A shopkeeper allows 23% commision on his advertised price and still makes a profit of 10%.
Question: A shopkeeper allows 23% commision on his advertised price and still makes a profit of 10%. If he gains Rs 56 on one item, find his advertised price. Solution: Let the CP of the item be Rs. $x$. Profit $=10 \%$ $\mathrm{SP}=\mathrm{CP}\left(\frac{100+\text { Profit } \%}{100}\right)$ $\mathrm{SP}=x\left(\frac{110}{100}\right)$ $\mathrm{SP}=$ Rs. $1.1 x$ Again, Profit $=\mathrm{SP}-\mathrm{CP}$ Therefore, Profit $=$ Rs. $(1.1 x-x)$ $=$ Rs. $0.1 x$ We get, $0.1 x=56$ $x=$ Rs. 560 Now, the...
Read More →If the length of the diagonal of a
Question: If the length of the diagonal of a cube is 63 cm, then the length of the edge of the cube is 3 cm. Solution: False Given, the length of the diagonal of a cube = 63 cm Let the edge (side) of a cube be a cm. Then, diagonal of a cube =a3 = 63 = a3 = a = 6 cm Hence, the edge of a cube is 6 cm....
Read More →A cone, a hemisphere and a cylinder
Question: A cone, a hemisphere and a cylinder stand on equal bases and have the same height. The ratio of their volumes is 1 : 2 : 3. Solution: True Let radius of hemisphere is r. Volume of a cone, V1= 1/3 r2h V1= 1/3r2(r) [h = r] = 1/3r3 Volume of a hemisphere, V2=2/3 r3 volume of cylinder, V3= r2h = r2x r = r3[ h=r] V1: V2:V3=1/2 r3: 2/3 r3: r3= 1 : 2 : 3 Hence, the ratio of their volumes is 1 : 2 : 3....
Read More →The radii of the circular ends of a bucket of height 15 cm
Question: The radii of the circular ends of a bucket of height 15 cm are 14 cm andrcm (r 14). If the volume of bucket is 5390 cm3, then find the value ofr. Solution: We have, Height, $h=15 \mathrm{~cm}$, Radius of the upper end, $R=14 \mathrm{~cm}$, Radius of lower end $=r$, As, Volume of the bucket $=5390 \mathrm{~cm}^{3}$ $\Rightarrow \frac{1}{3} \pi h\left(R^{2}+r^{2}+R r\right)=5390$ $\Rightarrow \frac{1}{3} \times \frac{22}{7} \times 15 \times\left(14^{2}+r^{2}+14 r\right)=5390$ $\Rightarro...
Read More →A cylinder and a right circular cone are
Question: A cylinder and a right circular cone are having the same base and same height. The volume of the cylinder is three times the volume of the cone. Solution: Let the radius of the base of a cylinder and a right circular cone be r and height be h. Then, Volume of a cylinder = r2h Volume of a cone = 1/3 r2h Volume of a cylinder = 3 x Volume of a cone Hence, the volume of a cylinder is three times the volume of the right circular cone....
Read More →Find the inverse of each of the following matrices by using elementary row transformations:
Question: Find the inverse of each of the following matrices by using elementary row transformations: Solution: $\left[\begin{array}{lll}1 1 2 \\ 3 1 1 \\ 2 3 1\end{array}\right]$ We know $A=I A$ $\Rightarrow\left[\begin{array}{lll}1 1 2 \\ 3 1 1 \\ 2 3 1\end{array}\right]=\left[\begin{array}{lll}1 0 0 \\ 0 1 0 \\ 0 0 1\end{array}\right] A$ $\Rightarrow\left[\begin{array}{ccc}1 1 2 \\ 0 -2 -5 \\ 0 1 -3\end{array}\right]=\left[\begin{array}{ccc}1 0 0 \\ -3 1 0 \\ -2 0 1\end{array}\right] A$ [Appl...
Read More →The volume of the largest right circular
Question: The volume of the largest right circular cone that can be fitted in a cube whose edge is 2r equals to the volume of a hemisphere radius r. Solution: True Given, edge of cube = 2r, then height of cube becomes h = 2r. Volume of a cone = 1/3 r2h = 1/3 r2(2r)= 2/3 r3 Volume of a hemisphere = 2/3 r3 Hence, the volume of a cone is equal to the volume of a hemisphere....
Read More →A cycle dealer offers a discount of 10% and still makes a profit of 26%.
Question: A cycle dealer offers a discount of 10% and still makes a profit of 26%. What is the actual cost to him of a cycle whose marked price is Rs 840? Solution: Given, MP of the cycle $=$ Rs. 840 Discount $=10 \%$ So, $\mathrm{SP}=\mathrm{MP} \times\left(\frac{100-\text { Discount } \%}{100}\right)$ $=840 \times\left(\frac{100-10}{100}\right)$ $=$ Rs. 756 Now, SP $=$ Rs. 756 and Gain $=26 \%$ So, $\mathrm{CP}=\frac{100}{100+\text { Gain } \%} \times 756$ $=\frac{100}{126} \times 756$ $=$ Rs....
Read More →The volume of a sphere is equal to
Question: The volume of a sphere is equal totwo-third of the volume of a cylinder whose height and diameter are equal to the diameter of the sphere. Solution: True Let the radius of the sphere and cylinder be r. Given, height of the cylinder = diameter of the base = h = 2r According to the given condition, Volume of sphere = (2/3) x Volume of cylinder 4/3 r3= (4/3) x r2x 2r 4/3r3= 4/3r3 Hence, the volume of a sphere is equal to two-third of the volume of a cylinder....
Read More →The lateral surface area of a
Question: The lateral surface area of a cube is 256 m2. The volume of the cube is (a)512 m3 (b)64 m3 (c)216 m3 (d)256 m3 Solution: (a)Given, lateral surface area of a cube = 256 m2 We know that, lateral surface area of a cube = 4 x (Side)2 = 256 =4 x (Side)2 = (Side)2= 256/4 = 64 = Side = 64 = 8 m [taking positive square root because side is always a positive quantity] Now, volume of a cube = (Side)3= (8)3= 8 x 8 x 8 = 512 m3 Hence, the volume of the cube is 512 m3....
Read More →A bucket is in the form of a frustum of a cone
Question: A bucket is in the form of a frustum of a cone and it can hold 28.49 litres of water. If the radii of its circular ends are 28 cm and 21 cm, then find the height of the bucket. Solution: We have, Radius of upper end, $R=28 \mathrm{~cm}$ and Radius of lower end, $r=21 \mathrm{~cm}$ Let the height of the bucket be $h$. Now, Volume of water the bucket can hold $=28.49 \mathrm{~L}$ $\Rightarrow$ Volume of bucket $=28490 \mathrm{~cm}^{3} \quad\left(\mathrm{As}, 1 \mathrm{~L}=1000 \mathrm{~c...
Read More →A shopkeeper marks his goods in such a way that after allowing a discount of 25% on the marked price,
Question: A shopkeeper marks his goods in such a way that after allowing a discount of 25% on the marked price, he still makes a profit of 50%. Find the ratio of the C.P. to the M.P. Solution: Let C.P be Rs $x$ and M.P be Rs $y$. Gain $\%=50$ We know that, S. $P=\left[\frac{(100+\text { Gain } \%)}{100} \times\right.$ C. $\left.P\right]$ $=\left[\frac{150}{100} \times x\right]$ $=\frac{3}{2} x$ Discount $\%=25$ Discount $=25 \%$ of $y$ $=$ Rs $0.25 y$ So, S.P = M.P - Discount $=y-0.25 y$ $=0.75 ...
Read More →In a cylinder, radius is doubled
Question: In a cylinder, radius is doubled and height is halved, then curved surface area will be (a)halved (b)doubled (c)same (d)four times Solution: (e)Let the radius be r and height be h of a cylinder., Curved surface area of cylinder = 2rh We have, radius = 2r, height = h/2 New curved surface area = 2 (2r) x (h/2) = 2 rh Hence, the curved surface area will be same....
Read More →Find the inverse of each of the following matrices by using elementary row transformations:
Question: Find the inverse of each of the following matrices by using elementary row transformations: $\left[\begin{array}{ccc}2 -1 3 \\ 1 2 4 \\ 3 1 1\end{array}\right]$ Solution: $A=\left[\begin{array}{ccc}2 -1 3 \\ 1 2 4 \\ 3 1 1\end{array}\right]$ We know $A=I A$ $\Rightarrow\left[\begin{array}{ccc}2 -1 3 \\ 1 2 4 \\ 3 1 1\end{array}\right]=\left[\begin{array}{lll}1 0 0 \\ 0 1 0 \\ 0 0 1\end{array}\right] A$ $\Rightarrow\left[\begin{array}{ccc}1 -\frac{1}{2} \frac{3}{2} \\ 1 2 4 \\ 3 1 1\end...
Read More →The total surface area of a cube
Question: The total surface area of a cube is 96cm2. The volume of the cube is (a)8 cm3 (b)512 cm3 (c)64 cm3 (d)27 cm3 Solution: (c)Surface area of a cube = 96 cm2 Surface area of a cube = 6 (Side)2= 96 = (Side)2= 16 = (Side) = 4 cm [taking positive square root because side is always a positive quantity] Volume of cube = (Side)3= (4)3= 64cm3 Hence, the volume of the cube is 64 cm3....
Read More →A container in the shape of a frustum of a cone having diameters of its two circular faces as 35 cm and 30 cm and vertical height 14 cm,
Question: A container in the shape of a frustum of a cone having diameters of itstwo circular faces as 35 cm and 30 cm and vertical height 14 cm,iscompletely filled with oil. If each cm3of oil has mass 1.2 g, then find thecost of oil in the container if it costs₹40 per kg. Solution: We have, Height, $h=14 \mathrm{~cm}$, Radius of upper end, $R=\frac{35}{2}=17.5 \mathrm{~cm}$ and Radius of lower end, $r=\frac{30}{2}=15 \mathrm{~cm}$ Now, Volume of the container $=\frac{1}{3} \pi h\left(R^{2}+r^{2...
Read More →A tradesman marks his goods at such a price that after allowing a discount of 15%,
Question: A tradesman marks his goods at such a price that after allowing a discount of 15%, he makes a profit of 20%. What is the marked price of an article whose cost price is Rs 170? Solution: Given, CP of the article $=R s .170$ Profit $=20 \%$ We know that, $\mathrm{SP}=\left[\frac{(100+\text { Gain } \%)}{100} \times \mathrm{CP}\right]$ $=\left[\frac{120}{100} \times 170\right]$ $=\frac{20400}{100}$ $=$ Rs. 204 Let the MP of the article be Rs. $x$. Discount $=15 \%$ There fore, Discount $=...
Read More →A bucket made up of a metal sheet is in the form of frustum of a cone.
Question: A bucket made up of a metal sheet is in the form of frustum of a cone. Its depth is 24 cm and the diameters of the top and bottom are 30 cm and 10 cm, respectively. Find the cost of completely filling the bucket with milk at the rate of Rs 20 per litre and the cost of metal sheet used if it costs Rs 10 per 100 cm2. Solution: Greater diameter of the bucket = 30 cmRadius of the bigger end of the bucket =R= 15 cmDiameter of the smaller end of the bucket = 10 cmRadius of the smaller end of...
Read More →A bucket made up of a metal sheet is in the form of frustum of a cone.
Question: A bucket made up of a metal sheet is in the form of frustum of a cone. Its depth is 24 cm and the diameters of the top and bottom are 30 cm and 10 cm, respectively. Find the cost of completely filling the bucket with milk at the rate of Rs 20 per litre and the cost of metal sheet used if it costs Rs 10 per 100 cm2. Solution: Greater diameter of the bucket = 30 cmRadius of the bigger end of the bucket =R= 15 cmDiameter of the smaller end of the bucket = 10 cmRadius of the smaller end of...
Read More →A shopkeeper allows 20% off on the marked price of goods and still gets a profit of 25%.
Question: A shopkeeper allows 20% off on the marked price of goods and still gets a profit of 25%. What is the actual cost to him of an article marked Rs 500? Solution: Given: MP of an article $=$ Rs. 500 Discount $=20 \%$ There fore, Discount $=20 \%$ of 500 $=0.20 \times 500$ $=100$ So, SP $=\mathrm{MP}-$ Discount $=$ Rs. $(500-100)$ $=$ Rs. 400 $\mathrm{CP}=\left[\frac{100}{(100+\text { Gain } \%)} \times \mathrm{SP}\right]$ $=\left[\frac{100}{(100+25)} \times 400\right]$ $=\frac{40000}{125}$...
Read More →A bucket made up of a metal sheet is in the form of a frustum of a cone of height 16 cm
Question: A bucket made up of a metal sheet is in the form of a frustum of a cone of height 16 cm and radii of its lower and upper ends are 8 cm and 20 cm, respectively. Find the cost of the bucket if the cost of metal sheet used is Rs 15 per 100 cm2. Solution: Greater radius of the frustum =R= 20 cmSmaller radius of the frustum =r= 8 cmHeight of the frustum =h= 16 cm Slant height,l,of the frustum $=\sqrt{h^{2}+\left(R^{2}-r^{2}\right)}$ $=\sqrt{16^{2}+(20-8)^{2}}$ $=\sqrt{256+(12)^{2}}$ $=\sqrt...
Read More →Find the inverse of each of the following matrices by using elementary row transformations:
Question: Find the inverse of each of the following matrices by using elementary row transformations: $\left[\begin{array}{ccc}1 2 0 \\ 2 3 -1 \\ 1 -1 3\end{array}\right]$ Solution: $A=\left[\begin{array}{ccc}1 2 0 \\ 2 3 -1 \\ 1 -1 3\end{array}\right]$ We know $A=I A$ $\Rightarrow\left[\begin{array}{ccc}1 2 0 \\ 2 3 -1 \\ 1 -1 3\end{array}\right]=\left[\begin{array}{lll}1 0 0 \\ 0 1 0 \\ 0 0 1\end{array}\right] A$ $\Rightarrow\left[\begin{array}{ccc}1 2 0 \\ 0 -1 -1 \\ 0 -3 3\end{array}\right]=...
Read More →A bucket made up of a metal sheet is in the form of a frustum of a cone of height 16 cm
Question: A bucket made up of a metal sheet is in the form of a frustum of a cone of height 16 cm and radii of its lower and upper ends are 8 cm and 20 cm, respectively. Find the cost of the bucket if the cost of metal sheet used is Rs 15 per 100 cm2. Solution: Greater radius of the frustum =R= 20 cmSmaller radius of the frustum =r= 8 cmHeight of the frustum =h= 16 cm Slant height,l,of the frustum $=\sqrt{h^{2}+\left(R^{2}-r^{2}\right)}$ $=\sqrt{16^{2}+(20-8)^{2}}$ $=\sqrt{256+(12)^{2}}$ $=\sqrt...
Read More →A shopkeeper allows his customers 10% off on the marked price of goods and still gets a profit of 25%.
Question: A shopkeeper allows his customers 10% off on the marked price of goods and still gets a profit of 25%. What is the actual cost to him of an article marked Rs 250? Solution: Let the CP of the article be Rs. $x$. MP of the article $=$ Rs. 250 Discount $=10 \%$ Discount $=10 \%$ of 250 $=0.10 \times 250$ $=$ Rs. 25 SP $=$ MP $-$ Discount $=250-25$ $=$ Rs. 225 Given, Profit $=25 \%$ $\mathrm{CP}=\left[\frac{100}{(100+\text { Gain } \%)} \times \mathrm{SP}\right]$ $x=\left[\frac{100}{(100+2...
Read More →A bucket is in the form of a frustum of a cone.
Question: A bucket is in the form of a frustum of a cone. Its depth is 15 cm and the diameters of the top and the bottom are 56 cm and 42 cm, respectively. Find how many litres of water can the bucket hold. Solution: Greater diameter of the frustum = 56 cmGreater radius of the frustum =R= 28 cmSmaller diameter of the frustum = 42 cmRadius of the smaller end of the frustum =r= 21 cmHeight of the frustum =h= 15 cmCapacity of the frustum $=\frac{1}{3} \pi h\left(R^{2}+r^{2}+R r\right)$ $=\frac{1}{3...
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