eSaral - JEE | NEET | Class 8-10 Exam Prep App

Find x, if

Question: Find $x$, if $\left[\begin{array}{lll}x -5 -1\end{array}\right]\left[\begin{array}{lll}1 0 2 \\ 0 2 1 \\ 2 0 3\end{array}\right]\left[\begin{array}{l}x \\ 4 \\ 1\end{array}\right]=O$ Solution: We have: $\left[\begin{array}{lll}x -5 -1\end{array}\right]\left[\begin{array}{lll}1 0 2 \\ 0 2 1 \\ 2 0 3\end{array}\right]\left[\begin{array}{l}x \\ 4 \\ 1\end{array}\right]=O$ $\Rightarrow[x+0-2 \quad 0-10+0 \quad 2 x-5-3]\left[\begin{array}{l}x \\ 4 \\ 1\end{array}\right]=O$ $\Rightarrow\left...

A small candle, 2.5 cm in size is placed at 27 cm in front of a concave mirror of radius of curvature 36 cm.

Question: A small candle, 2.5 cm in size is placed at 27 cm in front of a concave mirror of radius of curvature 36 cm. At what distance from the mirror should a screen be placed in order to obtain a sharp image? Describe the nature and size of the image. If the candle is moved closer to the mirror, how would the screen have to be moved? Solution: Size of the candle,h= 2.5 cm Image size =h Object distance,u= 27 cm Radius of curvature of the concave mirror,R= 36 cm Focal length of the concave mirr...

Why does boron trifluoride behave as a Lewis acid?

Question: Why does boron trifluoride behave as a Lewis acid? Solution: The electric configuration of boron isns2np1. It has three electrons in its valence shell. Thus, it can form only three covalent bonds. This means that there are only six electrons around boron and its octet remains incomplete. When one atom of boron combines with three fluorine atoms, its octet remains incomplete. Hence, boron trifluoride remains electron-deficient and acts as a Lewis acid....

Find a if the coefficients of x2 and x3 in the expansion of (3 + ax)9 are equal.

Question: Find $a$ if the coefficients of $x^{2}$ and $x^{3}$ in the expansion of $(3+a x)^{9}$ are equal. Solution: It is known that $(r+1)^{\text {th }}$ term, $\left(T_{r+1}\right)$, in the binomial expansion of $(a+b)^{n}$ is given by $T_{r+1}={ }^{n} C_{r} a^{n-t} b^{t}$. Assuming that $x^{2}$ occurs in the $(r+1)^{\text {th }}$ term in the expansion of $(3+a x)^{9}$, we obtain $\mathrm{T}_{\mathrm{r}+1}={ }^{9} \mathrm{C}_{\mathrm{r}}(3)^{9-t}(\mathrm{ax})^{\mathrm{r}}={ }^{9} \mathrm{C}_{...

If, show that

Question: If $A=\left[\begin{array}{rr}3 1 \\ -1 2\end{array}\right]$, show that $A^{2}-5 A+7 I=O$ Solution: It is given that $A=\left[\begin{array}{rr}3 1 \\ -1 2\end{array}\right]$ $\therefore A^{2}=A \cdot A=\left[\begin{array}{rr}3 1 \\ -1 2\end{array}\right]\left[\begin{array}{rr}3 1 \\ -1 2\end{array}\right]$ $=\left[\begin{array}{lr}3(3)+1(-1) 3(1)+1(2) \\ -1(3)+2(-1) -1(1)+2(2)\end{array}\right]$ $=\left[\begin{array}{lr}9-1 3+2 \\ -3-2 -1+4\end{array}\right]=\left[\begin{array}{rr}8 5 \...

For what values of

Question: For what values of $x,\left[\begin{array}{lll}1 2 1\end{array}\right]\left[\begin{array}{lll}1 2 0 \\ 2 0 1 \\ 1 0 2\end{array}\right]\left[\begin{array}{l}0 \\ 2 \\ x\end{array}\right]=O ?$ Solution: We have: $\left[\begin{array}{lll}1 2 1\end{array}\right]\left[\begin{array}{lll}1 2 0 \\ 2 0 1 \\ 1 0 2\end{array}\right]\left[\begin{array}{l}0 \\ 2 \\ x\end{array}\right]=0$ $\Rightarrow\left[\begin{array}{lll}1+4+1 2+0+0 0+2+2\end{array}\right]\left[\begin{array}{l}0 \\ 2 \\ x\end{arr...

Answer the following questions:

Question: Answer the following questions: (a)Long distance radio broadcasts use short-wave bands. Why? (b)It is necessary to use satellites for long distance TV transmission. Why? (c)Optical and radio telescopes are built on the ground but X-ray astronomy is possible only from satellites orbiting the earth. Why? (d)The small ozone layer on top of the stratosphere is crucial for human survival. Why? (e)If the earth did not have an atmosphere, would its average surface temperature be higher or low...

How can you explain higher stability of

Question: How can you explain higher stability of $\mathrm{BCl}_{3}$ as compared to $\mathrm{TICl}_{3}$ ? Solution: Boron and thallium belong to group 13 of the periodic table.In this group, the +1 oxidation state becomes more stable on moving down the group. BCl3is more stable than TlCl3 because the +3 oxidation state of B is more stable than the +3 oxidation state of Tl. In Tl, the +3 state is highly oxidising and it reverts back to the more stable +1 state....

Find a, b and n in the expansion of (a + b)n if the first three terms of the expansion are 729,

Question: Find $a, b$ and $n$ in the expansion of $(a+b)^{n}$ if the first three terms of the expansion are 729,7290 and 30375 , respectively. Solution: It is known that $(r+1)^{\text {th }}$ term, $\left(T_{r+1}\right)$, in the binomial expansion of $(a+b)^{n}$ is given by $\mathrm{T}_{r+1}={ }^{n} \mathrm{C}_{r} \mathrm{a}^{n-t} \mathrm{~b}^{t}$. The first three terms of the expansion are given as 729,7290 , and 30375 respectively. Therefore, we obtain $T_{1}={ }^{n} C_{0} a^{n-0} b^{0}=a^{n}=...

Given below are some famous numbers associated with electromagnetic radiations in different contexts in physics.

Question: Given below are some famous numbers associated with electromagnetic radiations in different contexts in physics. State the part of the electromagnetic spectrum to which each belongs. (a)21 cm (wavelength emitted by atomic hydrogen in interstellar space). (b)1057 MHz (frequency of radiation arising from two close energy levels in hydrogen; known as Lamb shift). (c)2.7 K [temperature associated with the isotropic radiation filling all space-thought to be a relic of the big-bang origin of...

Find the values of x, y, z if the matrix satisfy the equation

Question: Find the values of $x, y, z$ if the matrix $A=\left[\begin{array}{ccc}0 2 y z \\ x y -z \\ x -y z\end{array}\right]$ satisfy the equation $A^{\prime} A=I$ Solution: It is given that $A=\left[\begin{array}{ccc}0 2 y z \\ x y -z \\ x -y z\end{array}\right]$ $\therefore A^{\prime}=\left[\begin{array}{ccc}0 x x \\ 2 y y -y \\ z -z z\end{array}\right]$ Now, $A^{\prime} A=I$ $\Rightarrow\left[\begin{array}{ccc}0 x x \\ 2 y y -y \\ z -z z\end{array}\right]\left[\begin{array}{ccc}0 2 y z \\ x ...

Discuss the pattern of variation in the oxidation states of

Question: Discuss the pattern of variation in the oxidation states of (i) B to Tl and (ii) C to Pb. Solution: (i)B to Tl The electric configuration of group 13 elements isns2np1. Therefore, the most common oxidation state exhibited by them should be +3. However, it is only boron and aluminium which practically show the +3 oxidation state. The remaining elements, i.e., Ga, In, Tl, show both the +1 and +3 oxidation states. On moving down the group, the +1 state becomes more stable. For example, Tl...

Use the formula

Question: Use the formula mT= 0.29 cm K to obtain the characteristic temperature ranges for different parts of the electromagnetic spectrum. What do the numbers that you obtain tell you? Solution: A body at a particular temperature produces a continous spectrum of wavelengths. In case of a black body, the wavelength corresponding to maximum intensity of radiation is given according to Plancks law. It can be given by the relation, $\lambda_{\mathrm{m}}=\frac{0.29}{\mathrm{~T}} \mathrm{~cm} \mathr...

Which one of the alkaline earth metal carbonates is thermally the most stable?

Question: Which one of the alkaline earth metal carbonates is thermally the most stable? (a) MgCO3 (b) CaCO3 (c) SrCO3 (d) BaCO3 Solution: Thermal stability increases with the increase in the size of the cation present in the carbonate. The increasing order of the cationic size of the given alkaline earth metals is Mg Ca Sr Ba Hence, the increasing order of the thermal stability of the given alkaline earth metal carbonates is $\mathrm{MgCO}_{3}\mathrm{CaCO}_{3}\mathrm{SrCO}_{3}\mathrm{BaCO}_{3}$...

Find a positive value of m for which the coefficient of x2 in the expansion

Question: Find a positive value of $m$ for which the coefficient of $x^{2}$ in the expansion $(1+x)^{m}$ is 6 Solution: It is known that $(r+1)^{\text {th }}$ term, $\left(T_{r+1}\right)$, in the binomial expansion of $(a+b)^{n}$ is given by $\mathrm{T}_{\mathrm{r}+1}={ }^{n} \mathrm{C}_{\mathrm{r}} \mathrm{a}^{\mathrm{n}-\mathrm{r}} \mathrm{b}^{\mathrm{t}}$. Assuming that $x^{2}$ occurs in the $(r+1)^{\text {th }}$ term of the expansion $(1+x)^{m}$, we obtain $\mathrm{T}_{\mathrm{r}+1}={ }^{\ma...

About 5% of the power of a 100 W light bulb is converted to visible radiation.

Question: About 5% of the power of a 100 W light bulb is converted to visible radiation. What is the average intensity of visible radiation (a)at a distance of 1 m from the bulb? (b)at a distance of 10 m? Assume that the radiation is emitted isotropically and neglect reflection. Solution: Power rating of bulb,P= 100 W It is given that about 5% of its power is converted into visible radiation. $\therefore$ Power of visible radiation, $P^{\prime}=\frac{5}{100} \times 100=5 \mathrm{~W}$ Hence, the ...

Which one of the following alkali metals gives hydrated salts?

Question: Which one of the following alkali metals gives hydrated salts? (a) Li (b) Na (c) K (d) Cs Solution: Smaller the size of an ion, the more highly is it hydrated. Among the given alkali metals, Li is the smallest in size. Also, it has the highest charge density and highest polarising power. Hence, it attracts water molecules more strongly than the other alkali metals. As a result, it forms hydrated salts such as $\mathrm{LiCl} .2 \mathrm{H}_{2} \mathrm{O}$. The other alkali metals are lar...

Show that the matrix is symmetric or skew symmetric according as A is symmetric or skew symmetric.

Question: Show that the matrix $B^{\prime} A B$ is symmetric or skew symmetric according as $A$ is symmetric or skew symmetric. Solution: We suppose thatAis a symmetric matrix, then (1) Consider $\left(B^{\prime} A B\right)^{\prime}=\left\{B^{\prime}(A B)\right\}^{\prime}$ $\begin{array}{ll}=(A B)^{\prime}\left(B^{\prime}\right)^{\prime} {\left[(A B)^{\prime}=B^{\prime} A^{\prime}\right]} \\ =B^{\prime} A^{\prime}(B) {\left[\left(B^{\prime}\right)^{\prime}=B\right]}\end{array}$ $=B^{\prime}\left...

Which of the alkali metal is having least melting point?

Question: Which of the alkali metal is having least melting point? (a) Na (b) K (c) Rb (d) Cs Solution: Atomic size increases as we move down the alkali group. As a result, the binding energies of their atoms in the crystal lattice decrease. Also, the strength of metallic bonds decreases on moving down a group in the periodic table. This causes a decrease in the melting point. Among the given metals, Cs is the largest and has the least melting point....

Prove that the coefficient of xn in the expansion of (1 + x)2n is twice the coefficient of xn in the expansion of (1 + x)2n–1 .

Question: Prove that the coefinicient of $x^{n}$ in the expansion of $(1+x)^{2 n}$ is twice the coefficient of $x^{x}$ in the expansion of $(1+x)^{2 n-1}$ Solution: It is known that $(r+1)^{\text {th }}$ term $_{,}\left(T_{r+1}\right)$, in the binomial expansion of $(a+b)^{n}$ is given by $T_{r+1}={ }^{n} C_{r} a^{n-t} b^{t}$. Assuming that $x^{n}$ occurs in the $(r+1)^{\text {th }}$ term of the expansion of $(1+x)^{2 n}$, we obtain $\mathrm{T}_{\mathrm{r}+1}={ }^{2 \mathrm{n}} \mathrm{C}_{\math...

Suppose that the electric field part of an electromagnetic wave in vacuum

Question: Suppose that the electric field part of an electromagnetic wave in vacuum is $E=\{(3.1 \mathrm{~N} / \mathrm{C}) \cos [(1.8 \mathrm{rad} / \mathrm{m}) y+(5.4 \times$ $\left.\left.\left.10^{6} \mathrm{rad} / \mathrm{s}\right) t\right]\right\} \hat{i}$. (a)What is the direction of propagation? (b)What is the wavelength ? (c)What is the frequency? (d)What is the amplitude of the magnetic field part of the wave? (e)Write an expression for the magnetic field part of the wave. Solution: (a) ...

How would you explain the following observations?

Question: How would you explain the following observations? (i) BeO is almost insoluble but BeSO4in soluble in water, (ii) BaO is soluble but BaSO4is insoluble in water, (iii) LiI is more soluble than KI in ethanol. Solution: (i) $\mathrm{BeO}$ is almost insoluble in water and $\mathrm{BeSO}_{4}$ is soluble in water. $\mathrm{Be}^{2+}$ is a small cation with a high polarising power and $\mathrm{O}^{2-}$ is a small anion. The size compatibility of $\mathrm{Be}^{2+}$ and $\mathrm{O}^{2-}$ is high....

Write balanced equations for reactions between

Question: Write balanced equations for reactions between (a) Na2O2and water (b) KO2and water (c) Na2O and CO2 Solution: (a)The balanced chemical equation for the reaction between Na2O2and water is: $2 \mathrm{Na}_{2} \mathrm{O}_{2(s)}+2 \mathrm{H}_{2} \mathrm{O}_{(i)} \longrightarrow 4 \mathrm{NaOH}_{(a q)}+\mathrm{O}_{2(a q)}$ (b)The balanced chemical equation for the reaction between KO2and water is: $2 \mathrm{KO}_{2(s)}+2 \mathrm{H}_{2} \mathrm{O}_{(i)} \longrightarrow 2 \mathrm{KOH}_{(a q)}...

If A and B are symmetric matrices, prove that AB − BA is a skew symmetric matrix.

Question: IfAandBare symmetric matrices, prove thatABBAis a skew symmetric matrix. Solution: It is given thatAandBare symmetric matrices. Therefore, we have: $A^{\prime}=A$ and $B^{\prime}=B$ ....(1) Now, $(A B-B A)^{\prime}=(A B)^{\prime}-(B A)^{\prime} \quad\left[(A-B)^{\prime}=A^{\prime}-B^{\prime}\right]$ $\begin{array}{ll}=B^{\prime} A^{\prime}-A^{\prime} B^{\prime} {\left[(A B)^{\prime}=B^{\prime} A^{\prime}\right]} \\ =B A-A B {[U \operatorname{sing}(1)]} \\ =-(A B-B A) \end{array}$ $\the...

The coefficients of the (r – 1)th, rth and (r + 1)th terms in the expansion of

Question: The coefiicients of the $(r-1)^{\text {th }}, r^{\text {th }}$ and $(r+1)^{\text {th }}$ terms in the expansion of $(x+1)^{n}$ are in the ratio $1: 3.5$. Find $n$ and $r$. Solution: It is known that $(k+1)^{\text {th }}$ term, $\left(T_{k+1}\right)$, in the binomial expansion of $(a+b)^{n}$ is given by $T_{k+1}={ }^{n} C_{k} a^{n-k} b^{k}$. Therefore, $(r-1)^{\text {th }}$ term in the expansion of $(x+1)^{n}$ is $\mathrm{T}_{r-1}={ }^{n} \mathrm{C}_{r-2}(\mathrm{x})^{n-(r-2)}(1)^{(r-2)...

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