If f '(x) changes its sign from negative to positive as x increases t
Question: Iff'(x) changes its sign from negative to positive asxincreases throughcin the interval (c h, c + h), thenx=cis a point of ______________. Solution: First derivative test states that iff'(x) changes sign from negative to positive asxincreases throughc, thencis a point of local minima, andf(c) is local minimum value. Thus, iff'(x) changes its sign from negative to positive asxincreases throughcin the interval (c h, c + h), thenx=cis a point of local minimum. Iff'(x) changes its sign fro...
Read More →$\frac{a+b \sin x}{c+d \cos x}$
[question] Question. $\frac{a+b \sin x}{c+d \cos x}$ [/question] [solution] solution: Given $y=\frac{a+b \sin x}{c+d \cos x}$ Applying division rule or quotient rule of differentiation that is $\Rightarrow \frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{\mathrm{t}}{\mathrm{y}}\right)=\frac{\mathrm{y} \cdot \mathrm{dt}}{\mathrm{dx}}-\mathrm{t} \cdot \frac{\mathrm{dy}}{\mathrm{dx}}$ $\Rightarrow y=\frac{a+b \sin x}{c+d \cos x}$ $\Rightarrow \frac{d y}{d x}=\frac{(c+d \cos x) \frac{d}{d x}(a+b \sin x)-(a...
Read More →$\frac{x^{5}-\cos x}{\sin x}$
[question] Question. $\frac{x^{5}-\cos x}{\sin x}$ [/question] [solution] solution: Given $y=\frac{x^{5}-\cos x}{\sin x}$ $\mathrm{d} / \mathrm{dx}\left(\mathrm{x}^{5}-\cos \mathrm{x}\right) / \sin \mathrm{x}=\left[\sin \mathrm{x} \cdot \mathrm{d} / \mathrm{dx}\left(\mathrm{x}^{5}-\cos \right.\right.$ $\left.x)-\left(x^{5}-\cos x\right) \cdot d / d x(\sin x)\right] / \sin ^{2} x$ By using quotient rule, $=\left[\sin x\left(5 x^{4}+\sin x\right)-\right.$ $\left.\left(x^{5}-\cos x\right)(\cos x)\r...
Read More →$\frac{3 x+4}{5 x^{2}-7 x+9}$
[question] Question. $\frac{3 x+4}{5 x^{2}-7 x+9}$ [/question] [solution] solution: Given $\mathrm{y}=\frac{3 \mathrm{x}+4}{5 \mathrm{x}^{2}-7 \mathrm{x}+9}$ Applying quotient rule of differentiation that is $\Rightarrow \frac{d}{d x}\left(\frac{t}{y}\right)=\frac{y \cdot \frac{d t}{d x}-t \cdot \frac{d y}{d x}}{y^{2}}$ Applying the rule $\Rightarrow \frac{d y}{d x}=\frac{\left(5 x^{2}-7 x+9\right) \frac{d}{d x}(3 x+4)-(3 x+4) \frac{d}{d x}\left(5 x^{2}-7 x+9\right)}{\left(5 x^{2}-7 x+9\right)^{...
Read More →$x+\frac{1}{x}^{3}$
[question] Question. $x+\frac{1}{x}^{3}$ [/question] [solution] solution: Let $y=\left(x+\frac{1}{x}\right)^{3}$ Now differentiating y with respect to $x$ we get $\Rightarrow$$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}+\frac{1}{\mathrm{x}}\right)^{3}$ Expanding the equation using $(a+b)^{3}$ formula then we get $=\frac{d}{d x}\left(x^{3}+\frac{1}{x^{3}}+3 x+\frac{3}{x}\right)$ Splitting the differential we get $=\frac{d}{d x}\left(x^{3}\right)+\frac{d}{d x}\le...
Read More →$\frac{x^{4}+x^{3}+x^{2}+1}{x}$
[question] Question. $\frac{x^{4}+x^{3}+x^{2}+1}{x}$ [/question] [solution] solution: Let $y=\frac{x^{4}+x^{3}+x^{2}+1}{x}$ $\Rightarrow$$y=\frac{x^{4}+x^{3}+x^{2}+1}{x}$ Dividing by $x$ we get $\Rightarrow$$y=x^{3}+x^{2}+x+\frac{1}{x}$ Differentiating given equation with respect to $x$ $\Rightarrow$$\frac{d y}{d x}=\frac{d}{d x}\left(x^{3}+x^{2}+x+\frac{1}{x}\right)$ On differentiation we get $\Rightarrow$$\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}^{3}+\mathrm{x}^{2}+\mathrm{x}+\frac{1}{\ma...
Read More →$\lim _{x \rightarrow 0} \frac{\sin x-2 \sin 3 x+\sin 5 x}{x}$
[question] Question. $\lim _{x \rightarrow 0} \frac{\sin x-2 \sin 3 x+\sin 5 x}{x}$ [/question] [solution] solution: Given $\lim _{x \rightarrow 0} \frac{\sin x-2 \sin 3 x+\sin 5 x}{x}$ Now by splitting the limits in above equation we get $\Rightarrow$$\lim _{x \rightarrow 0} \frac{\sin x-2 \sin 3 x+\sin 5 x}{x}=\lim _{x \rightarrow 0} \frac{\sin x}{x}-\lim _{x \rightarrow 0} \frac{2 \sin 3 x}{x}+\lim _{x \rightarrow 0} \frac{\sin 5 x}{x}$ Taking constant term outside the limits we get $\Rightar...
Read More →$\lim _{x \rightarrow 0} \frac{\sqrt{2}-\sqrt{1+\cos x}}{\sin ^{2} x}$
[question] Question. $\lim _{x \rightarrow 0} \frac{\sqrt{2}-\sqrt{1+\cos x}}{\sin ^{2} x}$ [/question] [solution] solution: Given $\lim _{x \rightarrow 0} \frac{\sqrt{2}-\sqrt{1+\cos x}}{\sin ^{2} x}$ Multiply and divide the given equation by $\sqrt{2}-\sqrt{1+\cos x}$ Then we get $\Rightarrow$ $\lim _{x \rightarrow 0} \frac{\sqrt{2}-\sqrt{1+\cos x}}{\sin ^{2} x}=\lim _{x \rightarrow 0} \frac{\sqrt{2}-\sqrt{1+\cos x}}{\sin ^{2} x} \times\left(\frac{\sqrt{2}+\sqrt{1+\cos x}}{\sqrt{2}+\sqrt{1+\co...
Read More →$\lim _{x \rightarrow \frac{\pi}{6}} \frac{\cot ^{2} x-3}{\operatorname{cosec} x-2}$
[question] Question. $\lim _{x \rightarrow \frac{\pi}{6}} \frac{\cot ^{2} x-3}{\operatorname{cosec} x-2}$ [/question] [solution] solution: Given $\lim _{x \rightarrow \frac{\pi}{6}} \frac{\cot ^{2} x-3}{\operatorname{cosec} x-2}$ We know that $\cot ^{2} x=\operatorname{cosec}^{2} x-1$ By using this in given equation we get $\Rightarrow$ $\lim _{x \rightarrow \frac{\pi}{4}} \frac{\left(\operatorname{cosec}^{2} x-1\right)-3}{\operatorname{cosec} x-2}=\lim _{x \rightarrow \frac{\pi}{4}} \frac{\oper...
Read More →Find the value
Question: Evaluate $\lim _{x \rightarrow 0}\left(\frac{e^{x}-x-1}{x}\right)$ Solution: To evaluate: $\lim _{x \rightarrow 0} \frac{e^{x}-x-1}{x}$ 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 \r...
Read More →A team of medical students doing their internship
Question: A team of medical students doing their internship have to assist during surgeries at a city hospital. The probabilities of surgeries rated as very complex, complex, routine, simple or very simple are respectively, 0.15, 0.20, 0.31, 0.26, .08. Find the probabilities that a particular surgery will be rated. (a) complex or very complex; (b) neither very complex nor very simple; (c) routine or complex (d) routine or simple Solution: Let E1= event that surgeries are rated as very complex E2...
Read More →If A and B are mutually exclusive events,
Question: If A and B are mutually exclusive events, P (A) = 0.35 and P (B) = 0.45, find (a) P (A) (b) P (B) (c) P (AB) (d) P (AB) (e) P (AB) (f) P (A B) Solution: Given that P (A) = 0.35 and P (B) = 0.45 ∵The events A and B are mutually exclusive then P (A⋂B) = 0 (a) To find (a) P (A) We know that, P (A) + P (A) = 1 ⇒0.35 + P(A) = 1 [given] ⇒P (A) = 1 0.35 ⇒P (A) = 0.65 (b) To find (b) P (B) We know that, P (B) + P (B) = 1 ⇒0.45 + P (B) = 1 ⇒P (B) = 1 0.45 ⇒P (B) = 0.55 (c) To find (c) P (A⋃B) W...
Read More →In a large metropolitan area, the probabilities are .87, .36, .30
Question: In a large metropolitan area, the probabilities are .87, .36, .30 that a family (randomly chosen for a sample survey) owns a colour television set, a black and white television set, or both kinds of sets. What is the probability that a family owns either anyone or both kinds of sets? Solution: E1= Event that a family owns colour television E2= Event that the family owns black and white television Given that P (E1) = 0.87 P (E2) = 0.36 and P (E1⋂E2) = 0.30 Now, we have to find the proba...
Read More →A die is loaded in such a way that each odd number
Question: A die is loaded in such a way that each odd number is twice as likely to occur as each even number. Find P(G), where G is the event that a number greater than 3 occurs on a single roll of the die. Solution: Given that probability of odd numbers = 2 (Probability of even number) ⇒P (Odd) = 2 P (Even) Now, P (Odd) + P (Even) = 1 ⇒2 P (Even) + P (Even) = 1 ⇒3 P (Even) = 1 P (Even) = 1/3 So, $\mathrm{P}($ Odd $)=1-\frac{1}{3}=\frac{3-1}{3}=\frac{2}{3}$ Now, Total number occurs on a single r...
Read More →$\lim _{x \rightarrow a} \frac{\sin x-\sin a}{\sqrt{x}-\sqrt{a}}$
[question] Question. $\lim _{x \rightarrow a} \frac{\sin x-\sin a}{\sqrt{x}-\sqrt{a}}$ [/question] [solution] solution: Given $\lim _{x \rightarrow a} \frac{\sin x-\sin a}{\sqrt{x}-\sqrt{a}}$ Now we have to rationalize the denominator by multiplying the dividing by its rationalizing factor then we get $\Rightarrow$$\lim _{x \rightarrow a} \frac{\sin x-\sin a}{\sqrt{x}-\sqrt{a}}=\lim _{x \rightarrow a}\left[\frac{\sin x-\sin a}{\sqrt{x}-\sqrt{a}} \times \frac{\sqrt{x}+\sqrt{a}}{\sqrt{x}+\sqrt{a}}...
Read More →An experiment consists of rolling a die until a 2 appears.
Question: An experiment consists of rolling a die until a 2 appears. (i) How many elements of the sample space correspond to the event that the 2 appears on the kthroll of the die? (ii) How many elements of the sample space correspond to the event that the 2 appears not later than the kthroll of the die? Solution: Given number of outcomes when die is thrown = 6 (i) Given that 2 appears on the kthroll of the die. So, first (k 1)throll have 5 outcomes each and kthroll results 2 Number of outcomes ...
Read More →$\lim _{x \rightarrow 0} \frac{\sin 2 x+3 x}{2 x+\tan 3 x}$
[question] Question. $\lim _{x \rightarrow 0} \frac{\sin 2 x+3 x}{2 x+\tan 3 x}$ [/question] [solution] solution: Given $\lim _{x \rightarrow 0} \frac{\sin 2 x+3 x}{2 x+\tan 3 x}$ Multiply and divide the numerator of given equation by $2 x$ $\Rightarrow$$\lim _{x \rightarrow 0} \frac{\sin 2 x+3 x}{2 x+\tan 3 x}=\lim _{x \rightarrow 0} \frac{2 x(\sin 2 x) / 2 x+3 x}{2 x+3 x(\tan 3 x) / 3 x}$ Now by splitting the limits we get $\Rightarrow$$\lim _{x \rightarrow 0} \frac{2 x(\sin 2 x) / 2 x+3 x}{2 ...
Read More →Evaluate
Question: Evaluate $\lim _{x \rightarrow 0}\left(\frac{e^{3 x}-e^{2 x}}{x}\right)$ Solution: To evaluate: $\lim _{x \rightarrow 0} \frac{e^{3 x}-e^{2 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})}=\li...
Read More →Suppose an integer from 1 through 1000 is chosen at random,
Question: Suppose an integer from 1 through 1000 is chosen at random, find the probability that the integer is a multiple of 2 or a multiple of 9. Solution: We have integers 1, 2, 3 1000 Total number of outcomes, n(S) = 1000 Number of integers which are multiple of 2 are 2, 4, 6, 8, 10, 1000 Let p be the number of terms We know that, ap= a + (p 1) d Here, a = 2, d = 2 and ap= 1000 Putting the value, we get 2 + (p 1)2 = 1000 ⇒2 + 2p 2 = 1000 p = 1000/2 ⇒p = 500 Total number of integers which are ...
Read More →Evaluate
Question: Evaluate $\lim _{x \rightarrow 4}\left(\frac{e^{x}-e^{4}}{x-4}\right)$ Solution: To evaluate $\lim _{x \rightarrow 4} \frac{e^{x}-e^{4}}{x-4}$ 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 →$\lim _{x \rightarrow \frac{\pi}{6}} \frac{\sqrt{3} \sin x-\cos x}{x-\frac{\pi}{6}}$
[question] Question. $\lim _{x \rightarrow \frac{\pi}{6}} \frac{\sqrt{3} \sin x-\cos x}{x-\frac{\pi}{6}}$ [/question] [solution] solution: Given $\lim _{x \rightarrow \frac{\pi}{6}} \frac{\sqrt{3} \sin x-\cos x}{x-\frac{\pi}{6}}$ Consider $\sqrt{3} \sin x-\cos x=2\left(\frac{\sqrt{3} \sin x}{2}-\frac{\cos x}{2}\right)=2\left(\sin x \cos \left(\frac{\pi}{6}\right)-\cos x \sin \left(\frac{\pi}{6}\right)\right)$ On simplification the above equation can be written as $\Rightarrow$$2\left(\sin x \cos...
Read More →Six new employees, two of whom are married to each other,
Question: Six new employees, two of whom are married to each other, are to be assigned six desks that are lined up in a row. If the assignment of employees to desks is made randomly, what is the probability that the married couple will have nonadjacent desks? Solution: Total new employees = 6 So, they can be arranged in 6! Ways n (S) = 6! = 6 5 4 3 2 1 = 720 Two adjacent desks for married couple can be selected in 5 ways i.e. (1, 2), (2, 3), (3, 4), (4, 5), (5, 6) Married couple can be arranged ...
Read More →If the letters of the word ALGORITHM are arranged
Question: If the letters of the word ALGORITHM are arranged at random in a row what is the probability the letters GOR must remain together as a unit? Solution: Given word is ALGORITHM $\Rightarrow$ Total number of letters in algorithm $=9$ $\therefore$ Total number of words $=9 !$ $\mathrm{So}, \mathrm{n}(\mathrm{S})=9 !$ If 'GOR' remain together, then we consider it as one group. $\therefore$ Number of letters $=7$ Number of words, if ' $\mathrm{GOR}$ ' remain together in the order $=7 !$ So, ...
Read More →Evaluate
Question: Evaluate $\lim _{x \rightarrow 0}\left(\frac{e^{2+x}-e^{2}}{x}\right)$ Solution: To evaluate: $\lim _{x \rightarrow 0} \frac{e^{2+x}-e^{2}}{x}$ 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})}=\li...
Read More →Evaluate
Question: Evaluate $\lim _{x \rightarrow 0}\left(\frac{e^{4 x}-1}{x}\right)$ Solution: To evaluate: $\lim _{x \rightarrow 0} \frac{e^{4 x}-1}{x}$ 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 \r...
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