Assuming land to be uniformly fertile,
Question: Assuming land to be uniformly fertile, the area of land and the yield on it vary (a) directly with each other (b) inversely with each other (c) neither directly nor inversely with each other (d) sometimes directly and sometimes inversely with each other Solution: (a) If land to be uniformly fertile, then the area of land and the yield on it vary directly with each other. Hence, option (a) is correct. Note Two quantities x and y are said to be in direct proportion, if they increase or d...
Read More →If the solve the problem
Question: If $x=a(\theta-\sin \theta), y=a(1+\cos \theta)$ find $\frac{d^{2} y}{d x^{2}}$ Solution: Idea of parametric form of differentiation: If $y=f(\theta)$ and $x=g(\theta)$ i.e. $y$ is a function of $\theta$ and $x$ is also some other function of $\theta$. Then $d y / d \theta=f^{\prime}(\theta)$ and $d x / d \theta=g^{\prime}(\theta)$ We can write : $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\frac{\mathrm{dy}}{\mathrm{d} \theta}}{\frac{\mathrm{dx}}{\mathrm{d} \theta}}$ Given, $x=a(\theta-\sin...
Read More →Find the sum :
Question: Find the sum : $\frac{3}{5}+\frac{4}{5^{2}}+\frac{3}{5^{3}}+\frac{4}{5^{4}}+$ . To 2n terms Solution: We can split the above expression into 2 parts. We will split 2n terms into 2 parts also which will leave it as n terms and another n terms. $=\left(\frac{3}{5}+\frac{3}{5^{3}}+\ldots\right.$ to n terms $)+\left(\frac{4}{5}+\frac{4}{5^{2}}+\ldots\right.$ to n terms $)$ Sum of a G.P. series is represented by the formula $S_{n}=a \frac{1-r^{n}}{1-r}$ when |r|1. Sn represents the sum of t...
Read More →Both x and y vary inversely with each other.
Question: Both x and y vary inversely with each other. When x is 10, y is 6, which of the following is not a possible pair of corresponding values of x and y? (a) 12 and 5 (b) 15 and 4 (c) 25 and 2.4 (d) 45 and 1.3 Solution: (d) 45 and 1.3 Explanation: since x and y vary inversely, so xy = k (constant) Putting the value of x and y, we get; 106 = 60 Now if we observe the options available; Option (a) 12 and 5 125 = 60 Option (b) 15 and 4 154 = 60 Option (c) 25 and 2.4 252.4 = 60 Option (d) 45 and...
Read More →Both u and v vary directly with each other.
Question: Both u and v vary directly with each other. When u is 10, v is 15, which of the following is not a possible pair of corresponding values of u and v? (a)2 and 3 (b) 8 and 12 (c) 15 and 20 (d) 25 and 37.5 Solution: (c) Since, $u$ and $v$ vary directly, i.e. $u / v=k$ (constant) If $u=10$ and $v=15$, then, $\frac{u}{v}=\frac{10}{15}=\frac{2}{3}$ In option (b), $\frac{8}{12}=\frac{2}{3}$ In option (c), $\frac{15}{20}=\frac{3}{4}$ In option (d), $\frac{25}{37.5}=\frac{2}{3}$ So, option (c) ...
Read More →Find the sum to n terms of the sequence :
Question: Find the sum to n terms of the sequence : (i) $\left(x+\frac{1}{x}\right)^{2},\left(x^{2}+\frac{1}{x^{2}}\right)^{2},\left(x^{3}+\frac{1}{x^{3}}\right)^{2}$ ,.. to n terms (ii) $\left.(x+y), 9 x^{2}+x y+y^{2}\right),\left(x^{3}+x 2 y+x y^{2}+y^{3}\right), \ldots .$ to $n$ terms Solution: This can also be written as $=\left(x^{2}+\frac{1}{x^{2}}+2\right)+\left(x^{4}+\frac{1}{x^{4}}+2\right)+\left(x^{6}+\frac{1}{x^{6}}+2\right)+\ldots \ldots$ to $n$ term $\left(x^{2}+x^{4}+x^{6}+\ldots\r...
Read More →If the solve the problem
Question: If $x=a(\theta+\sin \theta), y=a(1+\cos \theta)$, prove that $\frac{d^{2} y}{d x^{2}}=-\frac{a}{y^{2}}$ Solution: Idea of parametric form of differentiation: If $y=f(\theta)$ and $x=g(\theta)$ i.e. $y$ is a function of $\theta$ and $x$ is also some other function of $\theta$. Then $d y / d \theta=f^{\prime}(\theta)$ and $d x / d \theta=g^{\prime}(\theta)$ We can write : $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\frac{d y}{d \theta}}{\frac{d x}{d \theta}}$ Given, $x=a(\theta+\sin \theta) \...
Read More →Sales tax is always calculated
Question: Sales tax is always calculated on the cost price of an item and is added to the value of the bill. Solution: False Sales tax is always calculated on the selling price of an item and is added to the value of the bill....
Read More →Simple interest on a given amount
Question: Simple interest on a given amount is always less than or equal to the compound interest on the same amount for the same time period and at the same rate of interest per annum. Solution: False For 1 yr, the simple interest and compound interest for same amount on same rate of interest are equal. But for 2 yr, the simple interest is less than the compound interest for same amount on same rate of interest....
Read More →C.P. = M.P. – Discount.
Question: C.P. = M.P. Discount. Solution: False The relation between marked price and discount is given by Selling price = Marked price Discount...
Read More →Compound interest is the interest
Question: Compound interest is the interest calculated on the previous years amount. Solution: True...
Read More →Discount is a reduction given
Question: Discount is a reduction given on cost price of an article. Solution: False Discount is a reduction given on marked price not on cost price....
Read More →If the solve the problem
Question: If $x=a\left(1-\cos ^{3} \theta\right), y=a \sin ^{3} \theta$, Prove that $\frac{d^{2} y}{d x^{2}}=\frac{32}{27 a} a t \theta=\frac{\pi}{6}$ Solution: Idea of parametric form of differentiation: If $y=f(\theta)$ and $x=g(\theta)$ i.e. $y$ is a function of $\theta$ and $x$ is also some other function of $\theta$. Then $d y / d \theta=f^{\prime}(\theta)$ and $d x / d \theta=g^{\prime}(\theta)$ We can write : $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\frac{\mathrm{dy}}{\mathrm{d} \theta}}{\f...
Read More →Additional expenses made after buying
Question: Additional expenses made after buying an article are included in the cost price and are known as Value Added Tax. Solution: False In the selling price (known as MRP) include the tax known as VAT (Value Added Tax). Hence, VAT is always included in selling price....
Read More →To calculate the growth of a bacteria
Question: To calculate the growth of a bacteria if the rate of growth is known, the formula for calculation of amount in compound interest can be used. Solution: True...
Read More →5% sales tax is charged on an article marked
Question: 5% sales tax is charged on an article marked Rs 200 after allowing a discount of 5%, then the amount payable is ____ Solution: Rs.199.50. Explanation: marked price = Rs. 200 Discount = 5% Selling price = 200 (5/100) 200 = 200-10 = 190 Selling price including 5% tax = 190+(5/100)190 = 190 + 9.5 = Rs. 199.5 In questions 46 to 65 state whether the statements are true (T) or false (F)....
Read More →Four times a number is a
Question: Four times a number is a ____ increase in the number. Solution: 300% Explanation: Let the number be x Four times of number = 4x 4x is greater than x by = 4x x = 3x Percentage increase in x = 3x/x 100 = 300%...
Read More →Prove the following
Question: 2500 is greater than 500 by ____. Solution: 400% Explanation: 2500 500 = 2000 Percentage increase in 500 to 2500 = (2000/500) 100 = 2000/5 = 400...
Read More →The cost of a tape-recorder is Rs 10,800
Question: The cost of a tape-recorder is Rs 10,800 inclusive of sales tax charged at 8%. The price of the tape-recorder before sales tax was charged is _____. Solution: Rs.10000 Explanation: Cost of tape recorder = Rs.10800 Say, the cost before including the tax = x Therefore, x + x(8/100) = 10800 (100x+8x)/100 = 10800 108x = 1080000 x = 10000...
Read More →Abida bought 100 pens at the rate of Rs 3.50
Question: Abida bought 100 pens at the rate of Rs 3.50 per pen and pays a sales tax of 4%. The total amount paid by Abida is ______. Solution: Rs.364 Explanation: Number of pens = 100 Rate of per pen = Rs.3.50 Cost of 100 pens = 100 3.50 = 350 Sales tax on pen = 4% Total amount paid = 350 (4/100) + 350 = 350 1/25 + 350 = 14 + 350 = 364...
Read More →The cost price of 10 tables is equal
Question: The cost price of 10 tables is equal to the sale price of 5 tables. The profit per cent in this transaction is _____ Solution: 100% Explanation: Say, the cost price of one table is Rs.1 Cost price of 10 tables = Sale price of 5 tables (Given) Sale price of 5 tables profit = cost price of 5 tables = Rs. 5 Profit percentage = Profit/CP 100 = 5/5 100 = 100%...
Read More →The loss per cent on selling 140
Question: The loss per cent on selling 140 geometry boxes at the loss of S.P. of 10 geometry boxes is equal to _____ Solution: 20/3% Explanation: Say, the selling price of one geometry box = Rs.1 So, the selling price of 140 geometry boxes = 1 140 = Rs.140 Selling price of 10 geometry boxes = Rs.10 Loss = Rs. 10 Loss percentage = Loss/CP 100 = 10/(140+10) 100 = 10/150 100 = 20/3%...
Read More →By selling an article for Rs 1,12,000
Question: By selling an article for Rs 1,12,000 a girl gains 40%. The cost price of the article was _______ Solution: Rs.80000 Explanation: Selling price of the article = ₹112000 Gain% = 40% Say, x is the cost price of the article. Since,cost price = selling price profit % on cost price Therefore, Selling price = cost price + profit % on cost price Hence, 112000 = x + x (40/100) 112000 = x + (2/5)x 112000 = 7x/5 x = (112000 5)/7 x = 80000...
Read More →Find the sum of the GP :
Question: Find the sum of the GP : $x(x+y)+x^{2}\left(x^{2}+y^{2}\right)+x^{3}\left(x^{3}+y^{3}\right)+\ldots .$ To $n$ terms Solution: The given expression can be written as $=\left(x^{2}+x y\right)+\left(x^{4}+x^{2} y^{2}\right)+\left(x^{6}+x^{3} y^{3}\right)+\ldots$ To $n$ terms $=\left(x^{2}+x^{4}+x^{6}+\ldots\right.$ to $n$ terms $)+\left(x y+x^{2} y^{2}+x^{3} y^{3}+\ldots\right.$ to $n$ terms $)$ Sum of a G.P. series is represented by the formula $S_{n}=a \frac{r^{n}-1}{r-1}$ when r1. Sn r...
Read More →If the solve the problem
Question: If $x=a \cos \theta, y=b \sin \theta$, show that $\frac{d^{2} y}{d x^{2}}=-\frac{b^{4}}{a^{2} y^{3}}$ Solution: Idea of parametric form of differentiation: If $y=f(\theta)$ and $x=g(\theta)$ i.e. $y$ is a function of $\theta$ and $x$ is also some other function of $\theta$. Then $\mathrm{dy} / \mathrm{d} \theta=\mathrm{f}^{\prime}(\theta)$ and $\mathrm{dx} / \mathrm{d} \theta=\mathrm{g}^{\prime}(\theta)$ We can write : $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\frac{d y}{d \theta}}{\frac{...
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