If the letters of the word ASSASSINATION
Question: If the letters of the word ASSASSINATION are arranged at random. Find the Probability that (a) Four Ss come consecutively in the word (b) Two Is and two Ns come together (c) All As are not coming together (d) No two As are coming together. Solution: Given word is ASSASSINATION Total number of letters in ASSASSINATION is 13 In word ASSASSINATION, there are 3As, 4Ss, 2Is, 2Ns, 1Ts and 1Os Total number of ways these letters can be arranged = $\mathrm{n}(\mathrm{S})=\frac{13 !}{3 ! 4 ! 2 !...
Read More →Evaluate the following limits:
Question: Evaluate the following limits: $\lim _{x \rightarrow 0} \frac{\sin x-2 \sin 3 x+\sin 5 x}{x}$ Solution: To Find: Limits NOTE: First Check the form of imit. Used this method if the limit is satisfying any one from 7 indeterminate form. In this Case, inderterminate Form is $\frac{0}{0}$ Formula used: $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$ So $\lim _{x \rightarrow 0} \frac{\sin x-2 \sin 3 x+\sin 5 x}{x}=\lim _{x \rightarrow 0}\left(\frac{\sin x}{x}-\frac{2 \sin 3 x}{x}+\frac{\sin 5 ...
Read More →Evaluate the following limits:
Question: Evaluate the following limits: $\lim _{x \rightarrow 0} \frac{\sin m x}{\tan n x}$ Solution: To Find: Limits NOTE: First Check the form of imit. Used this method if the limit is satisfying any one from 7 indeterminate form. In this Case, indeterminate Form is $\frac{0}{0}$ Formula used: $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$ and $\lim _{x \rightarrow 0} \frac{\operatorname{tanx}}{x}=1$ So $\lim _{x \rightarrow 0} \frac{\sin m x}{\tan n x}=\lim _{x \rightarrow 0}\left(\frac{\sin m...
Read More →Evaluate the following limits:
Question: Evaluate the following limits: $\lim _{x \rightarrow 0} \frac{\tan 3 x}{\sin 4 x}$ Solution: To Find: Limits NOTE: First Check the form of imit. Used this method if the limit is satisfying any one from 7 indeterminate form. In this Case, indeterminate Form is $\frac{0}{0}$ Formula used: $\lim _{x \rightarrow 0} \frac{\tan x}{x}=1$ and $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$ So $\lim _{x \rightarrow 0} \frac{\tan 3 x}{\sin 4 x}=\lim _{x \rightarrow 0}\left(\frac{\tan 3 x}{3 x}\righ...
Read More →A bag contains 8 red and 5 white balls.
Question: A bag contains 8 red and 5 white balls. Three balls are drawn at random. Find the Probability that (a) All the three balls are white (b) All the three balls are red (c) One ball is red and two balls are white Solution: Given that, number of red balls = 8 Number of white balls = 5 Total balls, n = 13 It is given that 3 balls are drawn at random ⇒ r = 3 n(S) =nCr=13C3 (a) All the three balls are white We know that, $P(A)=\frac{n(A)}{n(S)}=\frac{\text { Number of favourable outcomes }}{\t...
Read More →Evaluate the following limits:
Question: Evaluate the following limits: $\lim _{x \rightarrow 0} \frac{\tan 3 x}{\sin 4 x}$ Solution: To Find: Limits NOTE: First Check the form of imit. Used this method if the limit is satisfying any one from 7 indeterminate form. In this Case, indeterminate Form is $\frac{0}{0}$ Formula used: $\lim _{x \rightarrow 0} \frac{\tan x}{x}=1$ and $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$ So $\lim _{x \rightarrow 0} \frac{\tan 3 x}{\sin 4 x}=\lim _{x \rightarrow 0}\left(\frac{\tan 3 x}{3 x}\righ...
Read More →Evaluate the following limits:
Question: Evaluate the following limits: $\lim _{x \rightarrow 0} \frac{\sin 4 x}{\tan 7 x}$ Solution: To Find: Limits NOTE: First Check the form of imit. Used this method if the limit is satisfying any one from 7 indeterminate form. In this Case, indeterminate Form is $\frac{0}{0}$ Formula used: $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$ and $\lim _{x \rightarrow 0} \frac{\tan x}{x}=1$ So $\lim _{x \rightarrow 0} \frac{\sin 4 x}{\tan 7 x}=\lim _{x \rightarrow 0}\left(\frac{\sin 4 x}{4 x}\righ...
Read More →Evaluate the following limits
Question: Evaluate the following limits $\lim _{x \rightarrow 0} \frac{\tan \alpha x}{\tan \beta x}$ Solution: To Find: Limits NOTE: First Check the form of imit. Used this method if the limit is satisfied any one from 7 indeterminate form. In this Case, indeterminate Form is $\frac{0}{0}$ Formula used: $\lim _{x \rightarrow 0} \frac{\tan x}{x}=1$. So $\lim _{x \rightarrow 0} \frac{\tan \alpha x}{\tan \beta x}=\lim _{x \rightarrow 0}\left(\frac{\tan \alpha x}{\alpha x}\right) \times \frac{\beta ...
Read More →Evaluate the following limits:
Question: Evaluate the following limits: $\lim _{x \rightarrow 0} \frac{\tan 3 x}{\tan 5 x}$ Solution: To Find: Limits NOTE: First Check the form of imit. Used this method if the limit is satisfied any one from 7 indeterminate form. In this Case, indeterminate Form is $\frac{0}{0}$ Formula used: $\lim _{x \rightarrow 0} \frac{\tan x}{x}=1$ So $\lim _{x \rightarrow 0} \frac{\tan 3 x}{\tan 5 x}=\lim _{x \rightarrow 0}\left(\frac{\tan 3 x}{3 x}\right) \times \frac{5 x}{\sin 5 x} \times \frac{3 x}{5...
Read More →One urn contains two black balls (labelled B1 and B2)
Question: One urn contains two black balls (labelled B1 and B2) and one white ball. A second urn contains one black ball and two white balls (labelled W1 and W2). Suppose the following experiment is performed. One of the two urns is chosen at random. Next a ball is randomly chosen from the urn. Then a second ball is chosen at random from the same urn without replacing the first ball. (a) Write the sample space showing all possible outcomes (b) What is the probability that two black balls are cho...
Read More →Evaluate the following limits:
Question: Evaluate the following limits: $\lim _{x \rightarrow 0} \frac{\sin 5 x}{\sin 8 x}$ Solution: To Find: Limits NOTE: First Check the form of imit. Used this method if the limit is satisfied any one from 7 indeterminate form. In this Case, indeterminate Form is $\frac{0}{0}$ Formula used: $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$ So $\lim _{x \rightarrow 0} \frac{\sin 5 x}{\sin 8 x}=\lim _{x \rightarrow 0}\left(\frac{\sin 5 x}{5 x}\right) \times \frac{8 x}{\sin 8 x} \times \frac{5 x}{8...
Read More →Evaluate the following limits
Question: Evaluate the following limits $\lim _{x \rightarrow 0} \frac{\sin 4 x}{6 x}$ Solution: To Find: Limits NOTE: First Check the form of imit. Used this method if the limit is satisfied any one from 7 indeterminate forms. In this Case, indeterminate Form is $\frac{0}{0}$ Formula used: $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$ So $\lim _{x \rightarrow 0} \frac{\sin 4 x}{6 x}=\lim _{x \rightarrow 0}\left(\frac{\sin 4 x}{4 x}\right) \times \frac{4}{6}=\frac{4}{6}=\frac{2}{3}$ Therefore, $\...
Read More →The function
Question: The function $f(x)=a x+\frac{b}{x}, a, b, x0$ takes on the least value at $x$ equal to__________________ Solution: The given function is $f(x)=a x+\frac{b}{x}, a, b, x0$ $f(x)=a x+\frac{b}{x}$ Differentiating both sides with respect tox, we get $f^{\prime}(x)=a-\frac{b}{x^{2}}$ For maxima or minima, $f^{\prime}(x)=0$ $\Rightarrow a-\frac{b}{x^{2}}=0$ $\Rightarrow x^{2}=\frac{b}{a}$ $\Rightarrow x=\sqrt{\frac{b}{a}} \quad(x0)$ Now, $f^{\prime \prime}(x)=\frac{2 b}{x^{3}}$ At $x=\sqrt{\f...
Read More →One of the four persons John, Rita, Aslam
Question: One of the four persons John, Rita, Aslam or Gurpreet will be promoted next month. Consequently, the sample space consists of four elementary outcomes S = {John promoted, Rita promoted, Aslam promoted, Gurpreet promoted} You are told that the chances of Johns promotion is same as that of Gurpreet, Ritas chances of promotion are twice as likely as Johns. Aslams chances are four times that of John. (a) Determine P (John promoted) P (Rita promoted) P (Aslam promoted) P (Gurpreet promoted)...
Read More →The function
Question: The function $f(x)=a x+\frac{b}{x}, a, b, x0$ takes on the least value at $x$ equal to__________________ Solution: The given function is $f(x)=a x+\frac{b}{x}, a, b, x0$ $f(x)=a x+\frac{b}{x}$ Differentiating both sides with respect tox, we get $f^{\prime}(x)=a-\frac{b}{x^{2}}$ For maxima or minima, $f^{\prime}(x)=0$ $\Rightarrow a-\frac{b}{x^{2}}=0$ $\Rightarrow x^{2}=\frac{b}{a}$ $\Rightarrow x=\sqrt{\frac{b}{a}} \quad(x0)$ Now, $f^{\prime \prime}(x)=\frac{2 b}{x^{3}}$ At $x=\sqrt{\f...
Read More →Evaluate
Question: Evaluate $\lim _{x \rightarrow 0}\left(\frac{3^{2+x}-9}{x}\right)$ Solution: To evaluate: $\lim _{x \rightarrow 0} \frac{3^{2+x}-9}{x}$ Formula used: L'Hospital's rule Let $f(x)$ and $g(x)$ be two functions which are differentiable on an open interval I except at a point a where $\lim _{x \rightarrow a} \mathrm{f}(\mathrm{x})=\lim _{x \rightarrow a} \mathrm{~g}(\mathrm{x})=0$ or $\pm \infty$ then $\lim _{x \rightarrow a} \frac{\mathrm{f}(\mathrm{x})}{\mathrm{g}(\mathrm{x})}=\lim _{x \r...
Read More →Evaluate
Question: Evaluate $\lim _{x \rightarrow 0}\left(\frac{2^{x}-1}{x}\right)$ Solution: To evaluate: $\lim _{x \rightarrow 0} \frac{2^{x}-1}{x}$ Formula used: L'Hospital's rule Let $f(x)$ and $g(x)$ be two functions which are differentiable on an open interval I except at a point a where $\lim _{x \rightarrow a} \mathrm{f}(\mathrm{x})=\lim _{x \rightarrow a} \mathrm{~g}(\mathrm{x})=0$ or $\pm \infty$ then $\lim _{x \rightarrow a} \frac{\mathrm{f}(\mathrm{x})}{\mathrm{g}(\mathrm{x})}=\lim _{x \right...
Read More →The real number
Question: The real number which must exceeds its cube is _______________ Solution: Let the real number bex. The cube of the number is $x^{3}$. Differentiating both sides with respect tox, we get $f^{\prime}(x)=1-3 x^{2}$ For maxima or minima, $f^{\prime}(x)=0$ $\Rightarrow 1-3 x^{2}=0$ $\Rightarrow x^{2}=\frac{1}{3}$ $\Rightarrow x=\pm \frac{1}{\sqrt{3}}$ Now, $f^{\prime \prime}(x)=-6 x$ At $x=-\frac{1}{\sqrt{3}}$, we have $f^{\prime \prime}\left(-\frac{1}{\sqrt{3}}\right)=-6 \times\left(-\frac{...
Read More →The real number
Question: The real number which must exceeds its cube is _______________ Solution: Let the real number bex. The cube of the number is $x^{3}$. Differentiating both sides with respect tox, we get $f^{\prime}(x)=1-3 x^{2}$ For maxima or minima, $f^{\prime}(x)=0$ $\Rightarrow 1-3 x^{2}=0$ $\Rightarrow x^{2}=\frac{1}{3}$ $\Rightarrow x=\pm \frac{1}{\sqrt{3}}$ Now, $f^{\prime \prime}(x)=-6 x$ At $x=-\frac{1}{\sqrt{3}}$, we have $f^{\prime \prime}\left(-\frac{1}{\sqrt{3}}\right)=-6 \times\left(-\frac{...
Read More →Evaluate
Question: Evaluate $\lim _{x \rightarrow 0}\left(\frac{a^{x}-a^{-x}}{x}\right)$ Solution: To evaluate: $\lim _{x \rightarrow 0} \frac{a^{x}-a^{-x}}{x}$ Formula used: L'Hospital's rule Let $f(x)$ and $g(x)$ be two functions which are differentiable on an open interval I except at a point a where $\lim _{x \rightarrow a} \mathrm{f}(\mathrm{x})=\lim _{x \rightarrow a} \mathrm{~g}(\mathrm{x})=0$ or $\pm \infty$ then $\lim _{x \rightarrow a} \frac{f(x)}{g(x)}=\lim _{x \rightarrow a} \frac{f^{\prime}(...
Read More →The positive real number x when
Question: The positive real numberxwhen added to its reciprocal gives the minimum value of the sum when,x= __________________. Solution: LetS(x) be the sum of thepositive real numberx and its reciprocal. $\therefore S(x)=x+\frac{1}{x}$ Differentiating both sides with respect tox, we get $S^{\prime}(x)=1-\frac{1}{x^{2}}$ For maxima or minima, $S^{\prime}(x)=0$ $\Rightarrow 1-\frac{1}{x^{2}}=0$ $\Rightarrow x^{2}=1$ $\Rightarrow x=-1$ or $x=1$ Now, $S^{\prime \prime}(x)=\frac{2}{x^{3}}$ At $x=-1$,...
Read More →The positive real number x when
Question: The positive real numberxwhen added to its reciprocal gives the minimum value of the sum when,x= __________________. Solution: LetS(x) be the sum of thepositive real numberx and its reciprocal. $\therefore S(x)=x+\frac{1}{x}$ Differentiating both sides with respect tox, we get $S^{\prime}(x)=1-\frac{1}{x^{2}}$ For maxima or minima, $S^{\prime}(x)=0$ $\Rightarrow 1-\frac{1}{x^{2}}=0$ $\Rightarrow x^{2}=1$ $\Rightarrow x=-1$ or $x=1$ Now, $S^{\prime \prime}(x)=\frac{2}{x^{3}}$ At $x=-1$,...
Read More →Evaluate
Question: Evaluate $\lim _{x \rightarrow 0}\left(\frac{a^{x}-b^{x}}{x}\right)$ Solution: To evaluate: $\lim _{x \rightarrow 0} \frac{a^{x}-b^{x}}{x}$ Formula used: L'Hospital's rule Let $f(x)$ and $g(x)$ be two functions which are differentiable on an open interval I except at a point a where $\lim _{x \rightarrow a} \mathrm{f}(\mathrm{x})=\lim _{x \rightarrow a} \mathrm{~g}(\mathrm{x})=0$ or $\pm \infty$ then $\lim _{x \rightarrow a} \frac{\mathrm{f}(\mathrm{x})}{\mathrm{g}(\mathrm{x})}=\lim _{...
Read More →Evaluate
Question: Evaluate. $\lim _{x \rightarrow 0}\left(\frac{e^{b x}-e^{a x}}{x}\right), 0ab$ Solution: To evaluate: $\lim _{x \rightarrow 0} \frac{e^{b x}-e^{a x}}{x}$ Formula used L'Hospital's rule Let $f(x)$ and $g(x)$ be two functions which are differentiable on an open interval I except at a point a where $\lim _{x \rightarrow a} \mathrm{f}(\mathrm{x})=\lim _{x \rightarrow a} \mathrm{~g}(\mathrm{x})=0$ or $\pm \infty$ then $\lim _{x \rightarrow a} \frac{f(x)}{g(x)}=\lim _{x \rightarrow a} \frac{...
Read More →Four candidates A, B, C, D have applied for the assignment
Question: Four candidates A, B, C, D have applied for the assignment to coach a school cricket team. If A is twice as likely to be selected as B, and B and C are given about the same chance of being selected, while C is twice as likely to be selected as D, what are the probabilities that (a) C will be selected? (b) A will not be selected? Solution: Given that A is twice as likely to be selected as B i.e. P (A) = 2 P (B) 1 and C is twice as likely to be selected as D i.e. P (C) = 2 P (D) 2 Now, B...
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