The rain water from a roof of dimensions 22 m x 20 m

Question: The rain water from a roof of dimensions 22 m x 20 m drains into a cylindrical vessel having diameter of base 2 m and height 3.5 m. If the rain water collected from the roof just fill the cylindrical vessel, then find the rainfall (in cm). Solution: Given, length of roof = 22 m and breadth of roof = 20 m Let the rainfall be a cm. $\therefore$Volume of water on the roof $=22 \times 20 \times \frac{a}{100}=\frac{22 a}{5} \mathrm{~m}^{3}$ Also, we have radius of base of the cylindrical ve...

Tick (✓) the correct answer:

Question: Tick (✓) the correct answer: If $\left(x-\frac{1}{x}\right)=6$, then $\left(x^{2}+\frac{1}{x^{2}}\right)=?$ (a) 36 (b) 38 (c) 32 (d) $36 \frac{1}{36}$ Solution: (b) 38 $\left(x-\frac{1}{x}\right)=6$ $\Rightarrow$ Squaring both the sides: $\Rightarrow\left(x-\frac{1}{x}\right)^{2}=(6)^{2}$ $\Rightarrow\left(x^{2}+\frac{1}{x^{2}}-2(x)\left(\frac{1}{x}\right)\right)=36$ $\Rightarrow\left(x^{2}+\frac{1}{x^{2}}\right)-2=36$ $\Rightarrow\left(x^{2}+\frac{1}{x^{2}}\right)=36+2$ $\Rightarrow\l...

Differentiate the following functions with respect to x :

Question: Differentiate the following functions with respect to $x$ : $\sqrt{\frac{1-x^{2}}{1+x^{2}}}$ Solution: Let $y=\sqrt{\frac{1-x^{2}}{1+x^{2}}}$ On differentiating $y$ with respect to $x$, we get $\frac{d y}{d x}=\frac{d}{d x}\left(\sqrt{\frac{1-x^{2}}{1+x^{2}}}\right)$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left[\left(\frac{1-\mathrm{x}^{2}}{1+\mathrm{x}^{2}}\right)^{\frac{1}{2}}\right]$ We know $\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}^{\mathrm...

Tick (✓) the correct answer:

Question: Tick (✓) the correct answer: If $\left(x+\frac{1}{x}\right)=5$, then $\left(x^{2}+\frac{1}{x^{2}}\right)=?$ (a) 25 (b) 27 (c) 23 (d) $25 \frac{1}{25}$ Solution: (c) 23 $\left(x+\frac{1}{x}\right)=5$ $\Rightarrow$ 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\...

Water flows through a cylindrical pipe,

Question: Water flows through a cylindrical pipe, whose inner radius is 1 cm, at the rate of 80 cms-1in an empty cylindrical tank, the radius of whose base is 40 cm. What is the rise of water level in tank in half an hour? Solution: Given, radius of tank, r1= 40 cm Let height of water level in tank in half an hour = 1 cm. Also, given internal radius of cylindrical pipe, r2= 1 cm and speed of water = 80 cm/s i.e., in 1 water flow = 80 cm In 30 (min) water flow = 80x 60 x 30 = 144000 cm According ...

Tick (✓) the correct answer:

Question: Tick (✓) the correct answer: $\left(\frac{1}{x}+\frac{1}{y}\right)\left(\frac{1}{x}-\frac{1}{y}\right)=?$ (a) $\left(\frac{1}{x^{2}}-\frac{1}{y^{2}}\right)$ (b) $\left(\frac{1}{x^{2}}+\frac{1}{y^{2}}\right)$ (c) $\left(\frac{1}{x^{2}}+\frac{1}{y^{2}}-\frac{1}{x y}\right)$ (d) $\left(\frac{1}{x^{2}}-\frac{1}{y^{2}}+\frac{1}{x y}\right)$ Solution: (a) $\left(\frac{1}{x^{2}}-\frac{1}{y^{2}}\right)$ $\left(\frac{1}{x}+\frac{1}{y}\right)\left(\frac{1}{x}-\frac{1}{y}\right)$ $\Rightarrow$ Ac...

A solid right circular cone of height 120 cm

Question: A solid right circular cone of height 120 cm and radius 60 cm is placed in a right circular cylinder full of water of height 180 cm. Such that it touches the bottom. Find the volume of water left in the cylinder, if the radius of the cylinder is equal to the radius to the cone. Solution: (i) Whenever we placed a solid right circular cone in a right circular cylinder with full of water, then volume of a solid right circular cone is equal to the volume of water failed from the cylinder. ...

Tick (✓) the correct answer:

Question: Tick (✓) the correct answer: (a+ 1)(a 1)(a2+ 1) = ? (a) (a4 2a2 1) (b) (a4a2 1) (c) (a4 1) (d) (a4+ 1) Solution: (c) $\left(a^{4}-1\right)$ $(i)(a+1)(a-1)\left(a^{2}+1\right)$ $\Rightarrow\left((a)^{2}-(1)^{2}\right)\left(a^{2}+1\right)$ [according to the formula $a^{2}-b^{2}=(a+b)(a-b)$ ] $\Rightarrow\left(a^{2}-1\right)\left(a^{2}+1\right)$ $\Rightarrow\left(a^{2}\right)^{2}-\left(1^{2}\right)^{2}$ [according to the formula $a^{2}-b^{2}=(a+b)(a-b)$ ] $\Rightarrow a^{4}-1$...

Tick (✓) the correct answer

Question: Tick (✓) the correct answer: (x2 4x+ 4) (x 2) = ? (a) (x 2) (b) (x+ 2) (c) (2 x) (d) (2 +x+x2) Solution: (a) (x 2)...

Tick (✓) the correct answer

Question: Tick (✓) the correct answer: (x2 4x+ 4) (x 2) = ? (a) (x 2) (b) (x+ 2) (c) (2 x) (d) (2 +x+x2) Solution: (a) (x 2)...

Tick (✓) the correct answer:

Question: Tick (✓) the correct answer: (2x2+ 3x+ 1) (x+ 1) = ? (a) (x+ 1) (b) (2x+ 1) (c) (x+ 3) (d) (2x+ 3) Solution: (b) (2x+ 1)...

Tick (✓) the correct answer

Question: Tick (✓) the correct answer: 8a2b3 (2ab) = ? (a) 4ab2 (b) 4a2b (c) 4ab2 (d) 4a2b Solution: (c) $-4 a b^{2}$ $8 a^{2} b^{3} \div(-2 a b)$ $\Rightarrow\left(\frac{8}{-2}\right)\left(a^{2-1}\right)\left(b^{3-1}\right)$ $\Rightarrow-4 a b^{2}$...

A hemispherical bowl of internal radius 9 cm is full of liquid.

Question: A hemispherical bowl of internal radius 9 cm is full of liquid. The liquid is to be filled into cylindrical shaped bottles each of radius 1.5 cm and height 4 cm. How many bottles are needed to empty the bowl? Solution: Given, radius of hemispherical bowl, r = 9 cm and radius of cylindrical bottles, R = 1.5 cm and height, h = 4 cm $\therefore$ Number of required cylindrical bottles $=\frac{\text { Volume of hemispherical bowl }}{\text { Volume of one cylindrical bottle }}$ $=\frac{\frac...

Tick (✓) the correct answer:

Question: Tick (✓) the correct answer: (2x+ 5)(2x 5) = ? (a) (4x2+ 25) (b) (4x2 25) (c) (4x2 10x+ 25) (d) (4x2+ 10x 25) Solution: (b) $\left(4 x^{2}-25\right)$ $(2 x+5)(2 x-5)$ $\Rightarrow(2 x)^{2}-(5)^{2}$ (according to the formula $\left.(a+b)(a-b)=a^{2}-b^{2}\right)$ $\Rightarrow 4 x^{2}-25$...

A building is in the form of a cylinder surmounted

Question: A building is in the form of a cylinder surmounted by a hemispherical vaulted dome and contains $41 \frac{19}{21} \mathrm{~m}^{3}$ of air. If the internal diameter of dome is equal to its total height above the floor, find the height of the building? Solution: Let total height of the building = Internal diameter of the dome = 2r m $\therefore \quad$ Radius of building (or dome) $=\frac{2 r}{2}=r \mathrm{~m}$ Height of cylinder $=2 r-r=r \mathrm{~m}$ $\therefore \quad$ Volume of the cyl...

Differentiate the following functions with respect to x :

Question: Differentiate the following functions with respect to $x$ : $\sqrt{\frac{1+\sin x}{1-\sin x}}$ Solution: Let $y=\sqrt{\frac{1+\sin x}{1-\sin x}}$ On differentiating y with respect to $x$, we get $\frac{d y}{d x}=\frac{d}{d x}\left(\sqrt{\frac{1+\sin x}{1-\sin x}}\right)$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left[\left(\frac{1+\sin \mathrm{x}}{1-\sin \mathrm{x}}\right)^{\frac{1}{2}}\right]$ We know $\frac{d}{d x}\left(x^{n}\right)=n x^{n-1}$ $\Righ...

Tick (✓) the correct answer

Question: Tick (✓) the correct answer: (x 6)(x 6) = ? (a) (x2 36) (b) (x2+ 36) (c) (x2 6x+ 36) (d) (x2 12x+ 36) Solution: (d) $\left(x^{2}-12 x+36\right)$ $(x-6)(x-6)$ $\Rightarrow(x-6)^{2}$(according to the formula $\left.(a-b)^{2}=a^{2}-2 a b+b^{2}\right)$ $\Rightarrow\left(x^{2}\right)-2(x)(6)+(6)^{2}$ $\Rightarrow x^{2}-12 x+36$...

Question: Differentiate the following functions with respect to $x$ : $\sqrt{\frac{1+\sin x}{1-\sin x}}$ Solution: Let $y=\sqrt{\frac{1+\sin x}{1-\sin x}}$ On differentiating y with respect to $x$, we get $\frac{d y}{d x}=\frac{d}{d x}\left(\sqrt{\frac{1+\sin x}{1-\sin x}}\right)$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left[\left(\frac{1+\sin \mathrm{x}}{1-\sin \mathrm{x}}\right)^{\frac{1}{2}}\right]$ We know $\frac{d}{d x}\left(x^{n}\right)=n x^{n-1}$ $\Righ...

Tick (✓) the correct answer

Question: Tick (✓) the correct answer: (x+ 4)(x+ 4) = ? (a) (x2+ 16) (b) (x2+ 4x+ 16) (c) (x2+ 8x+ 16) (d) (x2+ 16x) Solution: (c) $\left(x^{2}+8 x+16\right)$ $(x+4)(x+4)$ $\Rightarrow(x+4)^{2}$ (according to the formula $\left.(a+b)^{2}=a^{2}+2 a b+b^{2}\right)$ $\Rightarrow\left(x^{2}\right)+2(x)(4)+(4)^{2}$ $\Rightarrow x^{2}+8 x+16$...

Tick (✓) the correct answer:

Question: Tick (✓) the correct answer: (x+ 5)(x 3) = ? (a)x2+ 5x 15 (b)x2 3x 15 (c)x2+ 2x+ 15 (d)x2+ 2x 15 Solution: (d)x2+ 2x 15 $(2 x+3)(3 x-1)$ $\Rightarrow(2 x)(3 x-1)+(3)(3 x-1)$ $\Rightarrow 6 x^{2}-2 x+9 x-3$ $\Rightarrow 6 x^{2}+7 x-3$...

A rocket is in the form of a right circular cylinder

Question: A rocket is in the form of a right circular cylinder closed at the lower end and surmounted by a cone with the same radius as that of the cylinder. The diameter and height of the cylinder are 6 cm and 12 cm, respectively. If the slant height of the conical portion is 5 cm, then find the total surface area and volume of the rocket, (use n = 3.14J) Solution: Since, rocket is the combination of a right circular cylinder and a cone. Given, diameter of the cylinder = 6 cm $\therefore \quad$...

Tick (✓) the correct answer:

Question: Tick (✓) the correct answer: (3q+ 7p2 2r3+ 4) (4p2 2q+ 7r3 3) = ? (a) (p2+ 2q+ 5r3+ 1) (b) (11p2+q+ 5r3+ 1) (c) (3p2 5q+ 9r3 7) (d) (3p2+ 5q 9r3+7) Solution: (d) (3p2+ 5q 9r3+7)...

Tick (✓) the correct answer:

Question: Tick (✓) the correct answer: The sum of (6a+ 4bc+ 3), (2b 3c+ 4), (11b 7a+ 2c 1) and (2c 5a 6) is (a) (4a 6b+ 2) (b) (3a+ 14b 3c+ 2) (c) (6a+ 17b) (d) (6a+ 6b+c4) Solution: (c) (6a+ 17b)...

If x − y = 7 and xy = 9,

Question: Ifxy= 7 andxy= 9, find the value of (x2+y2). Solution: $x-y=7$ $\Rightarrow$ On squaring both the sides : $\Rightarrow(x-y)^{2}=(7)^{2}$ $\Rightarrow x^{2}+y^{2}-2 x y=49$ $\Rightarrow x^{2}+y^{2}=49+2 x y$ Given: $x y=9$ $\Rightarrow x^{2}+y^{2}=49+2(9)$ $\Rightarrow x^{2}+y^{2}=49+18$ $\Rightarrow x^{2}+y^{2}=67$ Therefore, the value of $x^{2}+y^{2}$ is 67 ....

A cylindrical bucket of height 32 cm

Question: A cylindrical bucket of height 32 cm and base radius 18 cm is filled with sand. This bucket is emptied on the ground and a conical heap of sand is formed. If the height of the conical heap is 24 cm, find the radius and slant height of the heap. Solution: Given, radius of the base of the bucket = 18 cm Height of the bucket = 32 cm So, volume of the sand in cylindrical bucket = r2h= (18)2x 32 = 10368 Also, given height of the conical heap (h) = 24 cm Let radius of heap be r cm. Then, vol...