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Question: Tick (✓) the correct answer: x3 144x= ? (a)x(x 12)2 (b)x(x+ 12)2(c)x(x 12)(x+ 12) (d) none of these Solution: (c)x(x 12)(x+ 12) $x^{3}-144 x$ $=x\left(x^{2}-144\right)$ $=x(x-12)(x+12)$...
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Question: If $\mathrm{y}=\sqrt{\mathrm{x}+1}+\sqrt{\mathrm{x}-1}$, prove that $\sqrt{\mathrm{x}^{2}-1} \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{2} \mathrm{y}$. Solution: Given $\mathrm{y}=\sqrt{\mathrm{x}+1}+\sqrt{\mathrm{x}-1}$ On differentiating $y$ with respect to $x$, we get $\frac{d y}{d x}=\frac{d}{d x}(\sqrt{x+1}+\sqrt{x-1})$ $\Rightarrow \frac{d y}{d x}=\frac{d}{d x}(\sqrt{x+1})+\frac{d}{d x}(\sqrt{x-1})$ $\Rightarrow \frac{d y}{d x}=\frac{d}{d x}(x+1)^{\frac{1}{2}}+\frac{d}{d x}(x-1)^{\...
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Question: Tick (✓) the correct answer: (2x 32x3) = ? (a) 2(x 4)(x+ 4) (b) 2x(1 2x)2 (c) 2x(1 + 2x)2 (d) 2x(1 4x)(1 + 4x) Solution: (d) 2x(1 4x)(1 + 4x) $\left(2 x-32 x^{3}\right)$ $=2 x\left(1-16 x^{2}\right)$ $=2 x(1-4 x)(1+4 x)$...
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Question: Tick (✓) the correct answer: (7a2 63b2) = ? (a) (7a 9b)(9a+ 7b) (b) (7a 9b)(7a+ 9b) (c) 9(a 3b)(a+ 3b) (d) 7(a 3b)(a+ 3b) Solution: (d) 7(a 3b)(a+ 3b) $\left(7 a^{2}-63 b^{2}\right)=7\left(a^{2}-9 b^{2}\right)$ $=7(a-3 b)(a+3 b)$...
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Question: If $\mathrm{y}=\sqrt{\mathrm{x}+1}+\sqrt{\mathrm{x}-1}$, prove that $\sqrt{\mathrm{x}^{2}-1} \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{2} \mathrm{y}$. Solution: Given $\mathrm{y}=\sqrt{\mathrm{x}+1}+\sqrt{\mathrm{x}-1}$ On differentiating $y$ with respect to $x$, we get $\frac{d y}{d x}=\frac{d}{d x}(\sqrt{x+1}+\sqrt{x-1})$ $\Rightarrow \frac{d y}{d x}=\frac{d}{d x}(\sqrt{x+1})+\frac{d}{d x}(\sqrt{x-1})$ $\Rightarrow \frac{d y}{d x}=\frac{d}{d x}(x+1)^{\frac{1}{2}}+\frac{d}{d x}(x-1)^{\...
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Question: Factorise:7x2 19x 6 Solution: The given expression is $7 x^{2}-19 x-6$. Find two numbers that follow the conditions given below: Sum $=-19$ Product $=7 \times-6=-42$ Clearly, the numbers are $-21$ and 2 . $7 x^{2}-19 x-6=7 x^{2}-21 x+2 x-6$ $=7 x(x-3)+2(x-3)$ $=(7 x+2)(x-3)$...
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Question: Factorise:6x2 17x 3 Solution: The given expression is $6 x^{2}-17 x-3$. Find two numbers that follow the conditions given below : Sum $=-17$ Product $=6 \times-3=-18$ Clearly, the numbers are $-18$ and 1 . $6 x^{2}-17 x-3=6 x^{2}-18 x+x-3$ $=6 x(x-3)+1(x-3)$ $=(6 x+1)(x-3)$...
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Question: Factorise:4n2 8n+ 3 Solution: The given expression is $4 n^{2}-8 n+3$. Find two numbers that follow the conditions given below: Sum $=-8$ Product $=4 \times 3=12$ Clearly, the numbers are $-6$ and $-2$. $4 n^{2}-8 n+3=4 n^{2}-2 n-6 n+3$ $=2 n(2 n-1)-3(2 n-1)$ $=(2 n-1)(2 n-3)$...
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Question: Factorise:3m2+ 24m+ 36 Solution: The given expression is $3 m^{2}+24 m+36$. Find two numbers that follow the conditions given below: Sum $=24$ Product $=36 \times 3=108$ Clearly, the numbers are 18 and 6 . $3 m^{2}+24 m+36=3 m^{2}+18 m+6 m+36$ $=3 m(m+6)+6(m+6)$ $=(3 m+6)(m+6)=3(m+2)(m+6)$...
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Question: If $\mathrm{y}=\log \{\sqrt{\mathrm{x}-1}-\sqrt{\mathrm{x}+1}\}$, show that $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{-1}{2 \sqrt{\mathrm{x}^{2}-1}}$. Solution: Given $y=\log (\sqrt{x-1}-\sqrt{x+1})$ On differentiating $y$ with respect to $x$, we get $\frac{d y}{d x}=\frac{d}{d x}[\log (\sqrt{x-1}-\sqrt{x+1})]$ We know $\frac{\mathrm{d}}{\mathrm{dx}}(\log \mathrm{x})=\frac{1}{\mathrm{x}}$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{\sqrt{\mathrm{x}-1}-\sqrt{\mathrm{x}+1}} \frac{...
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Question: Factorise:6x2 5x 6 Solution: The given expression is $6 x^{2}-5 x-6$. Find two numbers that follow the conditions given below: Sum $=-5$ Product $=-6 \times 6=-36$ Clearly, the numbers are $-9$ and 4 . $6 x^{2}-5 x-6=6 x^{2}-9 x+4 x-6$ $=3 x(2 x-3)+2(2 x-3)$ $=(2 x-3)(3 x+2)$...
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Question: Factorise:3 + 23z 8z2 Solution: The given expression is $3+23 z-8 z^{2}$. Find two numbers that follow the conditions given below: Sum $=23$ Product $=3 \times-8=-24$ Clearly, the numbers are 24 and $-1$. $3+23 z-8 z^{2}=3+24 z-z-8 z^{2}$ $=3(1+8 z)-z(1+8 z)$ $=(1+8 z)(3-z)$...
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Question: Factorise:28 31x 5x2 Solution: The given expression is $28-31 x-5 x^{2}$. Find two numbers that follow the conditions given below: Sum $=-31$ Product $=28 \times-5=-140$ Clearly, the numbers are $-35$ and 4 . $28-31 x-5 x^{2}=28+4 x-35 x-5 x^{2}$ $=4(x+7)-5 x(7+x)$ $=(x+7)(4-5 x)$...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $\log \sqrt{\frac{x-1}{x+1}}$ Solution: Let $y=\log \sqrt{\frac{x-1}{x+1}}$ On differentiating $y$ with respect to $x$, we get $\frac{d y}{d x}=\frac{d}{d x}\left(\log \sqrt{\frac{x-1}{x+1}}\right)$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left[\log \left(\frac{\mathrm{x}-1}{\mathrm{x}+1}\right)^{\frac{1}{2}}\right]$ We know $\frac{d}{d x}(\log x)=\frac{1}{x}$ $\Rightarrow \frac{\mathrm{dy}}{...
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Question: Factorise:7y2 19y 6 Solution: The given expression is $7 y^{2}-19 y-6$. Find two numbers that follow the conditions given below: Sum $=-19$ Product $=7 \times-6=-42$ Clearly, the numbers are $-21$ and 2 . $7 y^{2}-19 y-6=7 y^{2}-21 y+2 y-6$ $=7 y(y-3)+2(y-3)$ $=(7 y+2)(y-3)$...
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Question: Factorise:2x2 17x 30 Solution: The given expression is $2 x^{2}-17 x-30$. Find two numbers that follow the conditions given below: Sum $=-17$ Product $=-30 \times 2=-60$ Clearly, the numbers are $-20$ and 3 . $2 x^{2}-17 x-30=2 x^{2}-20 x+3 x-30$ $=2 x(x-10)+3(x-10)$ $=(2 x+3)(x-10)$...
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Question: Differentiate the following functions with respect to $x$ : $\cos (\log x)^{2}$ Solution: Let $y=\cos (\log x)^{2}$ On differentiating y with respect to $x$, we get $\frac{d y}{d x}=\frac{d}{d x}\left[\cos (\log x)^{2}\right]$ We have $\frac{d}{d x}(\cos x)=-\sin x$ $\Rightarrow \frac{d y}{d x}=-\sin (\log x)^{2} \frac{d}{d x}\left[(\log x)^{2}\right]$ [using chain rule] We know $\frac{d}{d x}\left(x^{n}\right)=n x^{n-1}$ $\Rightarrow \frac{d y}{d x}=-\sin (\log x)^{2}\left[2(\log x)^{...
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Question: Factorise:6p2+ 11p 10 Solution: The given expression is $6 p^{2}+11 p-10$. Find two numbers that follow the conditions given below: Sum $=11$ Product $=6 \times-10=-60$ Clearly, the numbers are 15 and $-4$ $6 p^{2}+11 p-10=6 p^{2}+15 p-4 p-10$ $=3 p(2 p+5)-2(2 p+5)$ $=(2 p+5)(3 p-2)$...
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Question: Factorise:2x2+x 45 Solution: The given expression is $2 x^{2}+x-45$. Find two numbers that follow the conditions given below: Sum $=1$ Product $=-45 \times 2=-90$ Clearly, the numbers are 10 and $-9$. $2 x^{2}+x-45=2 x^{2}+10 x-9 x-45$ $=2 x(x+5)-9(x+5)$ $=(2 x-9)(x+5)$...
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Question: Factorise:3z2 10z+ 8 Solution: The given expression is $3 z^{2}-10 z+8$. Find two numbers that follow the conditions given below: Sum $=-10$ Product $=3 \times 8=24$ Clearly, the numbers are $-6$ and $-4$. $3 z^{2}-10 z+8=3 z^{2}-6 z-4 z+8$ $=3 z(z-2)-4(z-2)$ $=(3 z-4)(z-2)$...
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Question: Differentiate the following functions with respect to $x$ : $\log \left(\cos x^{2}\right)$ Solution: Let $y=\log \left(\cos x^{2}\right)$ On differentiating y with respect to $x$, we get $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left[\log \left(\cos \mathrm{x}^{2}\right)\right]$ We have $\frac{\mathrm{d}}{\mathrm{dx}}(\log \mathrm{x})=\frac{1}{\mathrm{x}}$ $\Rightarrow \frac{d y}{d x}=\frac{1}{\cos x^{2}} \frac{d}{d x}\left(\cos x^{2}\right)$ [using chain rule] We...
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Question: Factorise:3y2+ 14y+ 8 Solution: The given expression is $3 y^{2}+14 y+8$ Find two numbers that follow the conditions given below: Sum $=14$ Product $=24$ Clearly, the numbers are 12 and 2 . $3 y^{2}+14 y+8=3 y^{2}+12 y+2 y+8$ $=3 y(y+4)+2(y+4)$ $=(3 y+2)(y+4)$...
Read More →50 students enter for a school javelin
Question: 50 students enter for a school javelin throw competition. The distance (in metre) thrown are recorded below (i) Construct a cumulative frequency table. (ii) Draw a cumulative frequency curve (less than type) and calculate the median distance drawn by using this curve. (iii) Calculate the median distance by using the formula for median. (iv) Are the median distance calculated in (ii) and (iii) same? Solution: To draw less than type ogive, we plot the points (0, 0), (20, 6), (40,17), (60...
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Question: Factorise:3x2+ 10x+ 8 Solution: The given expression is $3 x^{2}+10 x+8$. Find two numbers that follow the conditions given below: Sum $=10$ Product $=3 \times 8=24$ Clearly, the numbers are 6 and 4 . $3 x^{2}+10 x+8=3 x^{2}+10 x+8$ $=3 x^{2}+6 x+4 x+8$ $=3 x(x+2)+4(x+2)$ $=(x+2)(3 x+4)$...
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Question: Differentiate the following functions with respect to $x$ : $e^{a x} \sec (x) \tan (2 x)$ Solution: Let $y=e^{a x} \sec (x) \tan (2 x)$ On differentiating $y$ with respect to $x$, we get $\frac{d y}{d x}=\frac{d}{d x}\left(e^{\operatorname{ax}} \sec x \tan 2 x\right)$ $\frac{d y}{d x}=\frac{d}{d x}\left[e^{\operatorname{ax}} \times(\sec x \tan 2 x)\right]$ We have (uv)' = vu' + uv' (product rule) $\Rightarrow \frac{d y}{d x}=\sec x \tan 2 x \frac{d}{d x}\left(e^{a x}\right)+e^{a x} \fr...
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