Expand:
Question: Expand: (i) (8a+ 3b)2 (ii) (7x+ 2y)2 (iii) (5x+ 11)2 (iv) $\left(\frac{a}{2}+\frac{2}{a}\right)^{2}$ (v) $\left(\frac{3 x}{4}+\frac{2 y}{9}\right)^{2}$ (vi) (9x 10)2 (vii) (x2yyz2)2 (viii) $\left(\frac{x}{y}-\frac{y}{x}\right)^{2}$ (ix) $\left(3 m-\frac{4}{5} n\right)^{2}$ Solution: We shall use the identities (a+b)2=a2+b2+2aband (a-b)2=a2+b2-2ab. (i) We have: $(8 a+3 b)^{2}$ $=(8 a)^{2}+2 \times 8 a \times 3 b+(3 b)^{2}$ $=64 a^{2}+48 a b+9 b^{2}$ (ii) We have: $(7 x+2 y)^{2}$ $=(7 x)...
Read More →The barrel of a fountain pen, cylindrical in shape,
Question: The barrel of a fountain pen, cylindrical in shape, is 7 cm long and 5 mm in diameter. A full barrel of ink in the pin is used up on writing 3300 words on an average. How many words can be written in a bottle of ink containing one-fifth of a litre? Solution: Given, length of the barrel of a fountain pen = 7 cm and diameter $=5 \mathrm{~mm}=\frac{5}{10} \mathrm{~cm}=\frac{1}{2} \mathrm{~cm}$ $\therefore$ Radius of the barrel $=\frac{1}{2 \times 2}=0.25 \mathrm{~cm}$ Volume of the barrel...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $2^{x^{3}}$ Solution: Let $y=2^{x^{3}}$ On differentiating $y$ with respect to $x$, we get $\frac{d y}{d x}=\frac{d}{d x}\left(2^{x^{3}}\right)$ We know $\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{a}^{\mathrm{x}}\right)=\mathrm{a}^{\mathrm{x}} \log \mathrm{a}$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=2^{\mathrm{x}^{3}} \log 2 \frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}^{3}\right)$ [using chain rule] We have $\frac{\mat...
Read More →How many cubic centimetres of iron is required
Question: How many cubic centimetres of iron is required to construct an open box whose external dimensions are 36 cm, 25 cm and 16.5 cm provided the thickness of the iron is 1.5 cm. If one cubic centimetre of iron weights 7.5 g, then find the weight of the box. Solution: Let the length $(l)$, breadth (b) and height ( $h$ ) be the external dimension of an open box and thickness be $x$. Given that, external length of an open box $(l)=36 \mathrm{~cm}$ external breadth of an open box $(b)=25 \mathr...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $\mathrm{x}$ : $\tan \left(5 x^{\circ}\right)$ Solution: Let $y=\tan \left(5 x^{\circ}\right)$ First, we will convert the angle from degrees to radians. We have $1^{\circ}=\left(\frac{\pi}{180}\right)^{c} \Rightarrow 5 \mathrm{x}^{\circ}=5 \mathrm{x} \times \frac{\pi}{180}^{c}$ $\Rightarrow y=\tan \left(5 x \times \frac{\pi}{180}\right)$ On differentiating $y$ with respect to $x$, we get $\frac{d y}{d x}=\frac{d}{d x}\left[\tan \lef...
Read More →A rectangular water tank of base 11 m x 6 m
Question: A rectangular water tank of base 11 m x 6 m contains water upto a height of 5 m. If the water in the tank is transferred to a cylindrical tank of radius 3.5 m, find the height of the water level in the tank. Solution: Given, dimensions of base of rectangular tank = 11 m x 6 m and height of water = 5 m Volume of the water in rectangular tank = 11 x 6 x 5 = 330 m3 Also, given radius of the cylindrical tank = 3.5 m Let height of water level in cylindrical tank be h. Then, volume of the wa...
Read More →Find each of the following products:
Question: Find each of the following products: (i) $(x-4)(x-4)$ (ii) $(2 x-3 y)(2 x-3 y)$ (iii) $\left(\frac{3}{4} x-\frac{5}{6} y\right)\left(\frac{3}{4} x-\frac{5}{6} y\right)$ (iv) $\left(x-\frac{3}{x}\right)\left(x-\frac{3}{x}\right)$ (v) $\left(\frac{1}{3} x^{2}-9\right)\left(\frac{1}{3} x^{2}-9\right)$ (vi) $\left(\frac{1}{2} y^{2}-\frac{1}{3} y\right)\left(\frac{1}{2} y^{2}-\frac{1}{3} y\right)$ Solution: (i) We have: $(x-4)(x-4)$ $=(x-4)^{2}$ $=x^{2}-2 \times x \times 4+4^{2}$ $\left[\ri...
Read More →A solid metallic hemisphere of radius 8 cm
Question: A solid metallic hemisphere of radius 8 cm is melted and recasted into a right circular cone of base radius 6 cm. Determine the height of the cone. Solution: Let height of the cone be h. Given, radius of the base of the cone = 6 cm $\therefore \quad$ Volume of circular cone $=\frac{1}{3} \pi r^{2} h=\frac{1}{3} \pi(6)^{2} h=\frac{36 \pi h}{3}=12 \pi h \mathrm{~cm}^{3}$ Also, given radius of the hemisphere $=8 \mathrm{~cm}$ $\therefore \quad$ Volume of the hemisphere $=\frac{2}{3} \pi r...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $\log _{7}(2 x-3)$ Solution: Let $y=\log _{7}(2 x-3)$ Recall that $\log _{a} b=\frac{\log b}{\log a}$ $\Rightarrow \log _{7}(2 x-3)=\frac{\log (2 x-3)}{\log 7}$ We know $\frac{\mathrm{d}}{\mathrm{dx}}(\log \mathrm{x})=\frac{1}{\mathrm{x}}$ $\Rightarrow \frac{d y}{d x}=\left(\frac{1}{\log 7}\right)\left(\frac{1}{2 x-3}\right) \frac{d}{d x}(2 x-3)$ [using chain rule] $\Rightarrow \frac{d y}{d x}=\frac{1}{(2 x-3) \log 7}\left[\fr...
Read More →Find the number of metallic circular disc with 1.5 cm
Question: Find the number of metallic circular disc with 1.5 cm base diameter and of height 0.2 cm to be melted to form a right circular cylinder of height 10 cm and diameter 4.5 cm. Solution: Given that, lots of metallic circular disc to be melted to form a right circular cylinder. Here, a circular disc work as a circular cylinder. Base diameter of metallic circular disc = 1.5 cm $\therefore$ Radius of metallic circular disc $=\frac{1.5}{2} \mathrm{~cm}$$[\because$ diameter $=2 \times$ radius $...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $\sin ^{2}(2 x+1)$ Solution: Let $y=\sin ^{2}(2 x+1)$ On differentiating $y$ with respect to $x$, we get $\frac{d y}{d x}=\frac{d}{d x}\left[\sin ^{2}(2 x+1)\right]$ We know $\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}^{\mathrm{n}}\right)=\mathrm{n} \mathrm{x}^{\mathrm{n}-1}$ $\Rightarrow \frac{d y}{d x}=2 \sin ^{2-1}(2 x+1) \frac{d}{d x}[\sin (2 x+1)]$ [using chain rule] $\Rightarrow \frac{d y}{d x}=2 \sin (2 x+1) \frac{d}...
Read More →Find each of the following products:
Question: Find each of the following products: (i) (x+ 6)(x+ 6) (ii) (4x+ 5y)(4x+ 5y) (iii) (7a+ 9b)(7a+ 9b) (iv) $\left(\frac{2}{3} x+\frac{4}{5} y\right)\left(\frac{2}{3} x+\frac{4}{5} y\right)$ (v) $\left(x^{2}+7\right)\left(x^{2}+7\right)$ (vi) $\left(\frac{5}{6} a^{2}+2\right)\left(\frac{5}{6} a^{2}+2\right)$ Solution: (i) We have: $(x+6)(x+6)$ $=(x+6)^{2}$ $=x^{2}+6^{2}+2 \times x \times 6$ $\left[\right.$ using $\left.(a+b)^{2}=a^{2}+b^{2}+2 a b\right]$ $=x^{2}+36+12 x$ (ii) We have: $(4 ...
Read More →A wall 24 m long, 0.4 m thick and 6 m high is constructed
Question: A wall $24 \mathrm{~m}$ long, $0.4 \mathrm{~m}$ thick and $6 \mathrm{~m}$ high is constructed with the bricks each of dimensions $25 \mathrm{~cm} \times 16$ $\mathrm{cm} \times 10 \mathrm{~cm}$. If the mortar occupies $\frac{1}{6}$ th of the volume of the wall, then find the number of bricks used in constructing the wall. Solution: Given that, a wall is constructed with the help of bricks and mortar. $\therefore$Number of bricks $=\frac{\text { (Nolume of wall) }-\left(\frac{1}{10} \te...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $e^{\tan x}$ Solution: Let $y=e^{\tan x}$ On differentiating y with respect to $x$, we get $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{e}^{\tan 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 x} \frac{d}{d x}(\tan x)$ [using chain rule] We have $\frac{\mathrm{d}}{\mathrm{dx}}(\tan \mathrm{x})=\s...
Read More →Write the quotient and remainder when we divide:
Question: Write the quotient and remainder when we divide:(8x4+ 10x3 5x2 4x+ 1) by (2x2+x 1) Solution: Therefore, the quotient is ( 4x2+ 3x-2) and the remainder is (x-1)....
Read More →Write the quotient and remainder when we divide:
Question: Write the quotient and remainder when we divide:(2x3 5x2+ 8x 5) by (2x2 3x+ 5) Solution: Therefore, the quotient is (x-1) and the remainder is 0....
Read More →Write the quotient and remainder when we divide:
Question: Write the quotient and remainder when we divide:(5x3 12x2+ 12x+ 13) by (x2 3x+ 4) Solution: Therefore, the quotient is ( 5x+ 3) and the remainder is (x+ 1)....
Read More →Write the quotient and remainder when we divide:
Question: Write the quotient and remainder when we divide:(x3 6x2+ 11x 6) by (x2 5x+ 6) Solution: Therefore, the quotient is $(x-1)$ and the remainder is 0 ....
Read More →Write the quotient and remainder when we divide:
Question: Write the quotient and remainder when we divide:(x4 2x3+ 2x2+x+ 4) by (x2+x+ 1) Solution: Therefore, the quotient is $\left(x^{2}-3 x+4\right)$ and remainder is 0 ....
Read More →Write the quotient and remainder when we divide:
Question: Write the quotient and remainder when we divide:(x3+ 1) by (x+ 1) Solution: Therefore, the quotient is $x^{2}-x+1$ and the remainder is 0 ....
Read More →How many spherical lead shots of diameter 4 cm
Question: How many spherical lead shots of diameter 4 cm can be made out of a solid cube of lead whose edge measures 44 cm. Solution: Given that, lots of spherical lead shots made out of a solid cube of lead. Number of spherical lead shots $=\frac{\text { Volume of a soiid cube of lead }}{\text { Volume of a spherical lead shot }}$...(i) Given that, diameter of a spherical lead shot i.e., sphere = 4cm $\Rightarrow \quad$ Radius of a spherical lead shot $(r)=\frac{4}{2}$ $r=2 \mathrm{~cm}$$[\beca...
Read More →Write the quotient and remainder when we divide:
Question: Write the quotient and remainder when we divide:(2x3+x2 5x 2) by (2x+ 3) Solution: Therefore, the quotient is $\left(x^{2}-x-1\right)$ and the remainder is 1 ....
Read More →Write the quotient and remainder when we divide:
Question: Write the quotient and remainder when we divide:(6x2 31x+ 47) by (2x 5) Solution: Therefore, the quotient is $(3 x-8)$ and the remainder is 7 ....
Read More →Write the quotient and remainder when we divide:
Question: Write the quotient and remainder when we divide:(14x2 53x+ 45) by (7x 9) Solution: Therefore, the quotient is $(2 x-5)$ and the remainder is 0 ....
Read More →Write the quotient and remainder when we divide:
Question: Write the quotient and remainder when we divide:(15x2+x 6) by (3x+ 2) Solution: Therefore, the quotient is $(5 x-3)$ and the remainder is 0 ....
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