Discuss the continuity of the function f(x) at the point x = 0, where
Question: Discuss the continuity of the functionf(x) at the pointx= 0, where $f(x)=\left\{\begin{array}{r}x, x0 \\ 1, x=0 \\ -x, x0\end{array}\right.$ Solution: Given: $f(x)=\left\{\begin{array}{c}x, x0 \\ 1, x=0 \\ -x, x0\end{array}\right.$ (LHL at $x=0$ ) $=\lim _{x \rightarrow 0^{-}} f(x)=\lim _{h \rightarrow 0} f(0-h)=\lim _{h \rightarrow 0} f(-h)=\lim _{h \rightarrow 0}-(-h)=0$ $(\mathrm{RHL}$ at $x=0)=\lim _{x \rightarrow 0^{+}} f(x)=\lim _{h \rightarrow 0} f(0+h)=\lim _{h \rightarrow 0} f...
Read More →A cylindrical vessel open at the top has diameter 20 cm and height 14 cm.
Question: A cylindrical vessel open at the top has diameter 20 cm and height 14 cm. Find the cost of tin-plating it on the inside at the rate of 50 paise per hundred square centimetre. Solution: Given: Diameter, $d=20 \mathrm{~cm}$ Radius, $r=10 \mathrm{~cm}$ Height, $h=14 \mathrm{~cm}$ Area inside the cylindrical vessel that is to be tin $-$ plated $=S$ $S=2 \pi r h+\pi r^{2}$ $=2 \pi \times 10 \times 14+\pi \times 10^{2}$ $=280 \pi+100 \pi$ $=380 \times \frac{22}{7} \mathrm{~cm}^{2}$ $=\frac{8...
Read More →Examine the continuity of the function
Question: Examine the continuity of the function $f(x)=\left\{\begin{array}{ll}3 x-2, x \leq 0 \\ x+1, x0\end{array}\right.$ at $x=0$ Also sketch the graph of this function. Solution: The given function can be rewritten as: $f(x)=\left\{\begin{array}{c}3 x-2, x0 \\ 3(0)-2, x=0 \\ x+1, x0\end{array}\right.$ $\Rightarrow f(x)=\left\{\begin{array}{c}3 x-2, x0 \\ -2, x=0 \\ x+1, x0\end{array}\right.$ We observe $(\mathrm{LHL}$ at $x=0)=\lim _{x \rightarrow 0^{-}} f(x)=\lim _{h \rightarrow 0} f(0-h)=...
Read More →Find the cost of plastering the inner surface of a well at Rs 9.50 per
Question: Find the cost of plastering the inner surface of a well at Rs 9.50 per m2, if it is 21 m deep and diameter of its top is 6 m. Solution: Given : Height, $h=21 \mathrm{~m}$ Diameter, $d=6 \mathrm{~m}$ Radius, $r=3 \mathrm{~m} Area of the inner surface of the well, $S=2 \pi r h$ $=2 \pi \times 3 \times 21 \mathrm{~m}^{2}$ $=2 \times \frac{22}{7} \times 3 \times 21 \mathrm{~m}^{2}$ $=396 \mathrm{~m}^{2}$ According to question, the cost per $\mathrm{m}^{2}$ is Rs $9.50$. $\therefore$ Inner ...
Read More →Find the value of 'a' for which the function f defined by
Question: Find the value of 'a' for which the functionf defined by $f(x)=\left\{\begin{array}{ll}a \sin \frac{\pi}{2}(x+1), x \leq 0 \\ \frac{\tan x-\sin x}{x^{3}}, x0\end{array}\right.$ is continuous at $x=0$ Solution: Given: $f(x)=\left\{\begin{array}{l}a \sin \frac{\pi}{2}(x+1), x \leq 0 \\ \frac{\tan x-\sin x}{x^{3}}, x0\end{array}\right.$ 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 \rightarrow 0} a \...
Read More →Find the value of 'a' for which the function f defined by
Question: Find the value of 'a' for which the functionf defined by $f(x)=\left\{\begin{array}{ll}a \sin \frac{\pi}{2}(x+1), x \leq 0 \\ \frac{\tan x-\sin x}{x^{3}}, x0\end{array}\right.$ is continuous at $x=0$ Solution: Given: $f(x)=\left\{\begin{array}{l}a \sin \frac{\pi}{2}(x+1), x \leq 0 \\ \frac{\tan x-\sin x}{x^{3}}, x0\end{array}\right.$ 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 \rightarrow 0} a \...
Read More →Write each of the sets given below in set7builder from:
Question: Write each of the sets given below in set7builder from: (i) $\mathrm{A}=\left\{1, \frac{1}{4}, \frac{1}{9}, \frac{1}{16}, \frac{1}{25}, \frac{1}{36}, \frac{1}{49}\right\}$ (ii) $\mathrm{B}=\left\{\frac{1}{2}, \frac{2}{5}, \frac{3}{10}, \frac{4}{17}, \frac{5}{26}, \frac{6}{37}, \frac{7}{50}\right\} .$ (iii) $C=\{53,59,61,67,71,73,79\}$. (iv) $D=\{-1,1\}$. (v) $E=\{14,21,28,35,42, \ldots ., 98\}$. Solution: $A=\left\{1, \frac{1}{4}, \frac{1}{9}, \frac{1}{16}, \frac{1}{25}, \frac{1}{36}, ...
Read More →The height of a right circular cylinder is 10.5 cm.
Question: The height of a right circular cylinder is 10.5 cm. If three times the sum of the areas of its two circular faces is twice the area of the curved surface area. Find the radius of its base. Solution: Let $r$ be the radius of the circular cylinder. Height, $h=10.5 \mathrm{~cm}$ Area of the curved surface, $S_{1}=2 \pi r h$ $S$ um of the areas of its two circular faces, $S_{2}=2 \pi r^{2}$ According to question: $3 S_{2}=2 S_{1}$ $3 \times 2 \pi r^{2}=2 \times 2 \pi r h$ $6 r=4 h$ $3 r=2 ...
Read More →Which term of the AP 53, 48, 43,
Question: Which term of the AP 53, 48, 43, is the first negative term? Solution: Given AP is 53, 48, 43, Whose, first term (a) = 53 and common difference (d) = 48 53 = 5 Let nth term of the AP be the first negative term. i.e., $\quad T_{n}^{-}0 \quad\left[\because n\right.$th termof an AP, $\left.T_{n}=a+(n-1) d\right]$ $(a+(n-1) d)0$ $\Rightarrow \quad 53+(n-1)(-5)0$ $\Rightarrow \quad 53-5 n+50$ $\Rightarrow \quad 58-5 n0 \Rightarrow 5 n58$ $\Rightarrow \quad n11.6 \Rightarrow n=12$ i.e., 12 t...
Read More →Show that
Question: Show that $f(x)=\left\{\begin{array}{cl}\frac{\sin 3 x}{\tan 2 x} , \text { if } x0 \\ \frac{3}{2} , \text { if } x=0 \text { is continuous at } x=0 \\ \frac{\log (1+3 x)}{e^{2 x}-1}, \text { if } x0\end{array}\right.$ Solution: Given: $f(x)=\left\{\begin{array}{c}\frac{\sin 3 x}{\tan 2 x}, \text { if } \mathrm{x}0 \\ \frac{3}{2}, \text { if } \mathrm{x}=0 \\ \frac{\log (1+3 x)}{\epsilon^{2 x}-1}, \text { if } \mathrm{x}0\end{array}\right.$ We observe $(\mathrm{LHL}$ at $x=0)=\lim _{x ...
Read More →The curved surface area of a cylinder is 1320 cm
Question: The curved surface area of a cylinder is 1320 cm2and its base has diameter 21 cm. Find the height of the cylinder. Solution: Let $h$ be the height of the cylinder. Given : Curved surface area, $S=1320 \mathrm{~cm}^{2}$ Diameter, $d=21 \mathrm{~cm}$ Radius, $r=10.5$ $\mathrm{S}=2 \pi r h$ $1320=2 \pi \times 10.5 \times h$ $h=\frac{1320}{2 \pi \times 10.5}$ $h=20 \mathrm{~cm}$...
Read More →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. Prove that its height and radius are equal. Solution: Let $S_{1}$ and $S_{2}$ be the curved surface area and total surface area of the circular cylinder, respectively. Then, $S_{1}=2 \pi r h, \mathrm{~S}_{2}=2 \pi r(r+h)$ According to the question: $S_{1}: S_{2}=1: 2$ $2 \pi r h: 2 \pi r(r+h)=1: 2$ $h:(r+h)=1: 2$ $\frac{h}{r+h}=\frac{1}{2}$ $2 h=r+h$ $h=r$ Therefore, the height a...
Read More →Find the 12th term from the end
Question: Find the 12th term from the end of the AP 2, 4, 6,.,-100. Solution: Given AP -2, -4, -6,,-100 Here, first term (a) = 2, common difference (d) = 4 (-2) = -2 and the last term (l) = -100. We know,that, the nth term a of an AP from the end is a = l (n 1)d, where l is the last term and d is the common difference, 12th term from the end, a12 =-100-(12-1)(-2) = -100+ (11) (2) = 100 + 22 = 78. Hence, the 12th term from the end is -78...
Read More →List all the elements of each of the sets given below
Question: List all the elements of each of the sets given below $H=\{x: x \in Z,|x| \leq 2\}$ Solution: Given $x \in Z$ and $|x| \leq 2$ Z is a set of integers Integers are -3, -2 , -1, 0, 1, 2, 3, Now, if we take x = -3 then we have to check that it satisfies the given condition |x| 2 $|-3|=32$ So, $-3 \notin H$ If $x=-2$ then $|-2|=2[$ satisfying $|x| \leq 2]$ So, $-2 \in H$ If $x=-1$ then $|-1|=1[$ satisfying $|x| \leq 2]$ $\therefore-1 \in H$ If $x=0$ then $|0|=0$ [satisfying $|x| \leq 2$ ] ...
Read More →Show that
Question: Show that $f(x)=\left\{\begin{array}{ll}1+x^{2}, \text { if } 0 \leq x \leq 1 \\ 2-x, \text { if } x1\end{array}\right.$ is discontinuous at $x=1$ Solution: Given: $f(x)=\left\{\begin{array}{c}1+x^{2}, \text { if } 0 \leq \mathrm{x} \leq 1 \\ 2-x, \text { if } x1\end{array}\right.$ We observe $(\mathrm{LHL}$ at $x=1)=\lim _{x \rightarrow 1^{-}} f(x)=\lim _{h \rightarrow 0} f(1-h)=\lim _{h \rightarrow 0}\left(1+(1-h)^{2}\right)=\lim _{h \rightarrow 0}\left(2+h^{2}-2 h\right)=2$ (RHL at ...
Read More →If sum of the 3rd and the 8th terms
Question: If sum of the 3rd and the 8th terms of an AP is 7 and the sum of the 7th and 14th terms is -3, then find the 10th term Solution: $a_{3}+a_{8}=7$ and $a_{7}+a_{14}=-3$ $\Rightarrow \quad a+(3-1) d+a+(8-1) d=7 \quad\left[\because a_{n}=a+(n-1) d\right]$ and $\quad a+(7-1) d+a+(14-1) d=-3$ $a+2 d+a+7 d=7$ and $\quad a+6 d+a+13 d=-3$ $2 a+9 d=7$ ...(i) and $2 a+19 d=-3$ $..... (ii) On subtracting Eq. (i) from Eq. (ii), we get $10 d=-10 \Rightarrow d=-1 \quad$ [from Eq. (i)] $2 a+9(-1)=7$ $...
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 curved surface areas. Solution: Let the radii of two cylinders be $2 r$ and $3 r$, respectively, and their heights $b e 5 h$ and $3 h$, respectively. Let $\mathrm{S}_{1}$ and $\mathrm{S}_{2}$ be the curved surface areas of the two cylinder. $\mathrm{S}_{1}=$ Curved surface area of the cylinder of height $5 h$ and radius $2 r$ $\mathrm{S}_{2}=$ Curved surface area of ...
Read More →List all the elements of each of the sets given below.
Question: List all the elements of each of the sets given below. $G=\left\{x: x=\frac{1}{(2 n-1)^{\prime}} n \in N\right.$ and $1 \leq n \leq 5$ Solution: Given: $x=\frac{1}{2 n-1}$ and $1 \leq n \leq 5$ So, n = 1, 2, 3, 4, 5 If $n=1$, then $x=\frac{1}{2 n-1}=\frac{1}{2(1)-1}=\frac{1}{1}=1$ If $n=2$, then $x=\frac{1}{2 n-1}=\frac{1}{2(2)-1}=\frac{1}{4-1}=\frac{1}{3}$ If $n=3$, then $x=\frac{1}{2 n-1}=\frac{1}{2(3)-1}=\frac{1}{6-1}=\frac{1}{5}$ If $n=4$, then $x=\frac{1}{2 n-1}=\frac{1}{2(4)-1}=\...
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 total surface area of the cylinder thus generated. Solution: The rectangular sheet of paper $44 \mathrm{~cm} \times 20 \mathrm{~cm}$ is rolled along its length to form a cylinder. The height of the cylinder is $20 \mathrm{~cm}$ and circumference is $44 \mathrm{~cm}$. We have: Height, $h=20 \mathrm{~cm}$ Circumference $=2 \pi r=44 \mathrm{~cm}$ $\therefore$ Total surface area is $S=2 \pi r...
Read More →Discuss the continuity of the following functions at the indicated point(s):
Question: Discuss the continuity of the following functions at the indicated point(s): (i) $f(x)=\left\{\begin{array}{cl}|x| \cos \left(\frac{1}{x}\right), x \neq 0 \\ 0 , x=0\end{array}\right.$ at $x=0$ (ii) $f(x)=\left\{\begin{array}{cl}x^{2} \sin \left(\frac{1}{x}\right), x \neq 0 \\ 0, x=0\end{array}\right.$ at $x=0$ (iii) $f(x)=\left\{\begin{array}{cl}(x-a) \sin \left(\frac{1}{x-a}\right), x \neq a \\ 0 , x=a\end{array}\right.$ at $x=a$ (iv) $f(x)=\left\{\begin{array}{cc}\frac{e^{x}-1}{\log...
Read More →Discuss the continuity of the following functions at the indicated point(s):
Question: Discuss the continuity of the following functions at the indicated point(s): (i) $f(x)=\left\{\begin{array}{cl}|x| \cos \left(\frac{1}{x}\right), x \neq 0 \\ 0 , x=0\end{array}\right.$ at $x=0$ (ii) $f(x)=\left\{\begin{array}{cl}x^{2} \sin \left(\frac{1}{x}\right), x \neq 0 \\ 0, x=0\end{array}\right.$ at $x=0$ (iii) $f(x)=\left\{\begin{array}{cl}(x-a) \sin \left(\frac{1}{x-a}\right), x \neq a \\ 0 , x=a\end{array}\right.$ at $x=a$ (iv) $f(x)=\left\{\begin{array}{cc}\frac{e^{x}-1}{\log...
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 total surface area of the solid thus generated. Solution: Since the rectangular strip of $25 \mathrm{~cm} \times 7 \mathrm{~cm}$ is rotated about the longer side, we have: Height, $h=25 \mathrm{~cm}$ Radius, $r=7 \mathrm{~cm}$ $\therefore$ Total surface area $=2 \pi r(r+h)=2 \pi(7)(25+7)$ $=14 \pi(32)$ $=448 \pi \mathrm{cm}^{2}$ $=448 \times \frac{22}{7} \mathrm{~cm}^{2}$ $=1408 \mathrm{~cm}^{2}$...
Read More →List all the elements of each of the sets given below
Question: List all the elements of each of the sets given below $F=\left\{x: x=\in Z\right.$ and $\left.-\frac{1}{2}X \frac{13}{2}\right\}$. Solution: Given x Z and $-\frac{1}{2}x\frac{13}{2}$ It can be seen that $-\frac{1}{2}=-0.5 \ \frac{13}{2}=6.5$ We know that, Z means Set of integers $\therefore F=\{0,1,2,3,4,5,6\}$...
Read More →If the nth terms of the two AP’s 9,7,5,
Question: If the nth terms of the two APs 9,7,5,.. and 24,21,18,. are the same, then find the value of n.Also,that term. Solution: Let the first term, common difference and number of terms of the AP 9, 7, 5,area1,d1and n1 respectively. i.e., first term (a1) = 9and common difference (d1)= 7 9 = 2. $\therefore$ Its nth term, $\quad T_{n_{1}}^{\prime}=a_{1}+\left(n_{1}-1\right) d_{1}$ $\Rightarrow \quad T_{n_{1}}^{\prime}=9+\left(n_{1}-1\right)(-2)$ $\Rightarrow \quad T_{n_{1}}^{\prime}=9-2 n_{1}+2...
Read More →The circumference of the base of a cylinder is 88 cm and its height is 15 cm.
Question: The circumference of the base of a cylinder is 88 cm and its height is 15 cm. Find its curved surface area and total surface area. Solution: Given : Height, $h=15 \mathrm{~cm}$ Circumference of the base of the cylinder $=88 \mathrm{~cm}^{2}$ Let $r$ be the radius of the cylinder. The circumference of the base of the cylinder $=2 \pi r$ $88=2 \times \frac{22}{7} \times \mathrm{r}$ $\mathrm{r}=\frac{88 \times 7}{2 \times 22}=14 \mathrm{~cm}$ Curved surface area $=2 \times \pi \times \mat...
Read More →