If A = {2, 3, 5, 7, 11} and B = ϕ, find:
Question: If $A=\{2,3,5,7,11\}$ and $B=\phi$, find: (i) $A \cup B$ (ii) $A \cap B$ Solution: Given; A = {2, 3, 5, 7, 11} and B = ϕ (i) $A \cup B=\{2,3,5,7,11\}$ (ii) $A \cap B=\phi$...
Read More →If R is the set of all real numbers and Q is the set of all rational numbers then
Question: If R is the set of all real numbers and Q is the set of all rational numbers then what is the set (R Q)? Solution: Given; R is the set of all real numbers and Q is the set of all rational numbers. Then (R Q) is the set of all irrational numbers....
Read More →Draw a pie-diagram for the following data of expenditure pattern in a family:
Question: Draw a pie-diagram for the following data of expenditure pattern in a family: Items Food Clothing Rent Education Unforeseen events Midicine Expenditure (in percent) 40% 20% 10% 10% 15% 5% Solution: We know: Central angle of a component = (component value/sum of component values360) Here, the total % of items = 100 Thus, central angle for each component can be calculated as follows: Item Expenditure Sector angle Food 40% 40/100360 = 144 Clothing 20% 20/100360 = 72 Rent 10% 10/100360 = 3...
Read More →Areas of two similar triangles are 36 cm2 and 100 cm2.
Question: Areas of two similar triangles are 36 cm2and 100 cm2. If the length of a side of the larger triangle is 20 cm. Find the length of the corresponding side of the smaller triangle. Solution: Given, area of smaller triangle = 36 cm2and area of larger triangle = 100 cm2 Also, length of a side of the larger triangle = 20 cm Let length of the corresponding side of the smaller triangle = x cm By property of area of similar triangle, $\frac{\operatorname{ar}(\text { larger triangle })}{\operato...
Read More →Solve this
Question: If $A=\left\{\frac{1}{x}: x \in N\right\}$ and $\left.x8\right\}$, and $B=\left\{\frac{1}{2 x}: x \in N\right.$ and $\left.x \leq 4\right\}$, find : (i) $A \cup B$ (ii) $\mathbf{A} \cap \mathbf{B}$ (iii) $A-B$ (vi) $\mathbf{B}-\mathbf{A}$ Solution: Given; $A=\left\{\frac{1}{x}: x \in N\right\}$ and $\mathrm{x}8$ and $B=\left\{\frac{1}{2 x}: x \in N\right\}$ and $\mathrm{x} \leq 4$ According to the given conditions; $A=\left\{1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{...
Read More →In a Δ PQR, N is a point on PR,
Question: In a Δ PQR, N is a point on PR, such that QN PR. If PN . NR = QN2, then prove that PQR = 90. Solution: Given ΔPQR, N is a point on PR, such that QN PR and $P N \cdot N R=Q N^{2}$ To prove $\angle P Q R=90^{\circ}$ $\begin{array}{ll}\text { Proof We have, } P N \cdot N R=Q N^{2}\end{array}$ $\Rightarrow \quad P N \cdot N R=Q N \cdot Q N$ $\Rightarrow \quad \frac{P N}{Q N}=\frac{Q N}{N R} \quad \ldots$ (i) In $\triangle Q N P$ and $\triangle R N Q$,$\frac{P N}{Q N}=\frac{Q N}{N R}$ and $...
Read More →Percentage of the different products of a village in a particular district are given below.
Question: Percentage of the different products of a village in a particular district are given below. Draw a pie-chart representing this information. Items Wheat Pulses Jwar Grounnuts Vegetables Total % 1253 1256 252 503 253 100 Solution: We know:Central angle of a component = (component value/sum of component values360) Here, the total % of items = 100 Thus, the central angle for each component can be calculated as follows: Item In % Sector angle Wheat 125/3 41.66 41.66/100360 = 149.97 Pulses 1...
Read More →Find the points of discontinuity, if any, of the following functions:
Question: Find the points of discontinuity, if any, of the following functions: (i) $f(x)=\left\{\begin{array}{cc}x^{3}-x^{2}+2 x-2, \text { if } x \neq 1 \\ 4 \text { if } x=1\end{array}\right.$ (ii) $f(x)=\left\{\begin{array}{cc}\frac{x^{4}-16}{x-2}, \text { if } x \neq 2 \\ 16 , \text { if } x=2\end{array}\right.$ (iii) $f(x)= \begin{cases}\frac{\sin x}{x}, \text { if } x0 \\ 2 x+3, x \geq 0\end{cases}$ (iv) $f(x)=\left\{\begin{array}{cc}\frac{\sin 3 x}{x}, \text { if } x \neq 0 \\ 4 , \text ...
Read More →Percentage of the different products of a village in a particular district are given below.
Question: Percentage of the different products of a village in a particular district are given below. Draw a pie-chart representing this information. Items Wheat Pulses Jwar Grounnuts Vegetables Total % 1253 1256 252 503 253 100 Solution: We know:Central angle of a component = (component value/sum of component values360) Here, the total % of items = 100 Thus, the central angle for each component can be calculated as follows: Item In % Sector angle Wheat 125/3 41.66 41.66/100360 = 149.97 Pulses 1...
Read More →Find the points of discontinuity, if any, of the following functions:
Question: Find the points of discontinuity, if any, of the following functions: (i) $f(x)=\left\{\begin{array}{cc}x^{3}-x^{2}+2 x-2, \text { if } x \neq 1 \\ 4 \text { if } x=1\end{array}\right.$ (ii) $f(x)=\left\{\begin{array}{cc}\frac{x^{4}-16}{x-2}, \text { if } x \neq 2 \\ 16 , \text { if } x=2\end{array}\right.$ (iii) $f(x)= \begin{cases}\frac{\sin x}{x}, \text { if } x0 \\ 2 x+3, x \geq 0\end{cases}$ (iv) $f(x)=\left\{\begin{array}{cc}\frac{\sin 3 x}{x}, \text { if } x \neq 0 \\ 4 , \text ...
Read More →If A = {a, b, c, d, e}, B = {a, c, e, g} and C = {b, e, f, g}, find:
Question: If A = {a, b, c, d, e}, B = {a, c, e, g} and C = {b, e, f, g}, find: (i) $A \cap(B-C)$ (ii) $A-(B \cup C)$ (iii) $A-(B \cap C)$ Solution: Given; A = {a, b, c, d, e}, B = {a, c, e, g} and C = {b, e, f, g} (i) $A \cap(B-C)=\{a, c\}$ (ii) $A-(B \cup C)=\{d\}$ (iii) $A-(B \cap C)=\{a, b, c, d\}$...
Read More →The following table shows the expenditure incurred by a publisher in publishing a book:
Question: The following table shows the expenditure incurred by a publisher in publishing a book: Items Paper Printing Binding Advertising Miscellaneous Expenditure (in%) 35% 20% 10% 5% 30% Present the above data in the form of a pie-chart. Solution: We know: Central angle of a component = (component value/sum of component values360) Here the total % of expenditures = 100% Thus the central angle for each component can be calculated as follows: Item Expenditure (in %) Sector angle Paper 35 35/100...
Read More →If A = {2, 4, 6, 8, 10, 12}, B = {3, 4, 5, 6, 7, 8, 10}, find:
Question: If A = {2, 4, 6, 8, 10, 12}, B = {3, 4, 5, 6, 7, 8, 10}, find: (i) $(A-B)$ (ii) $(B-A)$ (iii) $(A-B) \cup(B-A)$ Solution: Given; A = {2, 4, 6, 8, 10, 12}, B = {3, 4, 5, 6, 7, 8, 10} (i) $(A-B)=\{2,12\}$ (ii) $(B-A)=\{5,7\}$ (iii) $(\mathrm{A}-\mathrm{B}) \cup(\mathrm{B}-\mathrm{A})=\Phi$ or \{\}...
Read More →Corresponding sides of two similar triangles
Question: Corresponding sides of two similar triangles are in the ratio of 2 : 3. If the area of the smaller triangle is 48 cm2, then find the area of the larger triangle. Solution: Given, ratio of corresponding sides of two similar triangles $=2: 3$ or $\frac{2}{3}$ Area of smaller triangle = 48 cm2 By the property of area of two similar triangle, Ratio of area of both riangles = (Ratio of their corresponding sides)2 i.e.,$\frac{\operatorname{ar}(\text { smaller triangle })}{\operatorname{ar}(\...
Read More →If A = {2x : x ϵ N}, 1 ≤ x < 4}, B = {x + 2) : x ϵ N and 2 ≤ x < 5} and C = {x : x ϵ N and 4 < x < 8}, find:
Question: If $A=\{2 x: x \in N\}, 1 \leq x4\}, B=\{x+2): x \in N$ and $2 \leq x5\}$ and $C=\{x: x \in N$ and $4x8\}$, find: (i) $\mathbf{A} \cap \mathbf{B}$ (ii) $\mathbf{A \cup B}$ (iii) $(\mathrm{A} \cup \mathrm{B}) \cap \mathrm{C}$ Solution: Given; A = {2x : x ϵ N}, 1 x 4}, B = {x + 2) : x ϵ N and 2 x 5} and C = {x : x ϵ N and 4 x 8} According to the given conditions; A = {2, 4, 6}, B = {4, 5, 6} and C = {5, 6, 7} (i) $A \cap B=\{4,6\}$ (ii) $A \cup B=\{2,4,5,6\}$ (iii) $(A \cup B) \cap C=\{5...
Read More →The percentages of various categories of workers in a state are given in the following table.
Question: The percentages of various categories of workers in a state are given in the following table. Categoies Culti-vators Agricultural Labourers Industrial Workers Commercial Workers Others % of workers 40 25 12.5 10 12.5 Present the information in the form a pie-chart. Solution: We know: Central angle of a component = (component value/sum of component values360) Here, total percentage of workers = 100 Thus, the central angle for each component can be calculated as follows: Category Percent...
Read More →ABCD is a trapezium in which AB || DC and P,Q
Question: ABCD is a trapezium in which AB || DC and P,Q are points on AD and BC respectively, such that PQ || DC, if PD = 18 cm, BQ = 35 cm and QC = 15 cm, find AD. Solution: Given, a trapezium ABCD in which AB || DC. P and Q are points on AD and BC, respectively such that PQ || DC. Thus, AB || PQ || DC. Join BD. $\ln \triangle A B D$ $P O \| A B$ $[\because P Q \| A B]$ By basic proportionality theorem, $\frac{D P}{A P}=\frac{D O}{O B}$ $\ldots$ (i) $\ln \Delta B D C$, $O Q \| D C$ $[\because P...
Read More →The following data shows the expenditure of a person on different items during a month.
Question: The following data shows the expenditure of a person on different items during a month. Represent the data by a pie-chart. Items of expenditure Rent Education Food Clothing Others Amount (in Rs) 2700 1800 2400 1500 2400 Solution: We know: Central angle of a component = (component value/sum of component values360) Here, total amount = Rs 10800 Thus, the central angle for each component can be calculated as follows: Item Amount (in Rs) Sector angle Rent 2700 2700/10800360 = 90 Education ...
Read More →In one day the sales (in rupees) of different items of a baker's shop are given below:
Question: In one day the sales (in rupees) of different items of a baker's shop are given below: Items Ordinary bread Fruit bread Cakes and Pastries Biscuits Others Total Sales (in Rs) 260 40 100 60 20 480 Draw a pie-chart representing the above sales. Solution: We know: Central angle of a component = (component value/sum of component values360) Here, total sales = Rs 480 Thus, the central angle for each component can be calculated as follows: Item Sale (in Rs) Sector angle Ordinary bread 260 26...
Read More →In figure, if DE || BC,
Question: In figure, if DE || BC, then find the ratio of ar (Δ ADE) and ar (DECB). Solution: Given, $D E \| B C, D E=6 \mathrm{~cm}$ and $B C=12 \mathrm{~cm}$ In $\triangle A B C$ and $\triangle A D E$, $\angle A B C=\angle A D E$ [corresponding angle] $\angle A C B=\angle A E D$ [corresponding angle] $\begin{array}{lll}\text { and } \angle A=\angle A \text { [common side] }\end{array}$ $\therefore \quad \triangle A B C \sim \triangle A E D \quad$ [by AAA similarity criterion] Then, $\frac{\oper...
Read More →Employees of a company have been categorized according to their religions as given below:
Question: Employees of a company have been categorized according to their religions as given below: Religions Hindu Muslim Sikh Christian Total Number of workers 420 300 225 105 1080 Draw a pie-chart to represent the above information. Solution: We know: Central angle of a component = (component value / sum of component values360) Here, total number of workers = 1050 Thus, the central angle for each component can be calculated as follows: Religion Number of workers Sector angle Hindu 420 420/105...
Read More →If A = {x : x ϵ N}, B = {x : x ϵ N and x is even), C = {x : x ϵ N and x is odd} and D = {x : x ϵ N and x is prime} then find:
Question: If $A=\{x: x \in N\}, B=\{x: x \in N$ and $x$ is even), $C=\{x: x \in N$ and $x$ is odd $\}$ and $D=\{x: x \in N$ and $x$ is prime $\}$ then find: (i) $A \cap B$ (ii) $A \cap C$ (iii) $A \cap D$ (iv) $B \cap C$ (v) B $\cap$ D (vi) $C \cap D$ Solution: Given; A = {x : x ϵ N}, B = {x : x ϵ N and x is even), C = {x : x ϵ N and x is odd} and D = {x : x ϵ N and x is prime} (i) $A \cap B=\{x: x \in N$ and $x$ is even $\}$ (ii) $A \cap C=\{x: x \in N$ and $x$ is odd $\}$ (iii) $A \cap D=\{x: ...
Read More →If ΔABC ~ ΔDEF, AB = 4 cm, DE = 6, EF = 9 cm
Question: If ΔABC ~ ΔDEF, AB = 4 cm, DE = 6, EF = 9 cm and FD = 12 cm, then find the perimeter of Δ ABC. Solution: Given AB = 4cm, DE = 6cm and EF = 9cm and FD = 12 cm Also, $\triangle A B C \sim \triangle D E F$ $\therefore$ $\frac{A B}{E D}=\frac{B C}{E F}=\frac{A C}{D F}$ $\Rightarrow$ $\frac{4}{6}=\frac{B C}{9}=\frac{A C}{12}$ On taking first two terms, we get $\frac{4}{6}=\frac{B C}{9}$ $\Rightarrow$ $B C=\frac{4 \times 9}{6}=6 \mathrm{~cm}$ $=A C=\frac{6 \times 12}{9}=8 \mathrm{~cm}$ Now, ...
Read More →The number of hours, spent by a school boy on different activities in a working day, is given below:
Question: The number of hours, spent by a school boy on different activities in a working day, is given below: Activities Sleep School Home Play Others Total Number of hours 8 7 4 2 3 24 Present the information in the form of a pie-chart. Solution: We know: Central angle of a component = (component value / sum of component values360) Here, total number of hours = 24 Thus, the central angle for each component can be calculated as follows: Activity Number of hours Sector angle Sleep 8 8/24360 = 12...
Read More →Find the altitude of an equilateral triangle of side 8 cm.
Question: Find the altitude of an equilateral triangle of side 8 cm. Solution: Let ABC be an equilateral triangle of side 8 cm i.e., AB = BC = CA = 8 cm. Draw altitude AD which is perpendicular to BC. Then, D is the mid-point of BC. $\therefore \quad B D=C D=\frac{1}{2} B C=\frac{8}{2}=4 \mathrm{~cm}$ Now, $\quad A B^{2}=A D^{2}+B D^{2} \quad$ [by Pythagoras theorem] $\Rightarrow \quad(8)^{2}=A D^{2}+(4)^{2}$ $\Rightarrow \quad 64=A D^{2}+16$ $\Rightarrow \quad A D^{2}=64-16=48$ $\Rightarrow \qu...
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