Write the following sets in roster from:
Question: Write the following sets in roster from: E = {x : x is a prime number, which is a divisor of 42}. Solution: Prime number = Those number which is divisible by 1 and the number itself. Prime numbers are 2, 3, 5, 7, 11, 13, Divisor of 42 : $42=1 \times 42$ $42=2 \times 21$ $42=3 \times 14$ $42=6 \times 7$ So, divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42 The elements which are prime and divisor of 42 are 2, 3, 7 So, E = {2, 3, 7}...
Read More →Suppose that there are two cubes, having edges 2 cm and 4 cm, respectively.
Question: Suppose that there are two cubes, having edges 2 cm and 4 cm, respectively. Find the volumes V1and V2of the cubes and compare them. Solution: The edges of the two cubes are $2 \mathrm{~cm}$ and $4 \mathrm{~cm}$. Volume of the cube of side $2 \mathrm{~cm}, \mathrm{~V}_{1}=(\text { side })^{3}=(2)^{3}=8 \mathrm{~cm}^{3}$ Volume of the cube of side $4 \mathrm{~cm}, \mathrm{~V}_{2}=(\text { side })^{3}=(4)^{3}=64 \mathrm{~cm}^{3}$ We observe the following: $\mathrm{V}_{2}=64 \mathrm{~cm}^{...
Read More →Solve this
Question: Let $f(x)=\left\{\begin{array}{ccc}\frac{1-\cos x}{x^{2}}, \text { when } x \neq 0 \\ 1 , \text { when } x=0\end{array}\right.$ Show thatf(x) is discontinuous atx= 0. Solution: Given: $f(x)=\left\{\begin{array}{l}\frac{1-\cos x}{x^{2}}, \text { when } x \neq 0 \\ 1, \quad \text { when } x=0\end{array}\right.$ Consider: $\lim _{x \rightarrow 0} f(x)=\lim _{x \rightarrow 0}\left(\frac{1-\cos x}{x^{2}}\right)$ $\Rightarrow \lim _{x \rightarrow 0} f(x)=\lim _{x \rightarrow 0}\left(\frac{2 ...
Read More →An ice-cream brick measures 20 cm by 10 cm by 7 cm.
Question: An ice-cream brick measures 20 cm by 10 cm by 7 cm. How many such bricks can be stored in deep fridge whose inner dimensions are 100 cm by 50 cm by 42 cm? Solution: Dimension of an ice cream brick $=20 \mathrm{~cm} \times 10 \mathrm{~cm} \times 7 \mathrm{~cm}$ Its volume $=$ length $\times$ breadth $\times$ height $=(20 \times 10 \times 7) \mathrm{cm}^{3}=1400 \mathrm{~cm}^{3}$ Also, it is given that the inner dimension of the deep fridge is $100 \mathrm{~cm} \times 50 \mathrm{~cm} \ti...
Read More →Write the following sets in roster from:
Question: Write the following sets in roster from D = {x : x is an integer, x2 9} Solution: Integers = , -4, -3, -2, -1, 0, 1, 2, 3, 4, $x=-4, x^{2}=(-4)^{2}=169$ $x=-3, x^{2}=(-3)^{2}=9$ $x=-2, x^{2}=(-2)^{2}=4$ $x=-1, x^{2}=(-1)^{2}=1$ $x=0, x^{2}=(0)^{2}=0$ $x=1, x^{2}=(1)^{2}=1$ $x=2, x^{2}=(2)^{2}=4$ $x=3, x^{2}=(3)^{2}=9$ $x=4, x^{2}=(4)^{2}=16$ The elements of this set are -3, -2, -1, 0, 1, 2, 3 So, D = {-3, -2, -1, 0, 1, 2, 3}...
Read More →A cube A has side thrice as long as that of cube B.
Question: A cubeAhas side thrice as long as that of cubeB. What is the ratio of the volume of cubeAto that of cubeB? Solution: Suppose that the length of the side of cube B is $l \mathrm{~cm}$. Then, the length of the side of cube $\mathrm{A}$ is $3 \times l \mathrm{~cm}$. Now, ratio $=\frac{\text { volume of cube } \mathrm{A}}{\text { volume of cube } \mathrm{B}}=\frac{(3 \times l)^{3} \mathrm{~cm}^{3}}{(l)^{3} \mathrm{~cm}^{3}}=\frac{3^{3} \times l^{3}}{l^{3}}=\frac{27}{1}$ $\therefore$ The ra...
Read More →Solve this
Question: If $f(x)=\left\{\begin{array}{ll}e^{1 / x}, \text { if } x \neq 0 \\ 1, \text { if } x=0\end{array}\right.$ find whether $f$ is continuous at $x=0$ Solution: Given: $f(x)= \begin{cases}e^{\frac{1}{x}}, \text { if } x \neq 0 \\ 1, \text { if } x=0\end{cases}$ We observe $(\mathrm{LHL}$ 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} e^{\frac{-1}{h}}=\lim _{h \rightarrow 0}\left(\frac{1}{e^{\frac{1}{h}}}\ri...
Read More →A cuboidal block of solid iron has dimensions 50 cm, 45 cm and 34 cm.
Question: A cuboidal block of solid iron has dimensions 50 cm, 45 cm and 34 cm. How many cuboids of size 5 cm by 3 cm by 2 cm can be obtained from this block? Assume cutting causes no wastage. Solution: Dimension of the cuboidal iron block $=50 \mathrm{~cm} \times 45 \mathrm{~cm} \times 34 \mathrm{~cm}$ Volume of the iron block $=$ length $\times$ breadth $\times$ height $=(50 \times 45 \times 34) \mathrm{cm}^{3}=76500 \mathrm{~cm}^{3}$ It is given that the dimension of one small cuboids is $5 \...
Read More →Find the 20th term of the AP whose 7th
Question: Find the 20th term of the AP whose 7th term is 24 less than the 11th term, first term being 12. Solution: Let the first term, common difference and number of terms of an AP are a,d and n, respectively, Given that, first term (a) = 12. Now by condition 7 th term $\left(T_{7}\right)=11$ th term $\left(T_{11}\right)-24$ $\left[\because n\right.$th term of an AP, $\left.T_{n}=a+(n-1) d\right]$ $\Rightarrow \quad a+(7-1) d=a+(11-1) d-24$ $\Rightarrow \quad a+6 d=a+10 d-24$ $\Rightarrow \qua...
Read More →Find the number of cuboidal boxes measuring 2 cm
Question: Find the number of cuboidal boxes measuring 2 cm by 3 cm by 10 cm which can be stored in a carton whose dimensions are 40 cm, 36 cm and 24 cm. Solution: Dimension of one cuboidal box $=2 \mathrm{~cm} \times 3 \mathrm{~cm} \times 10 \mathrm{~cm}$ Volume $=(2 \times 3 \times 10) \mathrm{cm}^{3}=60 \mathrm{~cm}^{3}$ It is given that the dimension of a carton is $40 \mathrm{~cm} \times 36 \mathrm{~cm} \times 24 \mathrm{~cm}$, where the boxes can be sto red. $\therefore$ Volume of the carto...
Read More →The sum of the 5 th and the 7 th terms
Question: The sum of the 5 th and the 7 th terms of an AP is 52 and the 10 th term is 46 . Find the AP. Solution: Let the first term and common difference of AP are a and d, respectively. According to the question, $a_{5}+a_{7}=52$ and $a_{10}=46$ $\Rightarrow \quad a+(5-1) d+a+(7-1) d=52 \quad\left[\because a_{n}=a+(n-1) d\right]$ and $\quad a+(10-1) d=46$ $\Rightarrow \quad a+4 d+a+6 d=52$ and $\quad a+9 a=46$ $\Rightarrow \quad 2 a+10 d=52$ and$a+9 d=46$ $\Rightarrow$ $a+5 d=26$ ...(i) $a+9 d...
Read More →A cuboidal block of silver is 9 cm long, 4 cm broad and 3.5 cm in height.
Question: A cuboidal block of silver is 9 cm long, 4 cm broad and 3.5 cm in height. From it, beads of volume 1.5 cm3each are to be made. Find the number of beads that can be made from the block. Solution: Length of the cuboidal block of silver $=9 \mathrm{~cm}$ Breadth $=4 \mathrm{~cm}$ Height $=3.5 \mathrm{~cm}$ Now, volume of the cuboidal block $=$ length $\times$ breadth $\times$ height $=9 \times 4 \times 3.5$ $=126 \mathrm{~cm}^{3}$ $\therefore$ The required number of beads of volume $1.5 \...
Read More →Solve this
Question: If $f(x)=\left\{\begin{array}{cc}\frac{\sin 3 x}{x}, \text { when } \quad x \neq 0 \\ 1, \text { when } x=0\end{array}\right.$ Find whetherf(x) is continuous atx= 0. Solution: Given: $f(x)=\left\{\begin{array}{l}\frac{\sin 3 x}{x}, \text { when } x \neq 0 \\ 1, \quad \text { when } x=0\end{array}\right.$ We observe $(\mathrm{LHL}$ at $x=0)=\lim _{\mathrm{x} \rightarrow 0} f(x)=\lim _{h \rightarrow 0} f(0-h)=\lim _{h \rightarrow 0} f(-h)$ $=\lim _{h \rightarrow 0} \frac{\sin (-3 h)}{-h}...
Read More →How many wooden cubical blocks of side 25 cm can be cut from a log of wood of size 3 m by 75 cm by 50 cm,
Question: How many wooden cubical blocks of side 25 cm can be cut from a log of wood of size 3 m by 75 cm by 50 cm, assuming that there is no wastage? Solution: The dimension of the $\log$ of wood is $3 \mathrm{~m} \times 75 \mathrm{~cm} \times 50 \mathrm{~cm}$, i. e., $300 \mathrm{~cm} \times 75 \mathrm{~cm} \times 50 \mathrm{~cm}(\because 3 \mathrm{~m}=100 \mathrm{~cm})$. $\therefore$ Volume $=300 \mathrm{~cm} \times 75 \mathrm{~cm} \times 50 \mathrm{~cm}=1125000 \mathrm{~cm}^{3}$ It is given ...
Read More →Find the weight of solid rectangular iron piece of size 50 cm × 40 cm × 10cm,
Question: Find the weight of solid rectangular iron piece of size 50 cm 40 cm 10cm, if 1 cm3of iron weighs 8 gm. Solution: The dimension of the rectangular piece of iron is $50 \mathrm{~cm} \times 40 \mathrm{~cm} \times 10 \mathrm{~cm}$. i. e., volume $=50 \mathrm{~cm} \times 40 \mathrm{~cm} \times 10 \mathrm{~cm}=20000 \mathrm{~cm}^{3}$ It is given that the weight of $1 \mathrm{~cm}^{3}$ of iron is $8 \mathrm{gm}$. $\therefore$ The weight of the given piece of iron $=20000 \times 8 \mathrm{gm}$...
Read More →Solve this
Question: If $f(x)=\left\{\begin{array}{cc}\frac{x^{2}-1}{x-1} ; \text { for } \quad x \neq 1 \\ 2 \quad ; \text { for } x=1\end{array}\right.$ Find whether $f(x)$ is continuous at $x=1$. Solution: Given: $f(x)= \begin{cases}\frac{x^{2}-1}{x-1}, \text { if } x \neq 1 \\ 2, \text { if } x=1\end{cases}$ We observe $(\mathrm{LHL}$ at $x=1)=\lim _{x \rightarrow 1^{-}} f(x)=\lim _{h \rightarrow 0} f(1-h)$ $=\lim _{h \rightarrow 0} \frac{(1-h)^{2}-1}{(1-h)-1}=\lim _{h \rightarrow 0} \frac{1+h^{2}-2 h-...
Read More →Three cuboids of dimensions 5 cm × 6 cm × 7cm,
Question: Three cuboids of dimensions 5 cm 6 cm 7cm, 4cm 7cm 8 cm and 2 cm 3 cm 13 cm are melted and a cube is made. Find the side of cube. Solution: The dimensions of the three cuboids are $5 \mathrm{~cm} \times 6 \mathrm{~cm} \times 7 \mathrm{~cm}, 4 \mathrm{~cm} \times 7 \mathrm{~cm} \times 8 \mathrm{~cm}$ and $2 \mathrm{~cm} \times 3 \mathrm{~cm} \times 13 \mathrm{~cm}$. Now, a new cube is formed by melting the given cuboids. $\therefore$ Voulume of the cube $=$ sum of the volumes of the cub...
Read More →What will happen to the volume of a cuboid if its:
Question: What will happen to the volume of a cuboid if its: (i) Length is doubled, height is same and breadth is halved? (ii) Length is doubled, height is doubled and breadth is sama? Solution: (i) Suppose that the length, breadth and height of the cuboid are $l, b$ and $h$, respectively. Then, volume $=l \times b \times h$ When its length is doubled, its length becomes $2 \times l$. When its breadth is halved, its length becomes $\frac{b}{2}$. The height $h$ remains the same. Now, volume of th...
Read More →A function f(x) is defined as
Question: A functionf(x) is defined as $f(x)=\left\{\begin{array}{rrr}\frac{x^{2}-9}{x-3} ; \text { if } x \neq 3 \\ 6 ; \text { if } x=3\end{array}\right.$ Show that $f(x)$ is continuous at $x=3$ Solution: Given: $f(x)= \begin{cases}\frac{x^{2}-9}{x-3}, \text { if } x \neq 3 \\ 6, \text { if } x=3\end{cases}$ We observe $(\mathrm{LHL}$ at $x=3)=\lim _{x \rightarrow 3^{-}} f(x)=\lim _{h \rightarrow 0} f(3-h)$ $=\lim _{h \rightarrow 0} \frac{(3-h)^{2}-9}{(3-h)-3}=\lim _{h \rightarrow 0} \frac{3^{...
Read More →The 26 th, 11 th and the last terms of an AP are,
Question: The 26 th, 11 th and the last terms of an AP are, 0,3 and $\frac{-1}{6}$, respectively. Find the common difference and the number of terms. Solution: Let the first term, common difference and number of terms of an AP are a, d and n, respectively. We know that, if last term of an AP is known, then $l=a+(n-1) d$...(i) and $n$th term of an AP is $T_{n}=a+(n-1) d$ ...(ii) Given that, 26 th term of an $\mathrm{AP}=0$ $\Rightarrow \quad T_{26}=a+(26-1) d=0 \quad$ [from Eq. (i)] $\Rightarrow ...
Read More →What will happen to the volume of a cube, if its edge is
Question: What will happen to the volume of a cube, if its edge is (i) halved (ii) trebled? Solution: (i) Suppose that the length of the edge of the cube is $x$. Then, volume of the cube $=(\text { side })^{3}=x^{3}$ When the length of the side is halved, the length of the new edge becomes $\frac{x}{2}$. Now, volume of the new cube $=(\text { side })^{3}=\left(\frac{x}{2}\right)^{3}=\frac{x^{3}}{2^{3}}=\frac{x^{3}}{8}=\frac{1}{8} \times x^{3}$ It means that if the edge of a cube is halved, its n...
Read More →A cuboidal wooden block contains 36 cm
Question: A cuboidal wooden block contains 36 cm3wood. If it be 4 cm long and 3 cm wide, find its height. Solution: A cuboidal wooden block contains $36 \mathrm{~cm}^{3}$ of wood. i. e., volume $=36 \mathrm{~cm}^{3}$ Length of the block $=4 \mathrm{~cm}$ Breadth of block $=3 \mathrm{~cm}$ Suppose that the height of the block is $h \mathrm{~cm}$ Now, volume of a cuboid $=$ lenght $\times$ breadth $\times$ height $\Rightarrow 36=4 \times 3 \times h$ $\Rightarrow 36=12 \times h$ $\Rightarrow h=\fra...
Read More →A function f(x) is defined as,
Question: A functionf(x) is defined as, $f(x)=\left\{\begin{array}{rr}\frac{x^{2}-x-6}{x-3} ; \text { if } \quad x \neq 3 \\ 5 ; \text { if } x=3\end{array}\right.$ Show thatf(x) is continuous thatx= 3. Solution: Given: $f(x)=\left\{\begin{array}{l}\frac{x^{2}-x-6}{x-3}, x \neq 3 \\ 5, \quad x=3\end{array}\right.$ We observe $(\mathrm{LHL}$ at $x=3)=\lim _{x \rightarrow 3} f(x)=\lim _{h \rightarrow 0} f(3-h)$ $=\lim _{h \rightarrow 0} \frac{(3-h)^{2}-(3-h)-6}{(3-h)-3}=\lim _{h \rightarrow 0} \fr...
Read More →A milk container is 8 cm long and 50 cm wide.
Question: A milk container is 8 cm long and 50 cm wide. What should be its height so that it can hold 4 litres of milk? Solution: Length of the cuboidal milk container $=8 \mathrm{~cm}$ Breadth $=50 \mathrm{~cm}$ Let $h \mathrm{~cm}$ be the height of the container. It is given that the container can hold $4 \mathrm{~L}$ of milk. i. e., volume $=4 \mathrm{~L}=4 \times 1000 \mathrm{~cm}^{3}=4000 \mathrm{~cm}^{3}\left(\because 1 \mathrm{~L}=1000 \mathrm{~cm}^{3}\right)$ Now, volume of the container...
Read More →A cuboidal vessel is 10 cm long and 5 cm wide.
Question: A cuboidal vessel is 10 cm long and 5 cm wide. How high it must be made to hold 300 cm3of a liquid? Solution: Let $h \mathrm{~cm}$ be the height of the cuboidal vessel. Given : Length $=10 \mathrm{~cm}$ Breadth $=5 \mathrm{~cm}$ Volume of the vessel $=300 \mathrm{~cm}^{3}$ Now, volume of a cuboid $=$ length $\times$ breadth $\times$ height $\Rightarrow 300=10 \times 5 \times h$ $\Rightarrow 300=50 \times h$ $\therefore h=\frac{300}{50}=6 \mathrm{~cm}$...
Read More →