datavorous/entrance-exam-dataset huggingface dataset:
What is the standard cell potential for the reaction with $K=1$ (equilibrium constant)

Options:
A One
B Zero
C 2.303
D Infinity

The correct answer is:
Zero

Solution:
The relationship between cell potential and the equilibrium constant. We know.
$\Delta G=-R T \ln k=-n F E^{\circ}$
$\Delta G=-2.303 \mathrm{RT} \log k=-n F E^{\circ}$
$\log K_f=\frac{n F \times E^{\circ}}{R T \times 2.303}$
$\log (\mathrm{l})=\frac{n}{0.059} \times E^{\circ} \quad(n=1)$
$\Rightarrow \quad 0 \times 0.059=E^{\circ} \Rightarrow E^{\circ}=$ zero

Topic: Chemistry, Electrochemistry, TS EAMCET, TS EAMCET 2020 (11 Sep Shift 1)
What is the standard deviation of $7,9,11,13,15 ?$

Options:
A $2.4$
B $2.5$
C $2.7$
D $2.8$

The correct answer is:
$2.8$

Solution:
Given observations are $7,9,11,13,15$
$\bar{x}=\frac{7+9+11+13+15}{5}=\frac{55}{5}=11$
Now, Variance $=\frac{1}{n} \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}$
$=\frac{(7-11)^{2}+(9-11)^{2}+(11-11)^{2}+(13-11)^{2}+(15-11)}{2}$
$\frac{16+4+4+16}{5}=8$
$\mathrm{SD}=\sqrt{\mathrm{v}}=\sqrt{8}=2 \sqrt{2}=2.8$
$(\because \sqrt{2}=1.414)$

Topic: Mathematics, Statistics, NDA, NDA 2012 (Phase 2)
What is the standard deviation of numbers $7,9,11,13,15$

Options:
A 22
B 26
C $2.4$
D $2.8$

The correct answer is:
$2.8$

Solution:
Mean of given numbers $=\frac{7+9+11+13+15}{5}=\frac{55}{5}=11$
Now
$\mathrm{SD}=\sqrt{\frac{(7-11)^{2}+(9-11)^{2}+(11-11)^{2}+(13-11)^{2}+(15-11)^{2}}{5}}$
$=\sqrt{\frac{16+4+0+4+16}{5}}=\sqrt{8}=2.8($ aprox $)$

Topic: Mathematics, Statistics, NDA, NDA 2009 (Phase 1)
What is the standard electrode potential for the half-cell reaction involving fluorine?

Options:
A Highest
B Zero
C Negative
D Lowest

The correct answer is:
Highest

Solution:
The paragraph mentions that the standard electrode potential for fluorine is the highest indicating that fluorine gas (F2 ) has the maximum tendency to get reduced to fluoride ions (F- ) and therefore fluorine gas is the strongest oxidising agent and fluoride ion is the weakest reducing agent.

Topic: Chemistry, Electrochemistry, NEET
What is the standard electrode potential of copper ion (Cu2+) compared to hydrogen ion (H+)?

Options:
A Cu2+ has a higher standard electrode potential.
B Cu2+ has a lower standard electrode potential.
C Both have the same standard electrode potential.
D None of the above

The correct answer is:
Cu2+ has a higher standard electrode potential.

Solution:
The standard electrode potential of copper ion (Cu2+) is positive, indicating that it has a higher potential compared to hydrogen ion (H+).

Topic: Chemistry, Electrochemistry, NEET
What is the standard enthalpy of combustion defined as?

Options:
A The enthalpy change when a substance undergoes combustion in any state
B The enthalpy change when a substance undergoes combustion in its standard state at any temperature
C The enthalpy change when a substance undergoes combustion in its standard state at a specified temperature
D The enthalpy change when a substance undergoes combustion in its gaseous state

The correct answer is:
The enthalpy change when a substance undergoes combustion in its standard state at a specified temperature

Solution:
The standard enthalpy of combustion is defined as the enthalpy change per mole (or per unit amount) of a substance when it undergoes combustion, with all the reactants and products being in their standard states at a specified temperature.

Topic: Chemistry, Thermodynamics (C), NEET
What is the standard enthalpy of formation, ∆fH , a special case of?

Options:
A Standard molar enthalpy of fusion
B Standard molar enthalpy of vaporization
C Standard molar enthalpy of sublimation
D Standard molar enthalpy of reaction

The correct answer is:
Standard molar enthalpy of reaction

Solution:
The standard enthalpy of formation, ∆fH , is a special case of the standard molar enthalpy of reaction, ∆rH, where one mole of a compound is formed from its constituent elements.

Topic: Chemistry, Thermodynamics (C), NEET
What is the standard enthalpy of formation of an element in its reference state?

Options:
A Zero
B Positive
C Negative
D Undefined

The correct answer is:
Zero

Solution:
By convention, the standard enthalpy of formation of an element in its reference state is taken as zero.

Topic: Chemistry, Thermodynamics (C), NEET
What is the standard enthalpy of reaction (in kJ) when two moles of $\mathrm{Fe}_2 \mathrm{O}_3(s)$ reacts with $\mathrm{H}_2$ gas to give Fe metal?
$\Delta H_f^{\circ}$ of $\mathrm{Fe}_2 \mathrm{O}_3(s)$ and $\mathrm{H}_2 \mathrm{O}(l)$ are -824.2 and $-285.83 \mathrm{~kJ} \mathrm{~mol}^{-1}$ respectively.

Options:
A $-66.58$
B $-33.3$
C $-538.37$
D $-1110.03$

The correct answer is:
$-66.58$

Solution:
When $\mathrm{Fe}_2 \mathrm{O}_3(s)$ reacts with $\mathrm{H}_2(g)$, it will give $\mathrm{Fe}$
and $\mathrm{H}_2 \mathrm{O}$ as follows: $\mathrm{Fe}_2 \mathrm{O}_3+3 \mathrm{H}_2 \rightarrow 2 \mathrm{Fe}+3 \mathrm{H}_2 \mathrm{O}$ When 2 moles of $\mathrm{Fe}_2 \mathrm{O}_3$ are used, the reaction becomes as follows:
$2\left[\mathrm{Fe}_2 \mathrm{O}_3+3 \mathrm{H}_2 \rightarrow 2 \mathrm{Fe}+3 \mathrm{H}_2 \mathrm{O}\right]$
Given, $\Delta H_f^{\circ}$ for $\mathrm{Fe}_2 \mathrm{O}_3(s)=-824.2 \mathrm{~kJ} \mathrm{~mol}^{-1}$ and $\Delta H_f^{\circ}$ for $\mathrm{H}_2 \mathrm{O}(l)=-285.83 \mathrm{~kJ} \mathrm{~mol}^{-1}$
$\because$ Standard enthalpy of reaction
$\left(\Delta H_R^{\circ}\right)=\sum \Delta H_f^{\circ}$ (Product) $-\Sigma \Delta H_f^{\circ}$ (Reacant $)$
and $\left.\begin{array}{c}H_f^{\circ} \text { for } \mathrm{H}_2=0 \\ H_f^{\circ} \text { for } \mathrm{Fe}=0\end{array}\right]$ (as are present in free state)
Thus,
$\begin{aligned} \Delta H_R^{\circ} & =[6 \times-285.83]-[2 \times(-824.2)] \\ \Delta H_R^{\circ} & =(-1714.98)+(1648.4) \\ \Delta H_R^{\circ} & =-66.58 \mathrm{~kJ} / \mathrm{mol}\end{aligned}$
Hence, option (a) is the correct answer.

Topic: Chemistry, Thermodynamics (C), TS EAMCET, TS EAMCET 2019 (06 May Shift 1)
What is the standard enthalpy of reaction?

Options:
A The enthalpy change for a reaction under non-standard conditions
B The enthalpy change for a reaction when all substances are in their standard states
C The enthalpy change for a reaction carried out at high pressure
D The enthalpy change for a reaction at any given temperature

The correct answer is:
The enthalpy change for a reaction when all substances are in their standard states

Solution:
The standard enthalpy of reaction is the enthalpy change for a reaction when all the participating substances are in their standard states.

Topic: Chemistry, Thermodynamics (C), NEET
What is the standard enthalpy of sublimation?

Options:
A The heat change when a solid melts at a constant temperature
B The heat change when a solid vaporizes at a constant temperature
C The heat change when a solid sublimes at a constant temperature
D The heat change when a liquid vaporizes at a constant temperature

The correct answer is:
The heat change when a solid sublimes at a constant temperature

Solution:
The standard enthalpy of sublimation, ∆subH, is the heat change when one mole of a solid substance sublimes at a constant temperature and under standard pressure (1 bar).

Topic: Chemistry, Thermodynamics (C), NEET
What is the standard enthalpy of vaporization (∆vapH) for water?

Options:
A ∆vapH= + 40.79 kJ mol–1
B ∆vapH= – 393.5 kJ mol–1
C ∆vapH= – 2802.0 kJ mol–1
D ∆vapH= 435.0 kJ mol–1

The correct answer is:
∆vapH= + 40.79 kJ mol–1

Solution:
The standard enthalpy of vaporization (∆vapH) for water is ∆vapH = + 40.79 kJ mol–1. This represents the heat required to vaporize one mole of water at its boiling point under standard pressure.

Topic: Chemistry, Thermodynamics (C), NEET
What is the standard free energy change for the cell, having following cell reaction? $2 \mathrm{Ag}_{(\mathrm{aq} .)}^{+}+\mathrm{Cd}_{(\mathrm{s})} \longrightarrow 2 \mathrm{Ag}_{(\mathrm{s})}+\mathrm{Cd}_{(\mathrm{aq})}^{2+}, \mathrm{E}^{\circ} \mathrm{cell}=1 \cdot 20 \mathrm{~V}$

Options:
A $-231 \cdot 6 \mathrm{~kJ}$
B $-160 \cdot 8 \mathrm{~kJ}$
C $-115 \cdot 8 \mathrm{~kJ}$
D $-260 \cdot 8 \mathrm{K} \mathrm{j}$

The correct answer is:
$-231 \cdot 6 \mathrm{~kJ}$

Solution:
$2 \mathrm{Ag}_{(\mathrm{aq})}^{+}+\mathrm{Cd}_{(\mathrm{s})} \longrightarrow 2 \mathrm{Ag}_{(\mathrm{s})}+\mathrm{Cd}_{(\mathrm{aq})}^{2+}$
$\therefore \mathrm{n}=2$
$\Delta \mathrm{G}^{0}=-\mathrm{nFE}_{\mathrm{cell}}^{0}=-2 \times 96500 \times 1.20=-231600 \mathrm{~J}=-231.6 \mathrm{~kJ}$

Topic: Chemistry, Electrochemistry, MHT CET, MHT CET 2020 (15 Oct Shift 2)
What is the standard molar enthalpy of formation?

Options:
A The heat change when a compound is formed from its elements in their most stable states
B The heat change when a compound is dissolved in water
C The heat change when a compound is vaporized
D The heat change when a compound undergoes sublimation

The correct answer is:
The heat change when a compound is formed from its elements in their most stable states

Solution:
The standard molar enthalpy of formation is the heat change when one mole of a compound is formed from its elements in their most stable states of aggregation.

Topic: Chemistry, Thermodynamics (C), NEET
What is the standard potential of cell $\mathrm{Ni}\left|\mathrm{Ni}^{2+}{ }_{(1 \mathrm{M})} \| \mathrm{Cu}^{2+}{ }_{(1 \mathrm{M})}\right| \mathrm{Cu}$ If $\mathrm{E}^{\circ} \mathrm{Cu}=0 \cdot 337 \mathrm{~V}$ and $\mathrm{E}^{\circ} \mathrm{Ni}=-0 \cdot 236 \mathrm{~V}$

Options:
A $0 \cdot 101 \mathrm{~V}$
B $-0 \cdot 136 \mathrm{~V}$
C $0.573 \mathrm{~V}$
D $-0.753 \mathrm{~V}$

The correct answer is:
$0.573 \mathrm{~V}$

Solution:
$\begin{aligned} \mathrm{E}_{\text {ceil }}^{0} &=\mathrm{E}_{\text {cuthode }}^{0}-\mathrm{E}_{\text {anode }}^{0}=\mathrm{E}_{\mathrm{Cu}}^{0}-\mathrm{E}_{\mathrm{Ni}_{\mathrm{i}}}^{0} \\ &=0.337 \mathrm{~V}-(-0.236 \mathrm{~V})=0.573 \mathrm{~V} \end{aligned}$

Topic: Chemistry, Electrochemistry, MHT CET, MHT CET 2020 (13 Oct Shift 1)
What is the standard reduction potential Eo for Fe3+âFe?

Given that:

Fe2++2e-âFe;EFe2+/Feo= -0.47V

Fe3++e-âFe2+;EFe3+/Fe2+o=+0.77V

Options:
A -0.057 V
B +0.30Â V
C -0.30Â V
D +0.057Â V

The correct answer is:
-0.057 V

Solution:
â(i)Fe3+âFe E 1 o ;
'n' factor = 3
Î G 1 â=ââ3âÃFÃ E 1 o
(ii) F e 3+ + e â âF e 2+ E 2 o ;0.77
'n' factor = 1
Î G 2 â=ââ1âÃ0.77Ã f
(iii) F e 2+ +2 e â âFe E 3 o ;0.77
'n' factor = 2
Î G 3 â=â0.47âÃ2Ã f
Î G 1 =âÎ G 2 â+âÎ G 3
â3f E 1 o â=ââ1Ã0.77Ãf+2Ã0.47Ãf
E 1 o â=â â2Ã0.47+0.77 3 â=ââ0.057 V

Topic: Chemistry, Electrochemistry, NEET
What is the standard state (c0) for a solute when measuring concentrations with respect to it?

Options:
A 1 bar
B 1 Pa
C 1 kPa
D 1 molar solution

The correct answer is:
1 molar solution

Solution:
The standard state (c0) for a solute is 1 molar solution, and all concentrations can be measured with respect to it.

Topic: Chemistry, Chemical Equilibrium, NEET
What is the standard state for a pure gas when expressing equilibrium constants as dimensionless quantities?

Options:
A 1 atm
B 1 Pa
C 1 kPa
D 1 bar

The correct answer is:
1 bar

Solution:
For a pure gas, the standard state is 1 bar.

Topic: Chemistry, Chemical Equilibrium, NEET
What is the standard temperature at which data are usually taken for standard enthalpy changes?

Options:
A 0 K
B 25 K
C 273 K
D 298 K

The correct answer is:
298 K

Solution:
Data for standard enthalpy changes are usually taken at 298 K (25°

Topic: Chemistry, Thermodynamics (C), NEET
What is the structure of the vascular bundles in a dicot stem?

Options:
A Concentric
B Radial
C Conjoint
D Closed

The correct answer is:
Conjoint

Solution:
A large number of vascular bundles are arranged in a ring in a dicot stem and they are conjoint, open and with endarch protoxylem.

Topic: Biology, Anatomy of Flowering Plants, NEET
What is the sum of $\sqrt{3}+\frac{1}{\sqrt{3}}+\frac{1}{3 \sqrt{3}}+\ldots . . ?$

Options:
A $\frac{\sqrt{3}}{2}$
B $\frac{3 \sqrt{3}}{2}$
C $\frac{2 \sqrt{3}}{3}$
D $\sqrt{3}$

The correct answer is:
$\frac{3 \sqrt{3}}{2}$

Solution:
Given series is
$$
\sqrt{3}+\frac{1}{\sqrt{3}}+\frac{1}{3 \sqrt{3}}+\ldots \ldots \ldots
$$
Since, $\left(\frac{1}{\sqrt{3}}\right)^{2}=\sqrt{3} \times \frac{1}{3 \sqrt{3}}$
$\Rightarrow$ Given series is a G.P.
$\therefore \quad$ Sum upto $\infty=\frac{a}{1-r}$
where $\mathrm{a}=\sqrt{3}=$ first term and $\mathrm{r}=\frac{1}{3}=$ common ratio
$$
\therefore \quad S_{\infty}=\frac{\sqrt{3}}{1-\frac{1}{3}}=\frac{3 \sqrt{3}}{2}
$$

Topic: Mathematics, Sequences and Series, NDA, NDA 2011 (Phase 1)
What is the sum of all natural numbers between 200 and 400 which are divisible by $7 ?$

Options:
A 6729
B 8712
C 8729
D 9276

The correct answer is:
8729

Solution:
The numbers between 200 and 400 which are divisible by 7, are $203,210,217, \ldots \ldots, 399$
This is an A.P with first term $=\mathrm{a}=203$ and common difference $=d=7$ Now, let number of terms be n. Therefore from the $\mathrm{n}^{\text {th }}$ term of $\mathrm{A} . \mathrm{P}=\mathrm{a}+(\mathrm{n}-1) \mathrm{d}$ we have $399=203+(\mathrm{n}-1) 7$
$\Rightarrow \frac{196}{7}=(n-1) \Rightarrow n=29$
Required sum $=\frac{n}{2}[a+\ell]$ where $\ell=$ last term
Thus, required sum $=\frac{29}{2}[203+399]$
$=\frac{29 \times 602}{2}=8729$

Topic: Mathematics, Sequences and Series, NDA, NDA 2010 (Phase 1)
What is the sum of all the coefficients in the expansion of $(1+x)^{n} ?$

Options:
A $2^{n}$
B $2^{n}-1$
C $2^{n-1}$
D $2(n-1)$

The correct answer is:
$2^{n}$

Solution:
Given expansion is $(1+x)^{\mathrm{n}}$. Put $x=1$, we get Required sum $=(1+1)^{n}=2^{n}$

Topic: Mathematics, Binomial Theorem, NDA, NDA 2010 (Phase 2)
What is the sum of all three-digit numbers that can be formed using all the digits 3,4 and 5 , when repetition of digits is not allowed?

Options:
A 2664
B 3882
C 4044
D 4444

The correct answer is:
2664

Solution:
Number of 3 digit number made from digit 3,4 or 5
and having all distinct digit $=3 !=6$
and sum of such numbers are
$543+534+345+354+435+453=2664$

Topic: Mathematics, Permutation Combination, NDA, NDA 2018 (Phase 2)
What is the sum of first eight terms of the series
$1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\ldots . . ?$

Options:
A $\frac{89}{128}$
B $\frac{57}{384}$
C $\frac{85}{128}$
D None of the above

The correct answer is:
$\frac{85}{128}$

Solution:
$a=1$ $\mathrm{r}=-\frac{1}{2}( < 1)$
$\therefore \quad$ Sum of Ist 8 terms is :-
$\mathrm{S}_{8}=\frac{1\left[1-\left(\frac{-1}{2}\right)^{8}\right]}{1+\frac{1}{2}} \quad\left[\right.$ For $\left.\mathrm{G} . \mathrm{P} . \mathrm{S}_{\mathrm{n}}=\frac{\mathrm{a}\left(1-\mathrm{r}^{\mathrm{n}}\right)}{1-\mathrm{r}}\right]$
$=\frac{\frac{1}{1}-\frac{1}{256}}{\frac{3}{2}}=\frac{85}{128}$

Topic: Mathematics, Sequences and Series, NDA, NDA 2012 (Phase 2)
What is the sum of focal radii of any point on an ellipse equal to?

Options:
A Length oflatusrectum
B Length of major-axis
C Length of minor-axis
D Length of semi-latusrectum

The correct answer is:
Length of major-axis

Solution:
We know sum of the focal distances of a point on the ellipse is constant and is equal to the length of the major axis. Thus, we know that, the sum of focal radii of any point on ellipse is equal to length of major axis

Topic: Mathematics, Ellipse, NDA, NDA 2009 (Phase 1)
What is the sum of $\mathrm{n}$ terms of the series
$\sqrt{2}+\sqrt{8}+\sqrt{18}+\sqrt{32}+\ldots ?$

Options:
A $\frac{n(n-1)}{\sqrt{2}}$
B $\sqrt{2 n}(n+1)$
C $\frac{n(n+1)}{\sqrt{2}}$
D $\frac{\mathrm{n}(\mathrm{n}-1)}{2}$

The correct answer is:
$\frac{n(n+1)}{\sqrt{2}}$

Solution:
$\begin{aligned} & \sqrt{2}+\sqrt{8}+\sqrt{18}+\sqrt{32}+\ldots . . \\ &=\sqrt{2}+2 \sqrt{2}+3 \sqrt{2}+4 \sqrt{2}+\ldots . . \\ & \therefore \mathrm{S}_{\mathrm{n}}=\sqrt{2}[1+2+3+4+\ldots . \mathrm{n} \text { terms }] \\ &=\sqrt{2} \frac{\mathrm{n}(\mathrm{n}+1)}{2}=\frac{\mathrm{n}(\mathrm{n}+1)}{\sqrt{2}} \end{aligned}$

Topic: Mathematics, Sequences and Series, NDA, NDA 2015 (Phase 2)
What is the sum of the coefficients in the expansion of $(1+\mathrm{x})^{\mathrm{n}} ?$

Options:
A $2^{\mathrm{n}}$
B $2^{n}-1$
C $2^{n}+1$
D $\mathrm{n}+1$

The correct answer is:
$2^{\mathrm{n}}$

Solution:
Given expansion is $(1+\mathrm{x})^{\mathrm{n}}$.
Put $\mathrm{x}=1$, we get Sum of coefficient $=2^{\mathrm{n}}$.

Topic: Mathematics, Binomial Theorem, NDA, NDA 2013 (Phase 1)
What is the sum of the coefficients of all the terms in the expansion of $(45 x-49)^{4} ?$

Options:
A -256
B -100
C 100
D 256

The correct answer is:
256

Solution:
Given expansion is $(45 x-49)^{4}$. To find the sum of the coefficients of all the terms in the expansion, we have to put $x=1$ in the expansion. Thus, required sum of coefficients $=(45-49)^{4}$ $=(-4)^{4}=256$

Topic: Mathematics, Binomial Theorem, NDA, NDA 2010 (Phase 1)
What is the sum of the first 50 terms of the series $(1 \times 3)+(3 \times 5)+(5 \times 7)+\ldots . ?$

Options:
A $1,71,650$
B 26600
C 26650
D 26900

The correct answer is:
$1,71,650$

Solution:
The given series is $(1 \times 3)+(3 \times 5)+(5 \times 7)+\ldots \ldots \ldots . .$
Its general term is given by
$\mathrm{T}_{\mathrm{n}}=(2 \mathrm{n}-1)(2 \mathrm{n}+1)=4 \mathrm{n}^{2}-1$
Sum of series $=4 \Sigma \mathrm{n}^{2}-\Sigma 1$
$\mathrm{S}_{\mathrm{n}}=\frac{4 \mathrm{n}(\mathrm{n}+1)(2 \mathrm{n}+1)}{6}-\mathrm{n}$
$\mathrm{S}_{\mathrm{n}}=\mathrm{n}\left[\frac{2\left(2 \mathrm{n}^{2}+3 \mathrm{n}+1\right)}{3}-1\right]$
$\mathrm{S}_{\mathrm{n}}=\mathrm{n}\left[\frac{4 \mathrm{n}^{2}+6 \mathrm{n}+2-3}{3}\right]$
$\mathrm{S}_{\mathrm{n}}=\left[\frac{\mathrm{n}\left(4 \mathrm{n}^{2}+6 \mathrm{n}-1\right)}{3}\right]$
For sum of first 50 terms of the series, $\mathrm{n}=50$, $\mathrm{S}_{50}=\frac{50\left[4(50)^{2}+6(50)-1\right]}{3}$
$=\frac{50 \times(10000+300-1)}{3}=\frac{50 \times 10299}{3}=171650$

Topic: Mathematics, Sequences and Series, NDA, NDA 2006 (Phase 1)
What is the sum of the focal distances of a point of an
ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 ?$

Options:
A $a$
B $b$
C $2 a$
D $2 b$

The correct answer is:
$2 a$

Solution:
Bydefinition of ellipse,
we have $\mathrm{PS}+\mathrm{PS}^{\prime}=2 a$

Topic: Mathematics, Ellipse, NDA, NDA 2011 (Phase 1)
What is the sum of the major and minor axes of the ellipse whose eccentricity is $4 / 5$ and length of latus rectum is $14.4$ unit

Options:
A 32 units
B 48 units
C 64 units
D None of these

The correct answer is:
64 units

Solution:
Let $2 \mathrm{a}$ and $2 \mathrm{~b}$ be the length of major and minor axis respectively.
$\frac{\mathrm{b}^{2}}{\mathrm{a}^{2}}=\frac{9}{25}$
Also, $\frac{2 \mathrm{~b}^{2}}{\mathrm{a}}=14.4$
$\frac{\mathrm{b}^{2}}{\mathrm{a}}=7.2, \mathrm{~b}^{2}=7.2 \mathrm{a}$
Putting value of $\frac{\mathrm{b}^{2}}{\mathrm{a}}$ in equation (i)
$\frac{7.2}{\mathrm{a}}=\frac{9}{25} \Rightarrow \mathrm{a}=20$
$b^{2}=7.2 \times 20=144$
$b=12$
the sum of the major and minor axes $=2 a+2 b$
$=2(\mathrm{a}+\mathrm{b})=2(20+12)=64$ units

Topic: Mathematics, Ellipse, NDA, NDA 2014 (Phase 1)
What is the sum of the roots of the equation
$(2-\sqrt{3}) x^{2}-(7-4 \sqrt{3}) x+(2+\sqrt{3})=0 ?$

Options:
A $2-\sqrt{3}$
B $2+\sqrt{3}$
C $7-4 \sqrt{3}$
D 4

The correct answer is:
$2-\sqrt{3}$

Solution:
The given equation is $(2-\sqrt{3}) x^{2}-(7-4 \sqrt{3}) x+(2+\sqrt{3})=0$
Sum of roots $=\frac{(7-4 \sqrt{3})}{2-\sqrt{3}}$
$=\frac{(7-4 \sqrt{3})(2+\sqrt{3})}{(2-\sqrt{3})(2+\sqrt{3})}=\frac{14+7 \sqrt{3}-8 \sqrt{3}-12}{4-3}$
$=2-\sqrt{3}$

Topic: Mathematics, Quadratic Equation, NDA, NDA 2011 (Phase 1)
What is the sum of the series $1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\ldots .$ equal to?

Options:
A $\frac{1}{2}$
B $\frac{3}{2}$
C 2
D $\frac{2}{3}$

The correct answer is:
$\frac{2}{3}$

Solution:
Given series is a GP. with $a=1, r=-\frac{1}{2}$ $\therefore \mathrm{S}_{\infty}=\frac{\mathrm{a}}{1-\mathrm{r}}=\frac{1}{1-\left(\frac{-1}{2}\right)}=\frac{2}{3}$

Topic: Mathematics, Sequences and Series, NDA, NDA 2012 (Phase 1)
What is the sum of the series $1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\ldots . . ?$

Options:
A $\frac{1}{2}$
B $\frac{3}{4}$
C $\frac{3}{2}$
D $\frac{2}{3}$

The correct answer is:
$\frac{2}{3}$

Solution:
$1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\ldots$ can be written as
$1+\left(-\frac{1}{2}\right)+\left(-\frac{1}{2}\right)^{2}+\left(-\frac{1}{2}\right)^{3}+\ldots$
$[\because$ This is a GP with first term $=1$ and common ratio
$\left.=-\frac{1}{2}\right]$
So, sum of the series
$=\frac{1}{1-\left(-\frac{1}{2}\right)}=\frac{1}{1+\frac{1}{2}}=\frac{2}{3}$

Topic: Mathematics, Sequences and Series, NDA, NDA 2007 (Phase 1)
What is the sum of the series
$1+\frac{1}{8}+\frac{1.3}{8.16}+\frac{1.3 .5}{8.16 .24}+\ldots . \infty ?$

Options:
A $\frac{2}{\sqrt{3}}$
B $2 \sqrt{3}$
C $\frac{\sqrt{3}}{2}$
D $\frac{1}{2 \sqrt{3}}$

The correct answer is:
$\frac{2}{\sqrt{3}}$

Solution:
As given the series is
$\mathrm{S}=1+\frac{1}{8}+\frac{1.3}{8.16}+\frac{1.3 .5}{8.16 .24}+\ldots . \infty$
On comparing this series with
$\mathrm{S}=(1+\mathrm{x})^{\mathrm{n}}=1+\mathrm{nx}+\frac{\mathrm{n}(\mathrm{n}-1)}{2 !} \mathrm{x}^{2}+\ldots \infty$, we get
$\mathrm{nx}=\frac{1}{8}$ ...(i)
and $\frac{\mathrm{n}(\mathrm{n}-1)}{2 !} \mathrm{x}^{2}=\frac{1.3}{8.16}$ ...(ii)
From Eqs. (1) and (2), we get $\frac{\frac{n(n-1)}{2 !} x^{2}}{n^{2} x^{2}}=\frac{\frac{1.3}{8.16}}{\frac{1}{8} \cdot \frac{1}{8}}$
$\Rightarrow \quad \frac{n-1}{2 n}=\frac{3}{2}$
$\Rightarrow \quad n-1=3 n$
$\Rightarrow \quad n=-\frac{1}{2}$
On putting this value in Eq. (i) $\Rightarrow\left(-\frac{1}{2}\right) \mathrm{x}=\frac{1}{8}$
$\begin{aligned} \Rightarrow \quad x &=-\frac{1}{4} \\ \text { But } \quad & S=(1+x)^{n}=\left(1-\frac{1}{4}\right)^{-1 / 2} \\ &=\left(\frac{3}{4}\right)^{-1 / 2}=\frac{2}{\sqrt{3}} \end{aligned}$

Topic: Mathematics, Sequences and Series, NDA, NDA 2007 (Phase 2)
What is the sum of the series $0.3+0.33+0.333+\ldots . n$ terms?

Options:
A $\frac{1}{3}\left[\mathrm{n}-\frac{1}{9}\left(1-\frac{1}{10^{\mathrm{n}}}\right)\right]$
B $\frac{1}{3}\left[\mathrm{n}-\frac{2}{9}\left(1-\frac{1}{10^{\mathrm{n}}}\right)\right]$
C $\frac{1}{3}\left[\mathrm{n}-\frac{1}{3}\left(1-\frac{1}{10^{\mathrm{n}}}\right)\right]$
D $\frac{1}{3}\left[\mathrm{n}-\frac{1}{9}\left(1+\frac{1}{10^{\mathrm{n}}}\right)\right]$

The correct answer is:
$\frac{1}{3}\left[\mathrm{n}-\frac{1}{9}\left(1-\frac{1}{10^{\mathrm{n}}}\right)\right]$

Solution:
$\begin{aligned} & 0.3+0.33+0.333+\ldots \ldots \mathrm{n} \text { terms } \\ &=\frac{3}{10}+\frac{33}{100}+\frac{333}{1000}+\ldots \ldots . \\ &=3\left(\frac{1}{10}+\frac{11}{100}+\frac{111}{1000}+\ldots \ldots\right) \\ &=\frac{3}{9}\left(\frac{9}{10}+\frac{99}{100}+\frac{999}{1000}+\ldots . .\right) \\ &=\frac{1}{3}\left[\left(1-\frac{1}{10}\right)+\left(1-\frac{1}{100}\right)+\left(1-\frac{1}{1000}\right)+\ldots \ldots .\right] \\ &=\frac{1}{3}\left[\mathrm{n}-\left(\frac{1}{10}+\frac{1}{10^{2}}+\frac{1}{10^{3}}+\ldots \ldots \frac{1}{10^{\mathrm{n}}}\right)\right] \end{aligned}$
$=\frac{1}{3}\left[\mathrm{n}-\frac{\frac{1}{10}\left(1-\frac{1}{10^{\mathrm{n}}}\right)}{1-\frac{1}{10}}\right]$
$\left(\right.$ Since, Sum of $n$ terms of G.P $\left.=\frac{a\left(1-r^{n}\right)}{1-r}\right]$
$=\frac{1}{3}\left[n-\frac{10^{\mathrm{n}}-1}{9\left(10^{\mathrm{n}}\right)}\right]=\frac{1}{3}\left[\mathrm{n}-\frac{1}{9}\left(1-\frac{1}{10^{\mathrm{n}}}\right)\right]$

Topic: Mathematics, Sequences and Series, NDA, NDA 2017 (Phase 1)