What is the force between two small charged spheres having charges
Question: What is the force between two small charged spheres having charges of $2 \times 10^{-7} \mathrm{C}$ and $3 \times 10^{-7} \mathrm{C}$ placed $30 \mathrm{~cm}$ apart in air? Solution: Repulsive force of magnitude $6 \times 10^{-3} \mathrm{~N}$ Charge on the first sphere, $q_{1}=2 \times 10^{-7} \mathrm{C}$ Charge on the second sphere, $q_{2}=3 \times 10^{-7} \mathrm{C}$ Distance between the spheres, $r=30 \mathrm{~cm}=0.3 \mathrm{~m}$ Electrostatic force between the spheres is given by ...
Read More →Prove that: (sin 3x + sin x) sin x + (cos 3x – cos x) cos x = 0
Question: Prove that: $(\sin 3 x+\sin x) \sin x+(\cos 3 x-\cos x) \cos x=0$ Solution: L.H.S. $=(\sin 3 x+\sin x) \sin x+(\cos 3 x-\cos x) \cos x$ $=\sin 3 x \sin x+\sin ^{2} x+\cos 3 x \cos x-\cos ^{2} x$ $=\cos 3 x \cos x+\sin 3 x \sin x-\left(\cos ^{2} x-\sin ^{2} x\right)$ $=\cos (3 x-x)-\cos 2 x \quad[\cos (A-B)=\cos A \cos B+\sin A \sin B]$ $=\cos 2 x-\cos 2 x$ $=0$ $=\mathrm{RH} . \mathrm{S} .$...
Read More →Show that the square of any positive integer cannot be of the form 6m + 2 or 6m + 5 for any integer m.
Question: Show that the square of any positive integer cannot be of the form $6 m+2$ or $6 m+5$ for any integer $m$. Solution: Supposeabe any arbitrary positive integer, then by Euclid's division algorithm, corresponding to the positive integersaand 6, there exists non-negative integersaandrsuch that $a=6 q+r$, where $0 \leq r6$ $\Rightarrow a^{2}=(6 q+r)^{2}=36 q^{2}+r^{2}+12 q r$ $\Rightarrow a^{2}=6\left(6 q^{2}+2 q r\right)+r^{2} \quad \ldots \ldots(1) \quad$ where, $0 \leq r6$ Case: 1 Whenr...
Read More →Let * be the binary operation on N defined by a * b = H.C.F. of a and b.
Question: Let*be the binary operation onNdefined bya*b= H.C.F. ofaandb. Is*commutative? Is*associative? Does there exist identity for this binary operation onN? Solution: The binary operation * onNis defined as: a*b= H.C.F. ofaandb It is known that: H.C.F. of $a$ and $b=$ H.C.F. of $b$ and $a \ m n F o r E ; a, b \in \mathbf{N}$. $\therefore a^{*} b=b^{*} a$ Thus, the operation * is commutative. For $a, b, c \in \mathbf{N}$, we have: $\left(a^{*} b\right)^{*} c=(\text { H.C.F. of } a \text { and...
Read More →Find the general solution of the equation $sin x+sin 3 x+sin 5 x=0$
Question: Find the general solution of the equation $\sin x+\sin 3 x+\sin 5 x=0$ Solution: $\sin x+\sin 3 x+\sin 5 x=0$ $(\sin x+\sin 5 x)+\sin 3 x=0$ $\Rightarrow\left[2 \sin \left(\frac{x+5 x}{2}\right) \cos \left(\frac{x-5 x}{2}\right)\right]+\sin 3 x=0 \quad\left[\sin A+\sin B=2 \sin \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)\right]$ $\Rightarrow 2 \sin 3 x \cos (-2 x)+\sin 3 x=0$ $\Rightarrow 2 \sin 3 x \cos 2 x+\sin 3 x=0$ $\Rightarrow \sin 3 x(2 \cos 2 x+1)=0$ $\Rightarrow...
Read More →Is * defined on the set {1, 2, 3, 4, 5} by a * b = L.C.M.
Question: Is*defined on the set {1, 2, 3, 4, 5} bya*b= L.C.M. ofaandba binary operation? Justify your answer. Solution: The operation * on the set A = {1, 2, 3, 4, 5} is defined as a*b= L.C.M. ofaandb. Then, the operation table for the given operation * can be given as: It can be observed from the obtained table that: $3 * 2=2 * 3=6 \notin A, 5^{*} 2=2 * 5=10 \notin A, 3^{*} 4=4^{*} 3=12 \notin A$ $3 * 5=5^{*} 3=15 \notin A, 4 * 5=5 * 4=20 \notin A$ Hence, the given operation * is not a binary o...
Read More →A jar contains 24 marbles, some are green and others are blue.
Question: A jar contains 24 marbles, some are green and others are blue. If a marble is drawn at random from the jar, the probability that it is green is $\frac{2}{3}$. Find the number of blue balls in the jar. Solution: Total number of marbles = 24 Let the total number of green marbles bex. Then, total number of blue marbles = 24 xb $\mathrm{P}$ (getting a given marble) $=\frac{x}{24}$ According to the condition given in the question, $\frac{x}{24}=\frac{2}{3}$ $x=16$ Therefore, total number of...
Read More →The ionization constant of phenol is 1.0 × 10–10.
Question: The ionization constant of phenol is $1.0 \times 10^{-10}$. What is the concentration of phenolate ion in $0.05 \mathrm{M}$ solution of phenol? What will be its degree of ionization if the solution is also $0.01 \mathrm{M}$ in sodium phenolate? Solution: Ionization of phenol: $K_{a \pi}=\frac{\left[\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}^{-}\right]\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]}{\left[\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{OH}\right]}$ $K_{a}=\frac{x \times x}{0.05-x}$ As t...
Read More →A box contains 12 balls out of which x are black.
Question: A box contains 12 balls out of whichxare black. If one ball is drawn at random from the box, what is the probability that it will be a black ball? If 6 more black balls are put in the box, the probability of drawing a black ball is now double of what it was before. Findx. Solution: Total number of balls = 12 Total number of black balls =x $\mathrm{P}($ getting a black ball $)=\frac{x}{12}$ If 6 more black balls are put in the box, then Total number of balls = 12 + 6 = 18 Total number o...
Read More →A bag contains 5 red balls and some blue balls.
Question: A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball isdoublethat of a red ball, determine the number of blue balls in the bag. Solution: Let the number of blue balls bex. Number of red balls = 5 Total number of balls =x+ 5 $P($ getting a red ball $)=\frac{5}{x+5}$ $\mathrm{P}($ getting a blue ball $)=\frac{x}{x+5}$ Given that, $2\left(\frac{5}{x+5}\right)=\frac{x}{x+5}$ $10(x+5)=x^{2}+5 x$ $x^{2}-5 x-50=0$ $x^{2}-10 x+5 x-50=0$ $x(x-10)+5(x-10)=0$ ...
Read More →Find the general solution of the equation
Question: Find the general solution of the equation $\sec ^{2} 2 x=1-\tan 2 x$ Solution: $\sec ^{2} 2 x=1-\tan 2 x$ $\Rightarrow 1+\tan ^{2} 2 x=1-\tan 2 x$ $\Rightarrow \tan ^{2} 2 x+\tan 2 x=0$ $\Rightarrow \tan 2 x(\tan 2 x+1)=0$ $\Rightarrow \tan 2 x=0 \quad$ or $\quad \tan 2 x+1=0$ Now, $\tan 2 \mathrm{x}=0$ $\Rightarrow \tan 2 \mathrm{x}=\tan 0$ $\Rightarrow 2 \mathrm{x}=\mathrm{n} \pi+0$, where $\mathrm{n} \in \mathrm{Z}$ $\Rightarrow x=\frac{n \pi}{2}$, where $n \in Z$ $\tan 2 x+1=0$ $\R...
Read More →Let * be the binary operation on N given by a * b = L.C.M. of a and b. Find
Question: Let ${ }^{*}$ be the binary operation on $\mathbf{N}$ given by $a^{*} b=$ L.C.M. of $a$ and $b$. Find (i) $5^{*} 7,20^{*} 16$ (ii) Is * commutative? (iii) Is * associative? (iv) Find the identity of * in $\mathbf{N}$ (v) Which elements of $\mathbf{N}$ are invertible for the operation ${ }^{*}$ ? Solution: The binary operation * on $\mathbf{N}$ is defined as $a^{*} b=$ L.C.M. of $a$ and $b$. (i) $5^{*} 7=$ L.C.M. of 5 and $7=35$ $20 * 16=$ L.C.M of 20 and $16=80$ (ii) It is known that: ...
Read More →A die is numbered in such a way that its faces show the number 1, 2, 2, 3, 3, 6.
Question: A die is numbered in such a way that its faces show the number 1, 2, 2, 3, 3, 6. It is thrown two times and the total score in two throws is noted. Complete the following table which gives a few values of the total score on the two throws: What is the probability that the total score is (i) even? (ii) 6? (iii) at least 6? Solution: Total number of possible outcomes when two dice are thrown = 6 6 = 36 (i) Total times when the sum is even = 18 $P$ (getting an even number) $=\frac{18}{36}...
Read More →Find the general solution of the equation $sin 2 x+cos x=0$
Question: Find the general solution of the equation $\sin 2 x+\cos x=0$ Solution: $\sin 2 x+\cos x=0$ $\Rightarrow 2 \sin x \cos x+\cos x=0$ $\Rightarrow \cos x(2 \sin x+1)=0$ $\Rightarrow \cos x=0 \quad$ or $\quad 2 \sin x+1=0$ Now, $\cos x=0 \Rightarrow \cos x=(2 n+1) \frac{\pi}{2}$, where $n \in Z$ $2 \sin x+1=0$ $\Rightarrow \sin x=\frac{-1}{2}=-\sin \frac{\pi}{6}=\sin \left(\pi+\frac{\pi}{6}\right)=\sin \left(\pi+\frac{\pi}{6}\right)=\sin \frac{7 \pi}{6}$ $\Rightarrow \mathrm{x}=\mathrm{n} ...
Read More →The ionization constant of HF, HCOOH and HCN at 298K
Question: The ionization constant of $\mathrm{HF}, \mathrm{HCOOH}$ and $\mathrm{HCN}$ at $298 \mathrm{~K}$ are $6.8 \times 10^{-4}, 1.8 \times 10^{-4}$ and $4.8 \times 10^{-9}$ respectively. Calculate the ionization constants of the corresponding conjugate base. Solution: It is known that, $K_{b}=\frac{K_{w}}{K_{a}}$ Given, $K_{a}$ of HF $=6.8 \times 10^{-4}$ Hence, $K_{b}$ of its conjugate base $\mathrm{F}^{-}$ $=\frac{K_{w}}{K_{a}}$ $=\frac{10^{-14}}{6.8 \times 10^{-4}}$ $=1.5 \times 10^{-11}$...
Read More →Two customers Shyam and Ekta are visiting a particular shop in the same week (Tuesday to Saturday).
Question: Two customers Shyam and Ekta are visiting a particular shop in the same week (Tuesday to Saturday). Each is equally likely to visit the shop on any day as on another day. What is the probability that both will visit the shop on (i) the same day? (ii) consecutive days? (iii) different days? Solution: There are a total of 5 days. Shyam can go to the shop in 5 ways and Ekta can go to the shop in 5 ways. Therefore, total number of outcomes = 5 5 = 25 (i) They can reach on the same day in 5...
Read More →Find the general solution of the equation $cos 3 x+cos x-cos 2 x=0$
Question: Find the general solution of the equation $\cos 3 x+\cos x-\cos 2 x=0$ Solution: $\cos 3 x+\cos x-\cos 2 x=0$ $\Rightarrow 2 \cos \left(\frac{3 x+x}{2}\right) \cos \left(\frac{3 x-x}{2}\right)-\cos 2 x=0 \quad\left[\cos A+\cos B=2 \cos \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)\right]$ $\Rightarrow 2 \cos 2 x \cos x-\cos 2 x=0$ $\Rightarrow \cos 2 x(2 \cos x-1)=0$ $\Rightarrow \cos 2 x=0 \quad$ or $\quad 2 \cos x-1=0$ $\Rightarrow \cos 2 x=0 \quad$ or $\quad \cos x=\fra...
Read More →Which of the following arguments are correct and which are not correct?
Question: Which of the following arguments are correct and which are not correct? Give reasons for your answer. (i) If two coins are tossed simultaneously there are three possible outcomes--two heads, two tails or one of each. Therefore, for each of these outcomes, the probability is $\frac{1}{3}$. (ii) If a die is thrown, there are two possible outcomes--an odd number or an even number. Therefore, the probability of getting an odd number is $\frac{1}{2}$. Solution: (i) Incorrect When two coins ...
Read More →Find the general solution of the equation $cos 4 x=cos 2 x$
Question: Find the general solution of the equation $\cos 4 x=\cos 2 x$ Solution: $\cos 4 x=\cos 2 x$ $\Rightarrow \cos 4 x-\cos 2 x=0$ $\Rightarrow-2 \sin \left(\frac{4 x+2 x}{2}\right) \sin \left(\frac{4 x-2 x}{2}\right)=0$ $\left[\because \cos \mathrm{A}-\cos \mathrm{B}=-2 \sin \left(\frac{\mathrm{A}+\mathrm{B}}{2}\right) \sin \left(\frac{\mathrm{A}-\mathrm{B}}{2}\right)\right]$ $\Rightarrow \sin 3 x \sin x=0$ $\Rightarrow \sin 3 x=0 \quad$ or $\quad \sin x=0$ $\therefore 3 \mathrm{x}=\mathrm...
Read More →Show that any positive odd integer is of the form 6q + 1 or, 6q + 3 or, 6q + 5, where q is some integer.
Question: Show that any positive odd integer is of the form 6q+ 1 or, 6q+ 3 or, 6q+ 5, whereqis some integer. Solution: To Show: That any positive odd integer is of the form 6q+ 1 or 6q+ 3 or 6q+ 5 whereqis any some integer. Proof: Letabe any odd positive integer andb= 6. Then, there exists integersqandrsuch that $a=6 q+r, 0 \leq r6$ (by division algorithm) $\Rightarrow a=6 q$ or $6 q+1$ or $6 q+2$ or $6 q+3$ or $6 q+4$ But 6qor 6q+ 2 or 6q+ 4 are even positive integers. So, $a=6 q+1$ or $6 q+3$...
Read More →Let*′ be the binary operation on the set {1, 2, 3, 4, 5} defined by a *′ b = H.C.F.
Question: Let ${ }^{* \prime}$ be the binary operation on the set $\{1,2,3,4,5\}$ defined by $a^{* \prime} b=$ H.C.F. of $a$ and $b$. Is the operation $^{* \prime}$ same as the operation * defined in Exercise 4 above? Justify your answer. Solution: The binary operation *' on the set $\{1,2,34,5\}$ is defined as $a^{* \prime} b=$ H.C.F of $a$ and $b$. The operation table for the operation ${ }^{* \prime}$ can be given as: We observe that the operation tables for the operations ${ }^{*}$ and ${ }^...
Read More →A die is thrown twice. What is the probability that
Question: A die is thrown twice. What is the probability that (i) 5 will not come up either time? (ii) 5 will come up at least once? Solution: Total number of outcomes = 6 6 = 36 (i)Total number of outcomes when 5 comes up on either time are (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (1, 5), (2, 5), (3, 5), (4, 5), (6, 5) Hence, total number of favourable cases = 11 $P(5$ will come up either time $)=\frac{11}{36}$ $P(5$ will not come up either time $)=1-\frac{11}{36}=\frac{25}{36}$ (ii) Tot...
Read More →Consider a binary operation * on the set {1, 2, 3, 4, 5}
Question: Consider a binary operation * on the set $\{1,2,3,4,5\}$ given by the following multiplication table. (i) Compute $(2 * 3)^{*} 4$ and $2 *\left(3^{*} 4\right)$ (ii) Is * commutative? (iii) Compute $\left(2^{*} 3\right)^{*}\left(4^{*} 5\right)$. Solution: (i) $\left(2^{*} 3\right)^{*} 4=1^{*} 4=1$ $2^{*}\left(3^{*} 4\right)=2^{*} 1=1$ (ii) For every $a, b \in\{1,2,3,4,5\}$, we have $a^{*} b=b^{*} a$. Therefore, the operation * is commutative. (iii) $\left(2^{*} 3\right)=1$ and $\left(4^...
Read More →The pH of a sample of vinegar is 3.76. Calculate the concentration of hydrogen ion in it.
Question: The $\mathrm{oH}$ of a sample of vinegar is 376 Calculate the concentration of hydrogen ion in it Solution: Given, pH = 3.76 It is known that, $\mathrm{pH}=-\log \left[\mathrm{H}^{+}\right]$ $\Rightarrow \log \left[\mathrm{H}^{+}\right]=-\mathrm{pH}$ $\Rightarrow\left[\mathrm{H}^{+}\right]=\operatorname{antilog}(-\mathrm{pH})$ $=\operatorname{antilog}(-3.76)$ $=1.74 \times 10^{-4} \mathrm{M}$ Hence, the concentration of hydrogen ion in the given sample of vinegar is $1.74 \times 10^{-4...
Read More →A game consists of tossing a one rupee coin 3 times and noting its outcome each time.
Question: A game consists of tossing a one rupee coin 3 times and noting its outcome each time. Hanif wins if all the tosses give the same result i.e., three heads or three tails, and loses otherwise. Calculate the probability that Hanif will lose the game. Solution: The possible outcomes are {HHH, TTT, HHT, HTH, THH, TTH, THT, HTT} Number of total possible outcomes = 8 Number of favourable outcomes = 2 {i.e., TTT and HHH} $P$ (Hanif will win the game) $=\frac{2}{8}=\frac{1}{4}$ $\mathrm{P}$ (Ha...
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