The ratio between the curved surface area and the total surface area of a right circular cylinder is 1 : 2.
Question: The ratio between the curved surface area and the total surface area of a right circular cylinder is 1 : 2. Find the volume of the cylinder, if its total surface area is 616 cm2 Solution: Letrcm be the radius andhcm be the length of the cylinder. The curved surface area and the total surface area is 1:2. The total surface area is 616 cm2. The curved surface area is the half of 616 cm2, i.e. 308 cm2. Curved area = 2rh $r^{2}=\frac{308}{2 \times \frac{22}{7}}$ r= 7 cm $h=\frac{308}{2 \t...
Read More →Which of the following are pairs of equivalent sets?
Question: Which of the following are pairs of equivalent sets? (i) $A=\{-1,-2,0\}$ and $B=\{1,2,3,$, (ii) $C=\{x: x \in N, x3\}$ and $D=\{x: x \in W, x3\}$ (iii) $E=\{a, e, i, o, u\}$ and $F=\{p, q, r, s, t\}$ Solution: (i) Equivalent Sets can have different or same elements but have the same amount of elements. We have, $A=\{-1,-2,0\}$ and $B=\{1,2,3,$, A and B are equivalent sets because both have 3 elements in their set. (ii) Equivalent Sets can have different or same elements but have the sa...
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Question: If $f(x)=\left\{\begin{array}{rr}\frac{\cos ^{2} x-\sin ^{2} x-1}{\sqrt{x^{2}+1}-1}, x \neq 0 \\ k , x=0\end{array}\right.$ is continuous at $x=0$, find $k$. Solution: Given: $f(x)=\left\{\begin{array}{l}\frac{\cos ^{2} x-\sin ^{2} x-1}{\sqrt{x^{2}+1}-1}, x \neq 0 \\ k, x=0\end{array}\right.$ Iff(x) is continuous atx= 0, then $\lim _{x \rightarrow 0} f(x)=f(0)$ $\Rightarrow \lim _{x \rightarrow 0} \frac{\cos ^{2} x-\sin ^{2} x-1}{\sqrt{x^{2}+1}-1}=k$ $\Rightarrow \lim _{x \rightarrow 0...
Read More →The radii of two cylinders are in the ratio 2 : 3 and their heights are in the ratio 5 : 3.
Question: The radii of two cylinders are in the ratio 2 : 3 and their heights are in the ratio 5 : 3. Calculate the ratio of their volumes. Solution: Here,r1= Radius of cylinder 1 h1= Height of cylinder 1 r2= Radius of cylinder 2 h2= Height of cylinder 2 V1= Volume of cylinder 1 V2= Volume of cylinder 2Ratio of the radii of two cylinders = 2:3Ratio of the heights of two cylinders = 5:3 Volume of the cylinder =r2h V1/V2= (r12h1)/(r22h2) = ((2r)25h)/((3r)23h) V1/V2=(4r25h)/(9r23h) = 20 / 27 Hence,...
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Question: If $f(x)=\left\{\begin{array}{rr}\frac{\cos ^{2} x-\sin ^{2} x-1}{\sqrt{x^{2}+1}-1}, x \neq 0 \\ k , x=0\end{array}\right.$ is continuous at $x=0$, find $k$. Solution: Given: $f(x)=\left\{\begin{array}{l}\frac{\cos ^{2} x-\sin ^{2} x-1}{\sqrt{x^{2}+1}-1}, x \neq 0 \\ k, x=0\end{array}\right.$ Iff(x) is continuous atx= 0, then $\lim _{x \rightarrow 0} f(x)=f(0)$ $\Rightarrow \lim _{x \rightarrow 0} \frac{\cos ^{2} x-\sin ^{2} x-1}{\sqrt{x^{2}+1}-1}=k$ $\Rightarrow \lim _{x \rightarrow 0...
Read More →The volume and the curved surface area of a cylinder are 1650 cm
Question: The volume and the curved surface area of a cylinder are 1650 cm3and 660 cm2respectively. Find the radius and height of the cylinder. Solution: Curved surface area of the cylinder = 2rh=660 cm2 ... (1) Volume of the cylinder =r2h =1650 cm3 ... (2) From (1) and (2), we can calculate the radius (r) and the height of cylinder (h). We know the volume of the cylinder, i.e. 1650 cm3 1650 =r2h $h=\frac{1650}{\pi r^{2}}$ Substituting h into (1): 660 = 2rh $660=2 \pi r \times \frac{1650}{\pi r^...
Read More →How many terms of the AP – 15,
Question: How many terms of the AP 15, 13, -11, are needed to make the sum 55? Solution: Let n number of terms are needed to make the sum 55. Here, first term (a) = -15, common difference (d) = -13+15 = 2 $\because \quad$ Sum of $n$ terms of an AP, $S_{n}=\frac{n}{2}[2 a+(n-1) d]$ $\Rightarrow \quad-55=\frac{n}{2}[2(-15)+(n-1) 2] \quad\left[\because S_{n}=-55\right.$ (given)] $\Rightarrow \quad-55=-15 n+n(n-1)$ $\Rightarrow \quad n^{2}-16 n+55=0$ $\Rightarrow \quad n^{2}-11 n-5 n+55=0$[by factor...
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Question: If $f(x)=\left\{\begin{array}{cc}\frac{2^{x+2}-16}{4^{x}-16}, \text { if } \quad x \neq 2 \\ k \text {, if } x=2\end{array}\right.$ is continuous at $x=2$, find $k$. Solution: Given: $f(x)= \begin{cases}\frac{2^{x+2}-16}{4^{x}-16}, \text { if } x \neq 2 \\ k , \text { if } x=2\end{cases}$ If $f(x)$ is continuous at $x=2$, then $\lim _{x \rightarrow 2} f(x)=f(2)$ $\Rightarrow \lim _{x \rightarrow 2} \frac{2^{x+2}-16}{4^{x}-16}=f(2)$ $\Rightarrow \lim _{x \rightarrow 2} \frac{4\left(2^{x...
Read More →A rectangular sheet of paper, 44 cm × 20 cm,
Question: A rectangular sheet of paper, 44 cm 20 cm, is rolled along its length to form a cylinder. Find the volume of the cylinder so formed. Solution: The length $(l)$ and breadth $(b)$ of the rectangular sheet are $44 \mathrm{~cm}$ and $20 \mathrm{~cm}$ Now, the sheet is rolled along the length to form a cylinder. Let the radius of the cylinder be $r \mathrm{~cm}$. Height, $h=b=20 \mathrm{~cm}$ Circumference, $S=44 \mathrm{~cm}$ $2 \pi r=44 \mathrm{~cm}$ $2 \times \frac{22}{7} \times r=44 \ma...
Read More →Which of the following are pairs of equal sets?
Question: Which of the following are pairs of equal sets? $J=\{2,3\}$ and $K=\left\{x: x \in Z,\left(x^{2}+5 x+6\right)=0\right\}$ Solution: Equal Sets = Two sets A and B are said to be equal if they have exactly the same elements we write A = B We have, $J=\{2,3\}$ and $K=\left\{x: x \in Z,\left(x^{2}+5 x+6\right)=0\right\}$ Here, $x \in Z$ and $x^{2}+5 x+6=0$ The given equation can be solved as: $x^{2}+5 x+6=0$ $\Rightarrow x^{2}+2 x+3 x+6=0$ $\Rightarrow x(x+2)+3(x+2)=0$ $\Rightarrow(x+2)(x+3...
Read More →Find the sum of first seven numbers
Question: Find the sum of first seven numbers which are multiples of 2 as well as of 9. Solution: For finding, the sum of first seven numbers which are multiples of 2 as well as of 9. Take LCM of 2 and 9 which is 18. So, the series becomes 18, 36, 54, Here, first term (a) = 18, common difference (of) = 36 -18 = 18 $\therefore \quad S_{7}=\frac{n}{2}[2 a+(n-1) d]=\frac{7}{2}[2(18)+(7-1) 18]$ $=\frac{7}{2}[36+6 \times 18]=7(18+3 \times 18)$ $=7(18+54)=7 \times 72=504$...
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Question: Let $f(x)=\frac{\log \left(1+\frac{x}{a}\right)-\log \left(1-\frac{x}{b}\right)}{x}, x \neq 0 .$ Find the value of $f$ at $x=0$ so that $f$ becomes continuous at $x=0$. Solution: Given: $f(x)=\frac{\log \left(1+\frac{x}{a}\right)-\log \left(1-\frac{x}{b}\right)}{x}, x \neq 0$ Iff(x) is continuous atx= 0, then $\lim _{x \rightarrow 0} f(x)=f(0)$ $\Rightarrow \lim _{x \rightarrow 0}\left(\frac{\log \left(1+\frac{x}{a}\right)-\log \left(1-\frac{x}{b}\right)}{x}\right)=f(0)$ $\Rightarrow \...
Read More →A rectangular strip 25 cm × 7 cm is rotated about the longer side.
Question: A rectangular strip 25 cm 7 cm is rotated about the longer side. Find the volume of the solid, thus generated. Solution: Given : Rectangular strip has radius, $r=7 \mathrm{~cm}$ Height, $h=25 \mathrm{~cm}$ Volume of the solid, $\mathrm{V}=\pi r^{2} h$ $=\frac{22}{7} \times 7^{2} \times 25$ $=3850 \mathrm{~cm}^{3}$...
Read More →Find the sum of last ten terms
Question: Find the sum of last ten terms of the AP 8, 10, 12,, 126. Solution: For finding, the sum of last ten terms, we write the given AP in reverse order. i.e., 126,124,122 12,10,8 Here, first term (a) = 126, common difference, (d) = 124-126=-2 $\therefore \quad S_{10}=\frac{10}{2}[2 a+(10-1) d] \quad\left[\because S_{n}=\frac{n}{2}[2 a+(n-1) d]\right]$ $=5\{2(126)+9(-2)\}=5(252-18)$ $=5 \times 234=1170$...
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Question: Let $f(x)=\frac{\log \left(1+\frac{x}{a}\right)-\log \left(1-\frac{x}{b}\right)}{x}, x \neq 0 .$ Find the value of $f$ at $x=0$ so that $f$ becomes continuous at $x=0$. Solution: Given: $f(x)=\frac{\log \left(1+\frac{x}{a}\right)-\log \left(1-\frac{x}{b}\right)}{x}, x \neq 0$ Iff(x) is continuous atx= 0, then $\lim _{x \rightarrow 0} f(x)=f(0)$ $\Rightarrow \lim _{x \rightarrow 0}\left(\frac{\log \left(1+\frac{x}{a}\right)-\log \left(1-\frac{x}{b}\right)}{x}\right)=f(0)$ $\Rightarrow \...
Read More →Which of the following are pairs of equal sets?
Question: Which of the following are pairs of equal sets? $G=\{-1,1\}$ and $H=\left\{x: x \in Z, x^{2}-1=0\right\}$ Solution: Equal Sets = Two sets A and B are said to be equal if they have exactly the same elements we write A = B We have, $G=\{-1,1\}$ and $H=\left\{x: x \in Z, x^{2}-1=0\right\}$ Here, $x \in Z$ and $x^{2}-1=0$ The given equation can be solved as: $x^{2}-1=0$ $\Rightarrow \mathrm{x}^{2}=1$ $\Rightarrow \mathrm{x}=\sqrt{1}$ $\Rightarrow \mathrm{x}=\pm 1$ $\therefore \mathrm{x}=-1...
Read More →Find the volume of a cylinder, the diameter of whose base is 7 cm and height being 60 cm.
Question: Find the volume of a cylinder, the diameter of whose base is 7 cm and height being 60 cm. Also, find the capacity of the cylinder in litres. Solution: Given : Diameter, $d=7 \mathrm{~cm}$ Radius, $r=3.5 \mathrm{~cm}$ Height, $h=60 \mathrm{~cm}$ Volume of the cylinder, $\mathrm{V}=\pi r^{2} h$ $=\frac{22}{7} \times 3.5^{2} \times 60$ $=2310 \mathrm{~cm}^{3}$ Capacity of the cylinder in litres $=\frac{2310}{1000} \quad(1$ litre $=1000$ cubic $\mathrm{cm})$ $=2.31 \mathrm{~L}$...
Read More →Find the sum of all the 11 terms
Question: Find the sum of all the 11 terms of an AP whose middle most term is 30. Solution: Since, the total number of terms (n)=11[odd] $\therefore$ Middle most term $=\frac{(n+1)}{2}$ th term $=\left(\frac{11+1}{2}\right)$ th term $=6$ th term Given that,$a_{6}=30$ $\Rightarrow \quad a+(6-1) d=30$ $\Rightarrow \quad a+5 d=30$ ...(1) $\because$ Sum of $n$ terms of an AP, $S_{n}=\frac{n}{2}[2 a+(n-1) d]$ $\therefore$ $S_{11}=\frac{11}{2}[2 a+(11-1) d]$ $=\frac{11}{2}(2 a+10 d)=11(a+5 d) \quad$ [...
Read More →For what value of k is the function
Question: For what value ofkis the function $f(x)= \begin{cases}\frac{\sin 2 x}{x}, x \neq 0 \\ k , x=0\end{cases}$ continuous atx= 0? Solution: Given: $f(x)=\left\{\begin{array}{l}\frac{\sin 2 x}{x}, x \neq 0 \\ k, x=0\end{array}\right.$ If $f(x)$ is continuous at $x=0$, then $\lim _{x \rightarrow 0} f(x)=f(0)$ $\Rightarrow \lim _{x \rightarrow 0} \frac{\sin 2 x}{x}=k$ $\Rightarrow \lim _{x \rightarrow 0} \frac{2 \sin 2 x}{2 x}=k$ $\Rightarrow 2 \lim _{x \rightarrow 0} \frac{\sin 2 x}{2 x}=k$ $...
Read More →If sum of first 6 terms of an AP is 36
Question: If sum of first 6 terms of an AP is 36 and that of the first 16 terms is 256, then find the sum of first 10 terms Solution: Let a and d be the first term and common difference, respectively of an AP $\because$ Sum of $n$ terms of an AP, $S_{n}=\frac{n}{2}[2 a+(n-1) d]$$\ldots$ (i) Now, $S_{6}=36$ [given] $\Rightarrow \quad \frac{6}{2}[2 a+(6-1) d]=36$ $\Rightarrow$ $2 a+5 d=12$ (ii) $\Rightarrow \quad \frac{16}{2}[2 a+(16-1) d]=256$ $\Rightarrow \quad 2 a+15 d=32$ ...(iii) On subtracti...
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Question: If $f(x)= \begin{cases}\frac{x-4}{|x-4|}+a, \text { if } \quad x4 \\ a+b \quad, \text { if } \quad x=4 \text { is continuous at } x=4, \text { find } a, b . \\ \frac{x-4}{|x-4|}+b, \text { if } \quad x4\end{cases}$ Solution: Given: $f(x)=\left\{\begin{array}{c}\frac{x-4}{|x-4|}+a, \text { if } \mathrm{x}4 \\ a+b, \text { if } \mathrm{x}=4 \\ \frac{x-4}{|x-4|}+b, \text { if } \mathrm{x}4\end{array}\right.$ We observe $(\mathrm{LHL}$ at $x=4)=\lim _{x \rightarrow 4^{-}} f(x)=\lim _{h \ri...
Read More →Which of the following are pairs of equal sets?
Question: Which of the following are pairs of equal sets? $E=\left\{x: x \in Z, x^{2} \leq 4\right\}$ and $F=\left\{x: x \in Z, x^{2}=4\right\}$ Solution: Equal Sets = Two sets A and B are said to be equal if they have exactly the same elements we write A = B We have $E=\left\{x: x \in Z, x^{2} \leq 4\right\}$ Here, $x \in Z$ and $x^{2} \leq 4$ If $x=-2$, then $x^{2}=(-2)^{2}=4=4$ If $x=-1$, then $x^{2}=(-1)^{2}=14$ If $x=0$, then $x^{2}=(0)^{2}=04$ If $x=1$, then $x^{2}=(1)^{2}=14$ If $x=2$, th...
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Question: If $f(x)= \begin{cases}\frac{x-4}{|x-4|}+a, \text { if } \quad x4 \\ a+b \quad, \text { if } \quad x=4 \text { is continuous at } x=4, \text { find } a, b . \\ \frac{x-4}{|x-4|}+b, \text { if } \quad x4\end{cases}$ Solution: Given: $f(x)=\left\{\begin{array}{c}\frac{x-4}{|x-4|}+a, \text { if } \mathrm{x}4 \\ a+b, \text { if } \mathrm{x}=4 \\ \frac{x-4}{|x-4|}+b, \text { if } \mathrm{x}4\end{array}\right.$ We observe $(\mathrm{LHL}$ at $x=4)=\lim _{x \rightarrow 4^{-}} f(x)=\lim _{h \ri...
Read More →Which of the following are pairs of equal sets?
Question: Which of the following are pairs of equal sets? C = set of letters in the word, CATARACT. D = set of letters in the word, TRACT. Solution: Equal Sets = Two sets A and B are said to be equal if they have exactly the same elements we write A = B We have $C=$ set of letters in the word, 'CATARACT.' $\mathrm{C}=\{\mathrm{C}, \mathrm{A}, \mathrm{T}, \mathrm{R}\}$ and $D=$ set of letters in the word, 'TRACT.' $D=\{T, R, A, C\}$ Here, C = D because the elements in both the sets are equal. The...
Read More →The diameter of the base of a right circular cylinder is 42 cm and its height is 10 cm.
Question: The diameter of the base of a right circular cylinder is 42 cm and its height is 10 cm. Find the volume of the cylinder. Solution: Given : Diameter, $d=42 \mathrm{~cm}$ Radius, $r=\frac{d}{2}=21 \mathrm{~cm}$ Height, $h=10 \mathrm{~cm}$ Volume of the cylinder, $\mathrm{V}=\pi r^{2} h$ $=\frac{22}{7} \times 21^{2} \times 10$ $=13860 \mathrm{~cm}^{3}$...
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