In figure, if AB || DC and AC,
Question: In figure, if AB || DC and AC, PQ intersect each other at the point 0. Prove that OA . CQ = 0C . AP. Solution: Given AC and PQ intersect each other at the point O and AB || DC Prove that OA . CQ = 0C . AP Proof $\ln \Delta A O P$ and $\Delta C O Q$,$\angle A O P=\angle C O Q \quad$ [vertically opposite angles] $\angle A P O=\angle C Q O$ [since, $A B \| D C$ and $P Q$ is transversal, so alternate angles] $\therefore \quad \Delta A O P \sim \Delta C O Q \quad$ [by AAA similarity criteri...
Read More →Diagonals of a trapezium PQRS intersect
Question: Diagonals of a trapezium PQRS intersect each other at the point 0, PQ || RS and PQ = 3 RS. Find the ratio of the areas of Δ POQ and Δ ROS. Solution: Given PQRS is a trapezium in which PQ || PS and PQ = 3 RS $\Rightarrow$$\frac{P Q}{R S}=\frac{3}{1}$...(i) In $\triangle P O Q$ and $\triangle R O S$, $\angle S O R=\angle Q O P$ [vertically opposite angles] $\angle S R P=\angle R P Q$ [alternate angles] $\therefore \quad \Delta P O Q \sim \Delta R O S \quad$ [by AAA similarity criterion] ...
Read More →The weekly wages (in Rs.) of 30 workers in a factory are given:
Question: The weekly wages (in Rs.) of 30 workers in a factory are given: 830, 835, 890, 810, 835, 836, 869, 845, 898, 890, 820, 860, 832, 833, 855, 845, 804, 808, 812, 840, 885, 835, 835, 836, 878, 840, 868, 890, 806, 840 Mark a frequency table with intervals as 800-810, 810-820 and so on, using tally marks. Also, draw a histogram and answer the following questions: (i) Which group has the maximum number of workers? (ii) How many workers earn Rs 850 and more? (iii) How many workers earn less th...
Read More →The following histogram shows the frequency distribution f the ages of 22 teachers in a school:
Question: The following histogram shows the frequency distribution f the ages of 22 teachers in a school:(i) What is the number of eldest and youngest teachers in the school? (ii) Which age group teachers are more in the school and which least? (iii) What is the size of the classes? (iv) What are the class marks of the classes? Solution: (i) The eldest $(50-55$ years) $=1$ person This is because the height of the rectangle with class interval $50-55$ as base is 1 unit. The youngest $(20-25$ year...
Read More →In figure, if ∠1 =∠2 and ΔNSQ = ΔMTR,
Question: In figure, if 1 =2 and ΔNSQ = ΔMTR, then prove that ΔPTS ~ ΔPRQ. Solution: Given $\triangle N S Q \cong \triangle M T R$ and $\angle 1=\angle 2$ To prove $\triangle P T S \sim \Delta P R Q$ Proof Since, $\triangle N S Q \cong \triangle M T R$ So, $S Q=T R$ ...(i) Also, $\angle 1=\angle 2 \Rightarrow P T=P S$ ...(ii) [since, sides opposite to equal angles are also equal] From Eqs. (i) and (ii), $\frac{P S}{S Q}=\frac{P T}{T R}$ $\Rightarrow$ $S T \| Q R$ [by convense of basic proportion...
Read More →If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13} and C = {11, 13, 15}, and D = {15, 17}, find:
Question: If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13} and C = {11, 13, 15}, and D = {15, 17}, find: (i) $A \cap B$ (ii) $A \cap C$ (iii) $\mathbf{B} \cap \mathbf{C}$ (iv) $B \cap D$ (v) $B \cap(C \cup D)$ (vi) $A \cap(B \cup C)$ Solution: Given; A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13} and C = {11, 13, 15}, and D = {15, 17} (i) $A \cap B=\{7,9,11\}$ (ii) $A \cap C=\{11\}$ (iii) $\mathrm{B} \cap \mathrm{C}=\{11,13\}$ (iv) $B \cap D=\Phi$ or \{\} (v) $B \cap(C \cup D)=\{11,13\}$ (vi) $A \cap(B \cu...
Read More →Below is the histogram depicting marks obtained by 43 students of a class:
Question: Below is the histogram depicting marks obtained by 43 students of a class: (i) Write the number of students getting the highest marks. (ii) What is the class size? Solution: (i) In the given histogram, the interval with the highest marks is $90-100$. Three students are there in this interval because the height of the rectangle $(90-100)$ is 3 units. (ii) The class intervals are $10-20,20-30, \ldots, 90-100$. So, the class size is 10 ....
Read More →The following histogram shows the monthly wages (in Rs) of workers in a factory:
Question: The following histogram shows the monthly wages (in Rs) of workers in a factory:(i) In which wage-group the largest number of workers are being kept? What is their number? (ii) What wages are the least number of workers getting? What is the number of such workers? (iii) What is the total number of workers? (iv) What is the factory size? Solution: (i) In Fig $24.8$, the highest rectangle corresponds to the largest number of workers. The required interval is Rs $950-1000$. There are 8 wo...
Read More →Find the value of x for
Question: Find the value of $x$ for which $D E \| A B$ in given figure. Solution: Given, $D E \| A B$ $\therefore$ $\frac{C D}{A D}=\frac{C E}{B E}$ [by basic proportionality theorem] $\Rightarrow \quad \frac{x+3}{3 x+19}=\frac{x}{3 x+4}$ $\Rightarrow \quad(x+3)(3 x+4)=x(3 x+19)$ $\Rightarrow \quad 19 x-13 x=12$ $\Rightarrow \quad 6 x=12$ $\therefore$ $x=\frac{12}{6}=2$ Hence, the required value of $x$ is 2 ....
Read More →The following histogram shows the number of literate females in the age group of 10 to 40 years in a town:
Question: The following histogram shows the number of literate females in the age group of 10 to 40 years in a town:(i) Write the age group in which the number of literate female is the highest. (ii) What is the class width? (iii) What is the lowest frequency? (iv) What are the class marks of the classes? (v) In which age group literate females are the least? Solution: (i) The highest rectangle corresponds to the highest number of literate females, which is in the interval $15-20$ years. (ii) Th...
Read More →If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8} and C = {10, 11, 12, 13, 14}, find:
Question: If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8} and C = {10, 11, 12, 13, 14}, find (i) $A \cup B$ (ii) $\mathrm{B} \cup \mathrm{C}$ (iii) $\mathrm{A} \cup \mathrm{C}$ (iv) $B \cup D$ (v) $(A \cup B) \cup C$ (vi) $(A \cup B) \cap C$ (vii) $(A \cap B) \cup D$ (viii) $(A \cap B) \cup(B \cap C)$ (ix) $(A \cap C) \cap(C \cup D$ Solution: Given; A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8} and C = {10, 11, 12, 13, 14} (i) A B = {1, 2, 3, 4, 5, 6, 7, 8} (ii) $B \cup C=\{4,5,6,7,8,10,11,12,13,14\}$ (i...
Read More →In a ΔPQR, PR2 – PQ2 = QR2 and M is a point
Question: In a ΔPQR, PR2 PQ2= QR2and M is a point on side PR such that QM PR. Prove that QM2=PM MR. Solution: Given In A PQR, PR2 PQ2= QR2and QM PR To prove QM2= PM x MR Proof Since, PR2 PQ2= QR2 ⇒ PR2= PQ2+ QR2 So, $\triangle P Q R$ is right angled triangle at $Q$. In $\Delta Q M R$ and $\Delta P M Q$, $\angle M=\angle M$ [each 90] $\angle M Q R=\angle Q P M \quad$ [each equal to $90^{\circ}-\angle R$ ] $\therefore$ $\Delta Q M R \sim \Delta P M Q$ [by AAA similarity criterion] Now, using prope...
Read More →Draw a histogram to represent the following data:
Question: Draw a histogram to represent the following data: Monthly salary (in Rs) Number of teachers 56005700 8 57005800 4 58005900 3 59006000 5 60006100 2 61006200 3 62006300 1 63006400 2 Solution: Taking class intervals as bases and the corresponding frequencies as heights, the rectangles can be constructed to obtain the histogram of the given data. The class intervals are represented along thex-axis and the frequencies along they-axis on a suitable scale. The histogram representing the given...
Read More →Draw a histogram to represent the following data:
Question: Draw a histogram to represent the following data:/spanbr data-mce-bogus="1"/ppimg src="https://www.esaral.com/qdb/uploads/2022/01/31/image38116.png" alt="" Solution: Taking class intervals as bases and the corresponding frequencies as heights, the rectangles can be constructed to obtain the histogram of the given data. The class intervals are represented along thex-axis and the frequencies along they-axis on a suitable scale. The histogram representing the given data is shown below:...
Read More →Construct a histogram for the following data:
Question: Construct a histogram for the following data: Daily earnings (in Rs): 450500 500550 550600 600650 650700 Numbers of stores: 16 10 7 3 1 Solution: The class limits are represented along thex-axis and the frequencies along they-axis on a suitable scale. Taking class intervals as bases and corresponding frequencies as heights of the rectangles, the histogram of the given data can be obtained as shown in the figure below:...
Read More →Is it true to say that, if in two triangles,
Question: Is it true to say that, if in two triangles, an angle of one triangle is equal to an angle of another triangle and two sides of one triangle are proportional to the two sides of the other triangle, then the triangles are similar? Give reason for your answer. Solution: False Because, according to SAS similarity criterion, if one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the two triangles are similar. Here,...
Read More →Construct a histogram for the following data:
Question: Construct a histogram for the following data:/spanbr data-mce-bogus="1"/ppimg src="https://www.esaral.com/qdb/uploads/2022/01/31/image55688.png" alt="" Solution: The class limits are represented along thex-axis and the frequencies along they-axis on a suitable scale. Taking class intervals as bases and corresponding frequencies as heights of the rectangles, the histogram of the given data can be obtained as shown in the figure below:...
Read More →If A = {a, b, c, d, e, f}, B = {c, e, g, h} and C = {a, e, m, n}, find:
Question: If A = {a, b, c, d, e, f}, B = {c, e, g, h} and C = {a, e, m, n}, find: (i) $A \cup B$ (ii) $\mathrm{B} \cup \mathrm{C}$ (iii) $B \cup C$ (iv) $\mathrm{C} \cap \mathrm{A}$ (vi) $A \cap B$ Solution: Given; A = {a, b, c, d, e, f}, B = {c, e, g, h} and C = {a, e, m, n} (i) $A \cup B=\{a, b, c, d, e, f, g, h\}$ (ii) $B \cup C=\{a, c, e, g, h, m, n\}$ (iii) $B \cup C=\{a, c, e, g, h, m, n\}$ (iv) $C \cap A=\{a, e\}$ (vi) $A \cap B=\{c, e\}$...
Read More →In figure, if ∠D = ∠C,
Question: In figure, if D = C, then it is true that ΔADE ~ ΔACB? Why? Solution: True In ΔADE and ΔACB, A = A [common angle] D = C [given] ΔADE ~ ΔACB [by AAA similarity criterion]...
Read More →In a hypothetical sample of 20 people the amounts of money with them were found to be as follows:
Question: In a hypothetical sample of 20 people the amounts of money with them were found to be as follows: 114, 108, 100, 98, 101, 109, 117, 119, 126, 131, 136, 143, 156, 169, 182, 195, 207, 219, 235, 118. Draw the histogram of the frequency distribution (taking one of the class intervals as 50100). Solution: We first prepare the frequency table for the class intervals $50-100,100-150, \ldots, 200-250$, as shown below: The class limits are represented along thex-axis and the frequencies along t...
Read More →D is a point on side QR of ΔPQR
Question: D is a point on side QR of ΔPQR such that PD QR. Will it be correct to say that ΔPQD ~ ΔRPD? Why? Solution: False In ΔPQD and ΔRPD, PD = PD [common side] PDQ = PDR [each 90] Here, no other sides or angles are equal, so we can say that PQD is not similar to ΔRPD. But, if P = 90, then DPQ = PRD [each equal to 90 0 and by ASA similarity criterion, ΔPQD ~ΔRPD]...
Read More →Number of workshops organized by a school in different areas during the last five years are as follows:
Question: Number of workshops organized by a school in different areas during the last five years are as follows: Draw a histogram representing the above data. Solution: The class limits are represented along thex-axis and the frequencies are represented along they-axis on a suitable scale. Taking class intervals as bases and the corresponding frequencies as heights, the rectangles can be constructed to obtain histogram for the given frequency. The histogram is shown below:...
Read More →The ratio of the corresponding altitudes
Question: The ratio of the corresponding altitudes of two similar triangles is $\frac{3}{5}$. Is it correct to say that ratio of their areas is $\frac{6}{5}$ ? Why? Solution: False By the property of area of two similar triangles, $\left(\frac{\text { Area }_{1}}{\text { Area }_{2}}\right)=\left(\frac{\text { Altitude }_{1}}{\text { Altitude }_{2}}\right)^{2}$ $\Rightarrow$$\left(\frac{\text { Area }_{1}}{\text { Area }_{2}}\right)=\left(\frac{3}{5}\right)^{2}$$\left[\because \frac{\text { altit...
Read More →Draw a histogram of the following data:
Question: Draw a histogram of the following data: Class interval: 1015 1520 2025 2530 3035 3440 Frequency: 30 98 80 58 29 50 Solution: The class limits are represented along thex-axis and the frequencies are represented along they-axis on a suitable scale. Taking class intervals as bases and the corresponding frequencies as heights, the rectangles can be drawn to obtain the histogram of the given frequency distribution. The histogram is shown below:...
Read More →If in two right triangles,
Question: If in two right triangles, one of the acute angles of one triangle is equal to an acute angle of the other triangle. Can you say that two triangles will be similar? Why? Solution: True Let two right angled triangles be ΔABC and ΔPQR....
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