A wooden cylindrical pole is 7 m high and its base radius is 10 cm. Find its weight if the wood weighs 225 kg per cubic metre. Solution: Height $=7 \mathrm{~m}$ Radius $=10 \mathrm{~cm}=0.1 \mathrm{~m}$ Volume $=\pi \mathrm{r}^{2} \mathrm{~h}=\frac{22}{7} \times 0.1 \times 0.1 \times 7=0.22 \mathrm{~m}^{3}$ Weight of wood $=225 \mathrm{~kg} / \mathrm{m}^{3}$ $\therefore$ Weight of the pole $=0.22 \times 225=49.5 \mathrm{~kg}$...
Read More →A wooden cylindrical pole is 7 m high and its base radius is 10 cm.
Question: A wooden cylindrical pole is 7 m high and its base radius is 10 cm. Find its weight if the wood weighs 225 kg per cubic metre. Solution: Height $=7 \mathrm{~m}$ Radius $=10 \mathrm{~cm}=0.1 \mathrm{~m}$ Volume $=\pi \mathrm{r}^{2} \mathrm{~h}=\frac{22}{7} \times 0.1 \times 0.1 \times 7=0.22 \mathrm{~m}^{3}$ Weight of wood $=225 \mathrm{~kg} / \mathrm{m}^{3}$ $\therefore$ Weight of the pole $=0.22 \times 225=49.5 \mathrm{~kg}$...
Read More →A milk tank is in the form of a cylinder whose radius is 1.5 m and height is 10.5 m.
Question: A milk tank is in the form of a cylinder whose radius is 1.5 m and height is 10.5 m. Find the quantity of milk in litres that can be stored in the tank. Solution: $r=1.5 \mathrm{~m}$ $h=10.5 \mathrm{~m}$ Capacity of the tank $=$ volume of the $\operatorname{tank}=\pi r^{2} h=\frac{22}{7} \times 1.5 \times 1.5 \times 10.5=74.25 \mathrm{~m}^{3}$ We know that $1 \mathrm{~m}^{3}=1000 \mathrm{~L}$ $\therefore 74.25 \mathrm{~m}^{3}=74250 \mathrm{~L}$...
Read More →Ritwik draws a ball from a bag that contains
Question: Ritwik draws a ball from a bag that contains white and yellow balls. The probability of choosing a white ball is 2/9. If the total number of balls in the bag is 36, find the number of yellow balls. Solution: From the question is given that, Probability of choosing a white ball is = 2/9 The total number of balls in the bag is = 36 Number of white ball chosen = (2/9) 36 = 8 white balls Then, the number of yellow balls = Total balls in bag number of white balls = 36 8 = 28 yellow balls...
Read More →Find the volume, curved surface area and total surface area of each of the cylinders whose dimensions are:
Question: Find the volume, curved surface area and total surface area of each of the cylinders whose dimensions are: (i) radius of the base = 7 cm and height = 50 cm (ii) radius of the base = 5.6 m and height = 1.25 m (iii) radius of the base = 14 dm and height = 15 m Solution: Volume of a cylinder $=\pi r^{2} h$ Lateral surface $=2 \pi r h$ Total surface area $=2 \pi r(h+r)$ (i) Base radius $=7 \mathrm{~cm}$; height $=50 \mathrm{~cm}$ Now, we have the following: Volume $=\frac{22}{7} \times 7 \...
Read More →Classify the following statements under
Question: Classify the following statements under appropriate headings. (a) Getting the sum of angles of a triangle as 180. (b) India winning a cricket match against Pakistan. (c) Sun setting in the evening. (d) Getting 7 when a die is thrown. (e) Sun rising from the west. (f) Winning a racing competition by you. Solution: (a) It is certain to happen. Because, the sum of angles of a triangle as 180o. (b) It may or may not happen. Because, the result of match is unpredictable. (c) It is certain t...
Read More →A solid cubical block of fine wood costs Rs 256 at Rs 500 per m
Question: A solid cubical block of fine wood costs Rs 256 at Rs 500 per m2. Find its volume and the length of each side. Solution: Cost of wood = Rs $500 / \mathrm{m}^{3}$ Cost of the given block $=$ Rs 256 $\therefore$ Volume of the given block $=\mathrm{a}^{3}=\frac{256}{500}=0.512 \mathrm{~m}^{3}=512000 \mathrm{~cm}^{3}$ Also, length of its edge $=\mathrm{a}=\sqrt[3]{0.512}=0.8 \mathrm{~m}=80 \mathrm{~cm}$...
Read More →Find the modulus of each of the following complex numbers and hence
Question: Find the modulus of each of the following complex numbers and hence express each of them in polar form: $\frac{1-3 \mathrm{i}}{1+2 \mathrm{i}}$ Solution: $\frac{1-3 i}{1+2 i} \times \frac{1-2 i}{1-2 i}$ $=\frac{1+6 i^{2}-5 i}{1-4 i^{2}}$ $=\frac{-5 i-5}{5}$ $=-i-1$ Let $Z=-1-i=r(\cos \theta+i \sin \theta)$ Now , separating real and complex part , we get $-1=\operatorname{rcos} \theta \ldots \ldots \ldots . . \mathrm{eq} .1$ $-1=r \sin \theta \ldots \ldots \ldots \ldots . e q .2$ Squari...
Read More →A dice is rolled once.
Question: A dice is rolled once. What is the probability that the number on top will be (a) Odd (b) Greater than 5 (c) A multiple of 3 (d) Less than 1 (e) A factor of 36 (f) A factor of 6 Solution: A dice is rolled once, the possible outcomes are 1, 2, 3, 4, 5 and 6. Then, The probability that the number on top will be (a) Odd Odd numbers in the dice are 1, 3, 5 Then, Probability = total number of odd numbers/ Total number of outcomes = 3/6 [divide numerator and denominator by 3] = (b) Greater t...
Read More →If the length of each edge of a cube is doubled, how many times does its volume become?
Question: If the length of each edge of a cube is doubled, how many times does its volume become? How many times does its surface area become? Solution: Let $a$ be the length of the edge of a cube. Volume of the cube $=a^{3}$ Total surface area $=6 a^{2}$ If the length is doubled, then the new length becomes $2 a$. Now, new volume $=(2 a)^{3}=8 a^{3}$ Also, new surface area $=6(2 a)^{2}=6 \times 4 a^{2}=24 a^{2}$ $\therefore$ The volume is increased by a factor of 8 , while the surface area incr...
Read More →The dimensions of a metal block are 2.25 m by 1.5 m by 27 cm.
Question: The dimensions of a metal block are 2.25 m by 1.5 m by 27 cm. It is melted and recast into cubes, each of side 45 cm. How many cubes are formed? Solution: $1 \mathrm{~m}=100 \mathrm{~cm}$ Volume of the original block $=225 \times 150 \times 27=911250 \mathrm{~cm}^{3}$ Length of the edge of one cube $=45 \mathrm{~cm}$ Then volume of one cube $=45^{3}=91125 \mathrm{~cm}^{3}$ $\therefore$ Total number of blocks that can be cast $=\frac{\text { volume of the block }}{\text { volume of one ...
Read More →The volume of a cube is 729 cm
Question: The volume of a cube is 729 cm3. Find its surface area. Solution: Let $a$ be the length of the edge of the cube. Then volume $=a^{3}=729 \mathrm{~cm}^{3}$ Also, $a=\sqrt[3]{729}=9 \mathrm{~cm}$ $\therefore$ Surface area $=6 a^{2}=6 \times 9 \times 9=486 \mathrm{~cm}^{2}$...
Read More →The pie chart given below shows the result
Question: The pie chart given below shows the result of a survey carried out to find the modes of travel used by the children to go to school. Study the pie chart and answer the questions that follow. (a) What is the most common mode of transport? (b) What fraction of children travel by car? (c) If 18 children trfvelby car, how many children took part in the survey? (d) By which two modes of transport are equal number of children travelling? Solution: (a) The central angle is maximum for bus, he...
Read More →The surface area of a cube is 1176 cm
Question: The surface area of a cube is 1176 cm2. Find its volume. Solution: Let $a$ be the length of the edge of the cube. Total surface area $=6 a^{2}=1176 \mathrm{~cm}^{2}$ $\Rightarrow a=\sqrt{\frac{1176}{6}}=\sqrt{196}=14 \mathrm{~cm}$ $\therefore$ Volume $=a^{3}=14^{3}=2744 \mathrm{~cm}^{3}$...
Read More →Find the volume, lateral surface area and the total surface area of a cube each of whose edges measures:
Question: Find the volume, lateral surface area and the total surface area of a cube each of whose edges measures: (i) 7 m (ii) 5.6 cm (iii) 8 dm 5 cm Solution: (i) Length of the edge of the cube $=a=7 \mathrm{~m}$ Now, we have the following: Volume $=a^{3}=7^{3}=343 \mathrm{~m}^{3}$ Lateral surface area $=4 a^{2}=4 \times 7 \times 7=196 \mathrm{~m}^{2}$ Total Surface area $=6 a^{2}=6 \times 7 \times 7=294 \mathrm{~m}^{2}$ (ii) Length of the edge of the cube $=\mathrm{a}=5.6 \mathrm{~cm}$ Now, w...
Read More →A closed wooden box 80 cm long, 65 cm wide and 45 cm high, is made of 2.5-cm-thick wood.
Question: A closed wooden box 80 cm long, 65 cm wide and 45 cm high, is made of 2.5-cm-thick wood. Find the capacity of the box and its weight if 100 cm3of wood weighs 8 g. Solution: External length $=80 \mathrm{~cm}$ External width $=65 \mathrm{~cm}$ External height $=45 \mathrm{~cm}$ $\therefore$ External volume of the box $=80 \times 65 \times 45=234000 \mathrm{~cm}^{3}$ Thickness of the wood $=2.5 \mathrm{~cm}$ Then internal length $=80-(2.5 \times 2)=75 \mathrm{~cm}$ Internal width $=65-(2....
Read More →Given below is a pie chart showing the time
Question: Given below is a pie chart showing the time spend by a group of 350 children in different games. Observe it and answer the questions that follow. (a) How many children spend atleast one hour in playing games? (b) How many children spend more than 2 h in playing games? (c) How many children spend 3 or lesser hours in playing games? (d) Which is greater, number of children who spend 2 hours or more per . day or number of children who play for less than one hour? Solution: (a) Number of c...
Read More →Find the modulus of each of the following complex numbers and hence
Question: Find the modulus of each of the following complex numbers and hence express each of them in polar form: $\frac{1+3 i}{1-2 i}$ Solution: $=\frac{1+3 i}{1-2 i} \times \frac{1+2 i}{1+2 i}$ $=\frac{1+6 i^{2}+5 i}{1-4 i^{2}}$ $=\frac{5 i-5}{5}$ $=i-1$ Let $Z=1-i=r(\cos \theta+i \sin \theta)$ Now , separating real and complex part , we get $-1=\operatorname{rcos} \theta \ldots \ldots \ldots . . e q .1$ 1 = rsin eq.2 Squaring and adding eq.1 and eq.2, we get $2=r^{2}$ Since r is always a posi...
Read More →The external dimensions of a closed wooden box are 62 cm,
Question: The external dimensions of a closed wooden box are 62 cm, 30 cm and 18 cm. If the box is made of 2-cm-thick wood, find the capacity of the box. Solution: External length $=62 \mathrm{~cm}$ External width $=30 \mathrm{~cm}$ External height $=18 \mathrm{~cm}$ $\therefore$ External volume of the box $=62 \times 30 \times 18=33480 \mathrm{~cm}^{3}$ Thickness of the wood $=2 \mathrm{~cm}$ Now, internal length $=62-(2 \times 2)=58 \mathrm{~cm}$ Internal width $=30-(2 \times 2)=26 \mathrm{~cm...
Read More →A box with a lid is made of wood which is 3 cm thick.
Question: A box with a lid is made of wood which is 3 cm thick. Its external length, breadth and height are 56 cm, 39 cm and 30 cm respectively. Find the capacity of the box. Also find the volume of wood used to make the box. Solution: External length $=56 \mathrm{~cm}$ External width $=39 \mathrm{~cm}$ External height $=30 \mathrm{~cm}$ External volume of the box $=56 \times 39 \times 30=65520 \mathrm{~cm}^{3}$ Thickness of wood $=3 \mathrm{~cm}$ $\therefore$ Internal length $=56-(3 \times 2)=5...
Read More →Find the volume of iron required to make an open box whose external dimensions are 36 cm × 25 cm × 16.5 cm,
Question: Find the volume of iron required to make an open box whose external dimensions are 36 cm 25 cm 16.5 cm, the box being 1.5 cm thick throughout. If 1 cm3of iron weighs 8.5 grams, find the weight of the empty box in kilograms. Solution: External length $=36 \mathrm{~cm}$ External width $=25 \mathrm{~cm}$ External height $=16.5 \mathrm{~cm}$ External volume of the box $=36 \times 25 \times 16.5=14850 \mathrm{~cm}^{3}$ Thickness of iron $=1.5 \mathrm{~cm}$ $\therefore$ Internal length $=36-...
Read More →Find the volume of wood used to make a closed box of outer dimensions 60 cm × 45 cm × 32 cm,
Question: Find the volume of wood used to make a closed box of outer dimensions 60 cm 45 cm 32 cm, the thickness of wood being 2.5 cm all around. Solution: External length $=60 \mathrm{~cm}$ External width $=45 \mathrm{~cm}$ External height $=32 \mathrm{~cm}$ External volume of the box $=60 \times 45 \times 32=86400 \mathrm{~cm}^{3}$ Thickness of wood $=2.5 \mathrm{~cm}$ $\therefore$ Internal length $=60-(2.5 \times 2)=55 \mathrm{~cm}$ Internal width $=45-(2.5 \times 2)=40 \mathrm{~cm}$ Internal...
Read More →A swimming pool is 260 m long and 140 m wide.
Question: A swimming pool is 260 m long and 140 m wide. If 54600 cubic metres of water is pumped into it, find the height of the water level in it. Solution: Length of the pool $=260 \mathrm{~m}$ Width of the pool $=140 \mathrm{~m}$ Volume of water in the pool $=54600$ cubic metres $\therefore$ Height of water $=\frac{\text { volume }}{\text { length } \times \text { width }}=\frac{54600}{260 \times 140}=1.5$ metres...
Read More →Find the modulus of each of the following complex numbers and hence
Question: Find the modulus of each of the following complex numbers and hence express each of them in polar form: $\frac{1-\mathrm{i}}{1+\mathrm{i}}$ Solution: $=\frac{1-i}{1+i} \times \frac{1-i}{1-i}$ $=\frac{1+i^{2}-2 i}{1-i^{2}}$ $=-\frac{2 i}{2}$ $=-i$ Let $Z=-i=r(\cos \theta+i \sin \theta)$ Now, separating real and complex part, we get 0 = rcos.eq.1 -1 = rsin eq.2 Squaring and adding eq.1 and eq.2, we get r = 1, Hence its modulus is 1. Now, dividing eq.2 by eq.1 , we get, $\frac{r \sin \the...
Read More →The volume of a room is 378 m
Question: The volume of a room is 378 m3and the area of its floor is 84 m2. Find the height of the room. Solution: Volume $=$ height $\times$ area Given: Volume $=378 \mathrm{~m}^{3}$ Area $=84 \mathrm{~m}^{2}$ $\therefore$ Height $=\frac{\text { volume }}{\text { area }}=\frac{378}{84}=4.5 \mathrm{~m}$...
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