The volume of a cube is 729 cm3. Find its surface area.
Question: The volume of a cube is 729 cm3. Find its surface area. Solution: Let the edge of the cube be $a$. As, Volume of the cube $=729 \mathrm{~cm}^{3}$ $\Rightarrow a^{3}=729$ $\Rightarrow a=\sqrt[3]{729}$ $\Rightarrow a=9 \mathrm{~cm}$ Now, Surface area of the cube $=6 a^{2}$ $=6 \times 9 \times 9$ $=486 \mathrm{~cm}^{2}$ So, the surface area of the cube is 486 cm2....
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Question: If $A=\left[\begin{array}{lll}a 0 0 \\ 0 a 0 \\ 0 0 a\end{array}\right]$, then the value of $|\operatorname{adj} A|$ is (a) $a^{27}$ (b) $a^{9}$ (c) $a^{6}$ (d) $a^{2}$ Solution: (c) $a^{6}$ $A=\left[\begin{array}{lll}a 0 0\end{array}\right.$ $\therefore|A|=\mid a \quad 0 \quad 0$ and $n=3$ Thus, we have $|\operatorname{adj} A|=|A|^{n-1}=\left(a^{3}\right)^{2}=a^{6}$...
Read More →Rajat goes to a departmental store and buys the following articles:
Question: Rajat goes to a departmental store and buys the following articles: Calculate the total amount he has to pay to the store. Solution: Given, CP of 2 pair of shoes $=$ Rs. $800 \times 2=$ Rs. 1600 Rate of VAT $=5 \%$ So, $\mathrm{VAT}=5 \%$ of Rs. $1600=\frac{5}{100} \times 1600=$ Rs. 80 . Therefore, the amount Rajat needs to pa $y$ for 2 pair of shoes $=$ Rs. $(1600+80)=$ Rs. 1680 Again, CP of 1 sewing machine $=$ Rs. 1500 Rate of VAT $=6 \%$ So, VAT $=6 \%$ of Rs. $1500=\frac{6}{100} \...
Read More →Reena goes to a shop to buy a radio, costing Rs 2568.
Question: Reena goes to a shop to buy a radio, costing Rs 2568. The rate of VAT is 7%. She tells the shopkeeper to reduce the price of the radio such that she has to pay Rs 2568, inclusive of VAT. Find the reduction needed in the price of radio. Solution: Let the reduced price, excluding VAT, of the radio be Rs $x$. Then, VAT $=7 \%$ of Rs. $x=$ Rs. $\frac{7 x}{100}$ So, SP of the radio $=$ Rs. $\left(x+\frac{7 x}{100}\right)=$ Rs. $\frac{107 x}{100}$ But, $\mathrm{SP}=$ Rs. 2568 So, $\frac{107 ...
Read More →If A, B are two n × n non-singular matrices, then
Question: If $A, B$ are two $n \times n$ non-singular matrices, then (a) $A B$ is non-singular (b) $A B$ is singular (c) $(A B)^{-1} A^{-1} B^{-1}$ (d) $(A B)^{-1}$ does not exist Solution: (a) $A B$ is non-singular $A$ and $B$ are non-singular matrices of order $n \times n$. $\therefore|\mathrm{A}| \neq 0$ and $|\mathrm{B}| \neq 0 \quad \ldots(1)$ $A$ and $B$ are of the same order, so $A B$ is defined and is of the same order. Thus, $|\mathrm{AB}|=|\mathrm{A}||\mathrm{B}|$ $\Rightarrow|\mathrm{...
Read More →A colour TV is available for Rs 13440 inclusive of VAT.
Question: A colour TV is available for Rs 13440 inclusive of VAT. If the original cost of TV is Rs 12000, find the rate of VAT. Solution: Cost price of the TV including VAT $=$ Rs. 13440 Let the rate of VAT be $x \%$. Cost of the TV $=x \%$ of $12000+12000$ $13440-12000=120 x$ $120 x=1440$ $x=\frac{1440}{120}$ $=12$ Thus, the rate of VAT on the colour $T V$ is $12 \%$....
Read More →A refrigerator is available for Rs 13750 including VAT.
Question: A refrigerator is available for Rs 13750 including VAT. If the rate of VAT is 10%, find the original cost of the furniture. Solution: Cost of the refrigerator inclusive VAT $=$ Rs. 13,750 Let the original cost of the furniture be Rs. $x$. Cost of the furniture $=10 \%$ of $x+x$ $13,750=1.10 x$ $x=\frac{13,750}{1.10}$ $=$ Rs. 12,500 Thus, the original cost of the furniture is Rs. 12,500 ....
Read More →A river 1.5 m deep and 36 m wide is flowing at the rate of 3.5 km/hr.
Question: A river 1.5 m deep and 36 m wide is flowing at the rate of 3.5 km/hr. Find the amount of water (in cubic metres) that runs into the sea per minute. Solution: We have, Depth of the river, $h=1.5 \mathrm{~m}$, Width of the river, $b=36 \mathrm{~m}$, Speed of the flowing water, $l=3.5 \mathrm{~km} / \mathrm{hr}=\frac{3.5 \times 1000 \mathrm{~m}}{60 \mathrm{~min}}=\frac{175}{3} \mathrm{~m} / \mathrm{min}$ Now, The amount of water that runs into the sea per minute $=l b h$ $=\frac{175}{3} \...
Read More →The cost of furniture inclusive of VAT is Rs 7150.
Question: The cost of furniture inclusive of VAT is Rs 7150. If the rate of VAT is 10%, find the original cost of the furniture. Solution: Cost of the furniture inclusive VAT $=$ Rs. 7150 Let the original cost of the furniture be Rs. $x$. Cost of the furniture $=10 \%$ of $x+x$ $7150=1.10 x$ $x=\frac{7150}{1.10}$ $=$ Rs. 6500 Thus, the original cost of the furniture is Rs. 6500 ....
Read More →If A is a singular matrix, then adj A is
Question: IfAis a singular matrix, then adjAis(a) non-singular(b) singular(c) symmetric(d) not defined Solution: (b) singular $A$ is singular, so $|A|=0$. By definition, we have $A \operatorname{adj}(A)=O$ $\Rightarrow \mid A$ adj $(A)|=| O \mid$ $\Rightarrow|A||\operatorname{adj}(A)|=0$ $\Rightarrow|\operatorname{adj}(A)|=0$ Hence, adj $(A)$ is singular....
Read More →Sarita buys goods worth Rs 5500. She gets a rebate of 5% on it.
Question: Sarita buys goods worth Rs 5500. She gets a rebate of 5% on it. After getting the rebate if VAT at the rate of 5% is charged, find the amount she will have to pay for the goods. Solution: Price after getting a rebate of $5 \%$ on Rs $5500=5 \%$ of 5500 $=\frac{5}{100} \times 5500$ $=$ Rs 275 $\therefore$ new cost $=$ Rs $(5500-275)=$ Rs 5225 Now, VAT $=5 \%$ of 5225 $=\frac{5}{100} \times 5225$ $=$ Rs $261.25$ $\therefore$ a mount to be paid for the goods $=$ Rs $(5225+261.25)$ $=$ Rs ...
Read More →A right circular cone is divided into three parts by trisecting its height by two planes drawn parallel to the base.
Question: A right circular cone is divided into three parts by trisecting its height by two planes drawn parallel to the base. Show that the volume of the three portions starting from the top are in the ratio 1 : 7 : 19. Solution: Let ABC be a right circular cone of height 3hand base radiusr. This cone is cut by two planes such that AQ = QP = PO =h. Since $\Delta \mathrm{ABO} \sim \Delta \mathrm{AEP}$ (AA Similarity) $\therefore \frac{\mathrm{AO}}{\mathrm{AP}}=\frac{\mathrm{BO}}{\mathrm{EP}}$ $\...
Read More →if the solve this problem
Question: If $S=\left[\begin{array}{ll}a b \\ c d\end{array}\right]$, then adj $A$ is (a) $\left[\begin{array}{cc}-d -b \\ -c a\end{array}\right]$ (b) $\left[\begin{array}{cc}d -b \\ -c a\end{array}\right]$ (c) $\left[\begin{array}{ll}d b \\ c a\end{array}\right]$ (d) $\left[\begin{array}{ll}d c \\ b a\end{array}\right]$ Solution: (b) $\left[\begin{array}{cc}d -b \\ -c a\end{array}\right]$ Adjoint of a square matrix of order 2 is obtained by interchanging the diagonal elements and changing the s...
Read More →Swarna paid Rs 20 as VAT on a pair of shoes worth Rs 250.
Question: Swarna paid Rs 20 as VAT on a pair of shoes worth Rs 250. Find the rate of VAT. Solution: Given, Amount paid by Swarna for a pair of shoes $=R s .250$ VAT paid by her $=$ Rs. 20 Let the rate of VAT be $x \%$ Then, $x \%$ of $250=20$ $\frac{x}{100} \times 250=20$ $2.5 x=20$ $x=8$ Thus, Swarna paid $8 \%$ VAT on the pair of shoes....
Read More →Rani purchases a pair of shoes whose sale price is Rs 175.
Question: Rani purchases a pair of shoes whose sale price is Rs 175. If she pays VAT at the rate of 7%, how much amount does she poy as VAT? Also, find the net value of the pair of shoes. Solution: Given, SP of the pair of shoes $=$ Rs. 175 VAT $=7 \%$ Therefore, VAT $=7 \%$ of Rs. 175 $=\frac{7}{100} \times 175$ $=$ Rs. $12.25$ So, Rani has to pay Rs. $12.25$ as VAT. The net value of the pair of shoes $=$ Rs. $175+$ Rs. $12.25$ $=$ Rs. $187.25$...
Read More →Aman bought a shirt for Rs 374.50 which includes 7% VAT.
Question: Aman bought a shirt for Rs 374.50 which includes 7% VAT. Find the list price of the shirt. Solution: Let the list price of the shirt be Rs. $x$. Then, $x+\frac{7}{100} x=374.50$ $x+0.07 x=374.50$ $1.07 x=374.50$ $x=\frac{374.50}{1.07}$ $=R s .350$ Thus, the list price of the shirt is Rs. 350 ....
Read More →Vikram bought a watch for Rs 825.
Question: Vikram bought a watch for Rs 825. If this amount includes 10% VAT on the list price, what was the list price of the watch? Solution: Let the list price of the watch be Rs. $x$. Then, $x+\frac{10}{100} x=825$ $x+0.1 x=825$ $1.1 x=825$ $x=\frac{825}{1.1}$ $=$ Rs. 750 Thus, the list price of the watch is Rs. 750 ....
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Question: If $A=\left[\begin{array}{ll}3 4 \\ 2 4\end{array}\right], B=\left[\begin{array}{cc}-2 -2 \\ 0 -1\end{array}\right]$, then $(A+B)^{-1}=$ (a) is a skew-symmetric matrix (b) $A^{-1}+B^{-1}$ (c) does not exist (d) none of these Solution: (d) none of these We have $(A+B)=\left[\begin{array}{ll}1 2 \\ 2 3\end{array}\right]$ $\therefore|A+B|=-1 \neq 0$ Thus, $(A+B)^{-1}$ exists. Now, $(A+B)^{T}=\left[\begin{array}{ll}1 2 \\ 2 3\end{array}\right]$ Here, $(A+B)^{T} \neq-(A+B)$ Hence, it is not...
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Question: If $A=\left[\begin{array}{ll}3 4 \\ 2 4\end{array}\right], B=\left[\begin{array}{cc}-2 -2 \\ 0 -1\end{array}\right]$, then $(A+B)^{-1}=$ (a) is a skew-symmetric matrix (b) $A^{-1}+B^{-1}$ (c) does not exist (d) none of these Solution: (d) none of these We have $(A+B)=\left[\begin{array}{ll}1 2 \\ 2 3\end{array}\right]$ $\therefore|A+B|=-1 \neq 0$ Thus, $(A+B)^{-1}$ exists. Now, $(A+B)^{T}=\left[\begin{array}{ll}1 2 \\ 2 3\end{array}\right]$ Here, $(A+B)^{T} \neq-(A+B)$ Hence, it is not...
Read More →The list price of a refrigerator is Rs 9700.
Question: The list price of a refrigerator is Rs 9700. If a value added tax of 6% is to be charged on it, how much one has to pay to buy the refrigerator? Solution: List price of the refrigerator $=$ Rs. 9700 VAT $=6 \%$ So, $\mathrm{VAT}=6 \%$ of Rs. 9700 $=$ Rs. $\frac{6}{100} \times 9700$ $=$ Rs. 582 So, the total amount one has to pay $=$ Rs. $9700+$ Rs. 582 $=$ Rs. 10,282...
Read More →An oil funnel made of tin sheet consists of a 10 cm long cylindrical portion attached to a frustum of a cone.
Question: An oil funnel made of tin sheet consists of a10 cm long cylindrical portion attached to afrustum of a cone. If the total height is 22 cm,diameter of the cylindrical portion is 8 cm and the diameter of the top of the funnel is 18 cm, then find the area of the tin sheet required to make the funnel. Solution: We have, Height of the cylindrical portion, $h=10 \mathrm{~cm}$, Height of the frustum of cone portion, $H=22-10=12 \mathrm{~cm}$, Radius of the cylindical portion $=$ Radius of smal...
Read More →If A is an invertible matrix of order 3, then which of the following is not true
Question: IfAis an invertible matrix of order 3, then which of the following is not true (a) $|\operatorname{adj} A|=|A|^{2}$ (b) $\left(A^{-1}\right)^{-1}=A$ (c) If $B A=C A$, than $B \neq C$, where $B$ and $C$ are square matrices of order 3 (d) $(A B)^{-1}=B^{-1} A^{-1}$, where $B \neq\left[b_{i j}\right]_{3 \times 3}$ and $|B| \neq 0$ Solution: (c) If $B A=C A$, then $B \neq C$ where $B$ and $C$ are square matrices of order 3 . If $\mathrm{A}$ is an invertible matrix, then $A^{-1}$ exists. No...
Read More →If A is an invertible matrix of order 3, then which of the following is not true
Question: IfAis an invertible matrix of order 3, then which of the following is not true (a) $|\operatorname{adj} A|=|A|^{2}$ (b) $\left(A^{-1}\right)^{-1}=A$ (c) If $B A=C A$, than $B \neq C$, where $B$ and $C$ are square matrices of order 3 (d) $(A B)^{-1}=B^{-1} A^{-1}$, where $B \neq\left[b_{i j}\right]_{3 \times 3}$ and $|B| \neq 0$ Solution: (c) If $B A=C A$, then $B \neq C$ where $B$ and $C$ are square matrices of order 3 . If $\mathrm{A}$ is an invertible matrix, then $A^{-1}$ exists. No...
Read More →A lady shopkeeper allows her customers 10% discount on the marked price of the goods and still gets a profit of 25%.
Question: A lady shopkeeper allows her customers 10% discount on the marked price of the goods and still gets a profit of 25%. What is the cost price of a fan for her marked at Rs 1250? Solution: Given, MP of the fan $=$ Rs. 1250 Discount $=10 \%$ So, Discount $=10 \%$ of 1250 $=0.10 \times 1250$ $=$ Rs. 125 Since SP $=$ MP $-$ Discount, $S \mathrm{P}=$ Rs. $(1250-125)$ $=$ Rs. 1125 Now, $S P$ of the fan $=$ Rs. 1125 Profit $=25 \%$ $\mathrm{CP}=\left[\frac{100}{(100+\text { Profit } \%)} \times...
Read More →A fez, the cap used by the Turks, is shaped like the frustum of a cone.
Question: A fez, the cap used by the Turks, is shaped likethe frustum of a cone. If its radius on the openside is 10 cm, radius at the upper base is4 cm and its slant height is 15 cm, then find the area of material used for making it.$\left[\right.$ Use $\left.\pi=\frac{22}{7}\right]$ Solution: We have, Radius of open side, $R=10 \mathrm{~cm}$, Radius of upper base, $r=4 \mathrm{~cm}$ and Slant height, $l=15 \mathrm{~cm}$ Now, The area of material used $=\pi(R+r) l+\pi r^{2}$ $=\frac{22}{7} \tim...
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