Without repetition of the numbers,
Question: Without repetition of the numbers, four digit numbers are formed with the numbers 0, 2, 3, 5. The probability of such a number divisible by 5 is A. $\frac{1}{5}$ B. $\frac{4}{5}$ C. $\frac{1}{30}$ D. $\frac{5}{9}$ Solution: D. 5/9 Explanation: We have digits $0,2,3,5$. We know that, if unit place digit is ' 0 ' or ' 5 ' then the number is divisible by 5 If unit place is ' 0 ' Then first three places can be filled in $3 !$ ways $=3 \times 2 \times 1 \times 1=6$ If unit place is ' 5 ' Th...
Read More →Evaluate the following limits:
Question: Evaluate the following limits: $\lim _{x \rightarrow 0} \frac{\sin x \cos x}{3 x}$ 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: $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$ So, by using the above formula, we have $\lim _{x \rightarrow 0} \frac{\sin x \cos x}{3 x}=\lim _{x \rightarrow 0} \frac{\sin x}{x} \times \frac{\cos x}{3}=\...
Read More →If the function
Question: If the function $f(x)=x^{4}-62 x^{2}+a x+9$ attains a local maximum at $x=1$, then $a=$ _______________ Solution: It is given that, the function $f(x)=x^{4}-62 x^{2}+a x+9$ attains a local maximum at $x=1$ $\therefore f^{\prime}(x)=0$ at $x=1$ $f(x)=x^{4}-62 x^{2}+a x+9$ Differentiating both sides with respect tox, we get $f^{\prime}(x)=4 x^{3}-124 x+a$ Now, $f^{\prime}(1)=0$ $\Rightarrow 4 \times(1)^{3}-124 \times 1+a=0$ $\Rightarrow a=124-4=120$ Thus, the value ofais 120. At $x=1$, w...
Read More →If the function
Question: If the function $f(x)=x^{4}-62 x^{2}+a x+9$ attains a local maximum at $x=1$, then $a=$ _______________ Solution: It is given that, the function $f(x)=x^{4}-62 x^{2}+a x+9$ attains a local maximum at $x=1$ $\therefore f^{\prime}(x)=0$ at $x=1$ $f(x)=x^{4}-62 x^{2}+a x+9$ Differentiating both sides with respect tox, we get $f^{\prime}(x)=4 x^{3}-124 x+a$ Now, $f^{\prime}(1)=0$ $\Rightarrow 4 \times(1)^{3}-124 \times 1+a=0$ $\Rightarrow a=124-4=120$ Thus, the value ofais 120. At $x=1$, w...
Read More →Evaluate the following limits:
Question: Evaluate the following limits: $\lim _{x \rightarrow 0}(x \cot 2 x)$ 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 $0 \times \infty$ Formula used: $\lim _{x \rightarrow 0} \frac{x}{\tan x}=1$ So, by using the above formula, we have $\lim _{x \rightarrow 0} x \cot 2 x=\lim _{x \rightarrow 0} \frac{2 x}{2 \tan 2 x}=\frac{1}{2}$ Therefore, $\lim _{x \rightarro...
Read More →Seven persons are to be seated in a row.
Question: Seven persons are to be seated in a row. The probability that two particular persons sit next to each other is A. $\frac{1}{3}$ B. $\frac{1}{6}$ C. $\frac{2}{7}$ D. $\frac{1}{2}$ Solution: C. 2/7 Explanation: Given that 7 persons are to be seated in a row. If two persons sit next to each other, then consider these two persons as 1 group. Now we have to arrange 6 persons. $\therefore$ Number of arrangement $=2 ! \times 6 !$ Total number of arrangement of 7 persons $=7 !$ Probability $=\...
Read More →Evaluate the following limits:
Question: Evaluate the following limits: $\lim _{x \rightarrow 0} x \operatorname{cosec} x$ 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 are $0 \times \infty$ Formula used: $\lim _{x \rightarrow 0} \frac{x}{\sin x}=1$ So, by using the above formula, we have $\lim _{x \rightarrow 0} x \operatorname{cosec} x=\lim _{x \rightarrow 0} \frac{x}{\sin x}=1$ Therefore, $\lim _{...
Read More →The maximum value
Question: The maximum value of $f(x)=x e^{-x}$ is___________ Solution: The given function is $f(x)=x e^{-x}$. $f(x)=x e^{-x}$ Differentiating both sides with respect tox, we get $f^{\prime}(x)=x \times e^{-x} \times(-1)+e^{-x} \times 1$ $\Rightarrow f^{\prime}(x)=e^{-x}(-x+1)$ For maxima or minima, $f^{\prime}(x)=0$ $\Rightarrow e^{-x}(-x+1)=0$ $\Rightarrow-x+1=0$ $\left(e^{-x}0 \forall x \in \mathrm{R}\right)$ $\Rightarrow x=1$ Now, $f^{\prime \prime}(x)=e^{-x} \times(-1)+(-x+1) \times e^{-x} \...
Read More →Evaluate the following limits:
Question: Evaluate the following limits: $\lim _{x \rightarrow 0} \frac{\tan x-\sin x}{\sin ^{3} x}$ 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 $\frac{0}{0}$ NOTE $: \tan x-\sin x=\frac{\sin x}{\cos x}-\sin x=\frac{\sin x-\sin x \cos x}{\cos x}=\sin x\left(\frac{1-\cos x}{\cos x}\right)$ $\lim _{x \rightarrow 0} \frac{\tan x-\sin x}{\sin ^{2} x}=\lim _{x \rightarr...
Read More →While shuffling a pack of 52 playing cards,
Question: While shuffling a pack of 52 playing cards, 2 are accidentally dropped. Find the probability that the missing cards to be of different colours A. $\frac{29}{52}$ B. $\frac{1}{2}$ C. $\frac{26}{51}$ D. $\frac{27}{51}$ Solution: C. 26/51 Explanation: We know that, in a pack of 52 cards 26 are of red colour and 26 are of black colour. It is given that 2 cards are accidentally dropped So, Probability of dropping a red card first $=\frac{26}{52}$ Probability of dropping a red card second $=...
Read More →The maximum value
Question: The maximum value of $f(x)=x e^{-x}$ is___________ Solution: The given function is $f(x)=x e^{-x}$. $f(x)=x e^{-x}$ Differentiating both sides with respect tox, we get $f^{\prime}(x)=x \times e^{-x} \times(-1)+e^{-x} \times 1$ $\Rightarrow f^{\prime}(x)=e^{-x}(-x+1)$ For maxima or minima, $f^{\prime}(x)=0$ $\Rightarrow e^{-x}(-x+1)=0$ $\Rightarrow-x+1=0$ $\left(e^{-x}0 \forall x \in \mathrm{R}\right)$ $\Rightarrow x=1$...
Read More →Evaluate the following limits:
Question: Evaluate the following limits: $\lim _{x \rightarrow 0} \frac{x \cos x+\sin x}{x^{2}+\tan x}$ 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, by using the above formula, we have Divide numerator and denominator by $x$, $\lim _{x \right...
Read More →If the function
Question: If the function $f(x)=a \sin x+\frac{1}{3} \sin 3 x$ has an extremum at $x=\frac{\pi}{3}$ then $a=$_____________ Solution: It is given that, the function $f(x)=a \sin x+\frac{1}{3} \sin 3 x$ has an extremum at $x=\frac{\pi}{3}$. $\therefore f^{\prime}(x)=0$ at $x=\frac{\pi}{3}$ $f(x)=a \sin x+\frac{1}{3} \sin 3 x$ Differentiating both sides with respect tox, we get $f^{\prime}(x)=a \cos x+\frac{1}{3} \times 3 \cos 3 x$ $\Rightarrow f^{\prime}(x)=a \cos x+\cos 3 x$ Now, $f^{\prime}\left...
Read More →Three numbers are chosen from 1 to 20.
Question: Three numbers are chosen from 1 to 20. Find the probability that they are not consecutive A. $\frac{186}{190}$ B. $\frac{187}{190}$ C. $\frac{188}{190}$ D. $\frac{18}{{ }^{20} \mathrm{C}_{3}}$ Solution: B. 187/190 Explanation: Since, the set of three consecutive numbers from 1 to 20 are (1, 2, 3), (2, 3, 4), (3, 4, 5), , (18,19,20) Considering 3 numbers as a single digit the numbers will be 18 Now, we have to choose 3 numbers out of 20. This can be done in20C3ways n(S) =20C3 The desire...
Read More →If the function
Question: If the function $f(x)=a \sin x+\frac{1}{3} \sin 3 x$ has an extremum at $x=\frac{\pi}{3}$ then $a=$_____________ Solution: It is given that, the function $f(x)=a \sin x+\frac{1}{3} \sin 3 x$ has an extremum at $x=\frac{\pi}{3}$. $\therefore f^{\prime}(x)=0$ at $x=\frac{\pi}{3}$ $f(x)=a \sin x+\frac{1}{3} \sin 3 x$ Differentiating both sides with respect tox, we get $f^{\prime}(x)=a \cos x+\frac{1}{3} \times 3 \cos 3 x$ $\Rightarrow f^{\prime}(x)=a \cos x+\cos 3 x$ Now, $f^{\prime}\left...
Read More →In a non-leap year,
Question: In a non-leap year, the probability of having 53 Tuesdays or 53 Wednesdays isA. 1/7B. 2/7C. 3/7D. none of these Solution: B. 2/7 Explanation: We know that in a non-leap year, there are 365 days and we know that there are 7 days in a week 365 7 = 52 weeks + 1 day This 1 day can be Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday Total Outcomes = 7 If this day is a Tuesday or Wednesday, then the year will have 53 Tuesday or 53 Wednesday. P (non-leap year has 53 Tuesdays or ...
Read More →Determine the probability p,
Question: Determine the probability p, for each of the following events. (a) An odd number appears in a single toss of a fair die. (b) At least one head appears in two tosses of a fair coin. (c) A king, 9 of hearts, or 3 of spades appears in drawing a single card from a well shuffled ordinary deck of 52 cards. (d) The sum of 6 appears in a single toss of a pair of fair dice. Solution: (a) When a fair die is thrown, the possible outcomes are $S=\{1,2,3,4,5,6\}$ $\therefore$ total outcomes $=6$ an...
Read More →If the solve the problem
Question: If $y=a \log x+b x^{2}+x$ has its extreme values at $x=1$ and $x=2$, then $(a, b)=$ ____________ Solution: It is given that, $y=a \log x+b x^{2}+x$ has its extreme values at $x=1$ and $x=2$. $\therefore \frac{d y}{d x}=0$ at $x=1$ and $x=2$ $y=a \log x+b x^{2}+x$ Differentiating both sides with respect tox, we get $\frac{d y}{d x}=\frac{a}{x}+2 b x+1$ Now, $\left(\frac{d y}{d x}\right)_{x=1}=0$ $\Rightarrow a+2 b+1=0$ $\Rightarrow a+2 b=-1$ .....(1) Also, $\left(\frac{d y}{d x}\right)_...
Read More →If the solve the problem
Question: If $y=a \log x+b x^{2}+x$ has its extreme values at $x=1$ and $x=2$, then $(a, b)=$ ____________ Solution: It is given that, $y=a \log x+b x^{2}+x$ has its extreme values at $x=1$ and $x=2$. $\therefore \frac{d y}{d x}=0$ at $x=1$ and $x=2$ $y=a \log x+b x^{2}+x$ Differentiating both sides with respect tox, we get $\frac{d y}{d x}=\frac{a}{x}+2 b x+1$ Now, $\left(\frac{d y}{d x}\right)_{x=1}=0$ $\Rightarrow a+2 b+1=0$ $\Rightarrow a+2 b=-1$ .....(1) Also, $\left(\frac{d y}{d x}\right)_...
Read More →A sample space consists of 9 elementary
Question: A sample space consists of 9 elementary outcomes $e_{1}, e_{2}, \ldots, e_{9}$ whose probabilities are $P\left(e_{1}\right)=P\left(e_{2}\right)=.08, P\left(e_{3}\right)=P\left(e_{4}\right)=P\left(e_{5}\right)=.1$ $P\left(e_{6}\right)=P\left(e_{7}\right)=.2, P\left(e_{8}\right)=P\left(e_{9}\right)=.07$ Suppose $A=\left\{e_{1}, e_{5}, e_{8}\right\}, B=\left\{e_{2}, e_{5}, e_{8}, e_{9}\right\}$ (a) Calculate $P(A), P(B)$, and $P(A \cap B)$ (b) Using the addition law of probability, calcul...
Read More →A card is drawn from a deck of 52 cards.
Question: A card is drawn from a deck of 52 cards. Find the probability of getting a king or a heart or a red card. Solution: Given total number of playing cards $=52$ $\therefore \mathrm{n}(\mathrm{S})=52$ Total number of king cards $=4$ Total number of heart cards $=13$ Total number of red cards $=13+13=26$ $\therefore$ Favourable outcomes $=4+13+26-13-2$ $=28$ We know that, Probability $=\frac{\text { Number of favourable outcomes }}{\text { Total number of outcomes }}$ $\therefore$ Required ...
Read More →Evaluate the following limits:
Question: Evaluate the following limits: $\lim _{x \rightarrow 0} \frac{\left(x^{2}-\tan 2 x\right)}{\tan x}$ 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: $\lim _{x \rightarrow 0} \frac{\tan x}{x}=1$ or we can used $L$ hospital Rule, So, by using the above formula, we have Divide numerator and denominator by $x$, $\lim _{x \rightarrow 0} ...
Read More →Evaluate the following limits:
Question: Evaluate the following limits: $\lim _{x \rightarrow 0} \frac{(\tan 2 x-x)}{(3 x-\tan x)}$ 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: $\lim _{x \rightarrow 0} \frac{\tan x}{x}=1$ or we can used $L$ hospital Rule, So, by using the above formula, we have Divide numerator and denominator by $x$, $\lim _{x \rightarrow 0} \frac{\ta...
Read More →Evaluate the following limits:
Question: Evaluate the following limits: $\lim _{x \rightarrow 0} \frac{(\sin 2 x+3 x)}{(2 x+\sin 3 x)}$ 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: $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$ or we can used L hospital Rule, So, by using the above formula, we have Divide numerator and denominator by $x$, $\lim _{x \rightarrow 0} \frac{\...
Read More →Evaluate the following limits:
Question: Evaluate the following limits: $\lim _{x \rightarrow \pi / 6} \frac{\left(2 \sin ^{2} x+\sin x-1\right)}{\left(2 \sin ^{2} x-3 \sin x+1\right)}$ 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: $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$ or we can used $L$ hospital Rule, So, by using the rule, Differentiate numerator and denominato...
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