Find the product:
Question: Find the product:(x3 2x2+ 5) (4x 1) Solution: By horizontal method: $\left(x^{3}-2 x^{2}+5\right) \times(4 x-1)$ $=4 x\left(x^{3}-2 x^{2}+5\right)-1\left(x^{3}-2 x^{2}+5\right)$ $=4 x^{4}-8 x^{3}+20 x-x^{3}+2 x^{2}-5$ $=4 x^{4}-9 x^{3}+2 x^{2}+20 x-5$...
Read More →Differentiate the following functions with respect to $x$ :
Question: Differentiate the following functions with respect to $x$ : $\sin (3 x+5)$ Solution: Let $y=\sin (3 x+5)$ On differentiating $y$ with respect to $x$, we get $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}[\sin (3 \mathrm{x}+5)]$ We know $\frac{d}{d x}(\sin x)=\cos x$ $\Rightarrow \frac{d y}{d x}=\cos (3 x+5) \frac{d}{d x}(3 x+5)$ [using chain rule] $\Rightarrow \frac{d y}{d x}=\cos (3 x+5)\left[\frac{d}{d x}(3 x)+\frac{d}{d x}(5)\right]$ $\Rightarrow \frac{\mathrm{dy}}{...
Read More →Find the product: (x2 + xy + y2) × (x − y)
Question: Find the product:(x2+xy+y2) (xy) Solution: By horizontal method: $\left(x^{2}+x y+y^{2}\right) \times(x-y)$ $x\left(x^{2}+x y+y^{2}\right)-y\left(x^{2}+x y+y^{2}\right)$ $=x^{3}+x^{2} y+x y^{2}-x^{2} y-x y^{2}-y^{3}$ $=x^{3}-y^{3}$...
Read More →A backet is in the form of a frustum of a cone
Question: A backet is in the form of a frustum of a cone and holds 28.490 L of water. The radii of the top and bottom are 28 cm and 21 cm, respectively. Find the height of the bucket. Solution: Given, volume of the frustum = 28.49 L = 28.49 x 1000 cm3 [ 1 L = 1000 cm3] = 28490 cm3 and radius of the top (r1) = 28 cm radius of the bottom (r2) = 21 cm Let height of the bucket = h cm Now, volume of the bucket $=\frac{1}{3} \pi h\left(r_{1}^{2}+r_{2}^{2}+r_{1} r_{2}\right)=28490$ [given] $\Rightarrow...
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Question: Find the product:(x2xy+y2) (x+y) Solution: By horizontal method: $\left(x^{2}-x y+y^{2}\right) \times(x+y)$ $=x\left(x^{2}-x y+y^{2}\right)+y\left(x^{2}-x y+y^{2}\right)$ $=x^{3}-x^{2} y+y^{2} x+x^{2} y-x y^{2}+y^{3}$ $=x^{3}+y^{3}$...
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Question: Find the product:(3x2+ 5x 9) (3x 5) Solution: By horizontal method: $\left(3 x^{2}+5 x-9\right) \times(3 x-5)$ $=3 x\left(3 x^{2}+5 x-9\right)-5\left(3 x^{2}+5 x-9\right)$ $=9 x^{3}+15 x^{2}-27 x-15 x^{2}-25 x+45$ $=9 x^{3}-52 x+45$...
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Question: Find the product:(x2 3x + 7) (2x + 3) Solution: By horizontal method: $\left(x^{2}-3 x+7\right) \times(2 x+3)$ $=2 x\left(x^{2}-3 x+7\right)+3\left(x^{2}-3 x+7\right)$ $=2 x^{3}-6 x^{2}+14 x+3 x^{2}-9 x+21$ $=2 x^{3}-3 x^{2}+5 x+21$...
Read More →Differentiate each of the following functions from the first principal :
Question: Differentiate each of the following functions from the first principal : $\sin ^{-1}(2 x+3)$ Solution: We have to find the derivative of $\sin ^{-1}(2 x+3)$ with the first principle method, so, $f(x)=\sin ^{-1}(2 x+3)$ by using the first principle formula, we get, $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{\sin ^{-1}(2[x+h]+3)-\sin ^{-1}(2 x+3)}{h}$ Let $\sin ^{-1}[2(x+h)+3]=A$ and $\sin ^{-1}(2 x+3)=B$, so, $\sin A=[2(x+h...
Read More →How many shots each having diameter
Question: How many shots each having diameter 3 cm can be made from a cuboidal lead solid of dimensions 9 cm x 11 cm x 12 cm? Solution: Given, dimensions of cuboidal = 9 cm x 11 cm x 12 cm Volume of cuboidal = 9 x 11 x 12 = 1188 cm3 and diameter of shot = 3 cm $\therefore \quad$ Radius of shot, $r=\frac{3}{2}=1.5 \mathrm{~cm}$ Volume of shot $=\frac{4}{3} \pi r^{3}=\frac{4}{3} \times \frac{22}{7} \times(1.5)^{3}$ $=\frac{297}{21}=14.143 \mathrm{~cm}^{3}$ $\therefore \quad$ Required number of sho...
Read More →Differentiate each of the following functions from the first principal :
Question: Differentiate each of the following functions from the first principal : $\log \operatorname{cosec} x$ Solution: We have to find the derivative of $\log \operatorname{cosec} x$ with the first principle method, so, $f(x)=\log \operatorname{cosec} x$ by using the first principle formula, we get, $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{\log \operatorname{cosec}(x+h)-\log \operatorname{cosec} x}{h}$ $f^{\prime}(x)=\lim _{h ...
Read More →Three metallic solid cubes whose edges are 3 cm,
Question: Three metallic solid cubes whose edges are 3 cm, 4 cm and 5 cm are melted and formed into a single cube. Find the edge of the cube so formed. Solution: Given, edges of three solid cubes are 3 cm, 4 cm and 5 cm, respectively. Volume of first cube = (3)3= 27 cm3 Volume of second cube = (4)3= 64 cm3 and volume of third cube = (5)3= 125 cm3 Sum of volume of three cubes = (27 + 64 + 125) = 216 cm3 Let the edge of the resulting cube = R cm Then, volume of the resulting cube, R3=216 ⇒ R = 6cm...
Read More →An open metallic bucket is in the shape of a frustum of a cone,
Question: An open metallic bucket is in the shape of a frustum of a cone, mounted on a hollow cylindrical base made of the same metallic sheet. The surface area of the metallic sheet used is equal to curved surface area of frustum of a cone + area of circular base + curved surface area of cylinder. Solution: True Because the resulting figure is Here, ABCD is a frustum of a cone and CDEF is a hollow cylinder....
Read More →'The curved surface area of a frustum of a cone is
Question: 'The curved surface area of a frustum of a cone is $\pi l\left(r_{1}+r_{2}\right)$, where $l=\sqrt{h^{2}+\left(r_{1}+r_{2}\right)^{2}}, r_{1}$ and $r_{2}$ are the radii of the two ends of the frustum and $h$ is the vertical height. Solution: Fasle We know that, if $r_{1}$ and $r_{2}$ are the radii of the two ends of the frustum and $h$ is the vertical height, then curved surface area of a frustum is $\pi\left(r_{1}+r_{2}\right)$, where $l=\sqrt{h^{2}+\left(r_{1}-r_{2}\right)^{2}}$....
Read More →The capacity of a cylindrical vessel with a hemispherical
Question: The capacity of a cylindrical vessel with a hemispherical portion raised upward at the bottom as shown in the figure is $\frac{\pi r^{2}}{3}[3 h-2 r]$. Solution: True We know that, capacity of cylindrical vessel = r2h cm3 and capacity of hemisphere $=\frac{2}{3} \pi r^{3} \mathrm{~cm}$ From the figure, capacity of the cylindrical vessel $=\pi r^{2} h-\frac{2}{3} \pi r^{3}=\frac{1}{3} \pi r^{2}[3 h-2 r]$...
Read More →Find the product:
Question: Find the product: $\left(x^{4}+\frac{1}{x^{4}}\right) \times\left(x+\frac{1}{x}\right)$ Solution: By horizontal method: $\left(x^{4}+\frac{1}{x^{4}}\right) \times\left(x+\frac{1}{x}\right)$ $=x^{4}\left(x+\frac{1}{x}\right)+\frac{1}{x^{4}}\left(x+\frac{1}{x}\right)$ $=x^{5}+x^{3}+\frac{1}{x^{3}}+\frac{1}{x^{5}}$ i. e $x^{3}\left(x^{2}+1\right)+\frac{1}{x^{3}}\left(1+\frac{1}{x^{2}}\right)$...
Read More →Differentiate each of the following functions from the first principal:
Question: Differentiate each of the following functions from the first principal: $x^{2} e^{x}$ Solution: We have to find the derivative of $x^{2} e^{x}$ with the first principle method, so, $f(x)=x^{2} e^{x}$ by using the first principle formula, we get, $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{(x+h)^{2} e^{(x+h)}-x^{2} e^{x}}{h}$ $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{\left(x^{2}+h^{2}+2 h x\right) e^{(x+h)}-x^{2} e^{x}}{h...
Read More →The volume of the frustum of a cone is
Question: The volume of the frustum of a cone is $\frac{1}{3} \pi h\left[r_{1}^{2}+r_{2}^{2}-r_{1} r_{2}\right]$, where $h$ is vertical height of the frustum and $r_{1}, r_{2}$ are the radii of the ends. Solution: False Since, the volume of the frustum of a cone is $\frac{1}{3} \pi h\left[r_{1}^{2}+r_{2}^{2}+r_{1} r_{2}\right]$, where $h$ is vertical height of the frustum and $r_{1}, r_{2}$ are the radii of the ends....
Read More →Find the product: (x4 + y4) × (x2 − y2)
Question: Find the product:(x4+y4) (x2y2) Solution: By horizontal method: $\left(x^{4}+y^{4}\right) \times\left(x^{2}-y^{2}\right)$ $=x^{4}\left(x^{2}-y^{2}\right)+y^{4}\left(x^{2}-y^{2}\right)$ $=x^{6}-x^{4} y^{2}+y^{4} x^{2}-y^{6}$ $=\left(x^{6}-y^{6}\right)-x^{2} y^{2}\left(x^{2}-y^{2}\right)$...
Read More →A solid ball is exactly fitted inside
Question: A solid ball is exactly fitted inside the cubical box of side $a$. The volume of the ball is $\frac{4}{3} \pi a^{3}$. Solution: False Because solid ball is exactly fitted inside the cubical box of side a. So, a is the diameter for . the solid ball. $\therefore \quad$ Radius of the ball $=\frac{a}{2}$ $\mathrm{SO}$, volume of the ball $=\frac{4}{3} \pi\left(\frac{a}{2}\right)^{3}=\frac{1}{6} \pi a^{3}$...
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Question: Find the product:(x3y3) (x2+y2) Solution: By horizontal method: $\left(x^{3}-y^{3}\right) \times\left(x^{2}+y^{2}\right)$ $=x^{3}\left(x^{2}+y^{2}\right)-y^{3}\left(x^{2}+y^{2}\right)$ $=x^{5}+x^{3} y^{2}-x^{2} y^{3}-y^{5}$ $=\left(x^{5}-y^{5}\right)+x^{2} y^{2}(x-y)$...
Read More →Find the product:
Question: Find the product:(2x2 5y2) (x2+ 3y2) Solution: By horizontal method: $\left(2 x^{2}-5 y^{2}\right) \times\left(x^{2}+3 y^{2}\right)$ $=2 x^{2}\left(x^{2}+3 y^{2}\right)-5 y^{2}\left(x^{2}+3 y^{2}\right)$ $=2 x^{4}+6 x^{2} y^{2}-5 x^{2} y^{2}-15 y^{4}$ $=2 x^{4}+x^{2} y^{2}-15 y^{4}$...
Read More →A solid cone of radius r and height h
Question: A solid cone of radius $r$ and height $h$ is placed over a solid cylinder having same base radius and height as that of a cone The total surface area of thecombined solid is $\left[\sqrt{r^{2}+h^{2}}+3 r+2 h\right]$. Solution: False We know that, total surface area of a cone of radius, r and $\quad$ height, $h=$ Curved surface Area $+$ area of base $=\pi r l+\pi r^{2}$ where,$l=\sqrt{h^{2}+r^{2}}$ and total surface area of a cylinder of base radius, $r$ and height, $h$ $=$ Curved surfa...
Read More →Differentiate each of the following functions from the first principal :
Question: Differentiate each of the following functions from the first principal : $e^{\sqrt{\cot x}}$ Solution: We have to find the derivative of $\mathrm{e}^{\sqrt{\cot x}}$ with the first principle method, so, $f(x)=e^{\sqrt{\cot x}}$ by using the first principle formula, we get, $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{e^{\sqrt{\cot (x+h)}}-e^{\sqrt{\cot x}}}{h}$ $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{e^{\sqrt{\cot x}}\l...
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Question: Find the product:(3p2+q2) (2p2 3q2) Solution: By horizontal method: $\left(3 p^{2}+q^{2}\right) \times\left(2 p^{2}-3 q^{2}\right)$ $=3 p^{2}\left(2 p^{2}-3 q^{2}\right)+q^{2}\left(2 p^{2}-3 q^{2}\right)$ $=6 p^{4}-9 p^{2} q^{2}+2 p^{2} q^{2}-3 q^{4}$ i. $e 6 p^{4}-7 p^{2} q^{2}-3 q^{4}$...
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Question: Find the product:(x2y2) (x+ 2y) Solution: By horizontal method: $\left(x^{2}-y^{2}\right) \times(x+2 y)$ $=x^{2}(x+2 y)-y^{2}(x+2 y)$ $=x^{3}+2 x^{2} y-x y^{2}-2 y^{3}$ i. $e\left(x^{3}-2 y^{3}\right)+x y(2 x-y)$...
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