datavorous/entrance-exam-dataset huggingface dataset:
What is the number of ways that a cricket team of 11 players can be made out of 15 players?

Options:
A 364
B 1001
C 1365
D 32760

The correct answer is:
1365

Solution:
Number of ways that a cricket team of 11 players can
be made out of 15 players $={ }^{15} \mathrm{C}_{11}=\frac{15 !}{11 ! 4 !}$
$=\frac{15 \times 14 \times 13 \times 12 \times 11 !}{11 ! \times 1 \times 2 \times 3 \times 4}=1365$

Topic: Mathematics, Permutation Combination, NDA, NDA 2014 (Phase 2)
What is the number of words formed from the letters of the word 'JOKE' so that the vowels and consonants alternate?

Options:
A 4
B 8
C 12
D None of these

The correct answer is:
8

Solution:
Total number of letters $=4$ No. of vowels $=2$ No. of consonants $-2$ Possibilities of words formed from the letters of word "JOKE" are JOKE, KOJE, KEJO, JEKO, EJOK, EKOJ, OKEJ, OJEK Thus, required number of words $=8$

Topic: Mathematics, Permutation Combination, NDA, NDA 2009 (Phase 1)
What is the number ofplanespassing through threenon-collinear points?

Options:
A 3
B 2
C 1
D 0

The correct answer is:
1

Solution:
We know that the number of planes passing through the non-collinear points is 1 .

Topic: Mathematics, Three Dimensional Geometry, NDA, NDA 2009 (Phase 1)
What is the only solution of the initial value problem $\mathrm{y}^{\prime}=\mathrm{t}(1+\mathrm{y}), \mathrm{y}(0)=0 ?$

Options:
A $\mathrm{y}=-1+\mathrm{e}^{\mathrm{t}^{2} / 2}$
B $\mathrm{y}=1+\mathrm{e}^{\mathrm{t}^{2} / 2}$
C $\mathrm{y}=-\mathrm{t}$
D $\mathrm{y}=\mathrm{t}$

The correct answer is:
$\mathrm{y}=-1+\mathrm{e}^{\mathrm{t}^{2} / 2}$

Solution:
Given, equation is: $\mathrm{y}^{\prime}=\mathrm{t}(1+\mathrm{y})$
i.e., $\frac{\mathrm{dy}}{\mathrm{dt}}=\mathrm{t}(1+\mathrm{y})$
$\Rightarrow \int \frac{1}{1+y} d y=\int t d t \Rightarrow \log (1+y)=\frac{t^{2}}{2}+c$
As per initial conditions $\mathrm{y}(0)=0$ when $\mathrm{t}=0, \mathrm{y}=0$
$\Rightarrow \log 1=\mathrm{c} \Rightarrow \mathrm{c}=0$
$\therefore \log (1+\mathrm{y})=\frac{\mathrm{t}^{2}}{2} \Rightarrow 1+\mathrm{y}=\mathrm{e}^{t^{2} / 2}$
$\Rightarrow \quad y=-1+e^{t^{2} / 2}$
which is required solution.

Topic: Mathematics, Differential Equations, NDA, NDA 2007 (Phase 2)
What is the order of the differential equation$\left(\frac{d y}{d x}\right)^{2}+\frac{d y}{d x}-\sin ^{2} y=0 ?$

Options:
A 1
B 2
C 3
D Undefined

The correct answer is:
1

Solution:
Highest order derivative, present in the differential
equations is $\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)$, therefore its order is one.

Topic: Mathematics, Differential Equations, NDA, NDA 2013 (Phase 2)
What is the order of the differential equation?
$\frac{d y}{d x}+y=\frac{1}{\left(\frac{d y}{d x}\right)}$

Options:
A $-1$
B 0
C 1
D 2

The correct answer is:
1

Solution:
$\frac{d y}{d x}+y=\frac{1}{\left(\frac{d y}{d x}\right)} \Rightarrow\left(\frac{d y}{d x}\right)^{2}+y\left(\frac{d y}{d x}\right)=1$
Hence required order of differential equation $=1$

Topic: Mathematics, Differential Equations, NDA, NDA 2008 (Phase 2)
What is the order of the differential equation
$\frac{\mathrm{dx}}{\mathrm{dy}}+\int \mathrm{y} \mathrm{d} \mathrm{x}=\mathrm{x}^{3} ?$

Options:
A 1
B 2
C 3
D Cannot be determined

The correct answer is:
2

Solution:
$\frac{d x}{d y}+\int y \cdot d x=x^{3} \Rightarrow \int y \cdot d x=x^{3}-\frac{d x}{d y}$
$\Rightarrow 1+\frac{d y}{d x}\left(\int y \cdot d x\right)=x^{3} \cdot \frac{d y}{d x}$
Differentiate both sides w.e.t. $x$
$\Rightarrow 0+\frac{d y}{d x}(y)+\left(\int y \cdot d x\right)\left(\frac{d^{2} y}{d x^{2}}\right)=x^{3} \cdot \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}\left(2 x^{2}\right)$
$\Rightarrow y \cdot \frac{d y}{d x}+\frac{d^{2} y}{d x^{2}}\left[x^{3}-\frac{d x}{d y}\right]=x^{3} \cdot \frac{d^{2} y}{d x^{2}}+2 x^{2} \frac{d y}{d x}$
$\Rightarrow y \frac{d y}{d x}+x^{3} \frac{d^{2} y}{d x^{2}}-\left(\frac{d x}{d y}\right)\left(\frac{d^{2} y}{d x^{2}}\right)=x^{3} \frac{d^{2} y}{d x^{2}}+2 x^{2} \frac{d y}{d x}$
$\Rightarrow y \frac{d y}{d x}-\frac{d x}{d y} \cdot \frac{d^{2} y}{d x^{2}}=2 x^{2} \cdot \frac{d y}{d x}$
Multiplying both side by $\frac{d y}{d x}$
$y\left(\frac{d y}{d x}\right)^{2}-\frac{d^{2} y}{d x^{2}}=2 x^{2}\left(\frac{d y}{d x}\right)^{2}$
$\Rightarrow \frac{d^{2} y}{d x^{2}}+\left(2 x^{2}-y\right)\left(\frac{d y}{d x}\right)^{2}=0$
Order $=2$, degree $=1$

Topic: Mathematics, Differential Equations, NDA, NDA 2016 (Phase 2)
What is the order of the differential equation whose solution is $\mathrm{y}=\mathrm{a} \cos \mathrm{x}+\mathrm{b} \sin \mathrm{x}+\mathrm{ce}^{-\mathrm{x}}+\mathrm{d}$, where $\mathrm{a}, \mathrm{b}, \mathrm{c}$ and
$\begin{array}{lc}\text { d are arbitrary constants? } & {}\end{array}$

Options:
A 1
B 2
C 3
D 4

The correct answer is:
4

Solution:
Order $=4(\because$ No. of arbitrary constants $=4$ )

Topic: Mathematics, Differential Equations, NDA, NDA 2018 (Phase 2)
What is the order of the product
$\left[\begin{array}{lll}\mathrm{x} & \mathrm{y} & \mathrm{z}\end{array}\right]\left[\begin{array}{lll}\mathrm{a} & \mathrm{h} & \mathrm{g} \\ \mathrm{h} & \mathrm{b} & \mathrm{f} \\ \mathrm{g} & \mathrm{f} & \mathrm{c}\end{array}\right]\left[\begin{array}{l}\mathrm{x} \\ \mathrm{y} \\ \mathrm{z}\end{array}\right]$ ?

Options:
A $3 \times 1$
B $1 \times 1$
C $1 \times 3$
D $3 \times 3$

The correct answer is:
$1 \times 1$

Solution:
Given $\left[\begin{array}{lll}\mathrm{x} & \mathrm{y} & \mathrm{z}\end{array}\right]\left[\begin{array}{lll}\mathrm{a} & \mathrm{h} & \mathrm{g} \\ \mathrm{h} & \mathrm{b} & \mathrm{f} \\ \mathrm{g} & \mathrm{f} & \mathrm{c}\end{array}\right]\left[\begin{array}{l}\mathrm{x} \\ \mathrm{y} \\ \mathrm{z}\end{array}\right]$
Consider $\left[\begin{array}{lll}x & y & z\end{array}\right]_{1 \times 3}\left[\begin{array}{lll}a & h & g \\ h & b & f \\ g & f & c\end{array}\right]_{3 \times 3}$
$\Rightarrow$ order of product $=1 \times 3$
Now, order of product $=1 \times 3$ and order of $\left[\begin{array}{l}\mathrm{x} \\ \mathrm{y} \\ \mathrm{z}\end{array}\right]=3 \times 1$
$\therefore$ Required order $=1 \times 1$

Topic: Mathematics, Matrices, NDA, NDA 2012 (Phase 1)
What is the order of
$\left(\begin{array}{llll}\mathrm{x} \mathrm{y} & \mathrm{z}\end{array}\right)\left[\begin{array}{lll}\mathrm{a} & \mathrm{h} & \mathrm{g} \\ \mathrm{h} & \mathrm{b} & f \\ \mathrm{~g} & f & \mathrm{c}\end{array}\right]\left[\begin{array}{l}\mathrm{x} \\ \mathrm{y} \\ \mathrm{z}\end{array}\right] ?$

Options:
A $3 \times 1$
B $1 \times 1$
C $1 \times 3$
D $3 \times 3$

The correct answer is:
$1 \times 1$

Solution:

Topic: Mathematics, Matrices, NDA, NDA 2017 (Phase 1)
What is the outer boundary structure of a typical plant cell?

Options:
A Cell membrane
B Cell wall
C Nucleus
D Cytoplasm

The correct answer is:
Cell wall

Solution:
The outer boundary of a typical plant cell is a distinct cell wall.

Topic: Biology, Cell - The Unit of Life, NEET
What is the overall change represented by the standard electrode potential (Eo) for reducing power?

Options:
A Ionization enthalpy
B Hydration enthalpy
C Sublimation enthalpy
D Oxidation enthalpy

The correct answer is:
Hydration enthalpy

Solution:
The standard electrode potential (Eo) for reducing power represents the overall change in the process, which includes sublimation enthalpy, ionization enthalpy, and hydration enthalpy. Among these, the hydration enthalpy plays a significant role in determining the reducing power of alkali metals.

Topic: Chemistry, s Block Elements, NEET
What is the overall development of an individual with Klinefelter’s Syndrome?

Options:
A Only masculine development
B Only feminine development
C No secondary sexual characteristics
D Both masculine and feminine development

The correct answer is:
Both masculine and feminine development

Solution:
An individual with Klinefelter’s Syndrome has overall masculine development, but feminine development, such as the development of breasts (Gynaecomastia), is also expressed.

Topic: Biology, Principles of Inheritance and Variation, NEET
What is the perimeter of the triangle with vertices $\begin{array}{ll}\mathrm{A}(-4,2), \mathrm{B}(0,-1) \text { and } \mathrm{C}(3,3) ? & {}\end{array}$

Options:
A $7+3 \sqrt{2}$
B $10+5 \sqrt{2}$
C $11+6 \sqrt{2}$
D $5+\sqrt{2}$

The correct answer is:
$10+5 \sqrt{2}$

Solution:
By using Distance formula,
We have, $A B=\sqrt{(0+4)^{2}+(-1-2)^{2}}=\sqrt{16+9}=5$
$\mathrm{BC}=\sqrt{9+16}=5$
$\mathrm{CA}=\sqrt{49+(1)^{2}}=\sqrt{50}=5 \sqrt{2}$
Hence, Required Perimeter $=\mathrm{AB}+\mathrm{BC}+\mathrm{CA}$ $=10+5 \sqrt{2}$

Topic: Mathematics, Straight Lines, NDA, NDA 2012 (Phase 1)
What is the period of the function $\mathrm{f}(\mathrm{x})=\sin \mathrm{x} ?

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

The correct answer is:
$2 \pi$

Solution:
Period of the function, $\sin x$ is $2 \pi$.

Topic: Mathematics, Trigonometric Equations, NDA, NDA 2018 (Phase 1)
What is the perpendicular distance between the parallel lines $3 \mathrm{x}+4 \mathrm{y}=9$ and $9 \mathrm{x}+12 \mathrm{y}+28=0 ?$

Options:
A $\frac{7}{3}$ units
B $\frac{8}{3}$ units
C $\frac{10}{3}$ units
D $\frac{11}{3}$ units

The correct answer is:
$\frac{11}{3}$ units

Solution:
The given lines are:$3 x+4 y=9$
$\Rightarrow \quad y=\frac{9}{4}-\frac{3}{4} x$ ...(i)
and $9 x+12 y+28=0 \Rightarrow y=\frac{-7}{3}-\frac{3 x}{4}$ ...(ii)
We have,
$$
\mathrm{m}=\frac{-3}{4} ; \mathrm{C}_{1}=\frac{9}{4} ; \mathrm{C}_{2}=\frac{-7}{3}
$$
$$
\text { Now, Distance }=\frac{\left|C_{1}-C_{2}\right|}{\sqrt{1+m^{2}}}=\frac{\left|\frac{9}{4}-\left(-\frac{7}{3}\right)\right|}{\sqrt{1+\frac{9}{16}}}=\frac{11}{3} \text { units }
$$

Topic: Mathematics, Straight Lines, NDA, NDA 2012 (Phase 2)
What is the perpendicular distance of the point $(\mathrm{x}, \mathrm{y})$ from $\begin{array}{ll}\mathrm{x} \text { -axis? } & {}\end{array}$

Options:
A $\mathrm{x}$
B $\mathrm{y}$
C $|\mathrm{x}|$
D $|\mathrm{y}|$

The correct answer is:
$|\mathrm{y}|$

Solution:
$\mathrm{p}=\frac{|\mathrm{y}|}{\sqrt{1}}=|\mathrm{y}|$

Topic: Mathematics, Straight Lines, NDA, NDA 2012 (Phase 1)
What is the positive square root of $7+4 \sqrt{3}$ ?

Options:
A $\sqrt{3}-1$
B $\sqrt{3}+1$
C $\quad \sqrt{3}-2$
D $\sqrt{3}+2$

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

Solution:
$7+4 \sqrt{3}=(\sqrt{4})^{2}+(\sqrt{3})^{2}+2 \sqrt{4} \times \sqrt{3}$
$=(\sqrt{4}+\sqrt{3})^{2}$
$\sqrt{7+4 \sqrt{3}}=2+\sqrt{3}$

Topic: Mathematics, Basic of Mathematics, NDA, NDA 2013 (Phase 2)
What is the potential assigned to the standard hydrogen electrode at all temperatures?

Options:
A 0.34 V
B 1.10 V
C Zero potential
D 1 bar

The correct answer is:
Zero potential

Solution:
The standard hydrogen electrode is assigned a zero potential at all temperatures.

Topic: Chemistry, Electrochemistry, NEET
What is the prefix used to indicate a branched chain alkyl group that is derived from a fundamental name?

Options:
A but-
B neo-
C sec-
D tert-

The correct answer is:
neo-

Solution:
The prefixes iso- and neo- are used to indicate branched chain alkyl groups derived from fundamental names.

Topic: Chemistry, General Organic Chemistry, NEET
What is the primary advantage of using the triple-point of water as a standard fixed point in modern thermometry over using the melting point of ice and the boiling point of water, as originally done in the Celsius scale?

Options:
A The triple-point is more precise and reproducible.
B The Celsius scale has been updated to use the triple-point.
C The triple-point is easier to measure accurately.
D The melting point and boiling point are not well-defined.

The correct answer is:
The triple-point is more precise and reproducible.

Solution:
Correct Option is : (A)
The triple-point is more precise and reproducible.

Topic: Chemistry, Thermal Properties of Matter, NEET
What is the primary function of secondary valences in coordination compounds?

Options:
A To bond with negative ions
B To bond with neutral molecules
C To increase the coordination number
D To form ionic compounds

The correct answer is:
To bond with neutral molecules

Solution:
The primary function of secondary valences in coordination compounds is to bond with neutral molecules.

Topic: Chemistry, Coordination Compounds, NEET
What is the primary oocyte surrounded by?

Options:
A Secondary follicles
B A layer of granulosa cells
C The theca
D The zona pellucida

The correct answer is:
A layer of granulosa cells

Solution:
The primary oocyte is surrounded by a layer of granulosa cells.

Topic: Biology, Human Reproduction, NEET
What is the primary product obtained when aliphatic or aromatic primary and secondary amines react with acid chlorides, anhydrides, or esters?

Options:
A Alcohols
B Aldehydes
C Amides
D Alkenes

The correct answer is:
Amides

Solution:
The reaction of amines with acid chlorides, anhydrides, or esters results in the formation of amides.

Topic: Chemistry, Amines, NEET
What is the primary product of dehydrogenation when a secondary alcohol is passed over heated copper?

Options:
A Carboxylic acid
B Aldehyde
C Ketone
D Alkene

The correct answer is:
Alkene

Solution:
The primary product of dehydrogenation of a secondary alcohol is an alkene.

Topic: Chemistry, Alcohols Phenols and Ethers, NEET
What is the primary purpose of defining a boundary between the system and the surroundings in thermodynamics?

Options:
A To isolate the system from the surroundings
B To facilitate the movement of matter between the system and surroundings
C To control the temperature inside the system
D To prevent any interaction between the system and the surroundings

The correct answer is:
To isolate the system from the surroundings

Solution:
Defining a boundary allows for the isolation of the system from the surroundings, as mentioned in the paragraph.

Topic: Chemistry, Thermodynamics (C), NEET
What is the primary reason for the difficulty in separating Zr and Hf in nature?

Options:
A Their different chemical reactivity
B Their identical atomic numbers
C Their almost identical radii
D Their abundance in nature

The correct answer is:
Their almost identical radii

Solution:
The almost identical radii of Zr (160 pm) and Hf (159 pm), a consequence of the lanthanoid contraction, account for their occurrence together in nature and for the difficulty faced in their separation

Topic: Chemistry, d and f Block Elements, NEET
What is the primary standard used in volumetric analysis?

Options:
A Sodium chromate
B Sodium dichromate
C Potassium dichromate
D Chromite ore

The correct answer is:
Potassium dichromate

Solution:
Potassium dichromate is used as a primary standard in volumetric analysis.

Topic: Chemistry, d and f Block Elements, NEET
What is the principal argument of $(-1-\mathrm{i})$, where $\mathrm{i}=\sqrt{-1}$ ?

Options:
A $\frac{\pi}{4}$
B $-\frac{\pi}{4}$
C $-\frac{3 \pi}{4}$
D $\frac{3 \pi}{4}$

The correct answer is:
$-\frac{3 \pi}{4}$

Solution:
$\mathrm{z}=\mathrm{x}+\mathrm{iy}=(-1-\mathrm{i})$
$\therefore \mathrm{x}=-1, \mathrm{y}=-1$
This lies in $3^{\text {rd }}$ Quadrant. $\therefore \arg (\mathrm{z})=\theta-\pi$
$=\tan ^{-1}\left(\frac{\mathrm{y}}{\mathrm{x}}\right)-\pi$
$=\tan ^{-1}\left(\frac{-1}{-1}\right)-\pi$
$=\tan ^{-1}(1)-\pi$
$=\frac{\pi}{4}-\pi$
$=\frac{-3 \pi}{4}$

Topic: Mathematics, Complex Number, NDA, NDA 2018 (Phase 1)
What is the principal value of $\sec ^{-1}\left(\frac{2}{\sqrt{3}}\right) ?$

Options:
A $\frac{\pi}{2}$
B $\frac{\pi}{3}$
C $\frac{\pi}{4}$
D $\frac{\pi}{6}$

The correct answer is:
$\frac{\pi}{6}$

Solution:
Let $\sec ^{-1}\left(\frac{2}{\sqrt{3}}\right)=y$ where $0 < \mathrm{y} \leq \frac{\pi}{2}$
$\Rightarrow$ sec $y=\frac{2}{\sqrt{3}}$
$\Rightarrow \sec y=\sec \frac{\pi}{6} \Rightarrow y=\frac{\pi}{6}$
The principal value of $\sec ^{-1}\left(\frac{2}{\sqrt{3}}\right)=\frac{\pi}{6}$

Topic: Mathematics, Inverse Trigonometric Functions, NDA, NDA 2011 (Phase 2)
What is the principal value of $\sin ^{-1}\left(\sin \frac{2 \pi}{3}\right)$?

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

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

Solution:
$\sin ^{-1}\left(\sin \frac{2 \pi}{3}\right)=\sin ^{-1}\left(\sin \left(\pi-\frac{\pi}{3}\right)\right)$
$=\sin ^{-1}\left(\sin \frac{\pi}{3}\right)=\frac{\pi}{3}$.

Topic: Mathematics, Inverse Trigonometric Functions, NDA, NDA 2018 (Phase 1)
What is the principle value of $\operatorname{cosec}^{-1}(-\sqrt{2}) ?$

Options:
A $\frac{\pi}{4}$
B $\frac{\pi}{2}$
C $-\frac{\pi}{4}$
D 0

The correct answer is:
$-\frac{\pi}{4}$

Solution:
Let the principal value of $\operatorname{cosec}^{-1}(-\sqrt{2})=\theta$
$\Rightarrow-\sqrt{2}=\operatorname{cosec} \theta \Rightarrow-\sqrt{2}=\frac{1}{\sin \theta}$
$\Rightarrow \quad \sin \theta=-\frac{1}{\sqrt{2}} \Rightarrow \theta=-\frac{\pi}{4}$
Principal value of $\operatorname{cosec}^{-1}(-\sqrt{2})=-\frac{\pi}{4}$

Topic: Mathematics, Inverse Trigonometric Functions, NDA, NDA 2010 (Phase 1)
What is the probability of 5 Sundays in the month of December?

Options:
A $\frac{1}{7}$
B $\frac{2}{7}$
C $\frac{3}{7}$
D None of the above

The correct answer is:
$\frac{3}{7}$

Solution:
In month of December 31 days i.e. $(28+3)$ days. In 28 days will get 4 Sundays. If we get any Sunday in first 3 days of December than only we can get 5 Sundays in month. $n(5$ th Sunday $)=3 \quad[4$ weeks $+3$ days $]$ $n(5)=7$
Hence prob. of 5 Sundays in month of December $=\frac{3}{7}$.

Topic: Mathematics, Probability, NDA, NDA 2016 (Phase 1)
What is the probability of getting a sum of 7 with two dice?

Options:
A $\frac{1}{6}$
B $\frac{1}{3}$
C $\frac{1}{12}$
D $\frac{5}{36}$

The correct answer is:
$\frac{1}{6}$

Solution:
Total case $=6 \times 6=36$ Favourable $=(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)$
Probability $=\frac{6}{36}=\frac{1}{6}$

Topic: Mathematics, Probability, NDA, NDA 2013 (Phase 1)
What is the probability of getting five heads and seven tails in 12 flips of a balanced coin?

Options:
A $\mathrm{C}(12,5) /\left(2^{5}\right)$
B $\mathrm{C}(12,5) /\left(2^{7}\right)$
C $\mathrm{C}(12,5) /\left(2^{12}\right)$
D $\mathrm{C}(12,7) /\left(2^{6}\right)$

The correct answer is:
$\mathrm{C}(12,5) /\left(2^{12}\right)$

Solution:
Number of ways of selecting 5 heads cut of total 12
flips = $12_{\mathrm{C}_{5}}$
Probability of getting one head in a coin $=\frac{1}{2}$
Also, probability of getting one tail in a coin $=\frac{1}{2}$
Probability of getting 5 head $=\left(\frac{1}{2}\right)^{5}$
Probability of getting 7 tails $=\left(\frac{1}{2}\right)^{7}$
So, required probability
$=12_{\mathrm{C}_{5}}\left(\frac{1}{2}\right)^{5}\left(\frac{1}{2}\right)^{7}=12_{\mathrm{C}_{5}}\left(\frac{1}{2}\right)^{12}=\frac{12_{\mathrm{C}_{5}}}{2^{12}}$

Topic: Mathematics, Probability, NDA, NDA 2007 (Phase 1)
What is the probability of having 53 Sundays or 53 Mondays in a leap year?

Options:
A $2 / 7$
B $3 / 7$
C 4/7
D $5 / 7$

The correct answer is:
$3 / 7$

Solution:
A leap year has 366 days, in which 2 days may be any one of the following pairs. (Sunday, Monday), (Monday, Tuesday), (Tuesday, Wednesday), (Wednesday, Thursday), (Thursday, Friday) (Friday, Saturday) (Saturday, Sunday).
$\therefore \quad$ Required probability $=\frac{2}{7}+\frac{2}{7}-\frac{1}{7}=\frac{3}{7}$
(By using $\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})$

Topic: Mathematics, Probability, NDA, NDA 2010 (Phase 2)
What is the probability of having a knave and a queen when two cards are drawn from a pack of 52 cards?

Options:
A $\frac{16}{663}$
B $\frac{2}{663}$
C $\frac{4}{663}$
D $\frac{8}{663}$

The correct answer is:
$\frac{4}{663}$

Solution:
Required probability $=\frac{4}{52} \times \frac{4}{51}$
[Since, card is not replaced after first draw]
$=\frac{4}{13 \times 51}=\frac{4}{663}$

Topic: Mathematics, Probability, NDA, NDA 2006 (Phase 2)
What is the probability of two persons being born on the same day (ignoring date)

Options:
A $1 / 49$
B $1 / 365$
C $1 / 7$
D $2 / 7$

The correct answer is:
$1 / 365$

Solution:
Required probability $=\frac{365}{365} \times \frac{1}{365}=\frac{1}{365}$

Topic: Mathematics, Probability, NDA, NDA 2008 (Phase 2)
What is the probability that a leap year selected at random will contain 53 Mondays?

Options:
A $2 / 5$
B $2 / 7$
C $1 / 7$
D $5 / 7$

The correct answer is:
$2 / 7$

Solution:
Total no. of days in leap year $=366$ Favourable cases $=2$
Required prob $=\frac{2}{7}$

Topic: Mathematics, Probability, NDA, NDA 2012 (Phase 2)
What is the probability that an interior point in a circle is closer to the centre than to the circumference?

Options:
A $\frac{1}{4}$
B $\frac{1}{2}$
C $\frac{3}{4}$
D It cannot be determined

The correct answer is:
$\frac{1}{4}$

Solution:
Ifr is the radius of circle, $A=\pi r^{2}$
Probability $=\frac{\pi\left(\frac{r}{2}\right)^{2}}{\pi r^{2}}=\frac{\pi r^{2}}{4} \times \frac{1}{\pi r^{2}}=\frac{1}{4}$

Topic: Mathematics, Probability, NDA, NDA 2019 (Phase 1)
What is the probability that at least two persons out of a group of three persons were born in the same month (disregard year)?

Options:
A $\frac{33}{144}$
B $\frac{17}{72}$
C $\frac{1}{144}$
D $\frac{2}{9}$

The correct answer is:
$\frac{17}{72}$

Solution:
Probability that no one born in the same month $=\frac{12 \times 11 \times 10}{12 \times 12 \times 12}$
Probability that atleast two are born in same month
$=1-\frac{12 \times 11 \times 10}{12 \times 12 \times 12}$
$=\frac{144-110}{144}=\frac{17}{72}$

Topic: Mathematics, Probability, NDA, NDA 2017 (Phase 2)
What is the probability that in a family of 4 children there will be at least one boy?

Options:
A $\frac{15}{16}$
B $\frac{3}{8}$
C $\frac{1}{16}$
D $\frac{7}{8}$

The correct answer is:
$\frac{15}{16}$

Solution:
Total possibility of 4 children, either girl or boy is $2^{4}$ $=16$. Out of these there is one possibility in which there will be no boy and only girls. So, total possibility of at least one boy is $16-1=15$
$\Rightarrow \quad \mathrm{P}($ at least one boy $)=\frac{15}{16}$.

Topic: Mathematics, Probability, NDA, NDA 2008 (Phase 1)
What is the probability that the sum of any two different single digit natural numbers is a prime number?

Options:
A $\frac{5}{27}$
B $\frac{7}{18}$
C $\frac{1}{3}$
D None of the above

The correct answer is:
$\frac{7}{18}$

Solution:
Total no. of two different single digit natural number $={ }^{9} \mathrm{C}_{2}=36$
The number of prime number which is sum of two different single digit number. $(3,5,7,11,13 \& 17)=14$
$\therefore$ Required probability $=\frac{14}{36}=\frac{7}{18}$

Topic: Mathematics, Probability, NDA, NDA 2015 (Phase 2)
What is the process by which the secondary oocyte (ovum) is released from the ovary?

Options:
A Ovulation
B Meiosis
C Fertilization
D Degeneration

The correct answer is:
Ovulation

Solution:
The Graafian follicle ruptures to release the secondary oocyte (ovum) from the ovary through the process called ovulation.

Topic: Biology, Human Reproduction, NEET
What is the process of releasing the secondary oocyte from the ovary called?

Options:
A Ovulation
B Spermatogenesis
C Oogenesis
D Gamete mother cells

The correct answer is:
Ovulation

Solution:
The Graafian follicle ruptures to release the secondary oocyte (ovum) from the ovary by the process called ovulation.

Topic: Biology, Human Reproduction, NEET
What is the process that allows a secondary cell to be reused after use?

Options:
A Passing current through it in the same direction
B Discharging and charging alternately
C Replacing the electrolyte
D None of the above

The correct answer is:
Passing current through it in the same direction

Solution:
A secondary cell can be recharged by passing current through it in the opposite direction so that it can be used again.

Topic: Chemistry, Electrochemistry, NEET
What is the product formed when an amide containing three carbons undergoes Hoffmann bromamide degradation?

Options:
A A primary amine with three carbons
B A primary amine with two carbons
C A primary amine with one carbon
D A secondary amine with three carbons

The correct answer is:
A primary amine with two carbons

Solution:
In the Hoffmann bromamide degradation, the amine formed contains one carbon less than the amide. So, if the amide has three carbons, the resulting amine will have two carbons.

Topic: Chemistry, Amines, NEET