Draw a histogram of the following data:
Question: Draw a histogram of the following data:/spanbr data-mce-bogus="1"/ppimg src="https://www.esaral.com/qdb/uploads/2022/01/31/image31917.png" alt="" 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 →Two sides and the perimeter of one triangle
Question: Two sides and the perimeter of one triangle are respectively three times the corresponding sides and the perimeter of the other triangle. Are the two triangles similar? Why? Solution: True Here, the corresponding two sides and the perimeters of two triangles are proportional, then third side of both triangles will also in proportion....
Read More →Draw a histogram of the following data:
Question: Draw a histogram of the following data:br data-mce-bogus="1"/ppimg src="https://www.esaral.com/qdb/uploads/2022/01/31/image69205.png" alt="" 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 →Is the following statement true?
Question: Is the following statement true? Why? Two quadrilaterals are similar, if their corresponding angles are equal. Solution: False Two quadrilaterals are similar, if their corresponding angles are equal and corresponding sides must also be proportional....
Read More →In ΔPQR and ΔMST,
Question: In ΔPQR and ΔMST, P = 55, Q =25, M = 100 and S = 25. Is ΔQPR ~ ΔTSM? Why? Solution: FalseWe know that, the sum of three angles of a triangle is 180. In ΔPQR, P + Q + R = 180 ⇒ 55 + 25 + R = 180 ⇒ R = 180 (55 + 25)= 180 80 =100 In ΔTSM, T + S + M = 180 ⇒ T + 25+ 100 = 180 ⇒ T = 180-(25 +100) =180-125= 55 In ΔPQR and A TSM, and P = T, Q = S, and R = M ΔPQR ~ ΔTSM [since, all corresponding angles are equal] Hence, Δ QPR is not similar to ΔTSM, since correct correspondence is P T, Q r S an...
Read More →Draw a histogram of the following data:
Question: Draw a histogram of the following data:/spanbr data-mce-bogus="1"/ppimg src="https://www.esaral.com/qdb/uploads/2022/01/31/image47684.png" alt="" 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 →Draw a histogram of the following data:
Question: Draw a histogram of the following data:/spanbr data-mce-bogus="1"/ppimg src="https://www.esaral.com/qdb/uploads/2022/01/31/image99336.png" alt="" 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 →In figure, BD and CE intersect each
Question: In figure, BD and CE intersect each other at the point P. Is ΔPBC ~ ΔPDE? Why? Solution: True $\angle B P C=\angle E P D$ [vertically opposite angles] Now, $\frac{P B}{P D}=\frac{5}{10}=\frac{1}{2}$ $\ldots($ i) and $\frac{P C}{P E}=\frac{6}{12}=\frac{1}{2}$ ...(ii) From Eqs. (i) and (ii), $\frac{P B}{P D}=\frac{P C}{P E}$ Since, one angle of ΔPBC is equal to one angle of ΔPDE and the sides including these angles are proportional, so both triangles are similar. Hence, ΔPBC ΔPDE, by SAS...
Read More →Given below is the frequency distribution of the heights of 50 students of a class:
Question: Given below is the frequency distribution of the heights of 50 students of a class: Draw a histogram representing the above data. Solution: The class limits are represented along thex-axis on a suitable scale 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 the histogram of the given frequency distribution as shown in the figure below:...
Read More →State whether the given statement is true false:
Question: State whether the given statement is true false: (i) If $A \subset B$ and $x \notin B$ than $x \notin A$. (ii) If $A \subseteq \phi$ then $A=\phi$ (iii) If $A, B$ and $C$ are three sets such than $A \in B$ and $B \subset C$ then $A \subset C$. (iv) If $A, B$ and $C$ are three sets such than $A \subset B$ and $B \in C$ then $A \in C$. (v) If $A, B$ and $C$ are three sets such that $A \notin B$ and $B \notin C$ then $A \notin C$. (vi) If $A$ and $B$ are sets such that $x A$ and $A \in B$...
Read More →A and B are respectively the points on
Question: A and B are respectively the points on the sides PQ and PR of a ΔPQR such that PQ = 12.5 cm, PA = 5 cm, BR = 6 cm and PB = 4 cm. Is AB || QR ? Give reason for your answer. Solution: False Given, PQ = 12.5 cm, PA = 5 cm, BR = 6 cm and PB = 4 cm Then, $Q A=Q P-P A=12.5-5=7.5 \mathrm{~cm}$ Now, $\frac{P A}{A Q}=\frac{5}{7.5}=\frac{50}{75}=\frac{2}{3}$$\ldots$ (i) and $\frac{P B}{B R}=\frac{4}{6}=\frac{2}{3}$ ... (ii) From Eqs. (i) and (ii), $\frac{P A}{A Q}=\frac{P B}{B R}$ By converse of...
Read More →Construct a frequency distribution table for the following marks obtained by 25 students in a history test in class VIII of a school:
Question: Construct a frequency distribution table for the following marks obtained by 25 students in a history test in class VIII of a school: 9, 17, 12, 20, 9, 18, 25, 17, 19, 9, 12, 9, 12, 18, 17, 19, 20, 25, 9, 12, 17, 19, 19, 20, 9 (i) What is the range of marks? (ii) What is the highest mark? (iii) Which mark is occurring more frequently? Solution: The frequency distribution table is given below: (i) The range of marks is $25-9$, i.e. 16 . (ii) The highest mark is 25. (iii) The mark that o...
Read More →It is given that ΔDEF ~ ΔRPQ.
Question: It is given that ΔDEF ~ ΔRPQ. Is it true to say that D = R and F = P ? Why? Solution: False We know that, if two triangles are similar, then their corresponding angles are equal. D = R, E = P and F = Q...
Read More →Is the triangle with sides 25 cm,
Question: Is the triangle with sides 25 cm, 5 cm and 24 cm a right triangle? Give reason for your answer. Solution: False Let a = 25 cm, b = 5 cm and c =24 cm Now, b2+ c2= (5)2+ (24)2 = 25+ 576 = 601 (25)2 Hence, given sides do not make a right triangle because it does not satisfy the property of Pythagoras theorem....
Read More →Following figures relate to the weekly wages (in Rs) of 15 workers in a factory:
Question: Following figures relate to the weekly wages (in Rs) of 15 workers in a factory: 300, 250, 200, 250, 200, 150, 350, 200, 250, 200, 150, 300, 150, 200, 250 Prepare a frequency table. (i) What is the range in wages (in Rs)? (ii) How many workers are getting Rs 350? (iii) How many workers are getting the minimum wages? Solution: The frequency table for the number of accidents per day for a period of 30 days is given below: (i) The range of wages (in Rs) is $350-150$ i.e. $200 .$ (ii) From...
Read More →Discuss the continuity of the function
Question: Discuss the continuity of the function $f(x)=\left\{\begin{array}{cc}\frac{x}{|x|}, x \neq 0 \\ 0, x=0\end{array}\right.$. Solution: Given: $f(x)=\left\{\begin{array}{l}\frac{x}{|x|}, x \neq 0 \\ 0, x=0\end{array}\right.$ $|x|=\left\{\begin{array}{l}x, x \geq 0 \\ -x, x0\end{array}\right.$ $\Rightarrow f(x)=\left\{\begin{array}{c}1, x0 \\ -1, x0 \\ 0, x=0\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 \righta...
Read More →For any set A, prove that A⊆ϕ ⇔ A =ϕ
Question: For any set $A$, prove that $A \subseteq \phi \Leftrightarrow A=\phi$ Solution: Let $A \subseteq \phi$ A is a subset of the : set , then A is also an empty set. $\Rightarrow A=\phi$ Now, let $A=\phi$ ⇒ A is an empty set. Since, every set is a subset of itself $\Rightarrow A \subseteq \phi$ Hence, for any set $A, A \subseteq \phi \Leftrightarrow A=\phi$...
Read More →If S is a point on side PQ of a Δ PQR
Question: If S is a point on side PQ of a Δ PQR such that PS = QS = RS, then (a) $P R \cdot Q R=R S^{2}$ (b) $Q S^{2}+R S^{2}=Q R^{2}$ (c) $P R^{2}+Q R^{2}=P Q^{2}$ (d) $P S^{2}+R S^{2}=P R^{2}$ Solution: (c) Given, in $\triangle P Q R$, $P S=Q S=R S$ $\ldots$ (i) $\ln \Delta P S R_{1}$ $P S=R S$ [from Eq. (i)] $\Rightarrow$$\angle 1=\angle 2$ $\ldots$ (ii) Similarly, in $\Delta R S Q$, $\Rightarrow \quad \angle 3=\angle 4$ .....(iii) [corresponding angles of equal sides are equal] Now, in $\tri...
Read More →Prepare a frequency table of the following ages (in years) of 30 students of class VIII in your school:
Question: Prepare a frequency table of the following ages (in years) of 30 students of class VIII in your school: 13, 14, 13, 12, 14, 13, 14, 15, 13, 14, 13, 14, 16, 12, 14, 13, 14, 15, 16, 13, 14, 13, 12, 17, 13, 12, 13, 13, 13, 14 Solution: The frequency table of the ages of 30 students of class VII in the school is given below:...
Read More →In a study of number of accidents per day, the observations for 30 days were obtained as follows:
Question: In a study of number of accidents per day, the observations for 30 days were obtained as follows: Prepare a frequency distribution table. Solution: The frequency table for the number of accidents per day for a period of 30 days is given below:...
Read More →Prove that the function
Question: Prove that the function $f(x)=\left\{\begin{array}{ll}\frac{\sin x}{x}, x0 \\ x+1, x \geq 0\end{array}\right.$ is everywhere continuous. Solution: Whenx 0, we have $f(x)=\frac{\sin x}{x}$ We know that $\sin x$ as well as the identity function $x$ are everywhere continuous. So, the quotient function $\frac{\sin x}{x}$ is continuous at each $x0$. Now, Let us consider the point $x=0$. Given: $f(x)=\left\{\begin{array}{l}\frac{\sin x}{x}, x0 \\ x+1, x \geq 0\end{array}\right.$ We have $(\m...
Read More →Prove that A ⊆ B, B⊆ C and C⊆ A ⇒A = C.
Question: Prove that A B, B C and C A ⇒A = C. Solution: We have $A \subseteq B, B \subseteq C$ and $C \subseteq A$ Now , A is a subset of B and B is a subset of C, So A is a subset of C. Given that $C \subseteq A$. Hence, A = C....
Read More →A die was thrown 25 times and following scores were obtained:
Question: A die was thrown 25 times and following scores were obtained: Prepare a frequency table of the scores. Solution: The frequency of the scores of the die is shown below:...
Read More →Prove that the function
Question: Prove that the function $f(x)=\left\{\begin{array}{ll}\frac{\sin x}{x}, x0 \\ x+1, x \geq 0\end{array}\right.$ is everywhere continuous. Solution: Whenx 0, we have $f(x)=\frac{\sin x}{x}$ We know that $\sin x$ as well as the identity function $x$ are everywhere continuous. So, the quotient function $\frac{\sin x}{x}$ is continuous at each $x0$. Now, Let us consider the point $x=0$. Given: $f(x)=\left\{\begin{array}{l}\frac{\sin x}{x}, x0 \\ x+1, x \geq 0\end{array}\right.$ We have $(\m...
Read More →If A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 8} then find the universal set.
Question: If A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 8} then find the universal set. Solution: Elements of A+B+C = {1,3,5,2,4,6,0,8} Thus, the universal set for $A, B$ and $C=\{0,1,2,3,4,5,6,8\}$...
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