State in each case whether A ⊂ B or A ⊄ B.
Question: State in each case whether A B or A B. $A=\{x: x$ is an integer $\}, B=\{x: x$ is a rational number $\}$ Solution: $A=\{\ldots .-2,-1,0,1,2,3 \ldots\}$ $B=\{-\infty, \ldots \ldots \ldots 0, \ldots \ldots \infty\}$ A B as integers are contained in rational numbers....
Read More →Show that the sum of an AP
Question: Show that the sum of an AP whose first term is a, the second term b and the last term c, is equal to$\frac{(a+c)(b+c-2 a)}{2(b-a)}$ Solution: Given that, the AP is a, b..c Here, first term = a, common difference = b a and last term, $l=a_{n}=c$ $\because$ $a_{n}=l=a+(n-1) d$ $\Rightarrow \quad c=a+(n-1)(b-a)$ $\Rightarrow \quad(n-1)=\frac{c-a}{b-a}$ $\Rightarrow \quad n=\frac{c-a}{b-a}+1$ $\Rightarrow \quad n=\frac{c-a+b-a}{b-a}=\frac{c+b-2 a}{b-a}$$\ldots$ (i) $\therefore$ Sum of an A...
Read More →State in each case whether A ⊂ B or A ⊄ B.
Question: State in each case whether A B or A B. A = {x : x is an even natural number,}, B = {x : x is an integer} Solution: A B Explanation: we have, A = {2,4,6,8,...} and B = {. . ., 3, 2, 1, 0, 1, 2, 3, . . .}. since, even natural numbers are also integers, we observe that elements of A belongs to B. Thus , A B....
Read More →A hollow garden roller, 63 cm wide with a girth of 440 cm, is made of 4 cm thick iron.
Question: A hollow garden roller, 63 cm wide with a girth of 440 cm, is made of 4 cm thick iron. Find the volume of the iron. Solution: Here,R= Outer radius r= Inner radius t = Thickness = 4 cm w= Width = 63 cm Girth = 440 cm = 2R $\mathrm{R}=\frac{440}{2 \times \frac{22}{7}}=70 \mathrm{~cm}$ r=R t= 70 cm 4 cm = 66 cm Volume of the iron $=\pi\left(R^{2}-r^{2}\right) w=\frac{22}{7}-\left(70^{2}-66^{2}\right)-(63)=107712 \mathrm{~cm}^{3}$ Hence, volume of the iron is107712 cm3....
Read More →Solve this
Question: If $f(x)=\left\{\begin{array}{ll}2 x^{2}+k, \text { if } x \geq 0 \\ -2 x^{2}+k, \text { if } x0\end{array}\right.$, then what should be the value of $k$ so that $f(x)$ is continuous at $x=0$. Solution: The given function can be rewritten as $(x)=\left\{\begin{array}{l}2 x^{2}+k, \text { if } x \geq 0 \\ -2 x^{2}+k, \text { if } x0\end{array}\right.$ We have $(\mathrm{LHL}$ at $x=0)=\lim _{x \rightarrow 0^{-}} f(x)=\lim _{h \rightarrow 0} f(0-h)=\lim _{h \rightarrow 0} f(-h)=\lim _{h \...
Read More →State in each case whether A ⊂ B or A ⊄ B.
Question: State in each case whether A B or A B. $A=\left\{x: x \in Z, x^{2}=1\right\}, B=\left\{x: x \in N, x^{2}=1\right\}$ Solution: A B Explanation: we have, A = { -1,1} and B={1} Since, -1A and -1B thus A B ....
Read More →A well with 10 m inside diameter is dug 8.4 m deep.
Question: A well with 10 m inside diameter is dug 8.4 m deep. Earth taken out of it is spread all around it to a width of 7.5 m to form an embankment. Find the height of the embankment. Solution: Letrm be the radius anddm be the depth of the well that is dug. Volume of the well =r2d= (5 m)2(8.4 m) = 660m3 An embankment has the shape of hollow cylinder with thickness. Its inner radii is equal to the well's radii, i.e. r = 5 m, and its outer radii isR= (5 + 7.5 )= 12.5 cm. Then, the volume of the ...
Read More →The ratio of the 11th term to the 18th
Question: The ratio of the 11th term to the 18th term of an AP is 2 : 3. Find the ratio of the 5th term to the 21st term and also the ratio of the sum of the first five terms to the sum of the first 21 terms. Solution: Let a arid d be the first term and common difference of an AP Given that,$a_{11}: a_{18}=2: 3$ $\Rightarrow$ $\frac{a+10 d}{a+17 d}=\frac{2}{3}$ $\Rightarrow$ $3 a+30 d=2 a+34 d$ $\Rightarrow$ $a=4 d$$\ldots$ (i) Now, $a_{5}=a+4 d=4 d+4 d=8 d$ [from Eq. (i)] and $a_{n+1}=a+20 d=4 ...
Read More →State in each case whether A ⊂ B or A ⊄ B.
Question: State in each case whether A B or A B. $A=\{1,2,3,\}, B=\{1,2,4$, Solution: A B Explanation: A B since 3A and 3B....
Read More →2.2 cubic dm of brass is to be drawn into a cylindrical wire 0.25 cm in diameter.
Question: 2.2 cubic dm of brass is to be drawn into a cylindrical wire 0.25 cm in diameter. Find the length of the wire. Solution: Letrcm be the radius of the wire andhcm be the length of the wire. Volume of brass = Volume of the wire We know that the volume of brass = 2.2dm3= 2200 cm3 Volume of the wire= r2h= (0.125 cm)2(h) $h=\frac{2200 \mathrm{~cm}^{3}}{\pi(0.125 \mathrm{~cm})^{2}}=44800 \mathrm{~cm}=448 \mathrm{~m}$ Thus, length of the wire is 448 m....
Read More →Prove that:
Question: Prove that $f(x)=\left\{\begin{array}{ll}\frac{x-|x|}{x}, x \neq 0 \\ 2 , x=0\end{array}\right.$ is discontinuous at $x=0$ Solution: The given function can be rewritten as $f(x)= \begin{cases}\frac{x-x}{x}, \text { when } x0 \\ \frac{x+x}{x}, \text { when } x0 \\ 2, \text { when } x=0\end{cases}$ $\Rightarrow f(x)=\left\{\begin{array}{l}0, \text { when } x0 \\ 2, \text { when } x0 \\ 2, \text { when } x=0\end{array}\right.$ We have (LHL at $x=0$ ) $=\lim _{x \rightarrow 0^{-}} f(x)=\li...
Read More →State in each case whether A ⊂ B or A ⊄ B.
Question: State in each case whether A B or A B. $A=\phi, B=\{0\}$ Solution: A B Explanation: A is a : set. Since, ϕ is a subset of every set therefore A B....
Read More →Find the length of 13.2 kg of copper wire of diameter 4 mm,
Question: Find the length of 13.2 kg of copper wire of diameter 4 mm, when 1 cubic cm of copper weighs 8.4 gm. Solution: Since we know the weight and the volume of copper, we can calculate its density. density of copper $=\frac{\text { weight }}{\text { volume }}=\frac{8.4 \mathrm{gram}}{1 \mathrm{~cm}^{3}}=8.4 \frac{\mathrm{gram}}{\mathrm{cm}^{3}}$ If the weight of copper wire is 13.2 kg and the density of copper is 8.4 g/cm3, then: Volume = Weight / Density = 13.2 kg x 1000 gram/kg / 8.4 gram/...
Read More →State in each case whether A ⊂ B or A ⊄ B.
Question: State in each case whether A B or A B. $A=\{0,1,2,3,\}, B=\{1,2,3,4,5$, Solution: A B Explanation: A B since 0A and 0B....
Read More →Discuss the continuity of the f(x) at the indicated points:
Question: Discuss the continuity of thef(x) at the indicated points: (i) $f(x)=|x|+|x-1|$ at $x=0,1$. (ii) $f(x)=|x-1|+|x+1|$ at $x=-1,1$. Solution: (i) Given: $f(x)=|x|+|x-1|$ We have $(\mathrm{LHL}$ at $x=0)=\lim _{x \rightarrow 0^{-}} f(x)=\lim _{h \rightarrow 0} f(0-h)=\lim _{h \rightarrow 0}[|0-h|+|0-h-1|]=1$ $(\mathrm{RHL}$ at $x=0)=\lim _{x \rightarrow 0^{+}} f(x)=\lim _{h \rightarrow 0} f(0+h)=\lim _{h \rightarrow 0}[|0+h|+|0+h-1|]=1$ Also, $f(0)=|0|+|0-1|=0+1=1$ Now, $(\mathrm{LHL}$ at ...
Read More →Discuss the continuity of the f(x) at the indicated points:
Question: Discuss the continuity of thef(x) at the indicated points: (i) $f(x)=|x|+|x-1|$ at $x=0,1$. (ii) $f(x)=|x-1|+|x+1|$ at $x=-1,1$. Solution: (i) Given: $f(x)=|x|+|x-1|$ We have $(\mathrm{LHL}$ at $x=0)=\lim _{x \rightarrow 0^{-}} f(x)=\lim _{h \rightarrow 0} f(0-h)=\lim _{h \rightarrow 0}[|0-h|+|0-h-1|]=1$ $(\mathrm{RHL}$ at $x=0)=\lim _{x \rightarrow 0^{+}} f(x)=\lim _{h \rightarrow 0} f(0+h)=\lim _{h \rightarrow 0}[|0+h|+|0+h-1|]=1$ Also, $f(0)=|0|+|0-1|=0+1=1$ Now, $(\mathrm{LHL}$ at ...
Read More →Find the cost of sinking a tubewell 280 m deep,
Question: Find the cost of sinking a tubewell 280 m deep, having diameter 3 m at the rate of Rs 3.60 per cubic metre. Find also the cost of cementing its inner curved surface at Rs 2.50 per square metre. Solution: Cost of sinking a tube well $=$ Volume of the tube well $\times$ Cost of sinking a tube well per cubic metre $=\underline{22}_{7} \times\left(1.5^{2}\right) \times(280) \mathrm{m}^{3} \times$ Rs $3.6 / \mathrm{m}^{3}=$ Rs $7128 .$ Cost of cementing $=$ Inner surface area of the tube we...
Read More →Rewrite the following statements using the set notation:
Question: Rewrite the following statements using the set notation: (i) $\mathrm{a}$ is an element of set $\mathrm{A}$. (ii) $b$ is not an element of $A$. (iii) $A$ is an empty set and $B$ is a nonempty set. (iv) A number of elements in A is 6 . (v) 0 is a whole number but not a natural number. Solution: (i) Given: a is an element of set A this means a $\in \mathrm{A}$ (ii) Given: $b$ is not an element of $A$ this means $\mathrm{b} \notin \mathrm{A}$ (iii) Given: $A$ is an empty set and $B$ is a ...
Read More →Discuss the continuity of the f(x) at the indicated points:
Question: Discuss the continuity of thef(x) at the indicated points: (i) $f(x)=|x|+|x-1|$ at $x=0,1$. (ii) $f(x)=|x-1|+|x+1|$ at $x=-1,1$. Solution: (i) Given: $f(x)=|x|+|x-1|$ We have $(\mathrm{LHL}$ at $x=0)=\lim _{x \rightarrow 0^{-}} f(x)=\lim _{h \rightarrow 0} f(0-h)=\lim _{h \rightarrow 0}[|0-h|+|0-h-1|]=1$ $(\mathrm{RHL}$ at $x=0)=\lim _{x \rightarrow 0^{+}} f(x)=\lim _{h \rightarrow 0} f(0+h)=\lim _{h \rightarrow 0}[|0+h|+|0+h-1|]=1$ Also, $f(0)=|0|+|0-1|=0+1=1$ Now, $(\mathrm{LHL}$ at ...
Read More →A solid cylinder has a total surface area of
Question: A solid cylinder has a total surface area of $231 \mathrm{~cm}^{2}$. Its curved surface area is $\frac{2}{3}$ of the total surface area. Find the volume of the cylinder. Solution: We know that the total surface area of the cylinder is 231 cm2and the curved surface area is 2/3 of the total surface area. So, the curved surface area is: $2 / 3 \times\left(231 \mathrm{~cm}^{2}\right)=154 \mathrm{~cm}^{2}$ Then, the radius of the cylinder can be calculated in the following manner: Curved su...
Read More →Find the sum of the integers
Question: Find the sum of the integers between 100 and that are (i) divisible by 9. (ii) not divisible by 9. Solution: (i)The numbers (integers) between 100 and 200 which is divisible by 9 are 108, 117, 126, 198. Let n be the number of terms between 100 and 200 which is divisible by 9. Here, $a=108, d=117-108=9$ and $a_{n}=l=198$ $\because \quad a_{n}=l=a+(n-1) d$ $\Rightarrow \quad 198=108+(n-1) 9$ $\Rightarrow \quad 90=(n-1) 9$ $\rightarrow$ - $n-1=10$ $\Rightarrow \quad n=11$ $\therefore$ Sum...
Read More →State whether any given set is finite or infinite:
Question: State whether any given set is finite or infinite: L = set of all circles passing through the origin (0, 0) Solution: Infinitember of circles can pass through the origin, so the set will have infinite elements. So, the given set is infinite...
Read More →State whether any given set is finite or infinite:
Question: State whether any given set is finite or infinite:. $K=\{x: x \in N$ and $x$ is prime $\}$. Solution: The given set is the set of all prime numbers and since the set of prime numbers is infinite. Hence, the given set is infinite....
Read More →How many litres of water flow out of a pipe having an area of cross-section of
Question: How many litres of water flow out of a pipe having an area of cross-section of 5 cm2in one minute, if the speed of water in the pipe is 30 cm/sec? Solution: We know: Area of cross section = 5 cm2 Rate = 30 cm/s and Time =1 min So, the volume of water flow is: Volume = Volumetric rateTime = (30 cm/s)(5 cm2)(60 s/min) = 9000 cm3 = 9 litres Thus, 9 litres of water flows out of the pipe....
Read More →State whether any given set is finite or infinite:
Question: State whether any given set is finite or infinite: $J=\{x: x \in N$ and $x$ is prime $\}$ Solution: The given set is the set of all prime numbers and since the set of prime numbers is infinite. Hence, the given set is infinite....
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