If the solve the problem
Question: $f(x)=x^{3}-3 x$ Solution: Given : $f(x)=x^{3}-3 x$ $\Rightarrow f^{\prime}(x)=3 x^{2}-3$ For a local maximum or a local minimum, we must have $f^{\prime}(x)=0$ $\Rightarrow 3 x^{2}-3=0$ $\Rightarrow x^{2}-1=0$ $\Rightarrow x=\pm 1$ Since $f^{\prime}(x)$ changes from negative to positive as $x$ increases through $1, x=1$ is the point of local minima. The local minimum value of $f(x)$ at $x=1$ is given by $(1)^{3}-3(1)=-2$ Since $f^{\prime}(x)$ changes from positive to negative when $x$...
Read More →Eight chairs are numbered 1 to 8.
Question: Eight chairs are numbered 1 to 8. Two women and 3 men wish to occupy one chair each. First the women choose the chairs from amongst the chairs 1 to 4 and then men select from the remaining chairs. Find the total number of possible arrangements. Solution: We know that, nPr According to the question, W1 can occupy chairs marked 1 to 4 in 4 different way. W2can occupy 3 chairs marked 1 to 4 in 3 different ways. So, total no of ways in which women can occupy the chairs, 4P2= 4!/(4 2)! = (4...
Read More →If the solve the problem
Question: $f(x)=(x-5)^{4}$ Solution: Given : $f(x)=(x-5)^{4}$ $\Rightarrow f^{\prime}(x)=4(x-5)^{3}$ For a local maximum or a local minimum, we must have $f^{\prime}(x)=0$ $\Rightarrow 4(x-5)^{3}=0$ $\Rightarrow x=5$ Since $f^{\prime}(x)$ changes from negative to positive when $x$ increases through $5, x=5$ is the point of local minima. The local minimum value of $f(x)$ at $x=5$ is given by $(5-5)^{4}=0$...
Read More →Find the equation of the line passing through the intersection of the lines
Question: Find the equation of the line passing through the intersection of the lines $3 x-4 y+1=0$ and $5 x+y-1=0$ and which cuts off equal intercepts from the axes. Solution: Suppose the given two lines intersect at a point $P\left(x_{1}, y_{1}\right)$. Then, $\left(x_{1}, y_{1}\right)$ satisfies each of the given equations. $3 x-4 y+1=0 \ldots$ (i) $5 x+y-1=0 \ldots$ (ii) Now, we find the point of intersection of eq. (i) and (ii) Multiply the eq. (ii) by 4 , we get $20 x+4 y-4=0 \ldots$ (iii)...
Read More →Solve the following system
Question: Solve the following system of inequalities $\frac{2 x+1}{7 x-1}5, \frac{x+7}{x-8}2$ Solution: According to the question, $\frac{2 x+1}{7 x-1}5$ Subtracting 5 both side, we get $\frac{2 x+1}{7 x-1}-50$ $\Rightarrow \frac{2 x+1-35 x+5}{7 x-1}0$ $\Rightarrow \frac{6-33 x}{7 x-1}0$ For above fraction be greater than 0, either both denominator and numerator should be greater than 0 or both should be less than 0. ⇒6 33x 0 and 7x 1 0 ⇒33x 6 and 7x 1 ⇒x 2/11 and x 1/7 ⇒1/7 x 2/11 (i) Or ⇒6 33x...
Read More →In drilling world’s deepest hole it was
Question: In drilling worlds deepest hole it was found that the temperature T in degree Celsius, x km below the earths surface was given by T = 30 + 25 (x 3), 3 x 15. At what depth will the temperature be between 155C and 205C? Solution: According to the question, T = 30 + 25(x 3), 3 x 15; where, T = temperature and x = depth inside the earth The Temperature should be between 155C and 205C, So, we get, ⇒155 T 205 ⇒155 30 + 25(x 3) 205 ⇒155 30 + 25x 75 205 ⇒155 25x 45 205 Adding 45 to each term, ...
Read More →The longest side of a triangle is twice
Question: The longest side of a triangle is twice the shortest side and the third side is 2 cm longer than the shortest side. If the perimeter of the triangle is more than 166 cm then find the minimum length of the shortest side. Solution: Let the length of shortest side = x cm According to the question, The longest side of a triangle is twice the shortest side ⇒Length of largest side = 2x Also, the third side is 2 cm longer than the shortest side ⇒Length of third side = (x + 2) cm Perimeter of ...
Read More →A solution is to be kept between 40°C and 45°C.
Question: A solution is to be kept between 40C and 45C. What is the range of temperature in degree Fahrenheit, if the conversion formula is F = 9/5 C + 32? Solution: Let temperature in Celsius be C Let temperature in Fahrenheit be F According to the question, Solution should be kept between 40 C and 45C ⇒40 C 45 Multiplying each term by 9/5, we get ⇒72 9/5 C 81 Adding 32 to each term, we get ⇒104 9/5 C + 32 113 ⇒104 F 113 Hence, the range of temperature in Fahrenheit should be between 104 F and ...
Read More →Find the equation of the line through the intersection of the lines
Question: Find the equation of the line through the intersection of the lines $2 x+3 y-2$ = 0 and x 2y + 1 = 0 and having x-intercept equal to 3. Solution: Suppose the given two lines intersect at a point $P\left(x_{1}, y_{1}\right) .$ Then, $\left(x_{1}, y_{1}\right)$ satisfies each of the given equations. $2 x+3 y-2=0 \ldots$ (i) $x-2 y+1=0 \ldots$ (ii) Now, we find the point of intersection of eq. (i) and (ii) Multiply the eq. (ii) by 2, we get $2 x-4 y+2=0 \ldots$ (iii) On subtracting eq. (i...
Read More →Mark the correct alternative in the following:
Question: Mark the correct alternative in the following: The function $f(x)=x^{9}+3 x^{7}+64$ is increasing on A. $\mathrm{R}$ B. $(-\infty, 0)$ C. $(0, \infty)$ D. $R_{0}$ Solution: Formula:- (i) The necessary and sufficient condition for differentiable function defined on (a,b) to be strictly increasing on $(a, b)$ is that $f^{\prime}(x)0$ for all $x \in(a, b)$ Given:- $f(x)=x^{9}+3 x^{7}+64$ $\frac{\mathrm{d}(\mathrm{f}(\mathrm{x}))}{\mathrm{dx}}=9 \mathrm{x}^{8}+21 \mathrm{x}^{6}=\mathrm{f}^...
Read More →A solution of 9% acid is to be diluted
Question: A solution of 9% acid is to be diluted by adding 3% acid solution to it. The resulting mixture is to be more than 5% but less than 7% acid. If there is 460 litres of the 9% solution, how many litres of 3% solution will have to be added? Solution: According to the question, Let x litres of 3% solution is to be added to 460 liters of the 9% of solution Then, we get, Total solution = (460 + x) litres Total acid content in resulting solution = (460 9/100 + x 3/100) = (41.4 + 0.03x)% Accord...
Read More →The water acidity in a pool is considered normal
Question: The water acidity in a pool is considered normal when the average pH reading of three daily measurements is between 8.2 and 8.5. If the first two pH readings are 8.48 and 8.35, find the range of pH value for the third reading that will result in the acidity level being normal. Solution: According to the question, First reading = 8.48 Second reading = 8.35 Now, let the third reading be x Average pH should be between 8.2 and 8.5 Average pH = (8.48 + 8.35 + x)/3 $\Rightarrow 8.2\frac{8.48...
Read More →A company manufactures cassettes.
Question: A company manufactures cassettes. Its cost and revenue functions are C(x) = 26,000 + 30x and R(x) = 43x, respectively, where x is the number of cassettes produced and sold in a week. How many cassettes must be sold by the company to realise some profit? Solution: We know that, Profit = Revenue cost Requirement is, profit 0 According to the question, Revenue, R(x) = 43 x Cost, C(x) = 26,000 + 30 x; where x is number of cassettes ⇒Profit = 43x (26,000 + 30x) 0 ⇒13x 26,000 0 ⇒13x 26000 ⇒x...
Read More →Mark the correct alternative in the following:
Question: Mark the correct alternative in the following: If the function $f(x)=x^{3}-9 k x^{2}+27 x+30$ is increasing on $R$, then A. $-1 \leq k1$ B. $k-1$ or $k1$ C. $0\mathrm{k}1$ D. $-1k0$ Solution: Formula:- (i) $a x^{2}+b x+c0$ for all $x \Rightarrow a0$ and $b^{2}-4 a c0$ (ii) $a x^{2}+b x+c0$ for all $x \Rightarrow a0$ and $b^{2}-4 a c0$ (iii) The necessary and sufficient condition for differentiable function defined on $(a, b)$ to be strictly increasing on $(a, b)$ is that $f^{\prime}(x)...
Read More →Solve for x, the inequalities in 4x + 3 ≥ 2x + 17,
Question: Solve for x, the inequalities in 4x + 3 2x + 17, 3x 5 2. Solution: According to the question, 4x + 3 2x + 17 ⇒4x 2x 17 3 ⇒2x 14 ⇒x 7 (i) Also, 3x 5 2 ⇒3x 3 ⇒x 1 (2) Since, equations [i] and [ii] cannot be possible simultaneously, We conclude that x has no solution....
Read More →Solve for x, the inequalities in
Question: Solve for x, the inequalities in $-5 \leq \frac{2-3 \mathrm{x}}{4} \leq 9$ Solution: According to the question, $-5 \leq \frac{2-3 x}{4} \leq 9$ Multiplying each term by 4, we get ⇒-20 2 3x 36 Adding -2 each term, we get ⇒-22 -3x 34 Dividing each term by 3, we get ⇒-22/3 -x 34/3 We know that, Multiplication by -1 inverts the inequality. So, multiplying each term by -1, we get ⇒ -34/3 x 22/3...
Read More →Solve for x, the inequalities in |x – 1| ≤ 5,
Question: Solve for x, the inequalities in |x 1| 5, |x| 2 Solution: |x 1| 5 There are two cases, 1:- x 1 5 Adding 1 to LHS and RHS ⇒x 6 2:- ⇒-(x 1) 5 ⇒-x + 1 5 Subtracting 1 from LHS and RHS, ⇒-x 4 ⇒x -4 From cases 1 and 2, we have ⇒-4 x 6 [i] Also, |x| 2 ⇒x 2 and ⇒-x 2 ⇒x -2 ⇒x(, -2][2, ) [ii] Combining equation [i] and [ii], we get x [-4, -2] [2, 6]...
Read More →Find the equation of the line through the intersection of the lines
Question: Find the equation of the line through the intersection of the lines $2 x-3 y+1$ $=0$ and $x+y-2=0$ and drawn parallel to $y$-axis. Solution: Suppose the given two lines intersect at a point $P\left(x_{1}, y_{1}\right) .$ Then, $\left(x_{1}, y_{1}\right)$ satisfies each of the given equations. $2 x-3 y+1=0 \ldots$ (i) $x+y-2=0 \ldots$ (ii) Now, we find the point of intersection of eq. (i) and (ii) Multiply the eq. (ii) by 2 , we get $2 x+2 y-4=0 \ldots$ (iii) On subtracting eq. (iii) fr...
Read More →Solve for x, the inequalities in
Question: Solve for x, the inequalities in $\frac{1}{|x|-3} \leq \frac{1}{2}$ Solution: According to the question, $\frac{1}{|x|-3} \leq \frac{1}{2}$ $\Rightarrow \frac{1}{|x|-3}-\frac{1}{2} \leq 0$ $\Rightarrow \frac{2-|x|+3}{2(|x|-3)} \leq 0$ $\Rightarrow \frac{5-|x|}{(|x|-3)} \leq 0$ ⇒5 |x| 0 and |x| 3 0 or 5 |x| 0 and |x| 3 0 ⇒|x| 5 and |x| 3 or |x| 5 and |x| 3 ⇒|x| 5 or |x| 3 ⇒x(- , 5] or [5, ) or x( -3 , 3) ⇒ x (- , 5] ( -3 , 3) [5, )...
Read More →Solve for x, the inequalities in
Question: Solve for x, the inequalities in $\frac{|x-2|-1}{|x-2|-2} \leq 0$ Solution: According to the question, $\frac{|x-1|-1}{|x-2|-2} \leq 0$ Let $y=|x-2|$, then $\Rightarrow \frac{y-1}{y-2} \leq 0$ Now, if $y1$, then $\mathrm{y}-10$ and $\mathrm{y}-20$ and,$\frac{y-1}{y-2}0$ ,which is not required if $y2$, then $y-10$ and $y-20$ and, $\frac{y-1}{y-2}0$ , which is not required if $1 \leq \mathrm{y}2$, then $\mathrm{y}-1 \geq 0$ and $\mathrm{y}-20$ and, $\frac{y-1}{y-2}0$ , which is the requi...
Read More →Mark the correct alternative in the following:
Question: Mark the correct alternative in the following: The function $f(x)=-\frac{x}{2}+\sin x$ defined on $\left[-\frac{\pi}{3}, \frac{\pi}{3}\right]$ is A. increasing B. decreasing C. constant D. none of these Solution: Formula:- (i) The necessary and sufficient condition for differentiable function defined on (a,b) to be strictly increasing on $(a, b)$ is that $f^{\prime}(x)0$ for all $x \in(a, b)$ Given:- $f(x)=-\frac{x}{2}+\sin x$ $\frac{\mathrm{d}(\mathrm{f}(\mathrm{x}))}{\mathrm{dx}}=-\f...
Read More →Mark the correct alternative in the following:
Question: Mark the correct alternative in the following: If the function $f(x)=x^{2}-k x+5$ is increasing on $[2,4]$, then A. $k \in(2, \infty)$ B. $k \in(-\infty, 2)$ C. $k \in(4, \infty)$ D. $k \in(-\infty, 4)$ Solution: Formula:- (i) The necessary and sufficient condition for differentiable function defined on (a,b) to be strictly increasing on $(a, b)$ is that $f^{\prime}(x)0$ for all $x \in(a, b)$ $f(x)=x^{2}-k x+5$ $\frac{\mathrm{d}(\mathrm{f}(\mathrm{x}))}{\mathrm{dx}}=2 \mathrm{x}-\mathr...
Read More →Solve for x, the inequalities in
Question: Solve for x, the inequalities in $\frac{4}{x+1} \leq 3 \leq \frac{6}{x+1},(x0)$ Solution: According to the question, $\frac{4}{x+1} \leq 3 \leq \frac{6}{x+1}$ Multiplying each term by (x + 1) ⇒4 3(x + 1) 6 ⇒4 3x + 3 6 Subtracting each term by 3, we get, ⇒1 3x 3 Dividing each term by 3, we get, ⇒ (1/3) x 1...
Read More →Mark the correct alternative in the following:
Question: Mark the correct alternative in the following: Let $\phi(x)=f(x)+f(2 a-x)$ and $f^{\prime \prime}(x)0$ for all $x \in[0, a] .$ The, $\phi(x)$ A. increases on $[0, a]$ B. decreases on $[0, a]$ C. increases on $[-a, 0]$ D. decreases on $[a, 2 a]$ Solution: Formula:- (i) The necessary and sufficient condition for differentiable function defined on (a,b) to be strictly increasing on $(a, b)$ is that $f^{\prime}(x)0$ for all $x \in(a, b)$ $\phi(x)=f(x)+f(2 a-x)$ $\Rightarrow \phi^{\prime}(x...
Read More →Mark the correct alternative in the following:
Question: Mark the correct alternative in the following: Function $f(x)=\log _{a} x$ is increasing on $R$, if A. $0a1$ B. $a1$ C. $a1$ D. $a0$ Solution: Formula:- (i) The necessary and sufficient condition for differentiable function defined on (a,b) to be strictly increasing on $(a, b)$ is that $f^{\prime}(x)0$ for all $x \in(a, b)$ $f(x)=\log _{a} x$ $\frac{\mathrm{d}(\mathrm{f}(\mathrm{x}))}{\mathrm{dx}}=\frac{1}{\mathrm{x} \log _{\mathrm{e}} \mathrm{a}}=\mathrm{f}^{\prime}(\mathrm{x})$ For i...
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