Show that the relation R in the set R of real numbers, defined as
Question: Show that the relation R in the setRof real numbers, defined as $R=\left\{(a, b): a \leq b^{2}\right\}$ is neither reflexive nor symmetric nor transitive. Solution: $R=\left\{(a, b): a \leq b^{2}\right\}$ It can be observed that $\left(\frac{1}{2}, \frac{1}{2}\right) \notin \mathbf{R}$, since $\frac{1}{2}\left(\frac{1}{2}\right)^{2}=\frac{1}{4}$. R is not reflexive. Now, $(1,4) \in R$ as $14^{2}$ But, 4 is not less than $1^{2}$. $\therefore(4,1) \notin \mathrm{R}$ R is not symmetric. N...
Read More →Which of the following relations are functions? Give reasons. If it is a function
Question: Which of the following relations are functions? Give reasons. If it is a function, determine its domain and range. (i) $\{(2,1),(5,1),(8,1),(11,1),(14,1),(17,1)\}$ (ii) $\{(2,1),(4,2),(6,3),(8,4),(10,5),(12,6),(14,7)\}$ (iii) $\{(1,3),(1,5),(2,5)\}$ Solution: (i) $\{(2,1),(5,1),(8,1),(11,1),(14,1),(17,1)\}$ Since $2,5,8,11,14$, and 17 are the elements of the domain of the given relation having their unique images, this relation is a function. Here, domain $=\{2,5,8,11,14,17\}$ and rang...
Read More →Enthalpy of combustion of carbon to CO2 is –393.5 kJ mol–1.
Question: Enthalpy of combustion of carbon to $\mathrm{CO}_{2}$ is $393.5 \mathrm{~kJ} \mathrm{~mol}^{-1}$. Calculate the heat released upon formation of $35.2 \mathrm{~g}$ of $\mathrm{CO}_{2}$ from carbon and dioxygen gas. Solution: Formation of CO2from carbon and dioxygen gas can be represented as: $\mathrm{C}_{(s)}+\mathrm{O}_{2(g)} \longrightarrow \mathrm{CO}_{2(g)}$ $\Delta_{f} H=-393.5 \mathrm{~kJ} \mathrm{~mol}^{-1}$ $(1$ mole $=44 \mathrm{~g})$ Heat released on formation of $44 \mathrm{~...
Read More →Find the area of the shaded region in fig.,
Question: Find the area of the shaded region in fig., if ABCD is a square of side 14 cm and APD and BPC are semicircles. Solution: The area of the square $\mathrm{ABCD}=(14)^{2} \mathrm{~cm}^{2}=196 \mathrm{~cm}^{2}$ $(\because$ side of the square $14 \mathrm{~cm})$ The sum of the areas of the semicircles APD and BPC = 2 {area of semicircle APD} $(\because$ the areas of the two semicircles are equal) $=2 \times\left\{\frac{1}{2} \pi r^{2}\right\}=\pi \times\left(\frac{\text { AD }}{2}\right)^{2}...
Read More →Let R be the relation on Z defined by R = {(a, b): a, b ∈ Z, a – b is an integer}.
Question: Let $R$ be the relation on $Z$ defined by $R=\{(a, b): a, b \in Z, a-b$ is an integer\} $\}$ Find the domain and range of $R$. Solution: $\mathrm{R}=\{(a, b): a, b \in \mathbf{Z}, a-b$ is an integer $\}$ It is known that the difference between any two integers is always an integer. $\therefore$ Domain of $R=Z$ Range of $R=Z$...
Read More →Find the area of the shaded region in fig.,
Question: Find the area of the shaded region in fig., if radii of the two concentric circles with centre $\mathrm{O}$ are $7 \mathrm{~cm}$ and $14 \mathrm{~cm}$ respectively and $\angle \mathrm{AOC}=40^{\circ}$. Solution: Radius of the outer circle $=14 \mathrm{~cm}$ and $\theta=40^{\circ}$ $\therefore \quad$ Area of the sector $\mathrm{AOC}$ $=\frac{40^{\circ}}{360^{\circ}} \times \frac{22}{7} \times 14 \times 14 \mathrm{~cm}^{2}$ $=\frac{1}{9} \times 22 \times 2 \times 14 \mathrm{~cm}^{2}=\fra...
Read More →Determine whether each of the following relations are reflexive, symmetric and transitive:
Question: Determine whether each of the following relations are reflexive, symmetric and transitive: (i)Relation R in the setA= {1, 2, 313, 14} defined as $R=\{(x, y): 3 x-y=0\}$ (ii) Relation R in the setNof natural numbers defined as $R=\{(x, y): y=x+5$ and $x4\}$ (iii) Relation R in the setA= {1, 2, 3, 4, 5, 6} as $R=\{(x, y): y$ is divisible by $x\}$ (iv) Relation R in the setZof all integers defined as $R=\{(x, y): x-y$ is as integer $\}$ (v) Relation R in the setAof human beings in a town ...
Read More →Let A = {x, y, z} and B = {1, 2}.
Question: Let $A=\{x, y, z\}$ and $B=\{1,2\}$. Find the number of relations from $A$ to $B$. Solution: It is given that $A=\{x, y, z\}$ and $B=\{1,2\}$. $\therefore \mathrm{A} \times \mathrm{B}=\{(x, 1),(x, 2),(y, 1),(y, 2),(z, 1),(z, 2)\}$ Since $n(A \times B)=6$, the number of subsets of $A \times B$ is $2^{6}$. Therefore, the number of relations from $A$ to $B$ is $2^{6}$....
Read More →Calculate the enthalpy change on freezing of 1.0 mol of water at 10.0°C to ice
Question: Calculate the enthalpy change on freezing of $1.0 \mathrm{~mol}$ of water at $10.0^{\circ} \mathrm{C}$ to ice at $-10.0^{\circ} \mathrm{C} . \Delta_{\text {fus }} H=6.03 \mathrm{~kJ} \mathrm{~mol}^{-1}$ at $0^{\circ} \mathrm{C}$. $\mathrm{C}_{p}\left[\mathrm{H}_{2} \mathrm{O}(\mathrm{l})\right]=75.3 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$ $C_{\rho}\left[\mathrm{H}_{2} \mathrm{O}(\mathrm{s})\right]=36.8 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$ Solution: Total enthalpy c...
Read More →Write the relation R = {(x, x3): x is a prime number less than 10}
Question: Write the relation $\mathrm{R}=\left\{\left(x, x^{3}\right): x\right.$ is a prime number less than 10$\}$ in roster form. Solution: $\mathrm{R}=\left\{\left(x, x^{3}\right): x\right.$ is a prime number less than 10$\}$ The prime numbers less than 10 are $2,3,5$, and 7 . $\therefore R=\{(2,8),(3,27),(5,125),(7,343)\}$...
Read More →Determine the domain and range of the relation R defined by
Question: Determine the domain and range of the relation $R$ defined by $R=\{(x, x+5): x \in\{0,1,2,3,4,5\}\}$. Solution: $R=\{(x, x+5): x \in\{0,1,2,3,4,5\}\}$ $\therefore R=\{(0,5),(1,6),(2,7),(3,8),(4,9),(5,10)\}$ $\therefore$ Domain of $R=\{0,1,2,3,4,5\}$ Range of $R=\{5,6,7,8,9,10\}$...
Read More →Find the area of the shaded region in fig, if
Question: Find the area of the shaded region in fig, if PQ = 24, PR = 7 cm and O is the centre of the circle. Solution: In the figure, $\angle \mathrm{RPQ}=90^{\circ}$ (Angle subtended by a diameter on the circumference) Therefore, $\triangle R P Q$ is right angled at $P$, RP = 7 cm and PQ = 24 cm Then by Pythagoras Theorem, we have $\mathrm{QR}^{2}=\mathrm{RP}^{2}+\mathrm{PQ}^{2}$ $\mathrm{QR}^{2}=\mathrm{RP}^{2}+\mathrm{PQ}^{2}$ $=(7)^{2}+(24)^{2}=625$ $\Rightarrow \mathrm{QR}=25 \mathrm{~cm}$...
Read More →Let A = {1, 2, 3, 4, 6}. Let R be the relation on A defined by
Question: Let $A=\{1,2,3,4,6\}$. Let $R$ be the relation on $A$ defined by $\{(a, b): a, b \in A, b$ is exactly divisible by $a\}$. (i) Write R in roster form (ii) Find the domain of R (iii) Find the range of R. Solution: $\mathrm{A}=\{1,2,3,4,6\}, \mathrm{R}=\{(a, b): a, b \in \mathrm{A}, b$ is exactly divisible by $a\}$ (i) $R=\{(1,1),(1,2),(1,3),(1,4),(1,6),(2,2),(2,4),(2,6),(3,3),(3,6),(4,4),(6,6)\}$ (ii) Domain of $R=\{1,2,3,4,6\}$ (iii) Range of $R=\{1,2,3,4,6\}$...
Read More →Let A = {1, 2, 3, 4, 6}. Let R be the relation on A defined by
Question: Let $A=\{1,2,3,4,6\}$. Let $R$ be the relation on $A$ defined by $\{(a, b): a, b \in A, b$ is exactly divisible by $a\}$. (i) Write R in roster form (ii) Find the domain of R (iii) Find the range of R. Solution: $\mathrm{A}=\{1,2,3,4,6\}, \mathrm{R}=\{(a, b): a, b \in \mathrm{A}, b$ is exactly divisible by $a\}$ (i) $R=\{(1,1),(1,2),(1,3),(1,4),(1,6),(2,2),(2,4),(2,6),(3,3),(3,6),(4,4),(6,6)\}$ (ii) Domain of $R=\{1,2,3,4,6\}$ (iii) Range of $R=\{1,2,3,4,6\}$...
Read More →Calculate the number of kJ of heat necessary to raise the temperature of 60.0 g
Question: Calculate the number of $\mathrm{kJ}$ of heat necessary to raise the temperature of $60.0 \mathrm{~g}$ of aluminium from $35^{\circ} \mathrm{C}$ to $55^{\circ} \mathrm{C}$. $\mathrm{Molar}$ heat capacity of $\mathrm{Al}$ is $24 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$. Solution: From the expression of heat (q), $q=m . c \cdot \Delta T$ Where, c = molar heat capacity m= mass of substance $\Delta T=$ change in temperature Substituting the values in the expression ofq: $q=\left(\f...
Read More →The given figure shows a relationship between the sets P and Q. write this relation
Question: The given figure shows a relationship between the sets P and Q. write this relation (i) in set-builder form (ii) in roster form. What is its domain and range? Solution: According to the given figure, $P=\{5,6,7\}, Q=\{3,4,5\}$ (i) $\mathrm{R}=\{(x, y): y=x-2 ; x \in \mathrm{P}\}$ or $\mathrm{R}=\{(x, y): y=x-2$ for $x=5,6,7\}$ (ii) $R=\{(5,3),(6,4),(7,5)\}$ Domain of $R=\{5,6,7\}$ Range of $R=\{3,4,5\}$...
Read More →Tick the correct answer in the following :
Question: Tick the correct answer in the following : Area of a sector of angle p (in degree) of a circle with radius R is, (A) $\frac{\mathbf{P}}{\mathbf{1 8 0}} \times \mathbf{2} \pi \mathbf{R}$ (B) $\frac{\mathbf{p}}{\mathbf{1 8 0}} \times \pi \mathbf{R}^{\mathbf{2}}$ (C) $\frac{\mathbf{p}}{\mathbf{3 6 0}} \times \mathbf{2 \pi R}$ (D) $\frac{\mathbf{P}}{\mathbf{7 2 0}} \times \mathbf{2 \pi R}^{\mathbf{2}}$ Solution: (D) Here, radius (r) = R Angle of sector $(\theta)=p^{\circ}$ $\therefore \qua...
Read More →A = {1, 2, 3, 5} and B = {4, 6, 9}.
Question: $A=\{1,2,3,5\}$ and $B=\{4,6,9\}$. Define a relation $R$ from $A$ to $B$ by $R=\{(x, y)$ : the difference between $x$ and $y$ is odd; $x \in A, y \in B\}$. Write $R$ in roster form. Solution: $A=\{1,2,3,5\}$ and $B=\{4,6,9\}$ $\mathrm{R}=\{(x, y):$ the difference between $x$ and $y$ is odd; $x \in \mathrm{A}, y \in \mathrm{B}\}$ $\therefore R=\{(1,4),(1,6),(2,9),(3,4),(3,6),(5,4),(5,6)\}$...
Read More →The reaction of cyanamide, NH2CN(s),with dioxygen was carried out in a bomb calorimeter,
Question: The reaction of cyanamide, $\mathrm{NH}_{2} \mathrm{CN}_{(s)}$, with dioxygen was carried out in a bomb calorimeter, and $\Delta U$ was found to be $-742.7 \mathrm{~kJ}$ mol $^{-1}$ at $298 \mathrm{~K}$. Calculate enthalpy change for the reaction at $298 \mathrm{~K}$. $\mathrm{NH}_{2} \mathrm{CN}_{(\mathrm{s})}+\frac{3}{2} \mathrm{O}_{2}(\mathrm{~g}) \rightarrow \mathrm{N}_{2}(\mathrm{~g})+\mathrm{CO}_{2}(\mathrm{~g})+\mathrm{H}_{2} \mathrm{O}_{(\mathrm{l})}$ Solution: Enthalpy change ...
Read More →Define a relation R on the set N of natural numbers by R = {(x, y): y = x + 5,
Question: Define a relation $R$ on the set $N$ of natural numbers by $R=\{(x, y): y=x+5, x$ is a natural number less than $4 ; x, y \in N\}$. Depict this relationship using roster form. Write down the domain and the range. Solution: $\mathrm{R}=\{(x, y): y=x+5, x$ is a natural number less than $4, x, y \in \mathbf{N}\}$ The natural numbers less than 4 are 1, 2, and 3. $\therefore R=\{(1,6),(2,7),(3,8)\}$ The domain of R is the set of all first elementsof the ordered pairs in the relation. $\ther...
Read More →A round table cover has six equal designs as shown in fig.
Question: A round table cover has six equal designs as shown in fig. If the radius of the cover is $28 \mathrm{~cm}$, find the cost of making the designs at the rate of Rs $0.35$ per $\mathrm{cm}^{2}$. (Use $\sqrt{\mathbf{3}}=1.7$ ) Solution: Here, $r=28 \mathrm{~cm} . \theta=\frac{\mathbf{3 6 0}^{\circ}}{\mathbf{6}}=60^{\circ}$ In the figure OAB is equilateral having side 28 cm. The area of one shaded designed portion = The area of the sector OAB - The area of the $\Delta \mathrm{OAB}$ $=\left...
Read More →To warn ships for underwater rocks,
Question: To warn ships for underwater rocks, a lighthouse spreads a red coloured light over a sector of angle $80^{\circ}$ to a distance of $16.5 \mathrm{~km}$. Find the area of the sea over which the ships are warned. (Use $\pi=3.14$ ) Solution: Here, Radius (r) = 16.5 km and Sector angle $(\theta)=80^{\circ}$ $\therefore$ Area of the sea surface over which the ships are warned $=\frac{\theta}{\mathbf{3 6 0}^{\circ}} \times \pi^{2}=\frac{\mathbf{8 0}^{\circ}}{\mathbf{3 6 0}^{\circ}} \times \fr...
Read More →Let A = {1, 2, 3, … , 14}. Define a relation R from A to A by R = {(x, y): 3x – y = 0, where x, y ∈ A}.
Question: Let $A=\{1,2,3, \ldots, 14\}$. Define a relation $R$ from $A$ to $A$ by $R=\{(x, y): 3 x-y=0$, where $x, y \in A\}$. Write down its domain, codomain and range. Solution: The relation R from A to A is given as $R=\{(x, y): 3 x-y=0$, where $x, y \in A\}$ i.e., $R=\{(x, y): 3 x=y$, where $x, y \in A\}$ $\therefore R=\{(1,3),(2,6),(3,9),(4,12)\}$ The domain of R is the set of all first elementsof the ordered pairs in the relation. $\therefore$ Domain of $R=\{1,2,3,4\}$ The whole set A is t...
Read More →In a process, 701 J of heat is absorbed by a system and 394 J of work is done by the system.
Question: In a process, 701 J of heat is absorbed by a system and 394 J ofwork is done by the system. What is the change in internal energy for the process? Solution: Accordingto the first law of thermodynamics, ΔU=q+W(i) Where, ΔU= change in internal energy for a process q= heat W= work Given, q= + 701 J (Since heat is absorbed) W= 394 J (Since work is done by the system) Substituting the values in expression (i), we get ΔU= 701 J + (394 J) ΔU= 307 J Hence, the change in internal energy for the...
Read More →A car has two wipers which do not overlap.
Question: A car has two wipers which do not overlap. Each wiper has a blade of length 25 cm sweeping through an angle of 115. Find the total area cleaned at each sweep of the blades. Solution: Here, one blade of a wipe sweeps a sector area of a circle of radius 25 cm. The sector angle = 115 i.e., r = 25 cm and $\theta=115^{\circ}$ The area covered by one blade $=\frac{115}{360} \times \pi \times(25)^{2} \mathrm{~cm}^{2}$ Then, the area covered by two blades $=2 \times \frac{\mathbf{1 1 5}}{\math...
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