Factorise: Evaluate
Question: Factorise:Evaluate {(7.8)2 (2.2)2}. Solution: We have: $\left\{(7.8)^{2}-(2.2)^{2}\right\}=(7.8+2.2)(7.8-2.2)$ $=(10 \times 5.6)$ $=56$ $\therefore\left\{(7.8)^{2}-(2.2)^{2}\right\}=56$...
Read More →Factorise: Evaluate
Question: Factorise:Evaluate {(405)2 (395)2}. Solution: We have: $\left\{(405)^{2}-(395)^{2}\right\}=(405+395)(405-395)$ $=(800 \times 10)$ $=8000$ $\therefore\left\{(405)^{2}-(395)^{2}\right\}=8000$...
Read More →Two dice are thrown simultaneously.
Question: Two dice are thrown simultaneously. What is the probability that the sum of the numbers appearing on the dice is (i) 7 ? (ii) a prime number ? (iii) 1 ? Solution: Two dice are thrown simultaneously. [given] So, total number of possible outcomes = 36 (i) Sum of the numbers appearing on the dice is $7 .$ So, the possible ways are $(1,6),(2,5),(3,4),(4,3),(5,2)$ and $(6,1)$. Number of possible ways $=6$ $\therefore \quad$ Required probability $=\frac{6}{36}=\frac{1}{6}$ (ii) Sum of the nu...
Read More →Factorise:
Question: Factorise:100 (x 5)2 Solution: We have: $100-(x-5)^{2}=(10)^{2}-(x-5)^{2}$ $=\{10+(x-5)\}\{10-(x-5)\}$ $=(10+x-5)(10-x+5)$ $=(5+x)(15-x)$ $\therefore 100-(x-5)^{2}=(5+x)(15-x)$...
Read More →Two dice are thrown at the same time.
Question: Two dice are thrown at the same time. Find the probability of getting (i) same number on both dice. (ii) different number on both dice. Solution: Two dice are thrown at the same time. [given] So, total number of possible outcomes = 36 (i) We have, same number on both dice. So, possible outcomes are (1,1), (2, 2), (3, 3), (4, 4), (5, 5) and (6, 6). $\therefore$ Number of possible outcomes $=6$ Now, $\quad$ required probability $=\frac{6}{36}=\frac{1}{6}$ (ii) We have, different number o...
Read More →Factorise:
Question: Factorise:9a2b2+ 4b 4 Solution: We have: $9 a^{2}-b^{2}+4 b-4=9 a^{2}-\left(b^{2}-4 b+4\right)$ $=(3 a)^{2}-(b-2)^{2}$ $=\{3 a+(b-2)\}\{3 a-(b-2)\}$ $=(3 a+b-2)(3 a-b+2)$ $\therefore 9 a^{2}-b^{2}+4 b-4=(3 a+b-2)(3 a-b+2)$...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $\sqrt{\tan ^{-1}\left(\frac{x}{2}\right)}$ Solution: Let $y=\sqrt{\tan ^{-1} \frac{x}{2}}$ On differentiating $y$ with respect to $x$, we get $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left(\sqrt{\tan ^{-1} \frac{\mathrm{x}}{2}}\right)$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left[\left(\tan ^{-1} \frac{\mathrm{x}}{2}\right)^{\frac{1}{2}}\right]$ We know $\frac{\mathrm{...
Read More →The weight of coffee in 70 packets are shown in the following table
Question: The weight of coffee in 70 packets are shown in the following table Determine the model weight . Solution: In the given data, the highest frequency is 26, which lies in the interval 201-202 Here, $\quad l=201, f_{m}=26, f_{1}=12, f_{2}=20$ and (class width) $h=1$ $\because$Mode $=l+\left(\frac{f_{m}-f_{1}}{2 f_{m}-f_{1}-f_{2}}\right) \times h=201+\left(\frac{26-12}{2 \times 26-12-20}\right) \times 1$ $=201+\left(\frac{14}{52-32}\right)=201+\frac{14}{20}=201+0.7=201.7 \mathrm{~g}$ Hence...
Read More →Factorise:
Question: Factorise:25a2 4b2+ 28bc 49c2 Solution: We have: $25 a^{2}-4 b^{2}+28 b c-49 c^{2}=25 a^{2}-\left(4 b^{2}-28 b c+49 c^{2}\right)$ $=(5 a)^{2}-(2 b-7 c)^{2}$ $=\{5 a+(2 b-7 c)\}\{5 a-(2 b-7 c)\}$ $=(5 a+2 b-7 c)(5 a-2 b+7 c)$ $\therefore 25 a^{2}-4 b^{2}+28 b c-49 c^{2}=(5 a+2 b-7 c)(5 a-2 b+7 c)$...
Read More →Factorise:
Question: Factorise:25 a2b2 2ab Solution: We have: $25-a^{2}-b^{2}-2 a b=25-\left(a^{2}+b^{2}+2 a b\right)$ $=25-(a+b)^{2}$ $=(5)^{2}-(a+b)^{2}$ $=\{5+(a+b)\}\{5-(a+b)\}$ $=(5+a+b)(5-a-b)$ $\therefore 25-a^{2}-b^{2}-2 a b=(5+a+b)(5-a-b)$...
Read More →The monthly income of 100 families are given as below
Question: The monthly income of 100 families are given as below Caluculate the model income. Solution: In a given data, the highest frequency is 41, which lies in the interval 10000-15000. Here, $l=10000, f_{m}=41, f_{1}=26, f_{2}=16$ and $h=5000$ $\therefore$Mode $=l+\left(\frac{f_{m}-f_{1}}{2 f_{m}-t_{1}-f_{2}}\right) \times h$ $=10000+\left(\frac{41-26}{2 \times 41-26-16}\right) \times 5000$ $=10000+\left(\frac{15}{82-42}\right) \times 5000$ $=10000+\left(\frac{15}{40}\right) \times 5000$ $=1...
Read More →Factorise:
Question: Factorise:x2y2 2y 1 Solution: We have: $x^{2}-y^{2}-2 y-1=x^{2}-\left(y^{2}+2 y+1\right)$ $=(x)^{2}-(y+1)^{2}$ $=\{x+(y+1)\}\{x-(y+1)\}$ $=(x+y+1)(x-y-1)$ $\therefore x^{2}-y^{2}-2 y-1=(x+y+1)(x-y-1)$...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $e^{\tan ^{-1} \sqrt{x}}$ Solution: Let $y=e^{\tan ^{-1} \sqrt{x}}$ On differentiating $y$ with respect to $x$, we get $\frac{d y}{d x}=\frac{d}{d x}\left(e^{\tan ^{-1} \sqrt{x}}\right)$ We know $\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{e}^{\mathrm{x}}\right)=\mathrm{e}^{\mathrm{x}}$ $\Rightarrow \frac{d y}{d x}=e^{\tan ^{-1} \sqrt{x}} \frac{d}{d x}\left(\tan ^{-1} \sqrt{x}\right)$ [using chain rule] We have $\frac{\mathrm{...
Read More →The maximum bowling speeds,
Question: The maximum bowling speeds, in km per hour, of 33 players at a cricket coaching centre are given as follows Caluculate the median bowling speed. Solution: First we construct the cumulative frequency table It is given that, $n=33$ $\therefore$$\frac{n}{2}=\frac{33}{2}=16.5$ So, the median class is $100-115$. where, $\quad$ lower limit $(l)=100$, frequency $(t)=9$, cumulative frequency $(c f)=11$ and $\quad$ class width $(h)=15$ $\therefore$ Median $=l+\frac{\left(\frac{n}{2}-c f\right)}...
Read More →Factorise:
Question: Factorise:(3x 4y)2 25z2 Solution: We have: $(3 x-4 y)^{2}-25 z^{2}=(3 x-4 y)^{2}-(5 z)^{2}$ $=\{(3 x-4 y)+5 z\}\{(3 x-4 y)-5 z\}$ $=(3 x-4 y+5 z)(3 x-4 y-5 z)$ $\therefore(3 x-4 y)^{2}-25 z^{2}=(3 x-4 y+5 z)(3 x-4 y-5 z)$...
Read More →Factorise:
Question: Factorise:36c2 (5a+b)2 Solution: We have: $36 c^{2}-(5 a+b)^{2}=(6 c)^{2}-(5 a+b)^{2}$ $=\{(6 c)+(5 a+b)\}\{(6 c)-(5 a+b)\}$ $=(6 c+5 a+b)(6 c-5 a-b)$ $\therefore 36 c^{2}-(5 a+b)^{2}=(6 c+5 a+b)(6 c-5 a-b)$...
Read More →Weekly income of 600 families is tabulated below
Question: Weekly income of 600 families is tabulated below Compute the median income. Solution: First we construct a cumulative frequency table It is given that, $n=600$ $\therefore$$\frac{n}{2}=\frac{600}{2}=300$ Since, cumulative frequency 440 lies in the interval $1000-2000$. Here, (lower median class) $l=1000$, $f=190, c f=250$, (class width) $h=1000$ and (total observation) $n=600$ $\therefore$ Median $=l+\frac{\left\{\frac{n}{2}-c f\right\}}{f} \times n$ $=1000+\frac{(300-250)}{190} \times...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $\sin \left(2 \sin ^{-1} x\right)$ Solution: Let $y=\sin \left(2 \sin ^{-1} x\right)$ On differentiating $y$ with respect to $x$, we get $\frac{d y}{d x}=\frac{d}{d x}\left[\sin \left(2 \sin ^{-1} x\right)\right]$ We know $\frac{\mathrm{d}}{\mathrm{dx}}(\sin \mathrm{x})=\cos \mathrm{x}$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\cos \left(2 \sin ^{-1} \mathrm{x}\right) \frac{\mathrm{d}}{\mathrm{dx}}\left(2 \sin ^{-1} \mathr...
Read More →Factorise:
Question: Factorise:(2x+ 5y)2 1 Solution: We have: $(2 x+5 y)^{2}-1=(2 x+5 y)^{2}-(1)^{2}$ $=\{(2 x+5 y)+1\}\{(2 x+5 y)-1\}$ $=(2 x+5 y+1)(2 x+5 y-1)$ $\therefore(2 x+5 y)^{2}-1=(2 x+5 y+1)(2 x+5 y-1)$...
Read More →Given below is a cumulative frequency distribution
Question: Given below is a cumulative frequency distribution showing the marks secured by 50 students of a class Form the frequency distribution table for the data. Solution: Here, we observe that, 17 students have scored marks below 20 i.e., it lies between class interval 0-20 and 22 students have scored marks below 40, so 22 -17 = 5 students lies in the class interval 20-40 continuing in the same manner, we get the complete frequency distribution table for given data....
Read More →Factorise:
Question: Factorise:(l+m)2 (lm)2 Solution: We have: $(l+m)^{2}-(l-m)^{2}=\{(l+m)+(l-m)\}\{(l+m)-(l-m)\}$ $=(l+m+l-m)(l+m-l+m)$ $=(2 l)(2 m)$ $\therefore(l+m)^{2}-(l-m)^{2}=(2 l)(2 m)$...
Read More →The following are the ages of 300 patients
Question: The following are the ages of 300 patients getting medical treatment in a hospital on a particular day Form (i) less than type cumulative frequency distribution. (ii) More than type cumulative frequency distribution. Solution: (i) We observe that the number of patients which take medical treatment in a hospital on a particular day less than 10 is O. Similarly, less than 20 i nclude the number of patients which take medical treatment from 0-10 as well as the number of patients which tak...
Read More →Factorise:
Question: Factorise:(2a+ 3b)2 16c2 Solution: We have: $(2 a+3 b)^{2}-16 c^{2}=(2 a+3 b)^{2}-(4 c)^{2}$ $=\{(2 a+3 b)+4 c\}\{(2 a+3 b)-4 c\}$ $=(2 a+3 b+4 c)(2 a+3 b-4 c)$ $\therefore(2 a+3 b)^{2}-16 c^{2}=(2 a+3 b+4 c)(2 a+3 b-4 c)$...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $e^{\sin ^{-1} 2 x}$ Solution: Let $y=e^{\sin ^{-1} 2 x}$ On differentiating $y$ with respect to $x$, we get $\frac{d y}{d x}=\frac{d}{d x}\left(e^{\sin ^{-1} 2 x}\right)$ We know $\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{e}^{\mathrm{x}}\right)=\mathrm{e}^{\mathrm{x}}$ $\Rightarrow \frac{\mathrm{d} y}{\mathrm{dx}}=\mathrm{e}^{\sin ^{-1} 2 \mathrm{x}} \frac{\mathrm{d}}{\mathrm{dx}}\left(\sin ^{-1} 2 \mathrm{x}\right)$ [using...
Read More →Factorise:
Question: Factorise:1 (b c)2 Solution: We have: $1-(b-c)^{2}=(1)^{2}-(b-c)^{2}$ $=\{1+(b-c)\}\{1-(b-c)\}$ $=(1+b-c)(1-b+c)$ $\therefore 1-(b-c)^{2}=(1+b-c)(1-b+c)$...
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