Subtract: 2a − 5b + 2c − 9 from 3a − 4b − c + 6
Question: Subtract:2a 5b+ 2c 9 from 3a 4bc+ 6 Solution: Writing the terms of the given expressions (in the same order) in the form of rows with like terms below each other and subtracting column-wise:...
Read More →If two solid hemispheres of same base radius
Question: If two solid hemispheres of same base radius r are joined together along their bases, then curved surface area of this new solid is (a) 47r2 (b) 6r2 (c) 3r2 (d) 8r2 Solution: (a)Because curved surface area of a hemisphere is 2 w2and here, we join two solid hemispheres along their bases of radius r, from which we get a solid sphere. Hence, the curved surface area of new solid = 2 r2+ 2 r2= 4r2...
Read More →Differentiate the following functions from first principles :
Question: Differentiate the following functions from first principles : $e^{a x+b}$ Solution: We have to find the derivative of $\mathrm{e}^{\mathrm{ax}+\mathrm{b}}$ with the first principle method, so, $f(x)=e^{a x+b}$ by using the first principle formula, we get, $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{e^{a(x+h)+b}-e^{a x+b}}{h}$ $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{e^{a x+b}\left(e^{a h}-1\right) a}{a h}$ [By using $\l...
Read More →Subtract: −16p from −11p
Question: Subtract:16pfrom 11p Solution: Writing the terms of the given expressions (in the same order) in the form of rows with like terms below each other and subtracting column-wise:...
Read More →A medicine-capsule is in the shape of a cylinder
Question: A medicine-capsule is in the shape of a cylinder of diameter 0.5 cm with two hemispheres stuck to each of its ends. The length of entire capsule is 2 cm. The capacity of the capsule is (a) 0.36 cm3 (b) 0.35 cm3 (c) 0.34 cm3 (d) 0.33 cm3 Solution: (a)Given, diameter of cylinder = Diameter of hemisphere = 0.5 cm [since, both hemispheres are attach with cylinder] $\therefore$ Radius of cylinder $(r)=$ radius of hemisphere $(r)=\frac{0.5}{2}=0.25 \mathrm{~cm}$$[\because$ diameter $=2 \time...
Read More →Subtract: −2abc from −8abc
Question: Subtract:2abcfrom 8abc Solution: Writing the terms of the given expressions (in the same order) in the form of rows with like terms below each other and subtracting column-wise:...
Read More →Subtract: −8pq from 6pq
Question: Subtract: 8pq from 6pq Solution: Writing the terms of the given expressions (in the same order) in the form of rows with like terms below each other and subtracting column-wise:...
Read More →Subtract:
Question: Subtract:3a2bfrom 5a2b Solution: On arranging the terms of the given expressions in the descending powers of $x$ and subtracting:...
Read More →Differentiate the following functions from first principles:
Question: Differentiate the following functions from first principles: $e^{3 x}$ Solution: We have to find the derivative of $\mathrm{e}^{3 x}$ with the first principle method, so, $f(x)=e^{3 x}$ by using the first principle formula, we get, $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{e^{3(x+h)}-e^{3 x}}{h}$ $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{e^{2 x}\left(e^{2 h}-1\right)}{h}$ $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{e^...
Read More →The radii of the top and bottom of a bucket
Question: The radii of the top and bottom of a bucket of slant height 45 cm are 28 cm and 7 cm, respectively. The curved surface area of the bucket is (a) 4950 cm2 (b) 4951 cm2 (c) 4952 cm2 (d) 4953 cm2 Solution: (a)Given, the radius of the top of the bucket, R = 28 cm and the radius of the bottom of the bucket, r = 7 cm Slant height of the bucket, l= 45 cm Since, bucket is in the form of frustum of a cone. Curved surface area of the bucket = l (R + r) = x 45 (28 + 7) $[\because$ curved surface ...
Read More →Solve the following
Question: Add:4x2 7xy+ 4y2 3, 5 + 6y2 8xy+x2and 6 2xy+ 2x2 5y2 Solution: On arranging the terms of the given expressions in the descending powers of $x$ and adding column-wise:...
Read More →Twelve solid spheres of the same size
Question: Twelve solid spheres of the same size are made by melting a solid metallic cylinder of base diameter 2 cm and height 16 cm. The diameter of each sphere is (a) 4 cm (b) 3 cm (c) 2 cm (d) 6 cm Solution: (c)Given, diameter of the cylinder = 2 cm Radius = 1 cm and height of the cylinder = 16 cm [∵ diameter = 2 x radius] Volume of the cylinder = x (1)2x 16 = 16 cm3 $\left[\because\right.$ volume of cylinder $=\pi \times(\text { radius })^{2} \times$ height $]$ Now, let the radius of solid s...
Read More →Add: 6p + 4q − r + 3, 2r − 5p − 6, 11q − 7p + 2r − 1 and 2q − 3r + 4
Question: Add: 6p + 4q r + 3, 2r 5p 6, 11q 7p + 2r 1 and 2q 3r + 4 Solution: Writing the terms of the given expressions (in the same order) in the form of rows with like terms below each other and adding column-wise:...
Read More →Solve the following
Question: Add:2x3 9x2+ 8, 3x2 6x 5, 7x3 10x+ 1 and 3 + 2x 5x2 4x3 Solution: On arranging the terms of the given expressions in the descending powers of $x$ and adding column-wise:...
Read More →Differentiate the following functions from first principles :
Question: Differentiate the following functions from first principles : $e^{-x}$ Solution: We have to find the derivative of $\mathrm{e}^{-\mathrm{x}}$ with the first principle method, so, $f(x)=e^{-x}$ by using the first principle formula, we get, $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{e^{-(x+h)}-e^{-x}}{h}$ $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{e^{-x}\left(e^{-h}-1\right)}{h}$ $f^{\prime}(x)=\lim _{h \rightarrow 0} \fra...
Read More →A mason constructs a wall of dimensions
Question: A mason constructs a wall of dimensions $270 \mathrm{~cm} \times 300 \mathrm{~cm} \times 350 \mathrm{~cm}$ with the bricks each of size $22.5 \mathrm{~cm} \times$ $11.25 \mathrm{~cm} \times 8.75 \mathrm{~cm}$ and it is assumed that $\frac{1}{8}$ space is covered by the mortar. Then, the number of bricks used to construct the wall is (a) 11100 (b) 11200 (c) 11000 (d) 11300 Solution: (b) Volume of the wall $=270 \times 300 \times 350=28350000 \mathrm{~cm}^{3}$ $[\because$ volume of cuboi...
Read More →Add: 6ax − 2by + 3cz, 6by − 11ax − cz and 10cz − 2ax − 3by
Question: Add: 6ax 2by + 3cz, 6by 11ax cz and 10cz 2ax 3by Solution: Writing the terms of the given expressions (in the same order) in the form of rows with like terms below each other and adding column-wise, we get:...
Read More →Add: 5x − 8y + 2z, 3z − 4y − 2x, 6y − z − x and 3x − 2z − 3y
Question: Add: 5x 8y + 2z, 3z 4y 2x, 6y z x and 3x 2z 3y Solution: Writing the terms of the given expressions (in the same order) in the form of rows with like terms below each other and adding column-wise, we get:...
Read More →If a solid piece of iron in the form
Question: If a solid piece of iron in the form of a cuboid of dimensions 49 cm x 33 cm x 24 cm, is moulded to form a solid sphere. Then, radius of the sphere is (a) 21 cm (b) 23 cm (c) 25 cm (d)19cm Solution: (a)Given, dimensions of the cuboid = 49 cm x 33 cm x 24 cm Volume of the cuboid = 49 x 33 x 24 = 38808 cm3 $[\because$ volume of cuboid $=$ length $\times$ breadth $\times$ height $]$ Let the radius of the sphere is $r$, then Volume of the sphere $=\frac{4}{3} \pi r^{3}$ $\left[\because\rig...
Read More →Add: 3a − 4b + 4c, 2a + 3b − 8c, a − 6b + c
Question: Add: 3a 4b + 4c, 2a + 3b 8c, a 6b + c Solution: Writing the terms of the given expressions (in the same order) in the form of rows with like terms below each other and adding column-wise, we get:...
Read More →Add: 7x, −3x, 5x, −x, −2x
Question: Add: 7x, 3x, 5x, x, 2x Solution: Writing the terms of the given expressions (in the same order) in the form of rows with like terms below each other and adding column-wise, we get:...
Read More →Add: 8ab, −5ab, 3ab, −ab
Question: Add: 8ab, 5ab, 3ab, ab Solution: Writing the terms of the given expressions (in the same order) in the form of rows with like terms below each other and adding column-wise, we get:...
Read More →A metallic spherical shell of internal and external
Question: A metallic spherical shell of internal and external diameters 4 cm and 8 cm, respectively is melted and recast into the form a cone of base diameter 8 cm. The height of the cone is (a) 12 cm (b) 14 cm (c) 15 cm (d) 18 cm Solution: (b)Given, internal diameter of spherical shell = 4 cm and external diameter of shell = 8 cm $\therefore$ Internal radius of spherical shell, $r_{1}=\frac{4}{2} \mathrm{~cm}=2 \mathrm{~cm}$ " $\quad[\because$ diameter $=2 \times$ radius $]$ and external radius...
Read More →If 486*7 is divisible by 9, then the least value of * is
Question: If 486*7 is divisible by 9, then the least value of * is (a) 0 (b) 1 (c) 3 (d) 2 Solution: (d) 2 For a number to be divisible by 9, the sum of its digits must be divisible by 9. $4+8+6+^{*}+7=25+^{*}$ Now, $25+^{*}=27 \quad$ (if $^{*}=2$ and 27 is divisible by 9 )...
Read More →If a hollow cube of internal edge 22 cm
Question: If a hollow cube of internal edge 22 cm is filled with spherical marbles of diameter 0.5 cm and it is assumed that space of the cube remains unfilled. Then, the number of marbles that the cube can accomodate is (a) 142244 (b) 142344 (c) 142444 (d) 142544 Solution: (a)Given, edge of the cube = 22 cm $\therefore \quad$ Volume of the cube $=(22)^{3}=10648 \mathrm{~cm}^{3} \quad\left[\because\right.$ volume of cube $\left.=(\text { side })^{3}\right]$ Also, given diameter of marble $=0.5 \...
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