Factorise:
Question: Factorise:x2 4x 12 Solution: The given expression is $x^{2}-4 x-12$. Find two numbers that follow the conditions given below : Sum $=-4$ Product $=-12$ Clearly, the numbers are $-6$ and 2 . $x^{2}-4 x-12=x^{2}-6 x+2 x-12$ $=x(x-6)+2(x-6)$ $=(x-6)(x+2)$...
Read More →The following is the frequency distribution
Question: The following is the frequency distribution of duration for 100 calls made on a mobile phone. Solution: First, we calculate class marks as follows Here, (assumed mean) a = 170, and (class width) h = 30 By step deviation method, Average $(\bar{x})=a+\frac{\sum f_{i} u_{j}}{\sum f_{j}} \times h=170+\frac{1}{100} \times 30$ $=170+0.3=170.3$ Hence, average duration is 170.3s. For calculating median from a cumulative frequency curve We prepare less than type or more than type ogive We obser...
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Question: Factorise:z2 12z 45 Solution: The given expression is $z^{2}-12 z-45$. Find two numbers that follow the conditions given below: Sum $=-12$ Product $=-45$ Clearly, the numbers are $-15$ and 3 . $z^{2}-12 z-45=z^{2}-15 z+3 z-45$ $=z(z-15)+3(z-15)$ $=(z-15)(z+3)$...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $\log \left\{\cot \left(\frac{\pi}{4}+\frac{x}{2}\right)\right\}$ Solution: Let $y=\log \left\{\cot \left(\frac{\pi}{4}+\frac{x}{2}\right)\right\}$ On differentiating $y$ with respect to $x$, we get $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left[\log \left\{\cot \left(\frac{\pi}{4}+\frac{\mathrm{x}}{2}\right)\right\}\right]$ We know $\frac{\mathrm{d}}{\mathrm{dx}}(\log \mathrm{x})=\frac{1}{\mathrm{x}}$ $\...
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Question: Factorise:y2 6y 135 Solution: The given expression is $y^{2}-6 y-135$. Find two numbers that follow the conditions given below: Sum $=-6$ Product $=-135$ Clearly, the numbers are $-15$ and $9 .$ $y^{2}-6 y-135=y^{2}-15 y+9 y-135$ $=y(y-15)+9(y-15)$ $=(y-15)(y+9)$...
Read More →The annual rainful record of a city
Question: The annual rainful record of a cityfor 66 days is given in the following table. Calculate the median rainfall using ogives (or move than type and of less than type) Solution: We observe that, the annual rainfall record of a city less than 0 is 0. Similarly, less than 10 include the annual rainfall record of a city from 0 as well as the annual rainfall record of a city from 0-10.v So, the total annual rainfall record of a city for less than 10 cm is 0+ 22 =22 days. Continuing in this ma...
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Question: Factorise:x2 5x 24 Solution: The given expression is $x^{2}-5 x-24$. Find two numbers that follow the conditions given below: Sum $=-5$ Product $=-24$ Clearly, the numbers are $-8$ and 3 . $x^{2}-5 x-24=x^{2}-8 x+3 x-24$ $=x(x-8)+3(x-8)$ $=(x-8)(x+3)$...
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Question: Factorise:x2 11x 42 Solution: The given expression is $x^{2}-11 x-42$. Find two numbers that follow the conditions given below : Sum $=-11$ Product $=-42$ Clearly, the numbers are $-14$ and 3 . $x^{2}-11 x-42=x^{2}-14 x+3 x-42$ $=x(x-14)+3(x-14)$ $=(x-14)(x+3)$...
Read More →Size of agricultural holdings in a survey of 200
Question: Size of agricultural holdings in a survey of 200 families is given in the following Compute median and mode size of the holdings. Solution: (i) Here, $N=200$ Now, $\frac{N}{2}=\frac{200}{2}=100$, which lies in the interval 15-20. Lower limit, $l=15, h=5, f=80$ and $c f=55$ $\therefore \quad$ Median $=l+\left(\frac{\frac{N}{2}-c f}{f}\right) \times h=15+\left(\frac{100-55}{80}\right) \times 5$ $=15+\left(\frac{45}{16}\right)=15+2.81=17.81 \mathrm{hec}$ (ii) In a given table 80 is the hi...
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Question: Factorise:x2 7x 30 Solution: The given expression is $x^{2}-7 x-30$. Find two numbers that follow the conditions given below: Sum $=-7$ Product $=-30$ Clearly, the numbers are $-10$ and 3 . $x^{2}-7 x-30=x^{2}-10 x+3 x-30$ $=x(x-10)+3(x-10)$ $=(x-10)(x+3)$...
Read More →Factorise:
Question: Factorise:p2 4p 77 Solution: The given expression is $p^{2}-4 p-77$. Find two numbers that follow the conditions given below: Sum $=-4$ Product $=-77$ Clearly, the numbers are $-11$ and 7 . $p^{2}-4 p-77=p^{2}-11 p+7 p-77$ $=p(p-11)+7(p-11)$ $=(p-11)(p+7)$...
Read More →Factorise:
Question: Factorise:p2 4p 77 Solution: The given expression is $p^{2}-4 p-77$. Find two numbers that follow the conditions given below: Sum $=-4$ Product $=-77$ Clearly, the numbers are $-11$ and 7 . $p^{2}-4 p-77=p^{2}-11 p+7 p-77$ $=p(p-11)+7(p-11)$ $=(p-11)(p+7)$...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $\frac{x^{2}\left(1-x^{2}\right)^{3}}{\cos 2 x}$ Solution: Let $y=\frac{x^{2}\left(1-x^{2}\right)^{3}}{\cos 2 x}$ On differentiating $y$ with respect to $x$, we get $\frac{d y}{d x}=\frac{d}{d x}\left[\frac{x^{2}\left(1-x^{2}\right)^{3}}{\cos 2 x}\right]$ Recall that $\left(\frac{\mathrm{u}}{\mathrm{v}}\right)^{\prime}=\frac{\mathrm{vu}^{\prime}-\mathrm{uv}^{\prime}}{\mathrm{v}^{2}}$ (quotient rule) $\Rightarrow \frac{d y}{d x...
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Question: Factorise:a2+ 6a 91 Solution: The given expression is $a^{2}+6 a-91$. Find two numbers that follow the conditions given below : Sum $=6$ Product $=-91$ Clearly, the numbers are 13 and $-7$. $a^{2}+6 a-91=a^{2}+13 a-7 a-91$ $=a(a+13)-7(a+13)$ $=(a+13)(a-7)$...
Read More →The distribution of heights (in cm) of 96
Question: The distribution of heights (in cm) of 96 children is given below Draw a less than type cumulative frequency curve for this data and use it to compute median height of the children. Solution: To draw the less than type ogive, we plot the points (124, 0), (128, 5), (132, 13), (136, 30), (140, 54), (144, 70), (148, 82), (152, 88), (156, 92), (160, 95), (164, 96) and join all these point by free hand. Here,$\frac{N}{2}=\frac{96}{2}$ We take, y = 48 in Y-coordinate and draw a line parallel...
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Question: Factorise:y2+y 72 Solution: The given expression is $y^{2}+y-72$. Find two numbers that follow the conditions given below: Sum $=1$ Product $=-72$ Clearly, the numbers are 9 and $-8$. $y^{2}+y-72=y^{2}+9 y-8 y-72$ $=y(y+9)-8(y+9)$ $=(y+9)(y-8)$...
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Question: Factorise:z2+ 19z 150 Solution: The given expression is $z^{2}+19 z-150$. Find two numbers that follow the conditions given below: Sum $=19$ Product $=-150$ Clearly, the numbers are 25 and $-6$. $z^{2}+19 z-150=z^{2}+25 z-6 z-150$ $=z(z+25)-6(z+25)$ $=(z+25)(z-6)$...
Read More →The median of the following data is 50.
Question: The median of the following data is 50. Find the values of p and q, if the sum of all the frequencies is 90. Solution: Given $N=90$ $\therefore$ $\frac{N}{2}=\frac{90}{2}=45$ which lies in the interval $50-60$. Lower limit, $l=50, f=20, c f=40+p, h=10$ $\therefore$ Median $=l+\frac{\left(\frac{N}{2}-c f\right)}{f} \times h$ $=50+\frac{(45-40-\rho)}{20} \times 10$ $\Rightarrow$ $50=50+\left(\frac{5-p}{2}\right)$ $\Rightarrow$ $0=\frac{5-p}{2}$ $\therefore$ $p=5$ Also, $78+p+q=90$ [given...
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Question: Factorise:y2+ 7y 144 Solution: The given expression is $y^{2}+7 y-144$. Find two numbers that follow the conditions given below: Sum $=7$ Product $=-144$ Clearly, the numbers are 16 and $-9 .$ $y^{2}+7 y-144=y^{2}+16 y-9 y-144$ $=y(y+16)-9(y+16)$ $=(y+16)(y-9)$...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $\frac{x^{2}+2}{\sqrt{\cos x}}$ Solution: Let $y=\frac{x^{2}+2}{\sqrt{\cos x}}$ On differentiating y with respect to $x$, we get $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{\mathrm{x}^{2}+2}{\sqrt{\cos \mathrm{x}}}\right)$ Recall that $\left(\frac{\mathrm{u}}{\mathrm{v}}\right)^{\prime}=\frac{\mathrm{vu}^{\prime}-\mathrm{uv}^{\prime}}{\mathrm{v}^{2}}$ (quotient rule) $\Rightarrow \frac{d y}{d x}=...
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Question: Factorise:x2+ 5x 104 Solution: The given expression is $x^{2}+5 x-104$. Find two numbers that follow the conditions given below: Sum $=5$ Product $=-104$ Clearly, the numbers are 13 and $-8 .$ $x^{2}+5 x-104=x^{2}+13 x-8 x-104$ $=x(x+13)-8(x+13)$ $=(x+13)(x-8)$...
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Question: Factorise:x2+x 132Factorise: x2+x 132 Solution: The given expression is $x^{2}+x-132$. Find two numbers that follow the conditions given below: Sum $=1$ Product $=-132$ Clearly, the numbers are 12 and $-11$. $x^{2}+x-132=x^{2}+12 x-11 x-132$ $=x(x+12)-11(x+12)$ $=(x+12)(x-11)$...
Read More →The mean of the following frequency distribution
Question: The mean of the following frequency distribution is 50 but the frequencies f1andf2in classes 20-40 and 60-80, respectively are not known. Find these frequencies, if the sum of all the frequencies is 120. Solution: First we caluculate the class mark of given data Given that, sum of all frequencies $=120$ $\Rightarrow \quad \Sigma f_{1}=68+f_{1}+f_{2}=120$ $\Rightarrow \quad f_{1}+f_{2}=52$$\ldots(\mathrm{i})$ Here, $\quad$ (assumed mean) $a=50$ and (class width) $h=20$ By step deviation...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $3 e^{-3 x} \log (1+x)$ Solution: Let $y=3 e^{-3 x} \log (1+x)$ On differentiating $y$ with respect to $x$, we get $\frac{d y}{d x}=\frac{d}{d x}\left[3 e^{-3 x} \log (1+x)\right]$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=3 \frac{\mathrm{d}}{\mathrm{dx}}\left[\mathrm{e}^{-3 \mathrm{x}} \log (1+\mathrm{x})\right]$ We have (uv)' $=v u^{\prime}+u v^{\prime}$ (product rule) $\Rightarrow \frac{d y}{d x}=3\left[\log (1+x) \frac{...
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Question: Factorise:x2 9x+ 20 Solution: The given expression is $x^{2}-9 x+20$. Find two numbers that follow the conditions given below : Sum $=-9$ Product $=20$ Clearly, the numbers are $-5$ and $-4$. $x^{2}-9 x+20=x^{2}-5 x-4 x+20$ $=x(x-5)-4(x-5)$ $=(x-5)(x-4)$...
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