A well is dug 20 m deep and it has a diameter of 7 m.
Question: A well is dug 20 m deep and it has a diameter of 7 m. The earth which is so dug out is spread out on a rectangular plot 22 m long and 14 m broad. What is the height of the platform so formed? Solution: Height of the well =hm = 20 m Diameter of the well= dm =7 m Radius of the well =rm = 3.5 m Volume of the well = r2h =$\frac{22}{7}(3.5)^{2}(20) \mathrm{m}^{3}=770 \mathrm{~m}^{3}$ Volume of the well = Volume of the rectangular plot Length of the rectangular plot = 22 m Breadth of the rec...
Read More →The trunk of a tree is cylindrical and its circumference is 176 cm.
Question: The trunk of a tree is cylindrical and its circumference is 176 cm. If the length of the trunk is 3 m, find the volume of the timber that can be obtained from the trunk. Solution: Circumference of the tree = 176 cm = 2r Length of the trunk,h= 3 m =300 cm So, the radius(r) can be calculated by: $r=\frac{176}{2 \times \frac{22}{7}}=28 \mathrm{~cm}$ Thus, the volume (V)of the timber can be calculated using the following formula: V=r2(h) =22(28 )2(300) cm3= 739200 cm3= 0.74 m3...
Read More →How many cubic metres of earth must be dug-out to sink a well 21 m
Question: How many cubic metres of earth must be dug-out to sink a well 21 m deep and 6 m diameter? Solution: The volume of the earth that must be dug out is similar to the volume of the cylinder which is equal to r2h. Height of the well =21 m Diameter of the well= 6 m Volume of the earth that must be dug out = ( (32) (21)) m3= 594 m3...
Read More →The height of a right circular cylinder is 10.5 m.
Question: The height of a right circular cylinder is 10.5 m. Three times the sum of the areas of its two circular faces is twice the area of the curved surface. Find the volume of the cylinder. Solution: It is known that three times the sum of the areas of the two circular faces, of the right circular cylinder, is twice the area of the curved surface. Hence, it can be written using the following formula: 3 (2r2) = 2(2rh) 3r2= 2rh 3r= 2h It is known that the height of the cylinder (h)is 10.5 m. S...
Read More →Two circular cylinders of equal volumes have their heights in the ratio 1 : 2.
Question: Two circular cylinders of equal volumes have their heights in the ratio 1 : 2. Find the ratio of their radii. Solution: Here,V1= Volume of cylinder 1 V2= Volume of cylinder 2 r1= Radius of cylinder 1 r2= Radius of cylinder 2 h1= Height of cylinder 1 h2 = Height of cylinder 2 Volumes of cylinder 1 and 2 are equal. Height of cylinder 1 is half the height of cylinder 2. V1=V2 (r12h1) = (r22h2) (r12h) = (r222h) $\frac{r_{1}{ }^{2}}{r_{2}{ }^{2}}=\frac{2}{1}$ $\frac{r_{1}}{r_{2}}=\sqrt{\fr...
Read More →The curved surface area of a cylindrical pillar is 264 m
Question: The curved surface area of a cylindrical pillar is 264 m2and its volume is 924 m3. Find the diameter and the height of the pillar. Solution: Here,rm=radius of the cylinder hm= height of the cylinder Curved surface area of the cylinder = 2rh ... (1) Volume of the cylinder =r2h ... (2) 924 = r2h $h=\frac{924}{\pi r^{2}}$ Then, substitute h into equation (1): 264 =2rh $264=2 \pi r\left(\frac{924}{\pi r^{2}}\right)$ 264r= 2(924) $r=\frac{2 \times 924}{264}$ r= 7 m, sod= 14 m $h=\frac{924}{...
Read More →State whether any given set is finite or infinite:
Question: State whether any given set is finite or infinite: $G=\{x \in Z: x1] .$ Solution: Integers = -3, -2, -1, 0, 1, 2, 3, Integers less than 1 (x 1) = -4, -3, -2, -1, 0 There are infinite integers which are less than 1 the given set is infinite....
Read More →The sum of the first five terms of an AP
Question: The sum of the first five terms of an AP and the sum of the first seven terms of the same AP is 167. If the sum of the first ten terms of this AP is 235, find the sum of its first twenty terms. Solution: Let the first term, common difference and the number of terms of an AP are a, d and n, respectively. $\because$ Sum of first $n$ terms of an AP, $S_{n}=\frac{n}{2}[2 a+(n-1) d]$$\ldots$ (i) $\therefore$ Sum of first five terms of an AP, $S_{5}=\frac{5}{2}[2 a+(5-1) d]$ [from Eq. (i)] $...
Read More →State whether any given set is finite or infinite:
Question: State whether any given set is finite or infinite: $F=\{x \in R: 0x1] .$ Solution: R means set of Real numbers Real numbers include both rational and irrational numbers Real numbers between 0 and 1 are infinite. 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: E = set of all positive integers greater than 500 Solution: Positive Integers = 0, 1, 2, 3, 500 Positive Integers greater than 500 = 501, 502, 503, There are infinite positive integers which are greater than 500. 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: D = set of all leaves on a tree Solution: Here, the set is finite because infinite means never-ending. Definitely, number will be huge but by definition, it has to be finite....
Read More →The ratio between the radius of the base and the height of a cylinder is 2 : 3.
Question: The ratio between the radius of the base and the height of a cylinder is 2 : 3. Find the total surface area of the cylinder, if its volume is 1617 cm3. Solution: Letrcm be the radius andhcm be the height of the cylinder. It is given that the ratio ofrandhis 2:3, so h = 1.5rThe volume of the cylinder (V) is 1617 cm3. So, we can find the radius and the height of the cylinder from the equation given below: V=r2h 1617 =r2h 1617 =r2(1.5r) r3=343 r= 7 cm andh= 10.5 cm Total surface area = 2r...
Read More →Yasmeen saves ₹ 32 during the first month,
Question: Yasmeen saves ₹ 32 during the first month, ₹ 36 in the second month and ₹ 40 in the third month. If she continues to save in this manner, in how many months will she save ₹ 2000 ? Solution: Given that, Yasmeen, during the first month, saves = ₹ 32 During the second month, saves = ₹ 36 During the third month, saves = ₹ 40 Let Yasmeen saves ₹ 2000 during the n months. Here, we have arithmetic progression 32, 36, 40, First term (a) = 32, common difference (d) = 36 32 = 4 and she saves tot...
Read More →State whether any given set is finite or infinite:
Question: State whether any given set is finite or infinite: C = set of all lines parallel to the y-axis Solution: There are infinite lines parallel to y-axis, so the set will have infinite elements. So, the given set is infinite...
Read More →Find the value of k for which
Question: Find the value of $k$ for which $f(x)=\left\{\begin{array}{rr}\frac{1-\cos 4 x}{8 x^{2}}, \text { when } \\ k , \text { when } x=0\end{array}\right.$ is continuous at $x=0$; Solution: Given: $f(x)=\left\{\begin{array}{l}\frac{1-\cos 4 x}{8 x^{2}}, \text { when } x \neq 0 \\ k, \text { when } x=0\end{array}\right.$ Iff(x) is continuous atx= 0, then $\lim _{x \rightarrow 0} f(x)=f(0)$ $\Rightarrow \lim _{x \rightarrow 0} \frac{1-\cos 4 x}{8 x^{2}}=f(0)$ $\Rightarrow \lim _{x \rightarrow ...
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 volume of the cylinder. Solution: rcm = Radius of the cylinder hcm = Height of the cylinder Diameter of the cylinder is 21 cm. Thus, the radius is 10.5 cm. Since the curved surface area has been known, we can calculatehby the equation given below: The curved surface area of the cylinder = 2rh 1320 cm2= 2rh $1320 \mathrm{~cm}^{2}=2 \times \frac{22}{7} \times(10.5 \mathrm{~cm}) \times h$ h= 20 cm V...
Read More →Find the value of k for which
Question: Find the value of $k$ for which $f(x)=\left\{\begin{array}{rr}\frac{1-\cos 4 x}{8 x^{2}}, \text { when } \\ k , \text { when } x=0\end{array}\right.$ is continuous at $x=0$; Solution: Given: $f(x)=\left\{\begin{array}{l}\frac{1-\cos 4 x}{8 x^{2}}, \text { when } x \neq 0 \\ k, \text { when } x=0\end{array}\right.$ Iff(x) is continuous atx= 0, then $\lim _{x \rightarrow 0} f(x)=f(0)$ $\Rightarrow \lim _{x \rightarrow 0} \frac{1-\cos 4 x}{8 x^{2}}=f(0)$ $\Rightarrow \lim _{x \rightarrow ...
Read More →Kanika was given her pocket money on Jan 1st, 2008.
Question: Kanika was given her pocket money on Jan 1st, 2008. She puts ₹ 1 on day 1, ₹ 2 on day 2, ₹ 3 on day 3 and continued doing so till the end of the month, from this money into her piggy bank she also spent ₹ 204 of her pocket money, and found that at the end of the month she still had ₹ 100 with her. How much was her pocket money for the month? Solution: Let her pocket money be ₹ x. Now, she takes ₹ 1 on day 1, ₹ 2 on day 2, ₹ 3 on day 3 and so on till the end of the month, from this mone...
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 volume of the cylinder. Solution: rcm = Radius of the cylinder hcm = Height of the cylinder Diameter of the cylinder is 21 cm. Thus, the radius is 10.5 cm. Since the curved surface area has been known, we can calculatehby the equation given below: The curved surface area of the cylinder = 2rh 1320 cm2= 2rh 1320 cm2= 2 x22 x(10.5 cm) xh 7 h= 20 cm Volume of the cylinder (V) =r2h $V=\frac{22}{7}(1...
Read More →Solve this
Question: If $f(x)=\frac{2 x+3 \sin x}{3 x+2 \sin x}, x \neq 0$ is continuous at $x=0$, then find $f(0)$. Solution: Given: $f(x)=\frac{2 x+3 \sin x}{3 x+2 \sin x}, \quad x \neq 0$ If $f(x)$ is continuous at $x=0$, then $\lim _{x \rightarrow 0} f(x)=f(0)$ $\Rightarrow \lim _{x \rightarrow 0} \frac{2 x+3 \sin x}{3 x+2 \sin x}=f(0)$ $\Rightarrow \lim _{x \rightarrow 0} \frac{x\left(2+3 \frac{\sin x}{x}\right)}{x\left(3+2 \frac{\sin x}{x}\right)}=f(0)$ $\Rightarrow \lim _{x \rightarrow 0} \frac{\lef...
Read More →State whether any given set is finite or infinite:
Question: State whether any given set is finite or infinite: B = Set of all points on the circumference of a circle. Solution: There are infinite numbers of points on the circumference of the circle. so the set will have infinite elements. So, the given set is infinite....
Read More →Extend the definition of the following by continuity
Question: Extend the definition of the following by continuity $f(x)=\frac{1-\cos 7(x-\pi)}{5(x-\pi)^{2}}$ at the point $x=\pi$ Solution: Given: $f(x)=\frac{1-\cos 7(\mathrm{x}-\pi)}{5(\mathrm{x}-\pi)^{2}}, \quad x=\pi$ If $f(x)$ is continuous at $x=\pi$, then $\lim _{x \rightarrow \pi} f(x)=f(\pi)$ $\Rightarrow \lim _{x \rightarrow \pi} \frac{1-\cos 7(\mathrm{x}-\pi)}{5(\mathrm{x}-\pi)^{2}}=f(\pi)$ $\Rightarrow \frac{2}{5} \lim _{x \rightarrow \pi} \frac{\sin ^{2}\left(\frac{7(\mathrm{x}-\pi)}{...
Read More →Extend the definition of the following by continuity
Question: Extend the definition of the following by continuity $f(x)=\frac{1-\cos 7(x-\pi)}{5(x-\pi)^{2}}$ at the point $x=\pi$ Solution: Given: $f(x)=\frac{1-\cos 7(\mathrm{x}-\pi)}{5(\mathrm{x}-\pi)^{2}}, \quad x=\pi$ If $f(x)$ is continuous at $x=\pi$, then $\lim _{x \rightarrow \pi} f(x)=f(\pi)$ $\Rightarrow \lim _{x \rightarrow \pi} \frac{1-\cos 7(\mathrm{x}-\pi)}{5(\mathrm{x}-\pi)^{2}}=f(\pi)$ $\Rightarrow \frac{2}{5} \lim _{x \rightarrow \pi} \frac{\sin ^{2}\left(\frac{7(\mathrm{x}-\pi)}{...
Read More →The sum of the first n terms of an AP
Question: The sum of the first n terms of an AP whose first term is 8 and the common difference is 20 is equal to the sum of first 2n terms of another AP whose first term is 30 and the. common difference is 8. Find n. Solution: Given that, first term of the first AP (a) = 8 and common difference of the first AP (d) = 20 Let the number of terms in first AP be n. $\because$ Sum of first $n$ terms of an AP, $S_{n}=\frac{n}{2}[2 a+(n-1) d]$ $\therefore$ $S_{n}=\frac{n}{2}[2 \times 8+(n-1) 20]$ $\Rig...
Read More →State whether any given set is finite or infinite:
Question: State whether any given set is finite or infinite: A = Set of all triangles in a plane. Solution: The set of all triangles in a plane is an infinite set because in a plane there is an infinite number of triangles....
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