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Question: If $y=x \sin (a+y)$, prove that $\frac{d y}{d x}=\frac{\sin ^{2}(a+y)}{\sin (a+y)-y \cos (a+y)}$ Solution: We are given with an equation $y=x \sin (a+y)$, we have to prove that $\frac{d y}{d x}=\frac{\sin ^{2}(a+y)}{\sin (a+y)-y \cos (a+y)}$ by using the given equation we will first find the value of $\frac{d y}{d x}$ and we will put this in the equation we have to prove, so by differentiating the equation on both sides with respect to $x$, we get, $\frac{d y}{d x}=(1) \sin (a+y)+x \co...
Read More →The increasing order of reactivity of the metals is :
Question: The increasing order of reactivity of the metals is : Hg Ca Na K. Properties of the elements are given below. Where would you locate the following elements in the periodic table ? (a) A soft metal stored under kerosene (b) An element with variable (more than one) valency stored under water (c) An element which is tetravalent and forms the basis of organic chemistry (d) An element which is an inert gas with atomic number 2 (e) An element whose thin oxide layer is used to make other elem...
Read More →Meenakshi sells a pen for Rs 54 and loses
Question: Meenakshi sells a pen for Rs 54 and loses $\frac{1}{10}$ of her outlay. Find (i) the cost price of the pen, and (ii) the loss per cent. Solution: SP of the pen $=$ Rs. 54 Let Rs $x$ be the CP of the pen. Loss $=$ Rs. $\frac{x}{10}$ $\mathrm{SP}=\mathrm{CP}-$ Loss $=x-\frac{x}{10}$ $=$ Rs. $\frac{9 x}{10}$ Now, we have $\frac{9 \mathrm{x}}{10}=54$ $\Rightarrow x=54 \times \frac{10}{9}$ $\Rightarrow \mathrm{x}=60$ $\therefore$ CP of the pen $=$ Rs. 60 Now, loss $=\frac{\mathrm{x}}{10}$ N...
Read More →Find the domain and the range of each of the following real
Question: Find the domain and the range of each of the following real function: $f(x)=\frac{1}{\sqrt{x^{2}-1}}$ Solution: Given: $f(x)=\frac{1}{\sqrt{x^{2}-1}}$ Need to find: Where the functions are defined. The condition for the function to be defined, $x^{2}-10$ $\Rightarrow x^{2}1$ $\Rightarrow x1$ So, the domain of the function is the set of all the real numbers greater than 1 . The domain of the function, $\mathrm{D}_{\mathrm{f}(\mathrm{x})}=(1, \infty)$. Now putting any value of $x$ within...
Read More →Identify the elements with the following property
Question: Identify the elements with the following property and arrange them in increasing order of their reactivity (a) An metal which is soft and reactive (b) The metal which is an important constituent of limestone (c) The metal which exists in liquid state at room temperature Solution: (a) The metal may be either Na or K. (b) The metal is Ca. (c) The metal is Hg....
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Question: If $x y \log (x+y)=1$, prove that $\frac{d y}{d x}=\frac{y\left(x^{2} y+x+y\right)}{x\left(x y^{2}+x+y\right)}$. Solution: We are given with an equation $x y \log (x+y)=1$, we have to prove that $\frac{d y}{d x}=\frac{y\left(x^{2} y+x+y\right)}{x\left(y^{2} x+x+y\right)}$ by using the given equation we will first find the value of $\frac{d y}{d x}$ and we will put this in the equation we have to prove, so by differentiating the equation on both sides with respect to $x$, we get, By usi...
Read More →Arrange the following elements
Question: Arrange the following elements in the increasing order of their metallic character. , Mg, Ca, K, Ge, Ga Solution: In general, the metallic character increases down the group and decreases along a period. From the relative positions of the elements in the periodic table, the increasing order of metallic character is : Ge Ga Mg Ca K....
Read More →Write the formula of the compound formed
Question: Write the formula of the compound formed when the element A (atomic number 19) combines with the element B (atomic number 17). Draw its electronic dot structure. ^What is the nature of the bond formed ? (CBSE 2013) Solution: Electronic configuration of element A (Z = 19) is 2, 8, 8, 1 Electronic configuration of element B(Z = 17) is 2, 8, 7. The element A has one valence electron while the element B has seven electrons in its valence shell. One electron gets transferred from the atom o...
Read More →Find the domain and the range of each of the following real
Question: Find the domain and the range of each of the following real function: $f(x)=\sqrt{\frac{x-5}{3-x}}$ Solution: $f(x)=\sqrt{\frac{x-5}{3-x}}$ Need to find: Where the functions are defined The condition for the function to be defined, $3-x0$ $\Rightarrow^{x3}$ So, the domain of the function is the set of all the real numbers lesser than 3 . The domain of the function, $\mathrm{D} \mathrm{f}(x)=(-\infty, 3)$. The condition for the range of the function to be defined, $x-5 \geq 0 \^{3-x}0$ ...
Read More →Identify and name the metals out of the following
Question: Identify and name the metals out of the following elements whose electronic configurations are given below : (a) 2, 8, 2 (b) 2, 8, 1 (c) 2, 8, 7 (d) 2, 1. Solution: (a) Electronic configuration : 2, 8, 2 (Mg is a metal) (b) Electronic configuration : 2, 8, 1 (Na is a metal) (c) Electronic configuration : 2, 8, 7 (Cl is a non-metal) (d) Electronic configuration : 2, 1 (Li is a metal)...
Read More →A dealer sold a camera for Rs 1080 gaining
Question: A dealer sold a camera for Rs 1080 gaining $\frac{1}{8}$ of its cost price. Find (i) the cost price of the camera, and (ii) the gain per cent earned by the dealer. Solution: SP of the camera $=$ Rs. 1080 Let Rs $x$ be the CP. Gain $=$ Rs. $\frac{1}{8} x \quad \ldots(\mathrm{i})$ Also, gain $=\mathrm{SP}-\mathrm{CP}$ $=$ Rs. $(1080-x) \quad \ldots$ (ii) From (i) and (ii), we have : $\frac{1}{8} x=1080-x$ $\Rightarrow x=8640-8 x$ $\Rightarrow 9 x=8640$ $\Rightarrow \mathrm{x}=960$ $\ther...
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Question: If $\tan ^{-1}\left(\frac{x^{2}-y^{2}}{x^{2}+y^{2}}\right)=a$, prove that $\frac{d y}{d x}=\frac{x}{y} \frac{(1-\tan a)}{(1+\tan a)}$ Solution: We are given with an equation $\tan ^{-1}\left(\frac{x^{2}-y^{2}}{x^{2}+y^{2}}\right)=a$, we have to prove that $\frac{d y}{d x}=\frac{x}{y} \frac{(1-\tan a)}{(1+\tan a)}$ by using the given equation we will first find the value of $\frac{\mathrm{dy}}{\mathrm{dx}}$ and we will put this in the equation we have to prove, so by differentiating the...
Read More →Arrange the following elements in increasing
Question: Arrange the following elements in increasing order of their atomic radii. Solution: (a) All the elements belong to the same period (second). The atomic size or radius decreases along a period. Therefore, increasing order of atomic radii is : F N Be Li (b) The listed elements are present in group 17 (halogen family). The atomic size or radius increases down a group. Therefore, the correct increasing order of atomic radii is : Cl Br I At....
Read More →Compare the radii of two species X and Y.
Question: Compare the radii of two species X and Y. Give reasons for your answer. (a) X has 12 protons and 12 electrons (b) Y has 12 protons and 10 electrons Solution: The available information makes it clear that : (a) Species X with equal number of protons (12) and electrons (12) is an atom. (b) Species Y with two electrons less (10) than the number of protons (12), is a divalent cation y2+of species X. Therefore, the radius of species X is more as compared to that of y2+. Note : The radius of...
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Question: If $\sec \left(\frac{x+y}{x-y}\right)=a, p$ rove that $\frac{d y}{d x}=\frac{y}{x}$ Solution: We are given with an equation $\sec \left(\frac{x+y}{x-y}\right)=a$, we have to prove that $\frac{d y}{d x}=\frac{y}{x}$ by using the given equation we will first find the value of $\frac{\mathrm{dy}}{\mathrm{dx}}$ and we will put this in the equation we have to prove, so by differentiating the equation on both sides with respect to $x$, we get, $\sec \left(\frac{x+y}{x-y}\right) \tan \left(\f...
Read More →Find the domain and the range of each of the following real
Question: Find the domain and the range of each of the following real function: $f(x)=\sqrt{3 x-5}$ Solution: Given: $f(x)=\sqrt{3 x-5}$ Need to find: Where the functions are defined. The condition for the function to be defined, $3 x-5 \geq 0$ $\Rightarrow x \geq \frac{5}{3}$ So, the domain of the function is the set of all the real numbers greater than equals to $\frac{5}{3}$. The domain of the function, $\mathrm{D}_{\mathrm{f}(\mathrm{x})}=\left[\frac{5}{3}, \infty\right)$. Putting $\frac{5}{...
Read More →If an element X is placed in group 14,
Question: If an element X is placed in group 14, what will be the formula and the nature of bonding of its chloride ? Solution: The element X present in group 14 has four valence electrons in its atom. It can complete its octet by sharing four valence electrons with the electrons of other atoms. Therefore, it will form covalent bonds with the four atoms of chlorine. The formula of the chloride of the element X is....
Read More →Three elements A, B and C have 3, 4 and 2
Question: Three elements A, B and C have 3, 4 and 2 electrons respectively in their outermost shell. Give the group number to which they belong in the Modern Periodic Table. Also, give their valencies. Solution: The electrons present in the outermost shell are also known as valence electrons. The desired information is given :...
Read More →A man purchased some eggs at 3 for ₹ 16 and sold them at 5 for ₹ 36.
Question: A man purchased some eggs at 3 for ₹ 16 and sold them at 5 for ₹ 36. Thus, he gained ₹ 168 in all. How many eggs did he purchase? Solution: Let the number of eggs purchased be $x$. CP of 3 eggs $=₹ 16$ $\therefore \mathrm{CP}$ of $1 \mathrm{egg}=₹ \frac{16}{3}$ $\Rightarrow \mathrm{CP}$ of $x$ eggs $=₹ \frac{16}{3} x$ SP of 5 eggs $=₹ 36$ $\therefore \mathrm{SP}$ of $1 \mathrm{egg}=₹ \frac{36}{5}$ $\Rightarrow$ SP of $x$ eggs $=₹ \frac{36}{5} x$ Gain $=\mathrm{SP}-\mathrm{CP}=₹ 168$ $\...
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Question: If $\log \sqrt{x^{2}+y^{2}}=\tan ^{-1}\left(\frac{y}{x}\right)$, prove that $\frac{d y}{d x}=\frac{x+y}{x-y}$. Solution: We are given with an equation $\log \sqrt{x^{2}+y^{2}}=\tan ^{-1}\left(\frac{y}{x}\right)$, we have to prove that $\frac{d y}{d x}=\frac{y+x}{y-x}$ by using the given equation we will first find the value of $\frac{d y}{d x}$ and we will put this in the equation we have to prove, so by differentiating the equation on both sides with respect to $x$, we get, $\log \lef...
Read More →Write the formulae of chlorides of Eka-silicon
Question: Write the formulae of chlorides of Eka-silicon and Eka-aluminium, the ^elements predicted by Mendeleev. Solution: Eka-aluminium represents gallium (Ga) with valency three and Eka-silicon is for germanium (Ge) with valency four. The formulae of their respective chlorides are GaCl3and GeCl4....
Read More →A man bought apples at 10 for ₹ 75 and sold them at ₹ 75 per dozen.
Question: A man bought apples at 10 for ₹ 75 and sold them at ₹ 75 per dozen. Find his loss per cent. Solution: LCM of 10 and $12=60$ Let the number of apples bought be 60 . CP of 10 oranges $=₹ 75$ $\therefore$ CP of 1 orange $=₹ \frac{75}{10}$ $\Rightarrow$ CP of 60 orange $=60 \times \frac{75}{10}=₹ 450$ SP of 12 oranges $=₹ 75$ $\therefore$ SP of 1 orange $=₹ \frac{75}{12}$ $\Rightarrow$ SP of 60 oranges $=60 \times \frac{75}{12}=₹ 375$ Here, CP of 60 oranges $$ SP of 60 oranges. Loss $=$ CP...
Read More →“Hydrogen occupies a unique position
Question: Hydrogen occupies a unique position in Modern Periodic Table. Justify the statement. Solution: The position of the element hydrogen is still not clear even in the Modern Periodic Table. In electronic configuration, it resembles alkali metals of group 1. All of them have only one electron in the valence shell. Actually, hydrogen has only one shell (K-shell) which has one electron. However, it is not a metal whereas alkali-metals are metallic in nature. In characteristics, it resembles h...
Read More →Find the domain and the range of each of the following real function: f(x)
Question: Find the domain and the range of each of the following real function: f(x) $=\frac{a x-b}{c x-d}$ Solution: Given: $\mathrm{f}(\mathrm{x})=\frac{a x-b}{c x-d}$ Need to find: Where the functions are defined. Let, $f(x)=\frac{a x-b}{c x-d}=y$ ......(1) To find the domain of the function f(x) we need to equate the denominator of the function to 0. Therefore, $c x-d=0$ $\Rightarrow \mathrm{x}=\frac{d}{c}$ It means that the denominator is zero when $x=\frac{d}{c}$ So, the domain of the func...
Read More →In Mendeleev’s Periodic Table,
Question: In Mendeleevs Periodic Table, the elements were arranged in the increasing order of their atomic masses. However, cobalt with atomic mass of 58.93 amu was placed before nickel having an atomic mass of 58.71 amu. Give reason for the same. Solution: This is regarded as a defect in the Mendeleevs periodic table. However, the main reason for making the arrangement was that the elements with similar characteristics must be grouped together. In this case, Cobalt (Co) should be in the company...
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