Which of the following are sets? Justify your answer
Question: Which of the following are sets? Justify your answer The collection of all rich persons of Kolkata. Solution: As the collection of all rich persons of Kolkata may vary from person to person. Someone considers a person whose income is Rs 1 lakh per annum as a rich person, and someone considers a person whose income is Rs 1 crore per annum as a rich person. Here, the set is not well defined. , this is not a set...
Read More →Three equal cubes are placed adjacently in a row.
Question: Three equal cubes are placed adjacently in a row. Find the ratio of total surface area of the new cuboid to that of the sum of the surface areas of the three cubes. Solution: Suppose that the side of the cube $=\mathrm{x} \mathrm{cm}$ Surface area of the cube $=6 \times(\text { side })^{2}=6 \times \mathrm{x}^{2}=6 \mathrm{x}^{2} \mathrm{~cm}^{2}$ i.e., the sum of the surface areas of three such cubes $=6 \mathrm{x}^{2}+6 \mathrm{x}^{2}+6 \mathrm{x}^{2}=18 \mathrm{x}^{2} \mathrm{~cm}^{...
Read More →Solve the following for x and y:
Question: Solve the following forxandy: $\left[\begin{array}{rr}3 -4 \\ 9 2\end{array}\right]\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{r}10 \\ 2\end{array}\right]$ Solution: Here, $\left[\begin{array}{cc}3 -4 \\ 9 2\end{array}\right]\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{c}10 \\ 2\end{array}\right]$ $\Rightarrow\left[\begin{array}{l}3 x-4 y \\ 9 x-2 y\end{array}\right]=\left[\begin{array}{c}10 \\ 2\end{array}\right]$ $\Rightarrow 3 x-4 y=10 \q...
Read More →Which of the following are sets? Justify your answer.
Question: Which of the following are sets? Justify your answer. The collection of all those students of your class whose ages exceed 15 years. Solution: As the collection of all those students of your class whose ages exceed 15 years is known and can be counted, i.e. well defined. , this is a se...
Read More →A tank open at the top is made of iron sheet 4 m wide.
Question: A tank open at the top is made of iron sheet 4 m wide. If the dimensions of the tank are 12 m 8 m 6 m, find the cost of iron sheet at Rs 17.50 per metre. Solution: An open iron tank of dimensions $12 \mathrm{~m} \times 8 \mathrm{~m} \times 6 \mathrm{~m}$ is to be made. Surface area of the open tank $=$ (area of the base) $+$ (total area of the 4 walls) $=(12 \times 8)+2 \times(8 \times 6+12 \times 6)$ $=(96)+2 \times(48+72)$ $=336 \mathrm{~m}^{2}$ Also, it is given that the cost of the...
Read More →Which of the following are sets? Justify your answer.
Question: Which of the following are sets? Justify your answer. The collection of all short boys of your class. Solution: As the collection of all short boys of your class may vary to person to person Maybe someone consider short boys of height less than 120 cm and maybe someone consider short boys of height less than 90cm. Here, the set is not well defined. , this is not a set....
Read More →A closed iron tank 12 m long, 9 m wide and 4 m deep is to be made.
Question: A closed iron tank 12 m long, 9 m wide and 4 m deep is to be made. Determine the cost of iron sheet used at the rate of Rs 5 per metre sheet, sheet being 2 m wide. Solution: A closed iron tank of dimensions $12 \mathrm{~m}$ long, $9 \mathrm{~m}$ wide and $4 \mathrm{~m}$ deep is to be made. Surface area of the cuboidal tank $=2 \times($ length $\times$ breadth $+$ breadth $\times$ height $+$ length $\times$ height $)$ $=2 \times(12 \times 9+9 \times 4+12 \times 4)$ $=2 \times(108+36+48)...
Read More →Solve this
Question: If $\left[\begin{array}{lll}1 0 0 \\ 0 y 0 \\ 0 0 1\end{array}\right]\left[\begin{array}{r}x \\ -1 \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 1\end{array}\right]$, find $x, y$ and $z$ Solution: Here, $\left[\begin{array}{lll}1 0 0 \\ 0 y 0 \\ 0 0 1\end{array}\right]\left[\begin{array}{c}x \\ -1 \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 1\end{array}\right]$ $\Rightarrow\left[\begin{array}{c}x \\ -y \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 1\end{arr...
Read More →Which of the following are sets? Justify your answer.
Question: Which of the following are sets? Justify your answer. The collection of all interesting books. Solution: As the collection of all interesting books may vary to person to person. , this is not a set....
Read More →Verify that each of the following
Question: Verify that each of the following is an AP and then write its next three terms. (i) $0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, \ldots$ (ii) $5, \frac{14}{3}, \frac{13}{3}, 4, \ldots$ (iii) $\sqrt{3}, 2 \sqrt{3}, 3 \sqrt{3}, \ldots$ (iv) $a+b,(a+1)+b,(a+1)+(b+1), \ldots$ (v) $a, 2 a+1,3 a+2,4 a+3, \ldots$ Solution: (i) Here, $a_{1}=0, a_{2}=\frac{1}{4}, a_{3}=\frac{1}{2}$ and $a_{4}=\frac{3}{4}$ $a_{2}-a_{1}=\frac{1}{4}, a_{3}-a_{2}=\frac{1}{2}-\frac{1}{4}=\frac{1}{4}, a_{4}-a_{3}=\frac...
Read More →Solve this
Question: If $\left[\begin{array}{rrr}1 0 0 \\ 0 -1 0 \\ 0 0 -1\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 1\end{array}\right]$, find $x, y$ and $z$ Solution: Here, $\left[\begin{array}{ccc}1 0 0 \\ 0 -1 0 \\ 0 0 -1\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 1\end{array}\right]$ $\Rightarrow\left[\begin{array}{c}x \\ -y \\ -z\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 1\end{...
Read More →Which of the following are sets? Justify your answer
Question: Which of the following are sets? Justify your answer The collection of all the months of the year whose names begin with the letter M. Solution: Months of the Year = Jan, Feb, March, April, May, June, July, Aug, Sep, Oct, Nov, Dec Months of the year whose names begin with the letter M are: March May As, the collection of all the months of the year whose names begin with the letter M is known and can be counted .i.e. well defined , this is a set....
Read More →The breadth of a room is twice its height,
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: Suppose that the breadth of the room $=\mathrm{x}$ dm Since breadth is twice the height, breadth $=2 \times$ height So, height of the room $=\frac{\text { breadth }}{2}=\frac{x}{2}$ Also, it is given that the breadth is half the length. So, breadth $=\frac{1}{2} \times$ length i. e., length $=2 \times$ breadth $=2 \times \mathrm{x}$ Since volume of ...
Read More →Which of the following are sets? Justify your answer
Question: Which of the following are sets? Justify your answer The collection of all the months of the year whose names begin with the letter M. Solution: Months of the Year = Jan, Feb, March, April, May, June, July, Aug, Sep, Oct, Nov, Dec Months of the year whose names begin with the letter M are: March May...
Read More →Solve this
Question: If $\left[\begin{array}{lll}1 0 0 \\ 0 1 0 \\ 0 0 1\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{r}1 \\ -1 \\ 0\end{array}\right]$, find $x, y$ and $z$ Solution: Here, $\left[\begin{array}{lll}1 0 0 \\ 0 1 0 \\ 0 0 1\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{c}1 \\ -1 \\ 0\end{array}\right]$ $\Rightarrow\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{c}1 \\ -1 \\ 0\end{arr...
Read More →Cubes A, B, C having edges 18 cm,
Question: CubesA,B,Chaving edges 18 cm, 24 cm and 30 cm respectively are melted and moulded into a new cubeD. Find the edge of the bigger cubeD. Solution: We have the following: Length of the edge of cube $\mathrm{A}=18 \mathrm{~cm}$ Length of the edge of cube $\mathrm{B}=24 \mathrm{~cm}$ Length of the edge of cube $\mathrm{C}=30 \mathrm{~cm}$ The given cubes are melted and moulded into a new cube $\mathrm{D}$. Hence, volume of cube $\mathrm{D}=$ volume of cube $\mathrm{A}+$ volume of cube $\mat...
Read More →Which of the following are sets? Justify your answer.
Question: Which of the following are sets? Justify your answer. A team of 11 best cricket players of India Solution: As a collection of 11 best cricket players of India may vary from person to person. So, it is not well defined. , this is not a set....
Read More →A rectangular water reservoir contains 105 m
Question: A rectangular water reservoir contains 105 m3of water. Find the depth of the water in the reservoir if its base measures 12 m by 3.5 m. Solution: Length of the rectangular water reservoir $=12 \mathrm{~m}$ Breadth $=3.5 \mathrm{~m}$ Suppose that the height of the reservoir $=h \mathrm{~m}$ Also, it contains $105 \mathrm{~m}^{3}$ of water, i.e., its volume $=105 \mathrm{~m}^{3}$ Volume of the cuboidal water reservoir $=$ length $\times$ breadth $\times$ height $\Rightarrow 105=12 \times...
Read More →The set of values of k for which the system of equations
Question: The set of values of $k$ for which the system of equations $x+y+z=2,2 x+y-z=3,3 x+2 y+k z=4$ has a unique solution, is__________ Solution: The system of equations $x+y+z=2,2 x+y-z=3$ and $3 x+2 y+k z=4$ has a unique solution. $\therefore \Delta=\left|\begin{array}{ccc}1 1 1 \\ 2 1 -1 \\ 3 2 k\end{array}\right| \neq 0$ $\Rightarrow 1(k+2)-1(2 k+3)+1(4-3) \neq 0$ $\Rightarrow k+2-2 k-3+1 \neq 0$ $\Rightarrow k \neq 0$ Thus, the set of values of $k$ for which the given system of equations...
Read More →Which of the following are sets? Justify your answer.
Question: Which of the following are sets? Justify your answer. A collection of Hindi novels written by Munshi Prem Chand. Solution: As the collection of Hindi novels written by Munshi Prem Chand is known and can be counted, i.e. well defined. , this is a set....
Read More →The areas of three adjacent faces of a cuboid are x, y and z.
Question: The areas of three adjacent faces of a cuboid arex,yandz. If the volume isV, prove thatV2=xyz. Solution: The areas of three adjacent faces of a cuboid are $x, y$ and $z$. Volume of the cuboid $=\mathrm{V}$ Observe that $x=$ length $\times$ breadth $y=$ breadth $\times$ height $z=$ length $\times$ height Since volume of cuboid $V=$ length $\times$ breadth $\times$ height, we have : $V^{2}=V \times V$ $=($ length $\times$ breadth $\times$ height $) \times($ length $\times$ breadth $\time...
Read More →D. Which of the following are sets? Justify your answer.
Question: D. Which of the following are sets? Justify your answer. The collection of all the difficult chapters in this book Solution: As the collection of all difficult chapters in this book may vary from person to person. , this is not a set....
Read More →If V is the volume of a cuboid of dimensions a, b, c and S is its surface area,
Question: IfVis the volume of a cuboid of dimensionsa,b,candS is its surface area, then prove that$\frac{1}{V}=\frac{2}{S}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$ Solution: It is given that $V$ is the volume of a cuboid of length $=a$, breadth $=b$ and height $=c$. Also, $S$ is surface area of cuboid. Then, $V=a \times b \times c$ Surface area of the cuboid $=2 \times($ length $\times$ breadth $+$ breadth $\times$ height $+$ length $\times$ height $)$ $\Rightarrow S=2 \times(a \times b+...
Read More →The real value
Question: The real value of $\lambda$ for which the system of equations $\lambda x+y+z=0,-x+\lambda y+z=0,-x-y+\lambda z=0$ has a non-zero solution, is___________ Solution: The system of homogeneous equations $\lambda x+y+z=0,-x+\lambda y+z=0$ and $-x-y+\lambda z=0$ has a non-zero solution or an infinite many solutions. $\therefore \Delta=\left|\begin{array}{ccc}\lambda 1 1 \\ -1 \lambda 1 \\ -1 -1 \lambda\end{array}\right|=0$ $\Rightarrow \lambda\left(\lambda^{2}+1\right)-1(-\lambda+1)+1(1+\lam...
Read More →Match the AP’s given in column
Question: Match the APs given in column A with suitable common differences given in column B. Solution: A, $2,-2,-6,-10, \ldots$ Here, common difference. $d=-2-2=-4$ $A_{2} . \because$ $a_{t,} a+(n-1) d$ $\Rightarrow$ $0=-18+(10-1) d$ $18=9 d$ $\therefore$ Common difference, $d=2$ $A_{3} \cdot \because \quad a_{10}=6$ $\Rightarrow \quad a+(10-1) d=6$ $\Rightarrow \quad 0+9 d=6$ $[\because a=0$ (given) $]$ $9 d=6 \Rightarrow d=\frac{2}{3}$ $A_{4} \because \quad a_{2}=13$ $\Rightarrow \quad a+(2-1...
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