Which of the following experiments have equally likely outcomes?.

Question: Which of the following experiments have equally likely outcomes? Explain. (i) A driver attempts to start a car. The car starts or does not start. (ii) A player attempts to shoot a basketball. She/he shoots or misses the shot. (iii) A trial is made to answer a true-false question. The answer is right or wrong. (iv) A baby is born. It is a boy or a girl. Solution: (i) It is not an equally likely event, as it depends on various factors such as whether the car will start or not. And factor...

Describe the effect of:

Question: Describe the effect of: a) Addition of $\mathrm{H}_{2}$ b) Addition of $\mathrm{CH}_{3} \mathrm{OH}$ c) Removal of CO d) Removal of $\mathrm{CH}_{3} \mathrm{OH}$ on the equilibrium of the reaction: $2 \mathrm{H}_{2}(\mathrm{~g})+\mathrm{CO}(\mathrm{g}) \longleftrightarrow \mathrm{CH}_{3} \mathrm{OH}(\mathrm{g})$ Solution: (a) According to Le Chatelier's principle, on addition of $\mathrm{H}_{2}$, the equilibrium of the given reaction will shift in the forward direction. (b) On addition...

Question: Solution:...

Prove that if a positive integer is of the form 6q + 5, then it is of the form 3q + 2 for some integer q, but not conversely.

Question: Prove that if a positive integer is of the form 6q+ 5, then it is of the form 3q+ 2 for some integerq, but not conversely. Solution: To Prove: that if a positive integer is of the form 6q+ 5 then it is of the form 3q+ 2 for some integerq, but not conversely. Proof: Letn= 6q+ 5 Since any positive integer n is of the form of 3kor 3k+ 1, 3k+ 2 Ifq= 3k Then, $n=6 q+5$ $\begin{aligned} \Rightarrow \quad n =18 k+5(q=3 k) \\ \Rightarrow \quad n =3(6 k+1)+2 \\ \Rightarrow \quad n =3 m+2(\text ...

Dihydrogen gas is obtained from natural gas by partial oxidation with steam as per following endothermic reaction:

Question: Dihydrogen gas is obtained from natural gas by partial oxidation with steam as per following endothermic reaction: $\mathrm{CH}_{4}(\mathrm{~g})+\mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \longleftrightarrow \mathrm{CO}(\mathrm{g})+3 \mathrm{H}_{2}(\mathrm{~g})$ (a) Write as expression forKpfor the above reaction. (b) How will the values ofKpand composition of equilibrium mixture be affected by (i) Increasing the pressure (ii) Increasing the temperature (iii) Using a catalyst? Solution: (a)...

Complete the following statements:

Question: Complete the following statements: (i) Probability of an event E + Probability of the event not E = _______. (ii) The probability of an event that cannot happen is _________. Such as event is called _________. (iii) The probability of an event that is certain to happen is _________. Such as event is called ________. (iv) The sum of the probabilities of all the elementary events of an experiment is _________. (v) The probability of an event is greater than or equal to _______ and less t...

The equilibrium constant for the following reaction is 1.6 ×105 at 1024 K.

Question: The equilibrium constant for the following reaction is $1.6 \times 10^{5}$ at $1024 \mathrm{~K}$. $\mathrm{H}_{2}(\mathrm{~g})+\mathrm{Br}_{2}(\mathrm{~g}) \longleftrightarrow 2 \mathrm{HBr}(\mathrm{g})$ Find the equilibrium pressure of all gases if $10.0$ bar of $\mathrm{HBr}$ is introduced into a sealed container at $1024 \mathrm{~K}$. Solution: Given, $K_{\mathrm{p}}$ for the reaction i.e., $\mathrm{H}_{2(g)}+\mathrm{Br}_{2(g)} \longleftrightarrow 2 \mathrm{HBr}_{(g)}$ is $1.6 \time...

For any positive integer $n$, prove that $n^{3}-n$ divisible by 6 .

Question: For any positive integer $n$, prove that $n^{3}-n$ divisible by 6 . Solution: To Prove: For any positive integer $n, n^{3}-n$ is divisible by $6 .$ Proof: Letnbe any positive integer. $\Rightarrow n^{3}-n=(n-1)(n)(n+1)$ Since any positive integer is of the form 6qor 6q+ 1 or 6q+ 2 or 6q+ 3 or 6q+ 4, 6q+ 5 Ifn= 6q Then, $(n-1) n(n+1)=(6 q-1) 6 q(6 q+1)$ Ifn= 6q+ 1 Then, $(n-1) n(n+1)=(6 q+1)(6 q+2)(6 q+3)$ $\Rightarrow(n-1) n(n+1)=6(6 q+1)(3 q+1)(2 q+1)$ which is divisble by 6 Similarly...

The following table gives production yield per hectare of wheat of 100 farms of a village.

Question: The following table gives production yield per hectare of wheat of 100 farms of a village. Change the distribution to a more than type distribution and draw its ogive. Solution: Now, we will draw the ogive by plotting the points (50,100), (55,98), (60,90), (65,78), (70,54) and (75,16). Join these points by a freehand to get an ogive of 'more than' type....

Let f: X → Y be an invertible function.

Question: Let $f: X \rightarrow Y$ be an invertible function. Show that the inverse of $f^{-1}$ is $f$, i.e., $\left(f^{-1}\right)^{-1}=f .$ Solution: Let $f: X \rightarrow Y$ be an invertible function. Then, there exists a functiong:YXsuch thatgof= IXandfog= IY. Here, $f^{-1}=g$ Now, gof $=\left.\right|_{x}$ and fog $=\left.\right|_{y}$ $\Rightarrow f^{-1}$ of $=\left.\right|_{X}$ and $f \circ f^{-1}=\left.\right|_{Y}$ Hence, $f^{-1}: Y \rightarrow X$ is invertible and $f$ is the inverse of $f^...

During the medial check up of 35 students of a class,

Question: During the medial check up of 35 students of a class, their weights were recorded as follows Draw a less than type ogive for the given data. Hence obtain the median weight from the graph and verify the result by using the formula. Solution: To draw the 'less than' type ogive, we plot the points (38, 0), (40, 3), (42, 5), (44, 9), (46, 14), (48, 28), (50, 32) and (52, 35) on the graph. Median from the graph = 46.5 kg. median class is (46-48). (See in the table) We have $\ell=46, \mathrm...

Consider f: {1, 2, 3} → {a, b, c} given by f(1) = a, f(2) = b and f(3) = c.

Question: Consider $t:\{1,2,3\} \rightarrow\{a, b, c\}$ given by $t(1)=a, t(2)=b$ and $t(3)=c$. Find $f^{-1}$ and show that $\left(f^{-1}\right)^{-1}=f$. Solution: Function $f:\{1,2,3\} \rightarrow\{a, b, c\}$ is given by, $f(1)=a, f(2)=b$, and $f(3)=c$ If we define $g:\{a, b, c\} \rightarrow\{1,2,3\}$ as $g(a)=1, g(b)=2, g(c)=3$, then we have: Thus, the inverse of $f$ exists and $f^{-1}=g$. $\therefore f^{-1}:\{a, b, c\} \rightarrow\{1,2,3\}$ is given by, $f^{-1}(a)=1, f^{-1}(b)=2, f^{-1}(c)=3$...

The following distribution gives the daily income of 50 workers of a factory.

Question: The following distribution gives the daily income of 50 workers of a factory. Convert the distribution above to a less than type cumulative frequency distribution and draw its ogive. Solution: n = 50 gives n/2 = 25 On the graph, we will plot the points (120, 12), (140, 26), (160, 34), (180, 40), (200, 50)....

Which of the following reactions will get affected by increasing the pressure?

Question: Which of the following reactions will get affected by increasing the pressure? Also, mention whether change will cause the reaction to go into forward or backward direction. (i) $\mathrm{COCl}_{2}$ (g) $\longleftrightarrow \mathrm{CO}$ (g) $+\mathrm{Cl}_{2}$ (g) (ii) $\mathrm{CH}_{4}(\mathrm{~g})+2 \mathrm{~S}_{2}(\mathrm{~g}) \longleftrightarrow \mathrm{CS}_{2}(\mathrm{~g})+2 \mathrm{H}_{2} \mathrm{~S}(\mathrm{~g})$ (iii) $\mathrm{CO}_{2}(\mathrm{~g})+\mathrm{C}(\mathrm{s}) \longleftr...

Prove that the product of three consecutive positive integer is divisible by 6.

Question: Prove that the product of three consecutive positive integer is divisible by 6. Solution: To Prove: the product of three consecutive positive integers is divisible by 6. Proof: Letnbe any positive integer. Since any positive integer is of the form 6qor 6q+ 1 or 6q+ 2 or 6q+ 3 or 6q+ 4, 6q+ 5 Ifn= 6q $\Rightarrow n(n+1)(n+2)=6 q(6 q+1)(6 q+2)$, which is divisible by 6 Ifn= 6q+ 1 $\Rightarrow n(n+1)(n+2)=(6 q+1)(6 q+2)(6 q+3)$ $\Rightarrow n(n+1)(n+2)=6(6 q+1)(3 q+1)(2 q+1)$ Which is div...

Let f: X → Y be an invertible function. Show that f has unique inverse.

Question: Letf:XYbe an invertible function. Show thatfhas unique inverse. (Hint: supposeg1andg2are two inverses off. Then for allyY, fog1(y) = IY(y) =fog2(y). Use one-one ness off). Solution: Letf:XYbe an invertible function. Also, suppose $f$ has two inverses (say $g_{1}$ and $g_{2}$ ). Then, for all $y \in Y$, we have: Hence,fhas a unique inverse....

Does the number of moles of reaction products increase,

Question: Does the number of moles of reaction products increase, decrease or remain same when each of the following equilibria is subjected to a decrease in pressure by increasing the volume? (a) $\mathrm{PCl}_{5}(\mathrm{~g}) \longleftrightarrow \mathrm{PCl}_{3}(\mathrm{~g})+\mathrm{Cl}_{2}(\mathrm{~g})$ (b) $\mathrm{CaO}$ (s) $+\mathrm{CO}_{2}$ (g) $\longleftrightarrow \mathrm{CaCO}_{3}$ (s) (c) $3 \mathrm{Fe}(\mathrm{s})+4 \mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \longleftrightarrow \mathrm{Fe}...

The distribution below gives the weights of 30 students of a class.

Question: The distribution below gives the weights of 30 students of a class. Find the median weight of the students. Solution: $\frac{\mathbf{N}}{\mathbf{2}}=\frac{\mathbf{3 0}}{\mathbf{2}}=15$ Median class $=55-60$ Median $=\ell+\left\{\frac{\frac{\mathbf{N}}{\mathbf{2}}-\mathbf{c}}{\mathbf{f}}\right\} \times \mathbf{h}$ $=55+\left\{\frac{\mathbf{1 5}-\mathbf{1 3}}{\mathbf{6}}\right\} \times 5$ = 56.67...

Consider f: R+ → [−5, ∞) given by f(x) = 9x2 + 6x − 5.

Question: Consider $f: \mathbf{R}_{+} \rightarrow[-5, \infty)$ given by $f(x)=9 x^{2}+6 x-5 .$ Show that $f$ is invertible with $f^{-1}(y)=\left(\frac{(\sqrt{y+6})-1}{3}\right)$ Solution: $f: \mathbf{R}_{+} \rightarrow[-5, \infty)$ is given as $f(x)=9 x^{2}+6 x-5$ Let $y$ be an arbitrary element of $[-5, \infty)$. Let $y=9 x^{2}+6 x-5$ $\Rightarrow y=(3 x+1)^{2}-1-5=(3 x+1)^{2}-6$ $\Rightarrow(3 x+1)^{2}=y+6$ $\Rightarrow 3 x+1=\sqrt{y+6} \quad[$ as $y \geq-5 \Rightarrow y+60]$ $\Rightarrow x=\f...

100 surnames were randomly picked up from a local telephone directory

Question: 100 surnames were randomly picked up from a local telephone directory and the frequency distribution of the number of letters in the English alphabets in the surnames was obtained as follows: Determine the median number of letters in the surnames. Find the mean number of letters in the surnames? Also, find the modal size of the surnames. Solution: (i) Here, $\ell=7, \mathrm{n}=100, \mathrm{f}=40, \mathrm{cf}=36, \mathrm{~h}=3$ Median $=\ell+\left\{\frac{\frac{\mathbf{n}}{\mathbf{2}}-\m...

Calculate a) ΔG°and b) the equilibrium constant for the formation of NO2 from NO and O2 at 298 K

Question: Calculate a) $\triangle G^{\circ}$ and b) the equilibrium constant for the formation of $\mathrm{NO}_{2}$ from $\mathrm{NO}$ and $\mathrm{O}_{2}$ at $298 \mathrm{~K}$ $\mathrm{NO}(\mathrm{g})+1 / 2 \mathrm{O}_{2}(\mathrm{~g}) \longleftrightarrow \mathrm{NO}_{2}(\mathrm{~g})$ where $\triangle_{f} G^{\circ}\left(\mathrm{NO}_{2}\right)=52.0 \mathrm{~kJ} / \mathrm{mol}$ $\Delta_{f} G^{\circ}(\mathrm{NO})=87.0 \mathrm{~kJ} / \mathrm{mol}$ $\Delta_{f} G^{\circ}\left(\mathrm{O}_{2}\right)=0 \...

Prove that the product of two consecutive positive integers is divisible by 2.

Question: Prove that the product of two consecutive positive integers is divisible by 2. Solution: To Prove: that the product of two consecutive integers is divisible by 2. Proof: Letn 1 andnbe two consecutive positive integers. Then their product is $n(n-1)=n^{2}-n$ We know that every positive integer is of the form 2qor 2q+ 1 for some integerq. So letn= 2q So, $n^{2}-n=(2 q)^{2}-(2 q)$ $\Rightarrow n^{2}-n=(2 q)^{2}-2 q$ $\Rightarrow n^{2}-n=4 q^{2}-2 q$ $\Rightarrow n^{2}-n=2 q(2 q-1)$ $\Righ...

Prove that cot 4x (sin 5x + sin 3x) = cot x (sin 5x – sin 3x)

Question: Prove that $\cot 4 x(\sin 5 x+\sin 3 x)=\cot x(\sin 5 x-\sin 3 x)$ Solution: L.H.S $=\cot 4 x(\sin 5 x+\sin 3 x)$ $=\frac{\cos 4 x}{\sin 4 x}\left[2 \sin \left(\frac{5 x+3 x}{2}\right) \cos \left(\frac{5 x-3 x}{2}\right)\right]$ $\left[\because \sin A+\sin B=2 \sin \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)\right]$ $=\left(\frac{\cos 4 x}{\sin 4 x}\right)[2 \sin 4 x \cos x]$ $=2 \cos 4 x \cos x$ R.H.S. $=\cot x(\sin 5 x-\sin 3 x)$ $=\frac{\cos x}{\sin x}\left[2 \cos \le...

The following table gives the distribution of the life time of 400 neon lamps :

Question: The following table gives the distribution of the life time of 400 neon lamps : Find the median life time of a lamp. Solution: $\frac{\mathbf{N}}{\mathbf{2}}=\frac{\mathbf{3 9 9}}{\mathbf{2}}=199.5$ Median class $=3000-3500$ Median $=\ell+\left\{\frac{\frac{\mathbf{N}}{\mathbf{2}}-\mathbf{C}}{\mathbf{f}}\right\} \times \mathbf{h}$ $=3000+\left\{\frac{\mathbf{1 9 9 . 5}-\mathbf{1 3 0}}{\mathbf{8 5}}\right\} \times 500=3408.82$ Hence, median life time of a lamp 3408.82 hrs....

where R+ is the set of all non-negative real numbers.

Question: Consider $f: \mathbf{R}_{+} \rightarrow[4, \infty)$ given by $f(x)=x^{2}+4$. Show that $f$ is invertible with the inverse $f^{-1}$ of given $f$ by $f^{-1}(y)=\sqrt{y-4}$, where $\mathbf{R}_{+}$is the set of all non-negative real numbers. Solution: $f: \mathbf{R}_{+} \rightarrow[4, \infty)$ is given as $f(x)=x^{2}+4$ One-one: Letf(x) =f(y). $\Rightarrow x^{2}+4=y^{2}+4$ $\Rightarrow x^{2}=y^{2}$ $\Rightarrow x=y \quad\left[\right.$ as $\left.x=y \in \mathbf{R}_{+}\right]$ fis a one-one ...