If the solve the problem
Question: If $\log y=\tan ^{-1} x$, show that: $\left(1+x^{2}\right) y_{2}+(2 x-1) y_{1}=0$ Solution: Note: $y_{2}$ represents second order derivative i.e. $\frac{d^{2} y}{d x^{2}}$ and $y_{1}=d y / d x$ Given, $\log y=\tan ^{-1} x$ $\therefore y=e^{\tan ^{-1} x} \ldots \ldots$ equation 1 to prove : $\left(1+x^{2}\right) y_{2}+(2 x-1) y_{1}=0$ We notice a second order derivative in the expression to be proved so first take the step to find the second order derivative. Let's find $\frac{d^{2} y}{...
Read More →A GP consists of an even number of terms.
Question: A GP consists of an even number of terms. If the sum of all the terms is 5 times the sum of the terms occupying the odd places, find the common ratio of the GP. Solution: Let the terms of the G.P. be $a, a r, a r^{2}, a r^{3}, \ldots, a r^{n-2}, a r^{n-1}$ Sum of a G.P. series is represented by the formula, $\mathrm{S}_{\mathrm{n}}=\mathrm{a} \frac{\mathrm{r}^{\mathrm{n}}-1}{\mathrm{r}-1}$ when r1. Sn represents the sum of the G.P. series upto nth terms, a represents the first term, r ...
Read More →A cube of side 5 cm is painted on all its faces.
Question: A cube of side 5 cm is painted on all its faces. If it is sliced into 1 cubic centimetre cubes, then how many 1 cubic centimetre cubes will have exactly one of their faces painted? (a) 27 (b) 42 (c) 54 (d) 142 Solution: (c) Given, a cube of side $5 \mathrm{~cm}$ is painted on all its faces and is sliced into $1 \mathrm{~cm}^{3}$ cubes. Then, from figure, it is clear that there are 9 cubes available on face. Since, there are six faces available. Hence, total number of smaller cubes $=6 ...
Read More →If the solve the problem
Question: If $x=\sin \left(\frac{1}{a} \log y\right)$, show that $\left(1-x^{2}\right) y_{2}-x y_{1}-a^{2} y=0$ Solution: Note: $y_{2}$ represents second order derivative i.e. $\frac{d^{2} y}{d x^{2}}$ and $y_{1}=d y / d x$ Given, $x=\sin \left(\frac{1}{a} \log y\right)$ $(\log y)=a \sin ^{-1} x$ $y=e^{\operatorname{asin}^{-1} x} \ldots \ldots$ equation 1 to prove: $\left(1-x^{2}\right) y_{2}-x y_{1}-a^{2} y=0$ We notice a second order derivative in the expression to be proved so first take the ...
Read More →100 The variable x is inversely proportional to y.
Question: 100 The variable x is inversely proportional to y. If x increases by p%, then by what per cent will y decwsose? Solution: The variable x is inversely proportional to y. xy = k (constant) Since, we know that two quantities x and y are said to be in inverse proportion, if an increase in * cause a proportional decrease in y and vice- versa.So, we can say y decrease by p%....
Read More →Ms Anita has to drive from Jhareda to Ganwari.
Question: Ms Anita has to drive from Jhareda to Ganwari. She measures a distance of 3.5 cm between these village on the map. What is the actual distance between the villages, if the map scale is 1 cm = 10 km? Solution: The distance between Jhareda to Ganwari in the map = 3.5 cm Given scale, 1 cm = 10 km So, actual distance between the villages = 35 x 10 = 35 km...
Read More →The 4th and 7th terms of a GP are
Question: The $4^{\text {th }}$ and $7^{\text {th }}$ terms of a GP are $\frac{1}{27}$ and $\frac{1}{729}$ respectively. Find the sum of $n$ terms of the GP. Solution: $4^{\text {th }}$ term $=a r^{4-1}=a r^{3}=\frac{1}{27}$ $7^{\text {th }}$ term $=a r^{7-1}=a r^{6}=\frac{1}{729}$ Dividing the $7^{\text {th }}$ term by the $4^{\text {th }}$ term, $\frac{\mathrm{ar}^{6}}{\mathrm{ar}^{3}}=\frac{\frac{1}{729}}{\frac{1}{27}}$ $\Rightarrow r^{3}=\frac{1}{27}$ (i) $\therefore \mathrm{r}=\frac{1}{3}$ ...
Read More →A packet of sweets was distributed among
Question: A packet of sweets was distributed among 10 children and each of them received 4 sweets. If it is distributed among 8 children, how many sweets will each child get? Solution: The total number of children = 10 If each children received 4 sweets, then The total number of sweets = 10 x 4 = 40 sweets If 40 sweets distributed between 8 children, then each get 40/8 i.e. 5 sweets....
Read More →(i)The time taken by a train to cover a fixed
Question: (i)The time taken by a train to cover a fixed distance and the speed of the train. (ii) The distance travelled by CNG bus and the amount of CNG used. (iii) The number of people working and the time to complete a given work. (iv) Income tax and the income. (v) Distance travelled by an auto-rickshaw and time taken. Solution: (i) The time taken by a train to cover a fixed distance and the speed of the train are inversely proportional. e.g. Let a train cover 100 km in 1 h with speed 100 km...
Read More →If the solve the problem
Question: If $y=e^{2 x}(a x+b)$, show that $y_{2}-4 y_{1}+4 y=0 .$ Solution: Note: $y_{2}$ represents second order derivative i.e. $\frac{d^{2} y}{d x^{2}}$ and $y_{1}=d y / d x$ Given, $y=e^{2 x}(a x+b) \ldots \ldots$ equation 1 to prove: $y_{2}-4 y_{1}+4 y=0$ We notice a second order derivative in the expression to be proved so first take the step to find the second order derivative. Let's find $\frac{d^{2} y}{d x^{2}}$ As, $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\frac{\mathrm{d}}{\...
Read More →The area of cultivated land and
Question: The area of cultivated land and the crop harvested is a case of direct proportion. Solution: True The area of cultivated land and the crop harvested is a case of direct proportion. Since, the quantities of crop harvested is depend upon area of cultivated land....
Read More →The number of workers and the time
Question: The number of workers and the time to complete a job is a case of direct proportion. Solution: False The number of workers and the time to complete a job is a case of indirect proportion, e.g. If 60 workers can complete a work in 10 days. Then, 120 workers can complete the same work in 5 days....
Read More →When two quantities are related
Question: When two quantities are related in such a manner that if one increases and the other decreases, then they always vary inversely. Solution: True When, two quantities are related in such a manner that if one increases and the other decreases, then they always vary inversely. Above statement is correct for inverse proportion. It is a basic properties of inverse proportion....
Read More →The 2nd and 5th terms of a GP are
Question: The $2^{\text {nd }}$ and $5^{\text {th }}$ terms of a GP are $\frac{-1}{2}$ and $\frac{1}{16}$ respectively. Find the sum of $n$ terms GP up to 8 terms. Solution: $2^{\text {nd }}$ term $=a r^{2-1}=a r^{1}$ $5^{\text {th }}$ term $=a r^{5-1}=a r^{4}$ Dividing the $5^{\text {th }}$ term using the $3^{\text {rd }}$ term $\frac{a r^{4}}{a r}=\frac{\frac{1}{16}}{\frac{-1}{2}}$ $r^{3}=-\frac{1}{8}$ $\therefore r=-\frac{-1}{2}$ a = 1 Sum of a G.P. series is represented by the formula $S_{n}...
Read More →When two quantities are related
Question: When two quantities are related in such a manner that, if one increases, the other also increases, then they always vary directly. Solution: True When two quantities are related in such a manner that if, one increases the other also increases, then they always vary directly. Above statement is correct for direct proportion. It is a basic properties of direct proportion....
Read More →If p and q are in inverse proportion,
Question: If p and q are in inverse proportion, i.e. pq = k (constant), then (p + 2)and (q 2) are also in inverse proportion. Solution: False If p and q are in inverse proportion, then xy = k (constant) e.g. Let p = 3andq = 4 Then, pq = 34 = 12 Now, p+ 2 = 3+ 2 = 5 and q-2 = 4-2 =2 (p + 2) (q 2) = 5 x 2 = 10 [not in inverse proportion] Henc, (p+2) and (q -2)cannot be in inverse proportion....
Read More →If the solve the problem
Question: If $y=3 \cos (\log x)+4 \sin (\log x)$, prove that: $x^{2} y_{2}+x y_{1}+y=0$ Solution: Note: $y_{2}$ represents second order derivative i.e. $\frac{d^{2} y}{d x^{2}}$ and $y_{1}=d y / d x$ Given, $y=3 \cos (\log x)+4 \sin (\log x) \ldots \ldots$ equation 1 to prove: $x^{2} y_{2}+x y_{1}+y=0$ We notice a second order derivative in the expression to be proved so first take the step to find the second order derivative. Let's find $\frac{d^{2} y}{d x^{2}}$ As, $\frac{\mathrm{d}^{2} \mathr...
Read More →If x and y are in inverse proportion,
Question: If x and y are in inverse proportion, then (x +1) and (y +1) are also in inverse proportion. Solution: False If x and y are in inverse proportion, then xy = k (constant) e.g. Let x= 2 and y = 3 .-. xy = 2 x 3= 6. Now, x + 1=2 + 1 = 3 and y+ 1 = 3 + 1 = 4 Then, (x + 1)(y+1) = 3 x 4 = 12 [not in inverse proportion] Hence, (x+ 1)and (y + 1) cannot be in inverse proportion....
Read More →If x and y are in direct proportion,
Question: If x and y are in direct proportion, then (x 1) and (y 1) are also in direct proportion. Solution: False x and y are in direct proportion, then; x/y = k if x = 3 and y = 5, then k = 3/5 Now, x-1 = 2 and y-1=4 (x-1)/(y-1) = 2/4 = 1/2...
Read More →If a tree 24 m high casts a shadow of 15 m,
Question: If a tree 24 m high casts a shadow of 15 m, then the height of a pole that casts a shadow of 6 m under similar conditions is 9.6 m. Solution: True Height of tree = 24m Length of shadow of tree = 15m Let height of pole = x Length of shadow of pole = 6m Now, 24/15 = x/6 x = (246)/15 x=9.6m...
Read More →If d varies directly as
Question: If $d$ varies directly as $t^{2}$, then we can write $d t^{2}=k$, where $k$ is some constant. Solution: False If $d$ varies inversely as $t^{2}$, then we can write $d t^{2}=k$, where $k$ is some constant. Since, two quantities x and y are said to be in Inverse proportion, if an increases in x cause a proportional decreases in y and vice-versa, in such a manner that the product of their corresponding values remains constant....
Read More →The first term of a GP is 27, and its 8th term is
Question: The first term of a GP is 27 , and its $8^{\text {th }}$ term is $\frac{1}{81}$. Find the sum of its first 10 terms. Solution: 'Tn' represents the $n^{\text {th }}$ term of a G.P. series. $\mathrm{T}_{\mathrm{n}}=\mathrm{ar}^{\mathrm{n}-1}$ $\Rightarrow \frac{1}{81}=27 \times r^{8-1}$ $\Rightarrow \frac{1}{81}=27 \times r^{7}$ $\Rightarrow \frac{1}{81} \div \frac{1}{27}=r^{7}$ $\Rightarrow \frac{1}{2187}=r^{7}$ $\Rightarrow\left(\frac{1}{3}\right)^{7}=r^{7}$ $\therefore \quad r=\frac{1...
Read More →Length of a side of an equilateral triangle
Question: Length of a side of an equilateral triangle and its perimeter vary inversely with each other. Solution: False Length of a side of an equilateral triangle and its perimeter vary directly with each other, e.g. Let a be the side of an equilateral triangle. So, perimeter = 3 x (Side) = 3 x a = 3a . So, if we increase the length of side of the equilateral triangle, then their perimeter will also increases....
Read More →If the solve the problem
Question: If $y=e^{\tan -1 x}$, Prove that: $\left(1+x^{2}\right) y_{2}+(2 x-1) y_{1}=0$ Solution: Note: $y_{2}$ represents second order derivative i.e. $\frac{d^{2} y}{d x^{2}}$ and $y_{1}=d y / d x$ Given, $y=e^{\tan -1 x} \ldots \ldots$ equation 1 to prove : $\left(1+x^{2}\right) y_{2}+(2 x-1) y_{1}=0$ We notice a second order derivative in the expression to be proved so first take the step to find the second order derivative. Let's find $\frac{d^{2} y}{d x^{2}}$ $A S, \frac{d^{2} y}{d x^{2}}...
Read More →Length of a side of a square and
Question: Length of a side of a square and its area vary directly with each other. Solution: FalseLength of a side of a square and its area does not vary directly with each other, e.g. Let a be length of each side of a square. So, area of the square $=$ Side $^{2}=a^{2}$ So, if we increase the length of the side of a square, then their area increases but not directly....
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