The order of the differential equation
Question: The order of the differential equation $2 x^{2} \frac{d^{2} y}{d x^{2}}-3 \frac{d y}{d x}+y=0$ is (A)2(B)1(C)0(D)not defined Solution: $2 x^{2} \frac{d^{2} y}{d x^{2}}-3 \frac{d y}{d x}+y=0$ The highest order derivative present in the given differential equation is $\frac{d^{2} y}{d x^{2}}$. Therefore, its order is two. Hence, the correct answer is A....
Read More →There is a staircase as shown in the given figure,
Question: There is a staircase as shown in the given figure, connecting points A and B. Measurements of steps are marked in the figure. Find the straight line distance between A and B. Solution: We are given the following figure with the related information In the above figure complete the triangleABCwith right angled atC So AC= 2 + 2 + 2 + 2 = 8 and BC= 1 + 1.6 + 1.6 + 1.8 = 6 Using Pythagoras theorem for triangleABCto find $A B^{2}=A C^{2}+B C^{2}$ $=8^{2}+6^{2}$ $=100$ $\Rightarrow A B=10$ He...
Read More →The degree of the differential equation
Question: The degree of the differential equation $\left(\frac{d^{2} y}{d x^{2}}\right)^{3}+\left(\frac{d y}{d x}\right)^{2}+\sin \left(\frac{d y}{d x}\right)+1=0$ is (A) 3 (B) 2 (C) 1 (D) not defined Solution: $\left(\frac{d^{2} y}{d x^{2}}\right)^{3}+\left(\frac{d y}{d x}\right)^{2}+\sin \left(\frac{d y}{d x}\right)+1=0$ The given differential equation is not a polynomial equation in its derivatives. Therefore, its degree is not defined. Hence, the correct answer is D....
Read More →Determine order and degree(if defined) of differential equation
Question: Determine order and degree(if defined) of differential equation $y^{\prime \prime}+2 y^{\prime}+\sin y=0$ Solution: $y^{\prime \prime}+2 y^{\prime}+\sin y=0$ The highest order derivative present in the differential equation is $y^{\prime \prime}$. Therefore, its order is two. This is a polynomial equation in $y^{\prime \prime}$ and $y^{\prime}$ and the highest power raised to $y^{\prime \prime}$ is one. Hence, its degree is one....
Read More →Three metal cubes with edges 6cm, 8cm, 10cm respectively are melted together and formed into a single cube.
Question: Three metal cubes with edges 6cm, 8cm, 10cm respectively are melted together and formed into a single cube. Find the volume, surface area and diagonal of the new cube. Solution: Let a be the length of each edge of the new cube. Then $a^{3}=\left(6^{3}+8^{3}+10^{3}\right) \mathrm{cm}^{3}$ $\Rightarrow a^{3}=1728$ ⇒ a = 12 Therefore, Volume of the new cube $=a^{3}=1728 \mathrm{~cm}^{3}$ Surface area of the new cube $=6 \mathrm{a}^{2}=6 *(12)^{2}=864 \mathrm{~cm}^{2}$ Diagonal of the newl...
Read More →Water in a canal 30 dm wide and 12 dm deep, is flowing with a velocity of 100 km every hour.
Question: Water in a canal 30 dm wide and 12 dm deep, is flowing with a velocity of 100 km every hour. What much area will it irrigate in 30 minutes if 8 cm of standing water is desired? Solution: Given that, Water in the canal forms a cuboid of Width (b) = 30 dm = 3 m Height (h) = 12 dm = 1.2 m Cuboid length is equal to the distance traveled in 30 min with the speed of 100 km per hour. Therefore, Length of the cuboid = 10030/60 = 60 km = 50000 metres So, volume of water to be used for irrigatio...
Read More →A river 3m deep and 40 m wide is flowing at the rate of 2 km per hour.
Question: A river 3m deep and 40 m wide is flowing at the rate of 2 km per hour. How much water will fall into the sea in a minute? Solution: Radius of the water flow = 2 km per hour =(2000/60)m/min =(100/3)m/min Depth of the river (h) = 3 m Width of the river (b) = 40 m Volume of the water flowing in $1 \mathrm{~min}=100 / 3 * 40 * 3=4000^{3}$ Thus, 1 minute $4000 \mathrm{~m}^{3}=4000000$ litres of water will fall in the sea....
Read More →The breadth of a room is twice its height, one half of its length and the volume of the room is 512 cu.
Question: The breadth of a room is twice its height, one half of its length and the volume of the room is 512 cu. dm. Find its dimensions. Solution: Consider l, b and h are the length, breadth and height of the room. So,b = 2h and b = (1/2)l ⇒ l/2 = 2h ⇒ l = 4h ⇒ l = 4h, b = 2h Now, Volume $=512 \mathrm{dm}^{3}$ ⇒ 4h 2h h = 512 $\Rightarrow h^{3}=64$ ⇒ h = 4 So, Length of the room (l) = 4h = 4 * 4 = 16 dm Breadth of the room (b) = 2h = 2 * 4 = 8 dm And Height of the room (h) = 4 dm....
Read More →If the areas of three adjacent face of a cuboid are 8 cm3,
Question: If the areas of three adjacent face of a cuboid are $8 \mathrm{~cm}^{3}, 18 \mathrm{~cm}^{3}$ and $25 \mathrm{~cm}^{3}$. Find the volume of the cuboid. Solution: WKT, if x, y, z denote the areas of three adjacent faces of a cuboid. = x = l * b, y = b * h, z = l * h Volume (V) is given by V = l * b * h Now, $x y z=\mid b$ * $b h$ * $h \mid=v^{2}$ Here x = 8 y = 18 z = 25 Therefore, $v^{2}=8 * 18 * 25=3600$ $\Rightarrow \mathrm{V}=60 \mathrm{~cm}^{3}$...
Read More →The areas of three adjacent faces of a cuboid are x, y and z. If the volume is V,
Question: The areas of three adjacent faces of a cuboid are $x, y$ and $z$. If the volume is $v$, Prove that $v^{2}=x y z$. Solution: Let a, b and d be the length, breadth, and height of the cuboid. Then, x = ab y = bc z = ca and V = abc [V = l * b * h] $=x y z=a b^{*} b c^{*} c a=(a b c)^{2}$ and V = abc $v^{2}=(a b c)^{2}$ Therefore, $\mathrm{V}^{2}=(\mathrm{xyz})$...
Read More →If V is the volume of a cuboid of dimensions a, b, c and S is its surface area,
Question: If V is the volume of a cuboid of dimensions a, b, c and S is its surface area, then prove that 1/V = 2/S (1/a + 1/b + 1/c) Solution: Given Data: Length of the cube (l) = a Breadth of the cube (b) = b Height of the cube (h) = c Volume of the cube (V) = l * b * h = a * b * c = abc Surface area of the cube (S) = 2 (lb + bh + hl) = 2(ab + bc + ca) Now, $\frac{\mathrm{ab}+\mathrm{bc}+\mathrm{ca}}{\mathrm{abc}} \frac{2}{2(\mathrm{ab}+\mathrm{bc}+\mathrm{ca})}=\frac{2}{\mathrm{~S}}\left(\fra...
Read More →Find the cost of digging a cuboidal pit 8 m long,
Question: Find the cost of digging a cuboidal pit $8 \mathrm{~m}$ long, $6 \mathrm{~m}$ broad and $3 \mathrm{~m}$ deep at the rate of Rs 30 per $\mathrm{m}^{3}$ Solution: Given, Length of the cuboidal pit (l) = 8 m Breadth of the cuboidal pit (b) = 6 m Depth of the cuboidal pit (h) = 3 m Volume of the Cuboidal pit = l * b * h = 8 * 6 * 3 $=144 \mathrm{~m}^{3}$ Cost of digging $1 \mathrm{~m}^{3}=$ Rs. 30 Cost of digging $144 \mathrm{~m}^{3}=144^{*} 30=\mathrm{Rs} .4320$...
Read More →A cuboidal vessel is 10 m long and 8 m wide.
Question: A cuboidal vessel is 10 m long and 8 m wide. How high must it be made to hold 380 cubic meters of a liquid? Solution: Given that Length of the vessel (l) = 10 m Width of the Cuboidal vessel = 8 m Let 'h' be the height of the cuboidal vessel. Volume of the vessel $=380 \mathrm{~m}^{3}$ Therefore, $\mathrm{l} * \mathrm{~b} * \mathrm{~h}=380 \mathrm{~m}^{3}$ ⇒ 10 8 h = 380 $\Rightarrow \mathrm{h}=\frac{380}{10 * 8}$ ⇒ h = 4.75 m Therefore, height of the vessel should be 4.75 m....
Read More →A cuboidal water tank is 6 m long, 5 m wide and 4.5 m deep.
Question: A cuboidal water tank is 6 m long, 5 m wide and 4.5 m deep. How many liters of water can it hold? Solution: Given data: Length (l) = 6 m Breadth (b) = 5 m Height (h) = 4.5 m Volume of the tank = l * b * h = 6 * 5 * 4.5 $=135 \mathrm{~m}^{3}$ It is given that, $1 \mathrm{~m}^{3}=1000$ liters Therefore, $135 \mathrm{~m}^{3}=(135 * 1000)$ liters = 135000 liters The tank can hold 1, 35, 000 liters of water....
Read More →The paint in a certain container is sufficient to paint on an area equal to 9.375 m2,
Question: The paint in a certain container is sufficient to paint on an area equal to $9.375 \mathrm{~m}^{2}$. How many bricks of dimension $22.5 \mathrm{~cm} \times 10 \mathrm{~cm} \times 7.5 \mathrm{~cm}$ can be painted out of this container? Solution: The paint in the container can paint the area, $A=9.375 \mathrm{~m}^{2}$ $=93750 \mathrm{~cm}^{2}$ [Since $1 \mathrm{~m}=100 \mathrm{~cm}$ ] Dimensions of a single brick, Length (l) = 22.5 cm Breadth (b) = 10 cm Height (h) = 7.5 cm We need to fi...
Read More →A wooden bookshelf has external dimensions as follows: Height = 110 cm, Depth = 25 cm,
Question: A wooden bookshelf has external dimensions as follows: Height = 110 cm, Depth = 25 cm, Breadth = 85cm (See figure 18.5). The thickness of the plank is 5cm everywhere. The external faces are to be polished and the inner faces are to be painted. If the rate of polishing is 20 paise per cm2. Find the total expenses required for polishing and painting the surface of the bookshelf. Solution: External length of book shelf = 85 cm Breadth = 25 cm Height = 110 cm External surface area of shelf...
Read More →The length and breadth of a hall are in the ratio 4: 3 and its height is 5.5 meters.
Question: The length and breadth of a hall are in the ratio 4: 3 and its height is 5.5 meters. The cost of decorating its walls (including doors and windows) at Rs 6.60 per square meter is Rs 5082. Find the length and breadth of the room. Solution: Let the length be 4a and breadth be 3a Height = 5.5m [Given] As mentioned in the question, the cost of decorating 4 walls at the rate of Rs. 6.60 per m2is Rs. 5082. Area of four walls * rate = Total cost of Painting 2(l + b) * h * 6.6 = 5082 2(4a + 3a...
Read More →Determine order and degree(if defined) of differential equation
Question: Determine order and degree(if defined) of differential equation $y^{\prime \prime}+\left(y^{\prime}\right)^{2}+2 y=0$ Solution: $y^{\prime \prime}+\left(y^{\prime}\right)^{2}+2 y=0$ The highest order derivative present in the differential equation is $y^{\prime \prime}$. Therefore, its order is two. The given differential equation is a polynomial equation in $y^{\prime \prime}$ and $y^{\prime}$ and the highest power raised to $y^{\prime \prime}$ is one. Hence, its degree is one....
Read More →Determine order and degree(if defined) of differential equation
Question: Determine order and degree(if defined) of differential equation $y^{\prime}+y=e^{x}$ Solution: $y^{\prime}+y=e^{x}$ $\Rightarrow y^{\prime}+y-e^{x}=0$ The highest order derivative present in the differential equation is $y^{\prime}$. Therefore, its order is one. The given differential equation is a polynomial equation in $y^{\prime}$ and the highest power raised to $y^{\prime}$ is one. Hence, its degree is one....
Read More →Determine order and degree(if defined) of differential equation
Question: Determine order and degree(if defined) of differential equation $\left(y^{\prime \prime \prime}\right)^{2}+\left(y^{\prime \prime}\right)^{3}+\left(y^{\prime}\right)^{4}+y^{5}=0$ Solution: $\left(y^{\prime \prime \prime}\right)^{2}+\left(y^{\prime \prime}\right)^{3}+\left(y^{\prime}\right)+y^{5}=0$ The highest order derivative present in the differential equation is $y^{\prime \prime \prime}$. Therefore, its order is three. The given differential equation is a polynomial equation in $y...
Read More →In each of the figures given below, an altitude is drawn to the hypotenuse by a right-angled triangle.
Question: In each of the figures given below, an altitude is drawn to the hypotenuse by a right-angled triangle. The length of different line-segment are marked in each figure. Determinex,y,zin each case. Solution: (i) $\triangle A B C$ is right angled triangle right angled at $B$ $A B^{2}+B C^{2}=A C^{2}$ $x^{2}+z^{2}=(4+5)^{2}$ $x^{2}+z^{2}=9^{2}$ $x^{2}+z^{2}=81$......$(i)$ $\triangle B D A$ is right triangle right angled at $\mathrm{D}$ $B D^{2}+A D^{2}=A B^{2}$ $y^{2}+4^{2}=x^{2}$ $y^{2}+16...
Read More →Determine order and degree(if defined) of differential equation
Question: Determine order and degree(if defined) of differential equation $\frac{d^{2} y}{d x^{2}}=\cos 3 x+\sin 3 x$ Solution: $\frac{d^{2} y}{d x^{2}}=\cos 3 x+\sin 3 x$ $\Rightarrow \frac{d^{2} y}{d x^{2}}-\cos 3 x-\sin 3 x=0$ The highest order derivative present in the differential equation is $\frac{d^{2} y}{d x^{2}}$. Therefore, its order is two. It is a polynomial equation in $\frac{d^{2} y}{d x^{2}}$ and the power raised to $\frac{d^{2} y}{d x^{2}}$ is 1 . Hence, its degree is one....
Read More →Determine order and degree(if defined) of differential equation
Question: Determine order and degree(if defined) of differential equation $\left(\frac{d^{2} y}{d x^{2}}\right)^{2}+\cos \left(\frac{d y}{d x}\right)=0$ Solution: $\left(\frac{d^{2} y}{d x^{2}}\right)^{2}+\cos \left(\frac{d y}{d x}\right)=0$ The highest order derivative present in the given differential equation is $\frac{d^{2} y}{d x^{2}}$. Therefore, its order is 2 . The given differential equation is not a polynomial equation in its derivatives. Hence, its degree is not defined....
Read More →Determine order and degree(if defined) of differential equation
Question: Determine order and degree(if defined) of differential equation $\left(\frac{d s}{d t}\right)^{4}+3 s \frac{d^{2} s}{d t^{2}}=0$ Solution: $\left(\frac{d s}{d t}\right)^{4}+3 \frac{d^{2} s}{d t^{2}}=0$ The highest order derivative present in the given differential equation is $\frac{d^{2} s}{d t^{2}}$. Therefore, its order is two. It is a polynomial equation in $\frac{d^{2} s}{d t^{2}}$ and $\frac{d s}{d t}$. The power raised to $\frac{d^{2} s}{d t^{2}}$ is 1 . Hence, its degree is one...
Read More →Determine order and degree(if defined) of differential equation
Question: Determine order and degree(if defined) of differential equation $y^{\prime}+5 y=0$ Solution: The given differential equation is: $y^{\prime}+5 y=0$ The highest order derivative present in the differential equation is $y^{\prime}$. Therefore, its order is one. It is a polynomial equation in $y^{\prime}$. The highest power raised to $y^{\prime}$ is 1 . Hence, its degree is one....
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