A beam of wood is 5 m long and 36 cm thick.
Question: A beam of wood is 5 m long and 36 cm thick. It is made of 1.35 m3of wood. What is the width of the beam? Solution: Volume of the beam $=1.35 \mathrm{~m}^{3}$ Length $=5 \mathrm{~m}$ Thickness $=36 \mathrm{~cm}=0.36 \mathrm{~m}$ Width $==\frac{\text { volume }}{\text { thickness } \times \text { length }}=\frac{1.35}{5 \times 0.36}=0.75 \mathrm{~m}=75 \mathrm{~cm}$...
Read More →The top speeds of thirty different land
Question: The top speeds of thirty different land animals have been organised into a frequency table. Draw a histogram for the given data. Solution:...
Read More →A rectangular water tank is 90 cm wide and 40 cm deep.
Question: A rectangular water tank is 90 cm wide and 40 cm deep. If it can contain 576 litres of water, what is its length? Solution: Capacity of the water tank $=576$ litres $=0.576 \mathrm{~m}^{3}$ Width $=90 \mathrm{~cm}=0.9 \mathrm{~m}$ Depth $=40 \mathrm{~cm}=0.4 \mathrm{~m}$ Length $==\frac{\text { capacity }}{\text { width } \times \text { depth }}=\frac{0.576}{0.9 \times 0.4}=1.600 \mathrm{~m}$...
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 + isin) 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 $1=r^{2}$ Since r is always a positive no., therefore, r = 1, Hence its modulus is 1 Now, dividing eq...
Read More →A pit 5 m long and 3.5 m wide is dug to a certain depth.
Question: A pit 5 m long and 3.5 m wide is dug to a certain depth. If the volume of earth taken out of it is 14 m3, what is the depth of the pit? Solution: Let the depth of the pit be $d \mathrm{~m}$. Volume $=$ Length $\times$ width $\times$ depth $=5 \mathrm{~m} \times 3.5 \mathrm{~m} \times d m$ But, Given volume $=14 \mathrm{~m}^{3}$ $\therefore$ Depth $=d=\frac{\text { volume }}{\text { length } \times \text { width }}=\frac{14}{5 \times 3.5}=0.8 \mathrm{~m}=80 \mathrm{~cm}$...
Read More →A river 2 m deep and 45 m wide is flowing at the rate of 3 km/h.
Question: A river 2 m deep and 45 m wide is flowing at the rate of 3 km/h. Find the quantity of water that runs into the sea per minute. Solution: Area of the cross-section of river $=45 \times 2=90 \mathrm{~m}^{2}$ Rate of flow $=3 \mathrm{~km} / \mathrm{hr}=\frac{3 \times 1000}{60}=50 \frac{\mathrm{m}}{\mathrm{min}}$ Volume of water flowing through the cross-section in one minute $=90 \times 50=4500 \mathrm{~m}^{3}$ per minute...
Read More →The rainfall recorded on a certain day was 5 cm.
Question: The rainfall recorded on a certain day was 5 cm. Find the volume of water that fell on a 2-hectare field. Solution: Rainfall recorded $=5 \mathrm{~cm}=0.05 \mathrm{~m}$ Area of the field $=2$ hectare $=2 \times 10000 \mathrm{~m}^{2}=20000 \mathrm{~m}^{2}$ Total rain over the field = Area of the field $\times$ Height of the field $=0.05 \times 20000=1000 \mathrm{~m}^{3}$...
Read More →The volume of a block of gold is 0.5 m
Question: The volume of a block of gold is 0.5 m3. If it is hammered into a sheet to cover an area of 1 hectare, find the thickness of the sheet. Solution: Volume of the block $=0.5 \mathrm{~m}^{3}$ We know: 1 hectare $=10000 \mathrm{~m}^{2}$ Thickness $=\frac{\text { Volume }}{\text { Area }}=\frac{0.5}{10000}=0.00005 \mathrm{~m}=0.005 \mathrm{~cm}=0.05 \mathrm{~mm}$...
Read More →Find the capacity of a rectangular cistern in litres whose dimensions are 11.2 m × 6 m × 5.8 m.
Question: Find the capacity of a rectangular cistern in litres whose dimensions are 11.2 m 6 m 5.8 m. Find the area of the iron sheet required to make the cistern. Solution: Volume of the cistern $=11.2 \times 6 \times 5.8=389.76 \mathrm{~m}^{3}=389.76 \times 1000=389760$ litres Area of the iron sheet required to make this cistern $=$ Total surface area of the cistern $=2(11.2 \times 6+11.2 \times 5.8+6 \times 5.8)=2(67.2+64.96+34.8)=333.92 \mathrm{~cm}^{2}$...
Read More →A wall 15 m long, 30 cm wide and 4 m high is made of bricks, each measuring 22 cm × 12.5 cm × 7.5 cm.
Question: A wall $15 \mathrm{~m}$ long, $30 \mathrm{~cm}$ wide and $4 \mathrm{~m}$ high is made of bricks, each measuring $22 \mathrm{~cm} \times 12.5 \mathrm{~cm} \times 7.5 \mathrm{~cm}$. If $\frac{1}{12}$ of the total volume of the wall consists of mortar, how many bricks are there in the wall? Solution: Volume of the wall $=1500 \times 30 \times 400=18000000 \mathrm{~cm}^{3}$ Total quantity of mortar $=\frac{1}{12} \times 18000000=1500000 \mathrm{~cm}^{3}$ $\therefore$ Volume of the bricks $...
Read More →How many bricks, each of size 25 cm × 13.5 cm × 6 cm,
Question: How many bricks, each of size 25 cm 13.5 cm 6 cm, will be required to build a wall 8 m long, 5.4 m high and 33 cm thick? Solution: Volume of the brick $=25 \times 13.5 \times 6=2025 \mathrm{~cm}^{3}$ Volume of the wall $=800 \times 540 \times 33=14256000 \mathrm{~cm}^{3}$ Total number of bricks $=\frac{\text { Volume of the wall }}{\text { Volume of each brick }}=\frac{14256000}{2025}=7040$ bricks...
Read More →How many planks of size 2 m × 25 cm × 8 cm can be prepared from a wooden block 5 m long,
Question: How many planks of size 2 m 25 cm 8 cm can be prepared from a wooden block 5 m long, 70 cm broad and 32 cm thick, assuming that there is no wastage? Solution: Total volume of the block $=(500 \times 70 \times 32)=1120000 \mathrm{~cm}^{3}$ Total volume of each plank $=200 \times 25 \times 8=40000 \mathrm{~cm}^{3}=200 \times 25 \times 8=40000 \mathrm{~cm}^{3}$ $\therefore$ Total number of planks that can be made $=\frac{\text { Total volume of the block }}{\text { Volume of each plank }}...
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: $-3 \sqrt{2}+3 \sqrt{2} \mathrm{i}$ Solution: Let $Z=3 \sqrt{2} i-3 \sqrt{2}=r\left(\cos ^{\theta}+i \sin \theta\right)$ Now, separating real and complex part, we get $-3 \sqrt{2}=r \cos \theta$ $3 \sqrt{2}=r \sin \theta$ Squaring and adding eq.1 and eq.2, we get $36=r^{2}$ Since r is always a positive no., therefore, r = 6 Hence its modulus is 6. Now, dividing eq.2 by eq.1, we get, ...
Read More →The size of a matchbox is 4 cm × 2.5 cm × 1.5 cm. What is the volume of a packet containing 144 matchboxes?
Question: The size of a matchbox is 4 cm 2.5 cm 1.5 cm. What is the volume of a packet containing 144 matchboxes? How many such packets can be placed in a carton of size 1.5 m 84 cm 60 cm? Solution: Volume occupied by a single matchbox $=(4 \times 2.5 \times 1.5)=15 \mathrm{~cm}^{3}$ Volume of a packet containing 144 matchboxes $=(15 \times 144)=2160 \mathrm{~cm}^{3}$ Volume of the carton $=(150 \times 84 \times 60)=756000 \mathrm{~cm}^{3}$ Total number of packets is a carton $=\frac{\text { Vol...
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: $-4+4 \sqrt{3} i$ Solution: Let $Z=4 \sqrt{2} i-4=r(\cos \theta+i \sin \theta)$ Now, separating real and complex part, we get $-4=\operatorname{rcos} \theta \ldots \ldots \ldots . . \mathrm{eq} .1$ $4 \sqrt{3}=r \sin \theta$ Squaring and adding eq.1 and eq.2, we get $64=r^{2}$ Since r is always a positive no., therefore, r = 8 Hence its modulus is 8. Now, dividing eq.2 by eq.1, we ge...
Read More →A cardboard box is 1.2 m long, 72 cm wide and 54 cm high.
Question: A cardboard box is 1.2 m long, 72 cm wide and 54 cm high. How many bars of soap can be put into it if each bar measures 6 cm 4.5 cm 4 cm? Solution: Volume of the cardboard box $=(120 \times 72 \times 54)=466560 \mathrm{~cm}^{3}$ Volume of each bar of soap $=(6 \times 4.5 \times 4)=108 \mathrm{~cm}^{3}$ Total number of bars of soap that can be accommodated in that box $=\frac{\text { Volume of the box }}{\text { Volume of each soap }}=\frac{466560}{108}=4320$ bars...
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: 2 2i Solution: Let $Z=2-2 i=r(\cos \theta+i \sin \theta)$ Now , separating real and complex part , we get $2=r \cos \theta \ldots \ldots \ldots .$ eq. 1 $-2=r \sin \theta \ldots \ldots \ldots \ldots e q \cdot 2$ Squaring and adding eq.1 and eq.2, we get $8=r^{2}$ Since r is always a positive no. therefore, $r=2^{\sqrt{2}}$ Hence its modulus is $2 \sqrt{2}$. Now, dividing eq.2 by eq.1...
Read More →How many persons can be accommodated in a hall of length 16 m.
Question: How many persons can be accommodated in a hall of length 16 m. breadth 12.5 m and height 4.5 m, assuming that 3.6 m3of air is required for each person? Solution: Total volume of the hall $=(16 \times 12.5 \times 4.5)=900 m^{3}$ It is given that $3.6 \mathrm{~m}^{3}$ of air is required for each person. The total number of persons that can be accommodated in that hall $=\frac{\text { Total volume }}{\mathrm{Volume} \text { required by each person }}=\frac{900}{3.6}$ $=250$ people...
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: $1-\sqrt{3} \mathrm{i}$ Solution: Let $Z=-\sqrt{3} i+1=r(\cos \theta+i \sin \theta)$ Now, separating real and complex part, we get $1=\operatorname{rcos} \theta \ldots \ldots \ldots .$ eq. 1 $-\sqrt{3}=\operatorname{rsin} \theta \ldots \ldots \ldots \ldots$ eq. 2 Squaring and adding eq.1 and eq.2, we get $4=r^{2}$ Since r is always a positive no., therefore, r = 2, Hence its modulus ...
Read More →The area of a courtyard is 3750 m
Question: The area of a courtyard is 3750 m2. Find the cost of covering it with gravel to a height of 1 cm if the gravel costs Rs 6.40 per cubic metre. Solution: $1 \mathrm{~cm}=0.01 \mathrm{~m}$ Volume of the gravel used $=$ Area $\times H$ eight $=(3750 \times 0.01)=37.5 \mathrm{~m}^{3}$ Cost of the gravel is Rs $6.40$ per cubic meter. $\therefore$ Total cost $=(37.5 \times 6.4)=$ Rs 240...
Read More →A solid rectangular piece of iron measures 1.05 m × 70 cm × 1.5 cm.
Question: A solid rectangular piece of iron measures 1.05 m 70 cm 1.5 cm. Find the weight of this piece in kilograms if 1 cm3of iron weighs 8 grams. Solution: $1 m=100 \mathrm{~cm}$ $\therefore$ Dimensions of the iron piece $=105 \mathrm{~cm} \times 70 \mathrm{~cm} \times 1.5 \mathrm{~cm}$ Total volume of the piece of iron $=(105 \times 70 \times 1.5)=11025 \mathrm{~cm}^{3}$ $1 \mathrm{~cm}^{3}$ measures $8 \mathrm{gms}$. $\therefore$ Weight of the piece $=11025 \times 8=88200 \mathrm{~g}=\frac{...
Read More →The dimensions of a rectangular water tank are 2 m 75 cm by 1 m 80 cm by 1 m 40 cm.
Question: The dimensions of a rectangular water tank are 2 m 75 cm by 1 m 80 cm by 1 m 40 cm. How many litres of water does it hold when filled to the brim? Solution: $1 m=100 \mathrm{~cm}$ Therefore, dimensions of the tank are: $2 m 75 \mathrm{~cm} \times 1 m 80 \mathrm{~cm} \times 1 m 40 \mathrm{~cm}=275 \mathrm{~cm} \times 180 \mathrm{~cm} \times 140 \mathrm{~cm}$ $\therefore$ Volume = Length $\times$ Breadth $\times$ Height $=275 \times 180 \times 140=6930000 \mathrm{~cm}^{3}$ Also, $1000 \m...
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: $-1+\sqrt{3} \mathrm{i}$ Solution: Let $Z=\sqrt{3} i-1=r(\cos \theta+i \sin \theta)$ Now , separating real and complex part , we get $-1=r \cos \theta \ldots \ldots \ldots . . \mathrm{q} \cdot 1$ $\sqrt{3}=r \sin \theta \ldots \ldots \ldots \ldots .$ eq. 2 Squaring and adding eq.1 and eq.2, we get $4=r^{2}$ Since r is always a positive no., therefore r = 2, Hence its modulus is 2 Now...
Read More →Find the volume, lateral surface area and the total surface area of the cuboid whose dimensions are:
Question: Find the volume, lateral surface area and the total surface area of the cuboid whose dimensions are: (i) length = 22 cm, breadth = 12 cm and height = 7.5 cm (ii) length = 15 m, breadth = 6 m and height = 9 dm (iii) length = 24 m, breadth = 25 cm and height = 6 m (iv) length = 48 cm, breadth = 6 dm and height = 1 m Solution: Volume of a cuboid $=($ Length $\times$ Breadth $\times$ Height $)$ cubic units Total surface area $=2(l b+b h+l h)$ sq units Lateral surface area $=[2(l+b) \times ...
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: $\sqrt{3}+\mathrm{i}$ Solution: Let $Z=\sqrt{3}+i=r(\cos \theta+i \sin \theta)$ Now, separating real and complex part, we get $\sqrt{3}=\operatorname{rcos} \theta \ldots \ldots \ldots . . \mathrm{eq} .1$ $1=r \sin \theta \ldots \ldots \ldots \ldots .$ eq. 2 Squaring and adding eq.1 and eq.2, we get $4=r^{2}$ Since r is always a positive no., therefore, r =2 Hence its modulus is 2. No...
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