Prove the following
Question: If $\triangle A B C \sim \triangle Q R P, \frac{\operatorname{ar}(\triangle A B C)}{\operatorname{ar}(\Delta P Q R)}=\frac{9}{4}, A B=18 \mathrm{~cm}$ and $B C=15 \mathrm{~cm}$, then $P R$ is equal to (a) $10 \mathrm{~cm}$ (b) $12 \mathrm{~cm}$ (c) $\frac{20}{3} \mathrm{~cm}$ (d) $8 \mathrm{~cm}$ Solution: (a)Given, Δ ABC ~Δ QRP, AB = 18cm and BC = 15cm We know that, the ratio of area of two similar triangles is equal to the ratio of square of their corresponding sides. $\therefore$$\f...
Read More →Prepare a frequency table of the following scores obtained by 50 students in a test:
Question: Prepare a frequency table of the following scores obtained by 50 students in a test: Solution: The frequency table of 50 students is given below:...
Read More →If A = ϕ then find n{P(A)}.
Question: If $A=\phi$ then find $n\{P(A)\}$ Solution: We have, $A=\phi$, i.e. $A$ is a : set. Then, $n(A)=0$ $\therefore \mathrm{n}\{\mathrm{P}(\mathrm{A})\}=2^{\mathrm{m}}$, where $\mathrm{m}=\mathrm{n}(\mathrm{A})$ $\Rightarrow \mathrm{n}\{\mathrm{P}(\mathrm{A})\}=2^{0}=1$. Thus, $P(A)$ has one element....
Read More →Following data gives the number of children in 40 families:
Question: Following data gives the number of children in 40 families: , 2, 6, 5, 1, 5, 1, 3, 2, 6, 2, 3, 4, 2, 0, 0, 4, 4, 3, 2, 2, 0, 0, 1, 2, 2, 4, 3, 2, 1, 0, 5, 1, 2, 4, 3, 4, 1, 6, 2, 2. Represent it in the form of a frequency distribution. Solution: The data can be put in the form of frequency distribution in the following manner:...
Read More →The weights of new born babies (in kg) in a hospital on a particular day are as follows:
Question: The weights of new born babies (in kg) in a hospital on a particular day are as follows: 2.3, 2.2, 2.1, 2.7, 2.6, 3.0, 2.5, 2.9, 2.8, 3.1, 2.5, 2.8, 2.7, 2.9, 2.4 (i) Rearrange the weights in descending order. (ii) Determine the highest weight. (iii) Determine the lowest weight. (iv) Determine the range. (v) How many babies were born on that day? (vi) How many babies weigh below 2.5 kg? (vii) How many babies weigh more than 2.8 kg? (viii) How many babies weigh 2.8 kg? Solution: The fre...
Read More →If A = {1, {2, 3}}, find P(A) and n {P(A)}.
Question: If A = {1, {2, 3}}, find P(A) and n {P(A)}. Solution: Let {2,3} = x Now, $A=\{1, x\}$ Subsets of $A$ are $\phi,\{1\},\{x\},\{1, x\}$ $\Rightarrow$ Subsets of $A$ are $\phi,\{1\},\{2,3\},\{1,\{2,3\}\}$ Now, $n\{P(A)\}=2^{m}$, where $m=n(A)=2$ $\Rightarrow \mathrm{n}\{\mathrm{P}(\mathrm{A})\}=2^{2}=4$...
Read More →If in ΔABC and ΔDEF,
Question: If in $\triangle \mathrm{ABC}$ and $\triangle \mathrm{DEF}, \frac{A B}{D E}=\frac{B C}{F D}$, then they will be similar, when (a) B = E (b) A = D (c)B = D (d) A = F Solution: (c) Given, in $\triangle \mathrm{ABC}$ and $\triangle \mathrm{EDF}$, $\frac{A B}{D E}=\frac{B C}{F D}$ By converse of basic proportionality theorem, $\triangle A B C \sim \triangle E D F$ Then, $\angle B=\angle D, \angle A=\angle E$ and $\angle C=\angle F$...
Read More →Find the relationship between 'a' and 'b' so that the function 'f' defined by
Question: Find the relationship between 'a' and 'b' so that the function 'f' defined by $f(x)= \begin{cases}a x+1, \text { if } x \leq 3 \\ b x+3, \text { if } x \geq 3\end{cases}$ is continuous atx= 3. Solution: Given: $f(x)=\left\{\begin{array}{l}a x+1, \text { if } x \leq 3 \\ b x+3, \text { if } x3\end{array}\right.$ We have (LHL at $x=3$ ) $=\lim _{x \rightarrow 3^{-}} f(x)=\lim _{h \rightarrow 0} f(3-h)=\lim _{h \rightarrow 0} a(3-h)+1=3 a+1$ (RHL at $x=3$ ) $=\lim _{x \rightarrow 3^{+}} f...
Read More →If A = {a, b, c}, find P(A) and n{P(A)}.
Question: If A = {a, b, c}, find P(A) and n{P(A)}. Solution: The collection of all subsets of a set A is called the power set of A. It is denoted by P(A). Now, We know that $\phi$ is a subset of every set. So, $\phi$ is a subset of $\{a, b, c\}$. Also, $\{a\},\{b\},\{c\},\{a, b\},\{b, c\},\{a, c\}$ are also subsets of $\{a, b, c\}$ We know that every set is a subset of itself. So, $\{a, b, c\}$ is a subset of $\{a, b, c\}$. Thus, the set $\{a, b, c\}$ has, in all eight subsets, viz. $\phi,\{a\},...
Read More →The final marks in mathematics of 30 students are as follows:
Question: The final marks in mathematics of 30 students are as follows: (i) Arrange these marks in the ascending order, 30 to 39 one group, 40 to 49 second group etc. Now answer the following: (ii) What is the highest score? (iii) What is the lowest score? (iv) What is the range? (v) If 40 is the pass mark how many have failed? (vi) How many have scored 75 or more? (vii) Which observations between 50 and 60 have not actually appeared? (viii) How many have scored less than 50? Solution: The given...
Read More →If ΔABC ~ΔDFE, ∠A = 30°,
Question: If ΔABC ~ΔDFE, A = 30, C = 50, AB = 5 cm, AC = 8 cm and OF = 7.5 cm. Then, which of the following is true? (a) DE =12 cm, F =50 (b) DE = 12 cm, F =100 (c) EF = 12 cm, D = 100 (d) EF = 12 cm,D =30 Solution: (b)Given, AABC ~ ADFE,thenA = D = 30, C=E =50 $\therefore \quad \angle B=\angle F=180^{\circ}-\left(30^{\circ}+50^{\circ}\right)=100^{\circ}$ Also, $\quad A B=5 \mathrm{~cm}, A C=8 \mathrm{~cm}$ and $D F=7.5 \mathrm{~cm}$ $\therefore$ $\frac{A B}{D F}=\frac{A C}{D E}$ $\Rightarrow$ $...
Read More →if A = {3, {4, 5}, 6} find which of the following statements are true.
Question: if A = {3, {4, 5}, 6} find which of the following statements are true. (i) $\{4,5\} \nsubseteq \mathrm{A}$ (ii) $\{4,5\} \in A$ (iii) $\{\{4,5\}\} \subseteq A$ (iv) $4 \in A$ (v) $\{3\} \subseteq A$ (vi) $\{\phi\} \subseteq A$ (vii) $\phi \subseteq A$ (viii) $\{3,4,5\} \subseteq A$ (ix) $\{3,6\} \subseteq A$ Solution: (i) True Explanation: we have, $A=\{3,\{4,5\}, 6\}$ Let $\{4,5\}=x$ Now, $A=\{3, x, 6\}$ 4,5 is not in $A,\{4,5\}$ is an element of $A$ and element cannot be subset of se...
Read More →If the functions f(x), defined below is continuous at x=0, find the value of k.
Question: If the functions $f(x)$, defined below is continuous at $x=0$, find the value of $k$. $f(x)=\left\{\begin{array}{rr}\frac{1-\cos 2 x}{2 x^{2}}, x0 \\ k , x=0 \\ \frac{x}{|x|}, x0\end{array}\right.$ Solution: Given: $f(x)=\left\{\begin{array}{c}\frac{1-\cos 2 x}{2 x^{2}}, \quad \mathrm{x}0 \\ k, \quad x=0 \\ \frac{x}{|x|}, \quad x0\end{array}\right.$ $\Rightarrow f(x)=\left\{\begin{array}{c}\frac{1-\cos 2 x}{2 x^{2}}, \quad \mathrm{x}0 \\ k, \quad x=0 \\ 1, \quad x0\end{array}\right.$ W...
Read More →Prove the following
Question: If $\triangle A B C \sim \triangle P Q R$ with $\frac{B C}{Q R}=\frac{1}{3}$, then $\frac{\operatorname{ar}(\Delta P R Q)}{\operatorname{ar}(\Delta B C A)}$ is equal to (a) 9 (b) 3 (c) $\frac{1}{3}$ (d) $\frac{1}{9}$ Solution: (a) Given, $\triangle A B C \sim \Delta P Q R$ and $\frac{B C}{Q R}=\frac{1}{3}$ We know that, the ratio of the areas of two similar triangles is equal to square of the ratio of their corresponding sides. $\therefore$ $\frac{\operatorname{ar}(\Delta P R Q)}{\oper...
Read More →Define the following terms:
Question: Define the following terms: (i) Observations (ii) Raw data (iii) Frequency of an observation (iv) Frequency distribution (v) Discrete frequency distribution (vi) Grouped frequency distribution (vii) Class-interval (viii) Class-size (ix) Class limits (x) True class limits Solution: (i) Observation is the value at a particular period of a particular variable. (ii) Raw data is the data collected in its original form. (iii) Frequency of an observation is the number of times a certain value...
Read More →The daily minimum temperatures in degrees Celsius recorded in a certain Arctic region are as follows:
Question: The daily minimum temperatures in degrees Celsius recorded in a certain Arctic region are as follows: 12.5, 10.8, 18.6, 8.4, 10.8, 4.2, 4.8, 6.7, 13.2, 11.8, 2.3, 1.2, 2.6, 0, 2.4, 0, 3.2, 2.7, 3.4, 0, 2.4, 2.4, 0, 3.2, 2.7, 3.4,0, 2.4, 5.8, 8.9, 14.6, 12.3, 11.5, 7.8, 2.9 Represent them as frequency distribution table taking 19.9 to 15 as the first class interval. Solution: The frequency table of the daily minimum temperatures is given below:...
Read More →In Δ ABC and ΔDEF, ∠B = ∠E, ∠F = ∠C
Question: In Δ ABC and ΔDEF, B = E, F = C and AB = 30E Then, the two triangles are (a) congruent but not similar (b) similar but not congruent (c) neither congruent nor similar (d) congruent as well as similar Solution: (b)In ΔABC and ΔDEF, B = E, F = C and AB = 3DE We know that, if in two triangles corresponding two angles are same, then they are similar by AAA similarity criterion. Also, ΔA8C and ΔDEF do not satisfy any rule of congruency, (SAS, ASA, SSS), so both are not congruent....
Read More →Solve this
Question: Let $f(x)=\left\{\begin{aligned} \frac{1-\sin ^{3} x}{3 \cos ^{2} x}, \text { if } x\frac{\pi}{2} \\ a , \text { if } x=\frac{\pi}{2} . \text { If } f(x) \text { is continuous at } x=\frac{\pi}{2}, \text { find } a \text { and } b . \\ \frac{b(1-\sin x)}{(\pi-2 \mathrm{x})^{2}}, \text { if } x\frac{\pi}{2} \end{aligned}\right.$ Solution: Given: $f(x)=\left\{\begin{array}{c}\frac{1-\sin ^{3} x}{3 \cos ^{2} x}, \text { if } \mathrm{x}\frac{\pi}{2} \\ a, \text { if } x=\frac{\pi}{2} \\ \f...
Read More →Construct a frequency table with equal class intervals from the following
Question: Construct a frequency table with equal class intervals from the following data on the monthly wages (in rupees) of 28 labourers working in a factory, taking one of the class intervals as 210-230 (230 not included): 220, 268, 258, 242, 210, 268, 272, 242, 311, 290, 300, 320, 319, 304, 302, 318, 306, 292, 254, 278, 210, 240, 280, 316, 306, 215, 256, 236. Solution: The frequency table of the monthly wages of 28 labourers working in a factory is given below:...
Read More →Write each of the following intervals in the set-builder from:
Question: Write each of the following intervals in the set-builder from: (i) $A=(-2,3)$ (ii) $B=[4,10]$ (iii) $C=[-1,8)$ (iv) $D=(4,9]$ (v) $E=[-10,0)$ (vi) $F=(0,5]$ Solution: (i) $A=\{x: x \in R,-2x3\}$ (ii) $B=\{x: x \in R, 4 \leq x \leq 10\}$ (iii) $C=\{x: x \in R,-1 \leq x8\}$ (iv) $D=\{x: x \in R, 4x \leq 9\}$ (v) $E=\{x: x \in R,-10 \leq x0\}$ (vi) $F=\{x: x \in R, 0x \leq 5\}$...
Read More →If in two Δ DEF and Δ PQR,
Question: If in two Δ DEF and Δ PQR,D =Q and R = E,then which of the following is not true? (a) $\frac{E F}{P R}=\frac{D F}{P Q}$ (b) $\frac{D E}{P Q}=\frac{E F}{R P}$ (c) $\frac{D E}{Q R}=\frac{D F}{P Q}$ (d) $\frac{E F}{R P}=\frac{D E}{Q R}$ Solution: (b)Given,in ΔDEF,D =Q,R = E $\therefore$$\triangle D E F \sim \triangle Q R P$ [by AAA similarity criterion] $\Rightarrow$ $\angle F=\angle P$ [corresponding angles of similar triangles] $\therefore$ $\frac{D F}{Q P}=\frac{E D}{R Q}=\frac{F E}{P ...
Read More →The monthly wages of 30 workers in a factory are given below:
Question: The monthly wages of 30 workers in a factory are given below: 830, 835, 890, 810, 835, 836, 869, 845, 898, 890, 820, 860, 832, 833, 855, 845, 804, 808, 812, 840, 885, 835, 836, 878, 840, 868,890, 806, 840, 890. Represent the data in the form of a frequency distribution with class size 10. Solution: The frequency table of the monthly wages of 30 workers in a factory is given below:...
Read More →Express each of the following sets as an interval:
Question: Express each of the following sets as an interval: (i) $A=\{x: x \in R,-4x0\}$ (ii) $B=\{x: x \in R, 0 \leq x3\}$ (iii) $C=\{x: x \in R, 2x \leq 6\}$ (iv) $D=\{x: x \in R,-5 \leq x \leq 2\}$ (v) $E=\{x: x \in R,-3 \leq x2\}$ (vi) $F=\{x: x \in R,-2 \leq x0\}$ Solution: (i) A = (-4,0) Explanation: All the points between -4 and 0 belong to the open interval (-4,0) but -4 ,0 themselves do not belong to this interval. (ii) $B=[0,3)$ Explanation: $B=\{x: x \in R, 0 \leq x3\}$ is an open int...
Read More →The heights (in cm) of 30 students of class VIII are given below:
Question: The heights (in cm) of 30 students of class VIII are given below: 155, 158, 154, 158, 160, 148, 149, 150, 153, 159, 161, 148, 157, 153, 157, 162, 159, 151, 154, 156, 152, 156, 160, 152, 147, 155,163, 155, 157, 153. Prepare a frequency distribution table with 160-164 as one of the class intervals. Solution: The frequency table is given below:...
Read More →In figure, two line segments AC and BD intersect
Question: In figure, two line segments AC and BD intersect each other at the point P such that PA = 6 cm, PB = 3 cm, PC = 2.5 cm, PD = 5 cm, APB = 50 and CDP = 30. Then, PBA is equal to (a) 50 (b) 30 (c) 60 (d) 100 Solution: (d) $\ln \triangle A P B$ and $\triangle C P D$, $\angle A P B=\angle C P D=50^{\circ}$ [vertically opposite angles] $\frac{A P}{P D}=\frac{6}{5}$ ...(i) and $\frac{B P}{C P}=\frac{3}{2.5}=\frac{6}{5}$ ...(ii) From Eqs. (i) and (ii) $\frac{A P}{P D}=\frac{B P}{C P}$ $\theref...
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