Factorise:
Question: Factorise:1 (b c)2 Solution: We have: $1-(b-c)^{2}=(1)^{2}-(b-c)^{2}$ $=\{1+(b-c)\}\{1-(b-c)\}$ $=(1+b-c)(1-b+c)$ $\therefore 1-(b-c)^{2}=(1+b-c)(1-b+c)$...
Read More →Find the unknown entries o, b, c, d, e
Question: Find the unknown entries o, b, c, d, e and f in the following distribution of heights of students in a class Solution: On comparing last two tables, we get $a=12$ $\therefore \quad 12+b=25$ $\Rightarrow$ $b=25-12=13$ $22+b=c$ $\Rightarrow$ $c=22+13=35$ $22+b+d=43$ $\Rightarrow \quad 22+13+d=43$ $\Rightarrow \quad d=43-35=8$ and $\quad 22+b+d+e=48$ $\Rightarrow \quad 22+13+8+e=48$ $\Rightarrow \quad e=48-43=5$ and $\quad 24+b+d+e=f$ $\Rightarrow \quad 24+13+8+5=f$ $\therefore \quad f=50...
Read More →Factorise:
Question: Factorise:63a2b2 7 Solution: We have: $63 a^{2} b^{2}-7=7\left(9 a^{2} b^{2}-1\right)$ $=7\left\{(3 a b)^{2}-(1)^{2}\right\}$ $=7(3 a b+1)(3 a b-1)$ $\therefore 63 a^{2} b^{2}-7=7(3 a b+1)(3 a b-1)$...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $\tan ^{-1}\left(e^{x}\right)$ Solution: Let $y=\tan ^{-1}\left(e^{x}\right)$ On differentiating $y$ with respect to $x$, we get $\frac{d y}{d x}=\frac{d}{d x}\left(\tan ^{-1} e^{x}\right)$ We know $\frac{d}{d x}\left(\tan ^{-1} x\right)=\frac{1}{1+x^{2}}$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{1+\left(\mathrm{e}^{\mathrm{x}}\right)^{2}} \frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{e}^{\mathrm{x}}\right)$ [using ...
Read More →Factorise:
Question: Factorise:16p3 4p Solution: We have: $16 p^{3}-4 p=4 p\left(4 p^{2}-1\right)$ $=4 p\left\{(2 p)^{2}-(1)^{2}\right\}$ $=4 p(2 p+1)(2 p-1)$ $\therefore 16 p^{3}-4 p=4 p(2 p+1)(2 p-1)$...
Read More →From the frequency distribution table from the following data.
Question: From the frequency distribution table from the following data. Solution: Here, we observe that, all 34 students have scored marks more than or equal to 0. Since, 32 students have scored marks more than or equal to 10. So, 34- 32 =2 students lies in the interval 0-10 and so on. Now, we construct the frequency distribution table....
Read More →Factorise:
Question: Factorise:3x5 48x3 Solution: We have: $3 x^{5}-48 x^{3}=3 x^{3}\left(x^{2}-16\right)$ $=3 x^{3}\left\{(x)^{2}-(4)^{2}\right\}$ $=3 x^{3}(x+4)(x-4)$ $\therefore 3 x^{5}-48 x^{3}=3 x^{3}(x+4)(x-4)$...
Read More →The following table shows the cumulative frequency
Question: The following table shows the cumulative frequency distribution of marks of 800 students in an examination. Construct a frequency distribution table for the data above. Solution: Here, we observe that 10 students have scored marks below 10 i.e., it lies between class interval 0-10. Similarly, 50 students have scored marks below 20. So, 50 -10 = 40 students lies in the interval 10-20 and so on. The table of a frequency distribution for the given data is...
Read More →Factorise:
Question: Factorise:16x5 144x3 Solution: We have: $16 x^{5}-144 x^{3}=16 x^{3}\left(x^{2}-9\right)$ $=16 x^{3}\left\{(x)^{2}-(3)^{2}\right\}$ $=16 x^{3}(x+3)(x-3)$ $\therefore 16 x^{5}-144 x^{3}=16 x^{3}(x+3)(x-3)$...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $\log \left(\frac{x^{2}+x+1}{x^{2}-x+1}\right)$ Solution: Let $y=\log \left(\frac{x^{2}+x+1}{x^{2}-x+1}\right)$ On differentiating y with respect to $x$, we get $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left[\log \left(\frac{\mathrm{x}^{2}+\mathrm{x}+1}{\mathrm{x}^{2}-\mathrm{x}+1}\right)\right]$ We know $\frac{\mathrm{d}}{\mathrm{dx}}(\log \mathrm{x})=\frac{1}{\mathrm{x}}$ $\Rightarrow \frac{d y}{d x}=\f...
Read More →Factorise:
Question: Factorise:x3 64x Solution: We have: $x^{3}-64 x=x\left(x^{2}-64\right)$ $=x\left\{(x)^{2}-(8)^{2}\right\}$ $=x(x+8)(x-8)$ $\therefore x^{3}-64 x=x(x+8)(x-8)$...
Read More →The following is the distribution
Question: The following is the distribution of weights (in kg) of 40 persons. Construct a cumulative frequency distribution (of the less than type) table for the data above. Solution: The cumulative distribution (less than type) table is shown below...
Read More →Factorise:
Question: Factorise:12x2 27 Solution: We have: $12 x^{2}-27=3\left(4 x^{2}-9\right)$ $=3\left\{(2 x)^{2}-(3)^{2}\right\}$ $=3(2 x+3)(2 x-3)$ $\therefore 12 x^{2}-27=3(2 x+3)(2 x-3)$...
Read More →The mileage (km per litre) of 50 cars of
Question: The mileage (km per litre) of 50 cars of the same model was tested by a manufacturer and details are tabulated as given below Find the mean mileage. The manufacturer claimed that the mileage of the model was 16 kmL-1. Do you agree with this claim? Solution: Here, $\Sigma f_{i}=50$ and $\Sigma f_{i} x_{i}=724$ $\therefore$ Mean $\bar{x}=\frac{\sum f_{i} x_{i}}{\sum f_{j}}$ $=\frac{724}{50}=14.48$ Hence, mean mileage is $14.48 \mathrm{kmL}^{-1}$. No, the manufacturer is claiming mileage ...
Read More →Factorise:
Question: Factorise:20a2 45b2 Solution: We have: $20 a^{2}-45 b^{2}=5\left(4 a^{2}-9 b^{2}\right)$ $=5\left\{(2 a)^{2}-(3 b)^{2}\right\}$ $=5(2 a+3 b)(2 a-3 b)$ $\therefore 20 a^{2}-45 b^{2}=5(2 a+3 b)(2 a-3 b)$...
Read More →Factorise:
Question: Factorise:63a2 112b2 Solution: We have: $63 a^{2}-112 b^{2}=7\left(9 a^{2}-16 b^{2}\right)$ $=7\left\{(3 a)^{2}-(4 b)^{2}\right\}$ $=7(3 a+4 b)(3 a-4 b)$ $\therefore 63 a^{2}-112 b^{2}=7(3 a+4 b)(3 a-4 b)$...
Read More →The weights (in kg) of 50 wrestlers are recorded in the following table.
Question: The weights (in kg) of 50 wrestlers are recorded in the following table. Find the mean weight of the wrestlers. Solution: We first find the class mark of each class and then proceed as follows $\therefore$ Assumed mean $(a)=125$, Class width $(h)=10$ and total observation $(N)=50$ By assumed mean method. Mean $(\bar{x})=a+\frac{\Sigma f_{j} d_{i}}{\Sigma f_{i}}$ $=125+\frac{(-80)}{50}$ $=125-1.6-123.4 \mathrm{~kg}$...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $\frac{e^{e x}+e^{-2 x}}{e^{2 x}-e^{-2 x}}$ Solution: Let $y=\frac{e^{2 x}+e^{-2 x}}{e^{2 x}-e^{-2 x}}$ On differentiating $y$ with respect to $x$, we get $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{\mathrm{e}^{2 \mathrm{x}}+\mathrm{e}^{-2 \mathrm{x}}}{\mathrm{e}^{2 \mathrm{x}}-\mathrm{e}^{-2 \mathrm{x}}}\right)$ Recall that $\left(\frac{\mathrm{u}}{\mathrm{v}}\right)^{\prime}=\frac{\mathrm{vu}^{...
Read More →Factorise:
Question: Factorise:16a2 144 Solution: We have: $16 a^{2}-144=(4 a)^{2}-(12)^{2}$ $=(4 a+12)(4 a-12)$ $=4(a+3) 4(a-3)=16(a+3)(a-3)$ $\therefore 16 a^{2}-144=16(a+3)(a-3)$...
Read More →An aircraft has 120 passenger seats.
Question: An aircraft has 120 passenger seats. The number of seats occupied during 100 flights is given in the following table. Determine the mean number of seats occupied over the flights. Solution: We first, find the class mark xi, of each class and then proceed as follows. $\therefore \quad$ Assumed mean, $a=110$, Class width, $h=4$ and total observation, $N=100$ By assumed mean method, $\operatorname{Mean}(\bar{x})=a+\frac{\Sigma f_{j} d_{j}}{\Sigma f_{i}}$ $=110+\left(\frac{-8}{100}\right)=...
Read More →Factorise:
Question: Factorise:9a2b2 25 Solution: We have: $9 a^{2} b^{2}-25=(3 a b)^{2}-(5)^{2}$ $=(3 a b+5)(3 a b-5)$ $\therefore 9 a^{2} b^{2}-25=(3 a b+5)(3 a b-5)$...
Read More →Factorise:
Question: Factorise:16a2 225b2 Solution: We have: $16 a^{2}-225 b^{2}=(4 a)^{2}-(15 b)^{2}$ $=(4 a+15 b)(4 a-15 b)$ $\therefore 16 a^{2}-225 b^{2}=(4 a+15 b)(4 a-15 b)$...
Read More →The daily income of a sample
Question: The daily income of a sample of 50 employees are tabulated as follows. Find the mean daily income of employees. Solution: Since, given data is not continuous, so we subtract 0.5 from the lower limit and add 0.5 in the upper limit of each class. Now we first, find the class markxi, of each class and then proceed as follows $\therefore \quad$ Assumed mean, $a=300.5$ Class width, $h=200$ and total observations, $N=50$ By step deviation method, Mean $=a+h \times \frac{1}{N} \times \sum_{i=...
Read More →Factorise:
Question: Factorise:4x2 9y2 Solution: We have: $4 x^{2}-9 y^{2}=(2 x)^{2}-(3 y)^{2}$ $=(2 x+3 y)(2 x-3 y)$ $\therefore 4 x^{2}-9 y^{2}=(2 x+3 y)(2 x-3 y)$...
Read More →The following table gives the number of pages
Question: The following table gives the number of pages written by Sarika for completing her own book for 30 days. Find the mean number of pages written per day. Solution: Since, Since, given data is not continuous, so we subtract 0.5 from the lower limit and add 0.5 in the upper limit of each class. $\therefore$$\operatorname{Mean}(\bar{x})=\frac{\Sigma f_{j} x_{i}}{\Sigma f_{i}}=\frac{780}{30}=26$ Hence, the mean of pages written per day is 26....
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