eSaral - JEE | NEET | Class 8-10 Exam Prep App

Draw a pie-diagram representing the relative frequencies

Question: Draw a pie-diagram representing the relative frequencies (expressed as percentage) of the eight classes as given below: 12.6, 18.2, 17.5, 20.3, 2.8, 4.2, 9.8, 14.7 Solution: We know: Central angle of a component $=\left(\right.$ component value $/$ sum of component values $\left.\times 360^{\circ}\right)$ Here, total amount = 100.1% Thus, central angle for each component can be calculated as follows: Item Amount (in %) Sector angle Class I 12.6 12.6/100.1 x360 = 45.3 Class II 18.2 18.2...

If A = {a, b, c, d, e}, B = {a, c, e, g}, verify that:

Question: If A = {a, b, c, d, e}, B = {a, c, e, g}, verify that: (i) $A \cup B=B \cup A$ (ii) $A \cup C=C \cup A$ (iii) $B \cup C=C \cup B$ (iv) $A \cap B=B \cap A$ (v) $B \cap C=C \cap B$ (vi) $A \cap C=C \cap A$ (vii) $(A \cup B \cup C=A \cup(B \cup C)$ (viii) $(A \cap B) \cap C=A \cap(B \cap C)$ Solution: (i) LHS = A B $=\{a, b, c, d, e\} \cup\{a, c, e, g\}$ $=\{a, b, c, d, e, g\}$ $=\{a, c, e, g\} \cup\{a, b, c, d, e\}$ $=B \cup A$ $=R H S$ Hence proved. (ii) To prove: A C = C A Since the el...

Represent the following data with the help of a pie-diagram:

Question: Represent the following data with the help of a pie-diagram: Items Wheat Rice Tea Production (in metric tons) 3260 1840 900 Solution: We know: Central angle of a component $=\left(\right.$ component value $/$ sum of component values $\left.\times 360^{\circ}\right)$ Here, total production = 6000 (in metric tons)Thus, the central angle for each component can be calculated as follows: Item Production(in metric tons) Sector angle Wheat 3260 3260/6000 x 360 = 195.6 Rice 1840 1840/6000 x 36...

The function f(x) is defined as follows:

Question: The functionf(x) is defined as follows: $f(x)= \begin{cases}x^{2}+a x+b, 0 \leq x2 \\ 3 x+2 , \quad 2 \leq x \leq 4 \\ 2 a x+5 b \quad 4x \leq 8\end{cases}$ If $f$ is continuous on $[0,8]$, find the values of $a$ and $b$. Solution: Given: $f$ is continuous on $[0,8]$. $\therefore f$ is continuous at $x=2$ and $x=4$ At $x=2$, we have $\lim _{x \rightarrow 2^{-}} f(x)=\lim _{h \rightarrow 0} f(2-h)=\lim _{h \rightarrow 0}\left[(2-h)^{2}+a(2-h)+b\right]=4+2 a+b$ $\lim _{x \rightarrow 2^{+...

Following data gives the break up of the cost of production of a book:

Question: Following data gives the break up of the cost of production of a book: Printing Paper Binding charges Advertisement Royalty Miscellaneous 30% 15% 15% 20% 10% 15% Draw a pie- diagram depicting the above information. Solution: We know: Central angle of a component $=\left(\right.$ component value $/$ sum of component values $\left.\times 360^{\circ}\right)$ Here, total expenditures = 105% Thus, the central angle for each component can be calculated as follows: Total : 105%Now, the pie ch...

Prove that, if a line is drawn parallel to

Question: Prove that, if a line is drawn parallel to one side of a triangle to intersect the other two sides, then the two sides are divided in the same ratio. Solution: Let a ΔABC in which a line DE parallel to SC intersects AB at D and AC at E.To prove DE divides the two sides in the same ratio. i.e.,$\frac{A D}{D B}=\frac{A E}{E C}$ Construction Join $B E, C D$ and draw $E F \perp A B$ and $D G \perp A C$. Proof Here, $\frac{\operatorname{ar}(\Delta A D E)}{\operatorname{ar}(\Delta B D E)}=\f...

Represent the following data by a pie-diagram:

Question: Represent the following data by a pie-diagram: Solution: We know: Central angle of a component $=\left(\right.$ component value $/$ sum of component values $\left.\times 360^{\circ}\right)$ Here the total expenditure of family A = 10000 and family B = 11680 Thus the central angle for each component can be calculated as follows: Total expenditure of family A: 10000Total expenditure of family B: 11680 (not 16000)Now, the pie chat representing the given data can be constructed by followin...

Find the values of a and b so that the function f(x) defined by

Question: Find the values ofaandbso that the functionf(x) defined by $f(x)= \begin{cases}x+a \sqrt{2} \sin x , \quad \text { if } 0 \leq x\pi / 4 \\ 2 x \cot x+b , \quad \text { if } \pi / 4 \leq \mathrm{x}\pi / 2 \\ a \cos 2 x-b \sin x, \text { if } \pi / 2 \leq \mathrm{x} \leq \pi\end{cases}$ becomes continuous on [0, ]. Solution: Given: $f$ is continuous on $[0, \pi]$. $\therefore f$ is continuous at $x=\frac{\pi}{4}$ and $\frac{\pi}{2}$ At $x=\frac{\pi}{4}$, we have $\lim _{x \rightarrow \fr...

The following table shows how a student spends his pocket money during the course of a month.

Question: The following table shows how a student spends his pocket money during the course of a month. Represent it by a pie-diagram. Solution: We know: Central angle of a component $=$ (component value/sum of component values $\times 360^{\circ}$ ) Here, total expenditure = 100% Thus, central angle for each component can be calculated as follows: Now, the pie chat representing the given data can be constructed by following the steps below: Step 1 : Draw circle of an appropriate radius. Step 2 ...

It is given that ΔABC ~ ΔEDF such that AB = 5 cm,

Question: It is given that ΔABC ~ ΔEDF such that AB = 5 cm,AC = 7 cm, DF = 15 cm and DE = 12 cm. Find the lengths of the remaining sides of the triangles, Solution: Given, ΔABC ~ ΔEDF, so the corresponding sides of ΔASC and ΔEDF are in the same ratio. i.e.,$\frac{A B}{E D}=\frac{A C}{E F}=\frac{B C}{D F}$$\cdots$ (i) Also, $\quad A B=5 \mathrm{~cm}, A C=7 \mathrm{~cm}$ $D F=15 \mathrm{~cm}$ and $D E=12 \mathrm{~cm}$ On putting these values in Eq. (i), we get $\frac{5}{12}=\frac{7}{E F}=\frac{B C...

Solve this

Question: The function $f(x)=\left\{\begin{array}{cc}x^{2} / a, \text { if } 0 \leq x1 \\ a, \text { if } 1 \leq x\sqrt{2} \\ \frac{2 b^{2}-4 b}{x^{2}}, \text { if } \sqrt{2} \leq x\infty\end{array}\right.$ is continuous on $(0, \infty)$, then find the most suitable values of $a$ and $b$. Solution: Given: $f$ is continuous on $(0, \infty)$ $\therefore f$ is continuous at $x=1$ and $\sqrt{2}$ At $x=1$, we have $\lim _{x \rightarrow 1^{-}} f(x)=\lim _{h \rightarrow 0} f(1-h)=\lim _{h \rightarrow 0...

if U is the universal set and A U then fill in the blanks.

Question: if U is the universal set and A U then fill in the blanks. (i) $A \cup A^{\prime}=\ldots$ (ii) $\mathbf{A} \cap \mathbf{A}^{\prime}=\ldots$ (iii) $\phi$ ' $\cap \mathrm{A}=\ldots$ (iv) $U$ ' $\cap A=\ldots$ Solution: Given: $U$ is the universal set and $A \subset U$ (i) $A \cup A^{\prime}=U$ (ii) $A \cap A^{\prime}=\Phi$ or $\}$ (iii) $\phi$ ' $\cap \mathrm{A}=\Phi$ (iv) $U^{\prime} \cap A=\Phi$ or \{\}...

if U = {a, b, c} and A = {a, c, d, e} then verify that:

Question: if U = {a, b, c} and A = {a, c, d, e} then verify that: (i) $(A \cup B)^{\prime}=\left(A^{\prime} \cap B^{\prime}\right)$ (ii) $(A \cap B)^{\prime}=\left(A^{\prime} \cup B^{\prime}\right)$ Solution: Given; U = {a, b, c} and B = {a, c, d, e} (i) $(A \cup B)^{\prime}=\left(A^{\prime} \cap B^{\prime}\right)$ (ii) $(A \cap B)^{\prime}=\left(A^{\prime} \cup B^{\prime}\right)$...

In given figure, if ∠A = ∠C, AB = 6 cm,

Question: In given figure, if A = C, AB = 6 cm, BP = 15 cm, AP = 12 cm and CP = 4 cm, then find the lengths of PD and CD. Solution: Given,A = C, AS = 6cm, BP = 15cm, AP = 12 cm and CP = 4cm In ΔAPB and ΔCPD, A =C [given] APS = CPD [vertically opposite angles] $\therefore \quad \Delta A P D \sim \triangle C P D$ $\Rightarrow \quad \frac{A P}{C P}=\frac{P B}{P D}=\frac{A B}{C D}$ $\Rightarrow \quad \frac{12}{4}=\frac{15}{P D}=\frac{6}{C D}$ On taking first two terms, we get $\frac{12}{4}=\frac{15}...

The following data gives the amount spent on the construction of a house.

Question: The following data gives the amount spent on the construction of a house. Draw a pie diagram. Items Cement Timber Bricks Labour Steel Miscellaneous Expenditure(in thousand Rs) 60 30 45 75 45 45 Solution: We know: Central angle of a component $=$ (component value/sum of component values $\times 360^{\circ}$ ) Here. the total expenditures = 300 (in thousand Rs) Thus the central angle for each component can be calculated as follows: Item Expenditure(in thousandRs) Sector angle Cement 60 6...

In the following, determine the value(s) of constant(s) involved in the definition so that the given function is continuous:

Question: In the following, determine the value(s) of constant(s) involved in the definition so that the given function is continuous: (i) $f(x)=\left\{\begin{array}{cc}\frac{\sin 2 x}{5 x}, \text { if } x \neq 0 \\ 3 k \text {, if } x=0\end{array}\right.$ (ii) $f(x)= \begin{cases}k x+5, \text { if } x \leq 2 \\ x-1, \text { if } x2\end{cases}$ (iii) $f(x)=\left\{\begin{array}{cl}k\left(x^{2}+3 x\right), \text { if } x0 \\ \cos 2 x, \text { if } x \geq 0\end{array}\right.$ (iv) $f(x)=\left\{\beg...

If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A A = {1, 2, 3, 4,}, B = {2, 4, 6, 8} and = {1, 4, 5, 6}, find:

Question: If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A A = {1, 2, 3, 4,}, B = {2, 4, 6, 8} and = {1, 4, 5, 6}, find: (i) $\mathbf{A}^{\text {' }}$ (ii) B' (iii) $\mathbf{C}^{\text {' }}$ (iv) $\left(B^{\prime}\right)^{\prime}$ (v) $(\mathrm{A} \cup \mathrm{B})^{\prime}$ (vi) $(\mathbf{A} \cap \mathbf{C})^{\prime}$ (vii) $(\mathrm{B}-\mathrm{C})^{\prime}$ Solution: Given; U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4,}, B = {2, 4, 6, 8} and C = {1, 4, 5, 6} (i) $A^{\prime}=\{5,6,7,8,9\}$ (ii) $B^{\pr...

Foot of a 10 m long ladder leaning against

Question: Foot of a 10 m long ladder leaning against a vertical wall is 6 m away from the base of the wall. Find the height of the point on the wall where the top of the ladder reaches. Solution: Let AB be a vertical wall and AC = 10 m is a ladder. The top of the ladder reaches to A and distance of ladder from the base of the wall BC is 6 m. In right angled $\triangle A B C, \quad A C^{2}=A B^{2}+B C^{2} \quad$ [by Pythagoras theorem] $\Rightarrow \quad \cdot \quad(10)^{2}=A B^{2}+(6)^{2}$ $\Rig...

In the following, determine the value(s) of constant(s) involved in the definition so that the given function is continuous:

Question: In the following, determine the value(s) of constant(s) involved in the definition so that the given function is continuous: (i) $f(x)=\left\{\begin{array}{cc}\frac{\sin 2 x}{5 x}, \text { if } x \neq 0 \\ 3 k \text {, if } x=0\end{array}\right.$ (ii) $f(x)= \begin{cases}k x+5, \text { if } x \leq 2 \\ x-1, \text { if } x2\end{cases}$ (iii) $f(x)=\left\{\begin{array}{cl}k\left(x^{2}+3 x\right), \text { if } x0 \\ \cos 2 x, \text { if } x \geq 0\end{array}\right.$ (iv) $f(x)=\left\{\beg...

Which of the following sets are pairs of disjoint sets? Justify your answer

Question: Which of the following sets are pairs of disjoint sets? Justify your answer (i) $A=\{3,4,5,6\}$ and $B=\{2,5,7,9\}$ (ii) $C=\{1,2,3,4,5\}$ and $D=\{6,7,9,11\}$ (iii) $E=\{x: x \in N, x$ is even and $x8\}$ $F=\{x: x=3 n, n \in N$, and $x4\}$ (vi) $G=\{x: x \in N, x$ is even $\}$ and $H\{x: x \in N, x$ is prime $\}$ (v) $J=\{x: x \in N, x$ is even $\}$ and $K=\{x: x \in N, x$ is odd $\}$ Solution: Disjoint sets have their intersections as Φ (i) $A=\{3,4,5,6\}$ and $B=\{2,5,7,9\}$ Are pai...

Draw a pie-diagram of the areas of continents of the world given in the following table:

Question: Draw a pie-diagram of the areas of continents of the world given in the following table: Continents Asia U.S.S.R Africa Europe Noth America South America Australia Area(in million sq. km) 26.9 20.5 30.3 4.9 24.3 17.9 8.5 Solution: We know: Central angle of a component $=\left(\right.$ component value $/$ sum of component values $\left.\times 360^{\circ}\right)$ Here, total area in million sqkm = 133.3 Thus, the central angle for each component can be calculated as follows: Continent Ar...

A 15 high tower casts a shadow 24 Long

Question: A 15 high tower casts a shadow 24 Long at a certain time and at the same time, a telephone pole casts a shadow 16 long. Find the height of the telephone pole. Solution: Let BC = 15 m be the tower and its shadow AB is 24 m. At that time CAB = 8, Again, let EF = h be a telephone pole and its shadow DE = 16 m. At the same time EDF = 8 Here, ΔASC and ΔDEF both are right angled triangles. In $\triangle A B C$ and $\triangle D E F$, $\angle C A B=\angle E D F=\theta$ $\angle B=\angle E$[each...

Draw a pie-diagram of the areas of continents of the world given in the following table:

Question: Draw a pie-diagram of the areas of continents of the world given in the following table: Continents Asia U.S.S.R Africa Europe Noth America South America Australia Area(in million sq. km) 26.9 20.5 30.3 4.9 24.3 17.9 8.5 Solution: We know: Central angle of a component $=\left(\right.$ component value $/$ sum of component values $\left.\times 360^{\circ}\right)$ Here, total area in million sqkm = 133.3 Thus, the central angle for each component can be calculated as follows: Continent Ar...

If A and B are two sets such that A ⊆ B then find:

Question: If $A$ and $B$ are two sets such that $A \subseteq B$ then find: (i) $\mathbf{A} \cup \mathbf{B}$ (ii) $\mathbf{A} \cap \mathbf{B}$ Solution: Given; $A$ and $B$ are two sets such that $A \subseteq B$. (i) $A \cup B=A$ (ii) $A \cap B=B$...

In given figure, if ∠ACB = ∠CDA,

Question: In given figure, if ACB = CDA, AC = 8 cm and AD = 3 cm, then find BD. Solution: Given, AC = 8 cm, AD = 3cm and ACB = CDA From figure, CDA = 90 ACB = CDA = 90 In right angled $\triangle A D C$, $A C^{2}=A D^{2}+C D^{2}$ $\Rightarrow \quad(8)^{2}=(3)^{2}+(C D)^{2}$ $\Rightarrow \quad 64-9=C D^{2}$ $\Rightarrow \quad C D=\sqrt{55} \mathrm{~cm}$ In $\triangle C D B$ and $\triangle A D C$, $\angle B D C=\angle A D C$[each 90] $\angle D B C=\angle D C A \quad$ [each equal to $90^{\circ}-\ang...

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