If x + y = 12 and xy = 14,
Question: Ifx+y= 12 andxy= 14, find the value of (x2+y2). Solution: $x+y=12$ On squaring both the sides: $\Rightarrow(x+y)^{2}=(12)^{2}$ $\Rightarrow x^{2}+y^{2}+2 x y=144$ $\Rightarrow x^{2}+y^{2}=144-2 x y$ Given: $x y=14$ $\Rightarrow x^{2}+y^{2}=144-2(14)$ $\Rightarrow x^{2}+y^{2}=144-28$ $\Rightarrow x^{2}+y^{2}=116$ Therefore, the value of $x^{2}+y^{2}$ is 116 ....
Read More →Find the continued product:
Question: Find the continued product: (i) (x+ 1)(x 1)(x2+ 1) (ii) (x 3)(x+ 3)(x2+ 9) (iii) (3x 2y)(3x+ 2y)(9x2+ 4y2) (iv) (2p+ 3)(2p 3)(4p2+ 9) Solution: $(i)(x+1)(x-1)\left(x^{2}+1\right)$ $\Rightarrow\left(x^{2}-x+x-1\right)\left(x^{2}+1\right)$ $\Rightarrow\left(x^{2}-1\right)\left(x^{2}+1\right)$ $\Rightarrow\left(x^{2}\right)^{2}-\left(1^{2}\right)^{2}$ [according to the formula $\left.a^{2}-b^{2}=(a+b)(a-b)\right]$ $\Rightarrow x^{4}-1 .$ Therefore, the product of $(x+1)(x-1)\left(x^{2}+1\...
Read More →A milk container of height 16 cm is made of metal
Question: A milk container of height 16 cm is made of metal sheet in the form of a frustum of a cone with radii of its lower and upper ends as 8 cm and 20 cm, respectively. Find the cost of milk at the rate of ? 22 per L which the container can hold. Solution: Given that,height of milk container (h) = 16 cm, Radius of lower end of milk container (r) = 8 cm and radius of upper end of milk container (R) = 20 cm $\therefore$ Volume of the milk container made of metal sheet in the form of a frustum ...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $3^{x \log x}$ Solution: Let $y=3^{x \log x}$ On differentiating $y$ with respect to $x$, we get $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left(3^{x \log \mathrm{x}}\right)$ We know $\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{a}^{\mathrm{x}}\right)=\mathrm{a}^{\mathrm{x}} \log \mathrm{a}$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=3^{\mathrm{x} \log \mathrm{x}} \log 3 \frac{\mathrm{d}}{\mathrm{dx}}(\ma...
Read More →find the values of
Question: If $\left(x-\frac{1}{x}\right)=5$, find the values of (i) $\left(x^{2}+\frac{1}{x^{2}}\right)$ (ii) $\left(x^{4}+\frac{1}{x^{4}}\right)$. Solution: $(i)\left(x-\frac{1}{x}\right)=5$ $\Rightarrow$ Squaring both the sides: $\Rightarrow\left(x-\frac{1}{x}\right)^{2}=(5)^{2}$ $\Rightarrow\left(x^{2}+\frac{1}{x^{2}}-2(x)\left(\frac{1}{x}\right)\right)=25$ $\Rightarrow\left(x^{2}+\frac{1}{x^{2}}\right)-2=25$ $\Rightarrow\left(x^{2}+\frac{1}{x^{2}}\right)=25+2$ $\Rightarrow\left(x^{2}+\frac{1...
Read More →glass spheres each of radius 2 cm
Question: glass spheres each of radius 2 cm are packed into a cuboidal box of internal dimensions 16 cm x 8 cm x 8 cm and then the box is filled with water. Find the volume of water filled in the box. Solution: Given, dimensions of the cuboidal = 16 cm x 8 cm x 8 cm Volume of the cuboidal = 16 x 8 x 8 = 1024 cm3 Also, given radius of one glass sphere = 2 cm $\therefore \quad$ Volume of one glass sphere $=\frac{4}{3} \pi r^{3}=\frac{4}{3} \times \frac{22}{7} \times(2)^{3}$ $=\frac{704}{21}=33.523...
Read More →500 persons are taking a dip into a cuboidal
Question: 500 persons are taking a dip into a cuboidal pond which is 80 m long and 50 m broad. What is the rise of water level in the pond, if the average displacement of the water by a person is 0.04 m3? Solution: Let the rise of water level in the pond be hm, when 500 persons are taking a dip into a cuboidal pond. Given that Length of the cuboidal pond $=80 \mathrm{~m}$ Breadth of the cuboidal pond $=50 \mathrm{~m}$ Now, volume for the rise of water level in the pond $=$ Length $\times$ Breadt...
Read More →find the values of
Question: If $\left(x+\frac{1}{x}\right)=4$, find the values of (i) $\left(x^{2}+\frac{1}{x^{2}}\right)$ and (ii) $\left(x^{4}+\frac{1}{x^{4}}\right)$. Solution: (i) $\left(x+\frac{1}{x}\right)=4$ Squaring both the sides: $\Rightarrow\left(x+\frac{1}{x}\right)^{2}=(4)^{2}$ $\Rightarrow\left(x^{2}+\frac{1}{x^{2}}+2(x)\left(\frac{1}{x}\right)\right)=16$ $\Rightarrow\left(x^{2}+\frac{1}{x^{2}}\right)+2=16$ $\Rightarrow\left(x^{2}+\frac{1}{x^{2}}\right)=16-2$ $\Rightarrow\left(x^{2}+\frac{1}{x^{2}}\...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $\sqrt{\frac{a^{2}-x^{2}}{a^{2}+x^{2}}}$ Solution: Let $y=\sqrt{\frac{a^{2}-x^{2}}{a^{2}+x^{2}}}$ On differentiating $y$ with respect to $x$, we get $\frac{d y}{d x}=\frac{d}{d x}\left(\sqrt{\frac{a^{2}-x^{2}}{a^{2}+x^{2}}}\right)$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left[\left(\frac{\mathrm{a}^{2}-\mathrm{x}^{2}}{\mathrm{a}^{2}+\mathrm{x}^{2}}\right)^{\frac{1}{2}}\right]$ We know $\frac...
Read More →Find the value of the expression
Question: Find the value of the expression $\left(36 x^{2}+25 y^{2}-60 x y\right)$, when $x=\frac{2}{3}$ and $y=\frac{1}{5}$ Solution: $\left(36 x^{2}+25 y^{2}-60 x y\right)$ $\Rightarrow x=\frac{2}{3}, y=\frac{1}{5}$ $=(6 x)^{2}+(5 y)^{2}-2(6 x)(5 y)$ $=(6 x-5 y)^{2}$ $=\left(6\left(\frac{2}{3}\right)-5\left(\frac{1}{5}\right)\right)^{2}$ $=(4-1)^{2}$ $=(3)^{2}$ $\Rightarrow 9$...
Read More →A solid iron cuboidal block of dimensions 4.4 m x 2.6m x lm
Question: A solid iron cuboidal block of dimensions 4.4 m x 2.6m x lmis recast into a hollow cylindrical pipe of internal radius 30 cm and thickness 5 cm. Find the length of the pipe. Solution: Given that, a solid iron cuboidal block is recast into a hollow cylindrical pipe, Length of cuboidal pipe (l) = 4.4 m Breadth of cuboidal pipe (b) = 2.6 m and height of cuboidal pipe (h) = 1m So, volume of a solid iron cuboidal block $=l \cdot b \cdot h$ $=4.4 \times 2.6 \times 1=11.44 \mathrm{~m}^{3}$ Al...
Read More →Find the value of the expression
Question: Find the value of the expression $\left(64 x^{2}+81 y^{2}+144 x y\right)$, when $x=11$ and $y=\frac{4}{3}$ Solution: $\left(64 x^{2}+81 y^{2}+144 x y\right)$ Given: $x=11$ $y=\frac{4}{3}$ $\Rightarrow(8 x)^{2}+(9 y)^{2}+2(8 x)(9 y)$ $\Rightarrow(8 x+9 y)^{2}$ $\Rightarrow\left(8(11)+9\left(\frac{4}{3}\right)\right)^{2}$ $\Rightarrow(88+12)^{2}$ $\Rightarrow(100)^{2}$ $\Rightarrow 10000$ Therefore, the value of the expression $\left(64 x^{2}+81 y^{2}+144 x y\right)$, when $x=11$ and $y=...
Read More →Find the value of the expression
Question: Find the value of the expression (9x2+ 24x+ 16), whenx= 12. Solution: $\left(9 x^{2}+24 x+16\right)$ Given, $x=12$ $\Rightarrow(3 x)^{2}+2(3 x)(4)+(4)^{2}$ $\Rightarrow(3 x+4)^{2}$ $\Rightarrow(3(12)+4)^{2}$ $\Rightarrow(36+4)^{2}$ $\Rightarrow(40)^{2}=1600$ Therefore, the value of the expression (9x2+ 24x+ 16), whenx= 12, is 1600....
Read More →Water is flowing at the rate of 15 kmh-1 through
Question: Water is flowing at the rate of 15 kmh-1 througha pipe of diameter 14 cm into a cuboidal pond which is 50 m long and 44 m wide. In what time will the level of water in pond rise by 21 cm? Solution: Given, length of the pond= 50 m and width of the pond = 44 m Depth required of water $=21 \mathrm{~cm}=\frac{21}{100} \mathrm{~m}$ $\therefore \quad$ Volume of water in the pond $=\left(50 \times 44 \times \frac{21}{100}\right)^{3}=462 \mathrm{~m}^{3}$ Also, given radius of the pipe $=7 \mat...
Read More →Find the value of:
Question: Find the value of: (i) (82)2 (18)2 (ii) (128)2 (72)2 (iii) 197 203 (iv) $\frac{198 \times 198-102 \times 102}{96}$ (v) (14.7 15.3) (vi) (8.63)2 (1.37)2 Solution: We shall use the identity $(a-b)(a+b)=a^{2}-b^{2}$. (i) $(82)^{2}-(18)^{2}$ $=(82-18)(82+18)$ $=(64)(100)$ $=6400$ (ii) $(128)^{2}-(72)^{2}$ $=(128-72)(128+72)$ $=(56)(200)$ $=11200$ (iii) $(128)^{2}-(72)^{2}$ $=(128-72)(128+72)$ $=(56)(200)$ $=11200$ (iv) $\frac{198 \times 198-102 \times 102}{96}$ $=\frac{(198)^{2}-(102)^{2}}...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $3^{x^{2}+2 x}$ Solution: Let $y=3^{x^{2}+2 x}$ On differentiating $y$ with respect to $x$, we get $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left(3^{\mathrm{x}^{2}+2 \mathrm{x}}\right)$ We know $\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{a}^{\mathrm{x}}\right)=\mathrm{a}^{\mathrm{x}} \log \mathrm{a}$ $\Rightarrow \frac{d y}{d x}=3^{x^{2}+2 x} \log 3 \frac{d}{d x}\left(x^{2}+2 x\right)$ [using chain rule]...
Read More →A factory manufactures 120000 pencils daily.
Question: A factory manufactures 120000 pencils daily. The pencils are cylindrical in shape each of length 25 cm and circumference of base as 1.5 cm. Determine the cost of colouring the curved surfaces of the pencils manufactured in one day at ₹ 0.05 per dm2. Solution: Given, pencils are cylindrical in shape. Length of one pencil = 25 cm and circumference of base = 1.5 cm $\Rightarrow$$r=\frac{1.5 \times 7}{22 \times 2}=0.2386 \mathrm{~cm}$ Now curved surface area of one pencil $=2 \pi r h$ $=2 ...
Read More →Using the formula for squaring a binomial, evaluate the following:
Question: Using the formula for squaring a binomial, evaluate the following: (i) (69)2 (ii) (78)2 (iii) (197)2 (iv) (999)2 Solution: We shall use the identity (a-b)2=a2+b2-2ab. (i)$(69)^{2}$ $=(70-1)^{2}$ $=(70)^{2}-2 \times 70 \times 1+1$ $=4900-140+1$ $=4761$ (ii)$(78)^{2}$ $=(80-2)^{2}$ $=(80)^{2}-2 \times 80 \times 2+4$ $=6400-320+4$ $=6084$ (iii)$(197)^{2}$ $=(200-3)^{2}$ $=(200)^{2}-2 \times 200 \times 3+9$ $=40000-1200+9$ $=38809$ (iv)$(999)^{2}$ $=(1000-1)^{2}$ $=(1000)^{2}-2 \times 1000...
Read More →A heap of rice is in the form of a cone of diameter 9 m
Question: A heap of rice is in the form of a cone of diameter 9 m and height 3.5 m. Find the volume of the rice. How much canvas cloth is required to just cover heap? Solution: Given that, a heap of rice is in the form of a cone. Height of a heap of rice i.e., cone (h) = 3.5 m and diameter of a heap of rice i.e., cone = 9 m Radius of a heap of rice $i . \theta .$, cone $(r)=\frac{9}{2} \mathrm{~m}$ So, volume of rice $=\frac{1}{3} \pi \times r^{2} h$ $=\frac{1}{3} \times \frac{22}{7} \times \fra...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $\log _{x} 3$ Solution: Let $y=\log _{x} 3$ Recall that $\log _{a} b=\frac{\log _{b}}{\log a}$. $\Rightarrow \log _{x} 3=\frac{\log 3}{\log x}$ On differentiating $y$ with respect to $x$, we get $\frac{d y}{d x}=\frac{d}{d x}\left(\frac{\log 3}{\log x}\right)$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\log 3 \frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{1}{\log \mathrm{x}}\right)$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\l...
Read More →Using the formula for squaring a binomial, evaluate the following:
Question: Using the formula for squaring a binomial, evaluate the following: (i) (54)2 (ii) (82)2 (iii) (103)2 (iv) (704)2 Solution: We shall use the identity (a+b)2=a2+b2+2ab. (i)$(54)^{2}$ $=(50+4)^{2}$ $=(50)^{2}+2 \times 50 \times 4+(4)^{2}$ $=2500+400+16$ $=2916$ (ii)$(82)^{2}$ $=(80+2)^{2}$ $=(80)^{2}+2 \times 80 \times 2+(2)^{2}$ $=6400+320+4$ $(82)^{2}$ $=(80+2)^{2}$ $=(80)^{2}+2 \times 80 \times 2+(2)^{2}$ $=6400+320+4$ $=6724$$=6724$ (iii) $(103)^{2}$ $=(100+3)^{2}$ $=(100)^{2}+2 \time...
Read More →Find each of the following products:
Question: Find each of the following products: (i) (x+ 3)(x 3) (ii) (2x+ 5)(2x 5) (iii) (8 +x)(8 x) (iv) (7x+ 11y)(7x 11y) (v) $\left(5 x^{2}+\frac{3}{4} y^{2}\right)\left(5 x^{2}-\frac{3}{4} y^{2}\right)$ (vi) $\left(\frac{4 x}{5}-\frac{5 y}{3}\right)\left(\frac{4 x}{5}+\frac{5 y}{3}\right)$ (vii) $\left(x+\frac{1}{x}\right)\left(x-\frac{1}{x}\right)$ (viii) $\left(\frac{1}{x}+\frac{1}{y}\right)\left(\frac{1}{x}-\frac{1}{y}\right)$ Solution: (i) We have: $(x+3)(x-3)$ $=x^{2}-9$$\quad\left[\righ...
Read More →Find each of the following products:
Question: Find each of the following products: (i) (x+ 3)(x 3) (ii) (2x+ 5)(2x 5) (iii) (8 +x)(8 x) (iv) (7x+ 11y)(7x 11y) (v) $\left(5 x^{2}+\frac{3}{4} y^{2}\right)\left(5 x^{2}-\frac{3}{4} y^{2}\right)$ (vi) $\left(\frac{4 x}{5}-\frac{5 y}{3}\right)\left(\frac{4 x}{5}+\frac{5 y}{3}\right)$ (vii) $\left(x+\frac{1}{x}\right)\left(x-\frac{1}{x}\right)$ (viii) $\left(\frac{1}{x}+\frac{1}{y}\right)\left(\frac{1}{x}-\frac{1}{y}\right)$ Solution: (i) We have: $(x+3)(x-3)$ $=x^{2}-9$$\quad\left[\righ...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $3^{e^{x}}$ Solution: Let $y=3^{\mathrm{e}^{\mathrm{x}}}$ On differentiating y with respect to $x$, we get $\frac{d y}{d x}=\frac{d}{d x}\left(3^{e^{x}}\right)$ We know $\frac{d}{d x}\left(a^{x}\right)=a^{x} \log a$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=3^{\mathrm{e}^{\mathrm{x}}} \log 3 \frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{e}^{\mathrm{x}}\right)$ [using chain rule] We have $\frac{d}{d x}\left(e^{x}\right)=e^{x}$...
Read More →Water flows at the rate of 10 m min-1 through
Question: Water flows at the rate of 10 m min-1through a cylindrical pipe 5 mm in diameter. How long would it take to fill a conical vessel whose diameter at the base is 40 cm and depth 24 cm? Solution: Given, speed of water flow = 10 m min-1= 1000 cm/min and diameter of the pipe $=5 \mathrm{~mm}=\frac{\mathrm{s}}{10} \mathrm{~cm}$ $\therefore \quad$ Radius of the pipe $=\frac{5}{10 \times 2}=0.25 \mathrm{~cm}$ $\therefore$ Area of the face of pipe $=\pi r^{2}=\frac{22}{7} \times(0.25)^{2}=0.196...
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