Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $e^{\sin \sqrt{x}}$ Solution: Let $y=e^{\sin \sqrt{x}}$ On differentiating $y$ with respect to $x$, we get $\frac{d y}{d x}=\frac{d}{d x}\left(e^{\sin \sqrt{x}}\right)$ We know $\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{e}^{\mathrm{x}}\right)=\mathrm{e}^{\mathrm{x}}$ $\Rightarrow \frac{d y}{d x}=e^{\sin \sqrt{x}} \frac{d}{d x}(\sin \sqrt{x})$ [using chain rule] We have $\frac{d}{d x}(\sin x)=\cos x$ $\Rightarrow \frac{d y}{d...
Read More →How many spherical lead shots each of diameter 4.2 cm
Question: How many spherical lead shots each of diameter 4.2 cm can be obtained from a solid rectangular lead piece with dimensions 66 cm, 42 cm and 21 cm? Solution: Given that, lots of spherical lead shots made from a solid rectangular lead piece. Number of spherical lead shots $=\frac{\text { Volume of solid rectangular lead piece }}{\text { Volume of a spherical lead shot }}$..(i) Also, given that diameter of a spherical lead shot $i . e$, , sphere $=4.2 \mathrm{~cm}$ $\therefore$ Radius of a...
Read More →Write the quotient and remainder when we divide:
Question: Write the quotient and remainder when we divide:(x2+ 12x+ 35) by (x+ 7) Solution: (x2+ 12x+ 35) by (x+ 7) Therefore, the quotient is $(x+5)$ and the remainder is 0 ....
Read More →Write the quotient and remainder when we divide:
Question: Write the quotient and remainder when we divide:(x2 4) by (x+ 2) Solution: Therefore, the quotient is $x-2$ and the remainder is 0 ....
Read More →Write the quotient and remainder when we divide:
Question: Write the quotient and remainder when we divide:(x2 4x+ 4) by (x 2) Solution: Therefore, the quotient is $(x-2)$ and the remainder is 0 ....
Read More →Divide:
Question: Divide:(i) 5m3 30m2+ 45mby 5m(ii) 8x2y2 6xy2+ 10x2y3by 2xy(iii) 9x2y 6xy+ 12xy2by 3xy(iv) 12x4+ 8x3 6x2by 2x2 Solution: (i) 5m3 30m2+ 45mby 5m $\left(5 m^{3}-30 m^{2}+45 m\right) \div 5 m$ $\Rightarrow \frac{5 m^{3}}{5 m}-\frac{30 m^{2}}{5 m}+\frac{45 m}{5 m}$ $\Rightarrow m^{2}-6 m+9$ Therefore, the quotient ism26m +9 (ii) 8x2y2 6xy2+ 10x2y3by 2xy $\left(8 x^{2} y^{2}-6 x y^{2}+10 x^{2} y^{3}\right) \div 2 x y$ $\Rightarrow \frac{8 x^{2} y^{2}}{2 x y}-\frac{6 x y^{2}}{2 x y}+\frac{10 ...
Read More →Marbles of diameter 1.4 cm are dropped
Question: Marbles of diameter 1.4 cm are dropped into a cylindrical beaker of diameter 7 cm containing some water. Find the number of marbles that should be dropped into the beaker, so that the water level rises by 5.6 cm. Solution: Given, diameter of a marble = 1.4 cm $\therefore \quad$ Radius of marble $=\frac{1.4}{2}=0.7 \mathrm{~cm}$ So, $\quad$ volume of one marble $=\frac{4}{3} \pi(0.7)^{3}$ $=\frac{4}{3} \pi \times 0.343=\frac{1.372}{3} \pi \mathrm{cm}^{3}$ Also, given diameter of beaker ...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $\sin (\log x)$ Solution: Let $y=\sin (\log x)$ On differentiating $y$ with respect to $x$, we get $\frac{d y}{d x}=\frac{d}{d x}[\sin (\log x)]$ We know $\frac{d}{d x}(\sin x)=\cos x$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\cos (\log \mathrm{x}) \frac{\mathrm{d}}{\mathrm{dx}}(\log \mathrm{x})$ [using chain rule] However, $\frac{\mathrm{d}}{\mathrm{dx}}(\log \mathrm{x})=\frac{1}{\mathrm{x}}$ $\Rightarrow \frac{\mathrm{dy...
Read More →An ice-cream cone full of ice-cream
Question: An ice-cream cone full of ice-cream having radius 5 cm and height 10 cm as shown in figure Calculate the volume of ice-cream, provided that its $\frac{1}{6}$ part is left unfilled with ice-cream. Solution: Given, ice-cream cone is the combination of a hemisphere and a cone. Also , radius of hemisphere = 5 cm $\therefore \quad$ Volume of hemisphere $=\frac{2}{3} \pi r^{3}=\frac{2}{3} \times \frac{22}{7} \times(5)^{3}$ $=\frac{5500}{21}=261.90 \mathrm{~cm}^{3}$ Now, radius of the cone $=...
Read More →Divide:
Question: Divide: (i) 24x2y3by3xy (ii) 36xyz2by 9xz (iii) 72x2y2zby 12xyz (iv) 56mnp2by 7mnp Solution: (i) 24x2y3by 3xy $\frac{24 x^{2} y^{3}}{3 x y}$ $\Rightarrow\left(\frac{24}{3}\right)\left(x^{2-1}\right)\left(y^{3-1}\right)$ $\Rightarrow 8 x y^{2}$ Therefore, the quotient is8xy2. (ii) 36xyz2by 9xz $\frac{36 x y z^{2}}{-9 x z}$ $\Rightarrow\left(\frac{36}{-9}\right)\left(x^{1-1}\right)\left(y^{1-0}\right)\left(z^{2-1}\right)$ $\Rightarrow-4 y z$ Therefore, the quotient is4yz. (iii)$-72 x^{2}...
Read More →Two solid cones A and B are placed in a cylindrical
Question: Two solid cones A and B are placed in a cylindrical tube as shown in the figure. The ratio of their capacities is 2 : 1. Find the heights and capacities of cones. Also, find the volume of the remaining portion of the cylinder. Solution: Let volume of cone A be 2 V and volume of cone B be V. Again, let height of the cone A = h1cm, then height of cone B = (21 h1) cm Given, diameter of the cone $=6 \mathrm{~cm}$ $\therefore \quad$ Radius of the cone $=\frac{6}{2}=3 \mathrm{~cm}$ Now, volu...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $\tan \left(x^{\circ}+45^{\circ}\right)$ Solution: Let $y=\tan \left(x^{\circ}+45^{\circ}\right)$ First, we will convert the angle from degrees to radians. We have $1^{\circ}=\left(\frac{\pi}{180}\right)^{\mathrm{c}} \Rightarrow(\mathrm{x}+45)^{\circ}=\left[\frac{(\mathrm{x}+45) \pi}{180}\right]^{\mathrm{c}}$ $\Rightarrow y=\tan \left[\frac{(x+45) \pi}{180}\right]$ On differentiating $y$ with respect to $x$, we get $\frac{\mat...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $\tan \left(x^{\circ}+45^{\circ}\right)$ Solution: Let $y=\tan \left(x^{\circ}+45^{\circ}\right)$ First, we will convert the angle from degrees to radians. We have $1^{\circ}=\left(\frac{\pi}{180}\right)^{\mathrm{c}} \Rightarrow(\mathrm{x}+45)^{\circ}=\left[\frac{(\mathrm{x}+45) \pi}{180}\right]^{\mathrm{c}}$ $\Rightarrow y=\tan \left[\frac{(x+45) \pi}{180}\right]$ On differentiating $y$ with respect to $x$, we get $\frac{\mat...
Read More →Find the product:
Question: Find the product:(9x2x+ 15) (x2x 1) Solution: By horizontal method: $\left(9 x^{2}-x+15\right) \times\left(x^{2}-x-1\right)$ $=x^{2}\left(9 x^{2}-x+15\right)-x\left(9 x^{2}-x+15\right)-1\left(9 x^{2}-x+15\right)$ $=9 x^{4}-x^{3}+15 x^{2}-9 x^{3}+x^{2}-15 x-9 x^{2}+x-15$ $=9 x^{4}-10 x^{3}+7 x^{2}-14 x-15$...
Read More →Two cones with same base radius 8 cm
Question: Two cones with same base radius 8 cm and height 15 cm are joined together along their bases. Find the surface area of the shape so formed. Solution: If two cones with same base and height are joined together along their bases, then the shape so formed is look like as figure shown. Given that, radius of cone, $r=8 \mathrm{~cm}$ and height of cone, $h=15 \mathrm{~cm}$ So, surface area of the shape so formed $=$ Curved area of first cone + Curved surface area of second cone $=2$. Surface ...
Read More →Find the product:
Question: Find the product:(2x2+ 3x 7) (3x2 5x+ 4) Solution: By horizontal method: $\left(2 x^{2}+3 x-7\right) \times\left(3 x^{2}-5 x+4\right)$ $=2 x^{2}\left(3 x^{2}-5 x+4\right)+3 x\left(3 x^{2}-5 x+4\right)-7\left(3 x^{2}-5 x+4\right)$ $=6 x^{4}-10 x^{3}+8 x^{2}+9 x^{3}-15 x^{2}+12 x-21 x^{2}+35 x-28$ $=6 x^{4}-x^{3}-28 x^{2}+47 x-28$...
Read More →Find the product:
Question: Find the product:(x2 5x+ 8) (x2+ 2x 3) Solution: By horizontal method: $\left(x^{2}-5 x+8\right) \times\left(x^{2}+2 x-3\right)$ $=x^{2}\left(x^{2}-5 x+8\right)+2 x\left(x^{2}-5 x+8\right)-3\left(x^{2}-5 x+8\right)$ $=x^{4}-5 x^{3}+8 x^{2}+2 x^{3}-10 x^{2}+16 x-3 x^{2}+15 x-24$ $=x^{4}-3 x^{3}-5 x^{2}+31 x-24$...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $\mathrm{x}$ : $\tan ^{2} x$ Solution: Let $y=\tan ^{2} x$ On differentiating $y$ with respect to $x$, we get $\frac{d y}{d x}=\frac{d}{d x}\left(\tan ^{2} x\right)$ We know $\frac{d}{d x}\left(x^{n}\right)=n x^{n-1}$ $\Rightarrow \frac{d y}{d x}=2 \tan ^{2-1} x \frac{d}{d x}(\tan x)$ [using chain rule] $\Rightarrow \frac{d y}{d x}=2 \tan x \frac{d}{d x}(\tan x)$ However, $\frac{d}{d x}(\tan x)=\sec ^{2} x$ $\Rightarrow \frac{\mathr...
Read More →From a solid cube of side 7 cm,
Question: From a solid cube of side 7 cm, a conical cavity of height 7 cm and radius 3 cm is hollowed out. Find the volume of the remaining solid. Solution: Given that, side of a solid cube (a) = 7 cm Height of conical cavity i.e., cone, h = 7 cm Since, the height of conical cavity and the side of cube is equal that means the conical cavity fit vertically in the cube. Radius of conical cavity i.e., cone, r = 3 cm ⇒ Diameter = 2 x r = 2 x 3= 6 cm Since, the diameter is less than the side of a cub...
Read More →Find the product:
Question: Find the product:(3x+ 2y 4) (xy+ 2) Solution: By horizontal method: $(3 x+2 y-4) \times(x-y+2)$ $x(3 x+2 y-4)-y(3 x+2 y-4)+2(3 x+2 y-4)$ $=3 x^{2}+2 x y-4 x-3 x y-2 y^{2}+4 y+6 x+4 y-8$ $=3 x^{2}-2 y^{2}-x y+2 x+8 y-8$...
Read More →Find the product:
Question: Find the product:(x3 5x2+ 3x+ 1) (x3 3) Solution: By horizontal method: $\left(x^{3}-5 x^{2}+3 x+1\right) \times\left(x^{2}-3\right)$ $=x^{2}\left(x^{3}-5 x^{2}+3 x+1\right)-3\left(x^{3}-5 x^{2}+3 x+1\right)$ $=x^{5}-5 x^{4}+3 x^{3}+x^{2}-3 x^{3}+15 x^{2}-9 x-3$ $=x^{5}-5 x^{4}+16 x^{2}-9 x-3$...
Read More →Two identical cubes each of volume 64 cm3
Question: Two identical cubes each of volume 64 cm3are joined together end to end. What is the surface area of the resulting cuboid? Solution: Let the length of side of a cube = a cm Given, volume of the cube, $a^{3}=64 \mathrm{~cm}^{3} \Rightarrow a=4 \mathrm{~cm}$ On joining two cubes, we get a cuboid whose length, $l=2 \mathrm{acm}$ breadth, $b=a \mathrm{~cm}$ and height, $h=a \mathrm{~cm}$ Now, surface area of the resulting cuboid $=2(l b+b h+h l)$ $=2(2 a \cdot a+a \cdot a+a \cdot 2 a)$ $=2...
Read More →Find the product:
Question: Find the product:(x2 5x+ 8) (x2+ 2) Solution: By horizontal method: $\left(x^{2}-5 x+8\right) \times\left(x^{2}+2\right)$ $=x^{2}\left(x^{2}-5 x+8\right)+2\left(x^{2}-5 x+8\right)$ $=x^{4}-5 x^{3}+8 x^{2}+2 x^{2}-10 x+16$ $=x^{4}-5 x^{3}+10 x^{2}-10 x+16$...
Read More →A cone of radius 8 cm and height 12 cm
Question: A cone of radius 8 cm and height 12 cm is divided into two parts by a plane through the mid-point of its axis parallel to its base. Find the ratio of the volumes of two parts. Solution: Let ORN be the cone then given, radius of the base of the cone r1= 8cm and height of the cone, $(h) O M=12 \mathrm{~cm}$ Let $P$ be the mid-point of $O M$, then $O P=P M=\frac{12}{2}=6 \mathrm{~cm}$ Now, $\triangle O P D \sim \triangle O M N$ $\therefore \quad \frac{O P}{O M}=\frac{P D}{M N}$ $\Rightarr...
Read More →Find the product:
Question: Find the product:(9x2x+ 15) (x2 3) Solution: By horizontal method: $\left(9 x^{2}-x+15\right) \times\left(x^{2}-3\right)$ $=x^{2}\left(9 x^{2}-x+15\right)-3\left(9 x^{2}-x+15\right)$ $=9 x^{4}-x^{3}+15 x^{2}-27 x^{2}+3 x-45$ $=9 x^{4}-x^{3}-12 x^{2}+3 x-45$...
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