datavorous/entrance-exam-dataset huggingface dataset:
Which one of the following can be oxidised to the corresponding carbonyl compound?

Options:
A 2-Hydroxypropane
B Ortho-nitrophenol
C Phenol
D 2-Methyl-2-hydroxypropane

The correct answer is:
2-Hydroxypropane

Solution:
2-hydroxy propane or secondary alcohol is oxidised into propanone.
Caution
Primary alcohol on oxidation gives aldehyde while secondary alcohol on oxidation gives ketone. Aromatic hydroxyl compound cannot be oxidised to corresponding carbonyl compound.

Topic: Chemistry, Aldehydes and Ketones, NEET, NEET 2004
Which one of the following can be oxidised to the corresponding carbonyl compound?

Options:
A 2-hydroxypropane
B ortho-nitrophenol
C Phenol
D 2-methyl-2-hydroxypropane

The correct answer is:
2-hydroxypropane

Solution:
Secondary alcohol on oxidation gives a ketone containing the same number of carbon atoms.

Topic: Chemistry, Alcohols Phenols and Ethers, NEET
Which one of the following describes correctly the homologous structures?

Options:
A organs with anatomical similarities, but performing different functions
B organs with anatomical dissimilarities, but performing same function
C organs that have no function now, but had an important function in ancestors
D organs appearing only in embryonic stage and disappearing later in the adult

The correct answer is:
organs with anatomical similarities, but performing different functions

Solution:
The organs which have the same structure and origin but look different and perform different functions are called homologous organs. The organs which have the same function and are superficially alike but are quite different in their fundamental structure and embryonic origin are called analogous organs.

Topic: Biology, Evolution, NEET, NEET 2003
Which one of the following differential equations has a periodic solution?

Options:
A $\frac{\mathrm{d}^{2} \mathrm{x}}{\mathrm{dt}^{2}}+\mu \mathrm{x}=0$
B $\frac{\mathrm{d}^{2} \mathrm{x}}{\mathrm{dt}^{2}}-\mu \mathrm{x}=0$
C $\mathrm{x} \frac{\mathrm{dx}}{\mathrm{dt}}+\mu \mathrm{t}=0$
D $\frac{\mathrm{dx}}{\mathrm{dt}}+\mu \mathrm{xt}=0$
where $\mu>0$.

The correct answer is:
$\frac{\mathrm{d}^{2} \mathrm{x}}{\mathrm{dt}^{2}}+\mu \mathrm{x}=0$

Solution:
No Solution

Topic: Mathematics, Differential Equations, NDA, NDA 2018 (Phase 1)
Which one of the following differential equations is not linear? $\quad$

Options:
A $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}+4 \mathrm{y}=0$
B $\mathrm{x} \frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y}=\mathrm{x}^{3}$
C $(\mathrm{x}-\mathrm{y})^{2} \frac{\mathrm{dy}}{\mathrm{dx}}=9$
D $\cos ^{2} \mathrm{x} \frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y}=\tan \mathrm{x}$

The correct answer is:
$\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}+4 \mathrm{y}=0$

Solution:
Differential equation given in option (a) is not linear
because differential coefficient $\frac{\mathrm{dy}}{\mathrm{dx}}$ has exponent $2 .$

Topic: Mathematics, Differential Equations, NDA, NDA 2012 (Phase 1)
Which one of the following differential equations represents the family of straight lines which are at unit distance from the origin?

Options:
A $\left(\mathrm{y}-\mathrm{x} \frac{\mathrm{dy}}{\mathrm{dx}}\right)^{2}=1-\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^{2}$
B $\left(y+x \frac{d y}{d x}\right)^{2}=1+\left(\frac{d y}{d x}\right)^{2}$
C $\left(y-x \frac{d y}{d x}\right)^{2}=1+\left(\frac{d y}{d x}\right)^{2}$
D $\left(y+x \frac{d y}{d x}\right)^{2}=1-\left(\frac{d y}{d x}\right)^{2}$

The correct answer is:
$\left(y-x \frac{d y}{d x}\right)^{2}=1+\left(\frac{d y}{d x}\right)^{2}$

Solution:
$y=m x+c$ (Equation of straight line)
$\frac{d y}{d x}=m$ and $m x-y+c=0$ is at unit distance from origin.
$\therefore \frac{|m(0)-(0)+c|}{\sqrt{m^{2}+(-1)^{2}}}=1 \Rightarrow c=\sqrt{1+m^{2}}$
Now :
$\left[y-x \frac{d y}{d x}\right]^{2}=[m x+c-x m]^{2} \Rightarrow c^{2}=1+m^{2}$
also, :
$\left[y+x \frac{d y}{d x}\right]^{2}=[m x+c+m x]^{2}=\left[2 m x+\sqrt{1+m^{2}}\right]^{2}$
also, $1-\left(\frac{d y}{d x}\right)^{2}=1-m^{2}$ and $1+\left(\frac{d y}{d x}\right)^{2}=1+m^{2}$
$\Rightarrow\left[y-x \frac{d y}{d x}\right]^{2}=1+\left(\frac{d y}{d x}\right)^{2}$

Topic: Mathematics, Differential Equations, NDA, NDA 2016 (Phase 2)
Which one of the following differential equations represents the system of circles touching $\mathrm{y}$ -axis at the origin?

Options:
A $\frac{d y}{d x}=x^{2}-y^{2}$
B $\quad 2 x y \frac{d y}{d x}=y^{2}-x^{2}$
C $2 x y \frac{d y}{d x}=x^{2}-y^{2}$
D $\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{y}^{2}-\mathrm{x}^{2} \quad$

The correct answer is:
$\quad 2 x y \frac{d y}{d x}=y^{2}-x^{2}$

Solution:
Since, circle is touching y-axis at origin its center lies on $\mathrm{x}$ -axis. Let the centre be $(\mathrm{a}, 0)$. Its radius $=\mathrm{a}$ $(\mathrm{x}-\mathrm{a})^{2}+\mathrm{y}^{2}=\mathrm{a}^{2}$
$x^{2}+a^{2}-2 a x+y^{2}=a^{2}$
$\Rightarrow \quad x^{2}+y^{2}-2 a x=0$
$a=\frac{x^{2}+y^{2}}{2 x}$
Differentiating both sides
Now, $2 x+2 y \frac{d y}{d x}-2 a=0$
$\Rightarrow \quad 2 x+2 y \frac{d y}{d x}-\frac{x^{2}+y^{2}}{x}=0$
$\Rightarrow 2 x^{2}+2 x y \frac{d y}{d x}-x^{2}-y^{2}=0$
$\Rightarrow 2 x y \frac{d y}{d x}=y^{2}-x^{2}$

Topic: Mathematics, Differential Equations, NDA, NDA 2006 (Phase 1)
Which one of the following does not undergo iodoform reaction?

Options:
A Secondary butyl alcohol
B Iso-propyl alcohol
C Diethyl ketone
D Ethyl alcohol

The correct answer is:
Diethyl ketone

Solution:
Iodoform reaction is given by only those compounds that have either
or

Topic: Chemistry, Haloalkanes and Haloarenes, JIPMER, JIPMER 2014
Which one of the following factors does the expansions of
the determinant $\left(\begin{array}{ccc}\mathrm{x} & \mathrm{y} & 3 \\ \mathrm{x}^{2} & 5 \mathrm{y}^{3} & 9 \\ \mathrm{x}^{3} & 10 \mathrm{y}^{3} & 27\end{array}\right)$ contain?

Options:
A $\mathrm{x}-3$
B $x-y$
C $\mathrm{Y}-3$
D $x-3 y$

The correct answer is:
$\mathrm{x}-3$

Solution:
On Applying
$\mathrm{C}_{1} \rightarrow \mathrm{C}_{1}-\mathrm{C}_{3}$
$\left|\begin{array}{ccc}x-3 & y & 3 \\ x^{2}-9 & 5 y^{3} & 9 \\ x^{3}-27 & 10 y^{5} & 27\end{array}\right|$
$(x-3)\left|\begin{array}{ccc}1 & y & 3 \\ x+3 & 5 y^{3} & 9 \\ x^{2}+9-3 x & 10 y^{5} & 27\end{array}\right|$

Topic: Mathematics, Determinants, NDA, NDA 2018 (Phase 2)
Which one of the following features is common in silverfish, scorpion, dragonfly and prawn?

Options:
A Two pairs of legs and segmented body
B Living chitinous cuticle and two pairs of antennae
C Jointed appendages and chitinous exoskeleton
D Closed blood carcular system

The correct answer is:
Jointed appendages and chitinous exoskeleton

Solution:
Silverfish, scorpion, dragonfly and prawn all fall in phylum arthropods which have jointed appendages and chitinous exoskeleton as the characteristic feature.

Topic: Biology, Animal Kingdom, AIIMS, AIIMS 2005
Which one of the following features is common to cockroach, earthworm and mosquito?

Options:
A Presence of chitinous exoskeleton
B Presence of jointed appendages
C Presence of two pair of wings
D Presence of schizocoelom as a true coelom

The correct answer is:
Presence of schizocoelom as a true coelom

Solution:
The chitinous exoskeleton, jointed appendages and two pairs of wings are characteristic features of class Insecta of Arthropoda (examples: cockroach and mosquito). Earthworm belongs to Annelida in which first-time true coelom appeared, called schizocoelom. This type of coelom is present in Arthropoda and Mollusca also.

Topic: Biology, Animal Kingdom, NEET
Which one of the following functions $f: R \rightarrow R$ is injective?

Options:
A $\quad f(x)=|x|$ for all $x \in R$
B $f(x)=x^{2}$ for all $x \in R$
C $f(x)=11$ for all $x \in R$
D $f(x)=-x$ for all $x \in R$

The correct answer is:
$f(x)=-x$ for all $x \in R$

Solution:
A injective function means one-one. Consider $f(x)=-x$ Let $f(x)=f(y) \forall x, y \in R$
$\Rightarrow-x=-y \Rightarrow x=y$
For every values of $x$, we get a different values of $f$. Hence, it is injective.

Topic: Mathematics, Functions, NDA, NDA 2009 (Phase 2)
Which one of the following functions is differentiable for all real values of $\mathrm{x}$ ?

Options:
A $\frac{\mathrm{x}}{|\mathrm{x}|}$
B $\mathrm{x}|\mathrm{x}|$
C $\frac{1}{|\mathrm{x}|}$
D $\frac{1}{x}$

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

Solution:
Since $\frac{\mathrm{x}}{\mathrm{x}}$ is not continuous function $\therefore$ it is not
differentiable also. Also, L.H.D. and R.H.D. at $\mathrm{x}=0$ not equal. Thus, only function given in option 'b' gives differentiability for all real values of $\mathrm{x}$.

Topic: Mathematics, Continuity and Differentiability, NDA, NDA 2012 (Phase 1)
Which one of the following gives a foul-smelling substance when treated with chloroform and alcoholic $\mathrm{KOH}$ ?

Options:
A

B

C

D

The correct answer is:

Solution:
Aliphatic and aromatic primary amines on heating with chloroform and alcoholic $\mathrm{KOH}$ form isocyanides or carbylamines which are foul smelling substances. Secondary and tertiary amines do not show this reaction. This reaction is known as carbylamine reaction or isocyanide test. Hence, the compound should be benzylamine.

Topic: Chemistry, Amines, TS EAMCET, TS EAMCET 2023 (14 May Shift 2)
Which one of the following graph represents the function $f(\mathrm{x})=\frac{\mathrm{x}}{\mathrm{x}}, \mathrm{x} \neq 0 ?$

Options:
A
B
C
D None of the above

The correct answer is:

Solution:
$\mathrm{f}(\mathrm{x})=\frac{\mathrm{x}}{\mathrm{x}}, \mathrm{x} \neq 0$.
$\therefore$ The graph is discontinious at $\mathrm{x}=0$, and correctly shown in option (e).

Topic: Mathematics, Functions, NDA, NDA 2017 (Phase 2)
Which one of the following groups of three animals each are correctly matched with their characteristic morphological feature?

Options:
A

Animals	Morphological feature
Centipede, Prawn, Sea urchin	Joined appendages

B

Animals	Morphological feature
Scorpion, Spider, Cockroach	Ventral solid central nervous system

C

Animals	Morphological feature
Cockroach, Locust , Taenia	Metameric segmentation

D

Animals	Morphological feature
Liver fluke, Sea anemone,Sea cucumber	Bilateral symmetry

The correct answer is:

Animals	Morphological feature
Scorpion, Spider, Cockroach	Ventral solid central nervous system

Solution:
Scorpion, Spider and Cockroaches are Arthropods and thus are Non-chordates. All non-chordates show presence of ventral nerve cord. In option (B), Centipede and prawn show jointed appendages as these are Arthropods but Sea urchin is an Echinoderm. In option (C), Cockroaches and Locust are Arthropods and hence body is divisible into head, thorax and abdomen while Taenia are flat worms and belong to Phylum Platyhelminthes and thus segmentation is absent. In option (D), Liver fluke shows bilateral symmetry. Sea anemone is Coelenterate and thus shows radial symmetry while sea cucumber belongs to Echinodermata which shows bilateral symmetry in larval stage and radial symmetry in adult stage.

Topic: Biology, Animal Kingdom, NEET
Which one of the following groups of three animals each is correctly matched with their one characteristic morphological feature?

Options:
A

Animal	Morphological features

Liver fluke, Sea cucumber	Bilateral symmetry

B

Animal	Morphological features

Centipede, Prawn, Sea cucumber	Jointed appendages

C

Animal	Morphological features

Scorpion, Spider, Cockroach	Ventral solid central nervous system

D

Animal	Morphological features

Cockroach, locust, Taenia	Metameric segmentation

The correct answer is:

Animal	Morphological features

Scorpion, Spider, Cockroach	Ventral solid central nervous system

Solution:
Scorpion, spider and cockroach have a ventral solid central nervous system, all belonging to phyla Arthropoda.
Bilateral symmetry is found in liver fluke, but sea cucumber has radial symmetry.
Metamerism is observed in cockroach and locust (both Arthropoda) but not in Taenia (Platyhelminthes)
Joint appendages are a characteristic feature of Arthropoda and sea cucumber is excluded.

Topic: Biology, Animal Kingdom, NEET
Which one of the following has the largest population in a food chain?
(a) Producers
(b) Primary consumers
(c) Secondary consumers
(d) Decomposers

Options:

None

Solution:
(d)

Topic: Biology, Ecosystem
Which one of the following has the largest population in a food chain?

Options:
A Producers
B Primary consumers
C Secondary consumers
D Decomposers

The correct answer is:
Decomposers

Solution:
Decomposers form the largest population in food chain.

Topic: Biology, Ecosystem, NEET
Which one of the following is a linear differential equation?

Options:
A $\frac{d x}{d y}+y^2=e^{e^x}$
B $d r+\left(2 r^2 \cot \theta+\sin 2 \theta\right) d \theta=0$
C $\frac{d y}{d x}=e^{x-y}\left(e^x-e^{-y}\right)$
D $x^2 d y+x y d x-1=0$

The correct answer is:
$x^2 d y+x y d x-1=0$

Solution:
The standard form of linear differential equation is
$P \frac{d y}{d x}+Q y=R$ or $P \frac{d x}{d y}+Q x=R$ where $P, Q$ and $R$ are the function of $x$ only.
Option (d) can be written as
$\begin{aligned} & x^2 d y+x y d x-d x=0 \\ & x^2 \frac{d y}{d x}+x y-1=0 \\ & \Rightarrow x^2 \frac{d y}{d x}+x y=1\end{aligned}$
Which follows the standard form of linear differential equation.

Topic: Mathematics, Differential Equations, AP EAMCET, AP EAMCET 2023 (18 May Shift 1)
Which one of the following is a measure of dispersion?

Options:
A Mean
B Median $\quad$
C Mode
D Standard deviation

The correct answer is:
Standard deviation

Solution:
Standard Deviation is a measure of dispersion.

Topic: Mathematics, Statistics, NDA, NDA 2012 (Phase 2)
Which one of the following is a null set?

Options:
A $\{0\}$
B $\{\{f\}\}$
C $\{\{\}\}$
D $\left\{x \mid x^{2}+1=0, x \in R\right\}$

The correct answer is:
$\left\{x \mid x^{2}+1=0, x \in R\right\}$

Solution:
Consider the set given in option 'd' $\left\{\mathrm{x} \mid \mathrm{x}^{2}+1=0, \mathrm{x} \in \mathrm{R}\right\}$
Let $\mathrm{x}^{2}+1=0 \Rightarrow \mathrm{x}^{2}=-1 \Rightarrow \mathrm{x}=\pm \mathrm{i}$ which iscomplex
But $\mathrm{x} \in \mathrm{R}$. Hence for, any $\mathrm{x} \in \mathrm{R}, \mathrm{x}^{2}+1$ can not be zero

Topic: Mathematics, Sets and Relations, NDA, NDA 2013 (Phase 1)
Which one of the following is a pair of endangered species?

Options:
A Garden lizard and Mexican poppy
B Rhesus monkey and Sal tree
C Indian peacock and carrot grass
D Hornbill and Indian aconite

The correct answer is:
Hornbill and Indian aconite

Solution:
The species or taxa, which are in dangers of extinction and which may not survive if the adverse factors continue to operate or the species whose numbers have been reduced to a critical level or their habitats have been drastically reduced in such a way that they are in instant danger of extinction are called endangered species. In India, aconite and horn bill are endangered species.

Topic: Biology, Biodiversity and its Conservation, AIIMS, AIIMS 2004
Which one of the following is a secondary alcohol?

Options:
A 2-methyl-1 propanol
B 2-methyl-2 propanol
C 2-butanol
D 1-butanol

The correct answer is:
2-butanol

Solution:
2- butanol has following structure

Topic: Chemistry, General Organic Chemistry, TS EAMCET, TS EAMCET 2004
Which one of the following is a secondary alcohol?

Options:
A 2-methyl-2-propanol
B 1-propanol
C 1-butanol
D 2-pentanol

The correct answer is:
2-pentanol

Solution:

Topic: Chemistry, General Organic Chemistry, AP EAMCET, AP EAMCET 2002
Which one of the following is a secondary alcohol?

Options:
A 2-methyl-1 propanol
B 2-methyl-2 propanol
C 2-butanol
D 1-butanol

The correct answer is:
2-butanol

Solution:
2- butanol has following structure

Topic: Chemistry, General Organic Chemistry, AP EAMCET, AP EAMCET 2004
Which one of the following is a secondary alcohol?

Options:
A 2-methyl-2-propanol
B 1-propanol
C 1-butanol
D 2-pentanol

The correct answer is:
2-pentanol

Solution:

Topic: Chemistry, General Organic Chemistry, TS EAMCET, TS EAMCET 2002
Which one of the following is an endangered plant species of India?

Options:
A Rauwolfia serpentina
B Santalum album (Sandal wood)
C Cycas beddomei
D All of the above

The correct answer is:
All of the above

Solution:
Endangered (ER): It is facing a high risk of extinction in the wild in the near future if conservation measures are not promptly taken.

Topic: Biology, Biodiversity and its Conservation, NEET
Which one of the following is an endangered plant species of India?

Options:
A
Rauwolfia serpentina

B
Santalum album (sandal wood)

C
Cycas beddonei

D
All of the above

The correct answer is:
All of the above

Solution:
All of the above plants are endangered plant species of India. Rauwolfia serpentina (sarpgandha), Santalum album (sandal wood) and Cycas beddonei are facing the threat of extinction because of their medicinal and commercial importance.

Topic: Biology, Biodiversity and its Conservation
Which one of the following is an example of non-empty
set ?

Options:
A Set of all even prime numbers
B $\left(x: x^{2}-2=0\right.$ and $x$ is rational)
C $\{x: x$ is a natural number, $x < 8$ and simultaneously $x>12$
D $\{x: x$ is a point common to any two parallel lines

The correct answer is:
Set of all even prime numbers

Solution:
$\mathrm{S}$ be the set of all even prime numbers $\mathrm{S}=2$ in an even prime number $=($ Non-empty set $)$

Topic: Mathematics, Sets and Relations, NDA, NDA 2013 (Phase 2)
Which one of the following is an infinite set?

Options:
A The set of human beings on the earth
B The set of water drops in a glass of water
C The set of trees in a forest
D The set of all primes

The correct answer is:
The set of all primes

Solution:
In the given sets, the set of all primesisan infiniteset.

Topic: Mathematics, Sets and Relations, NDA, NDA 2006 (Phase 2)
Which one of the following is correct? $\left(1+\cos 67 \frac{1^{\circ}}{2}\right)\left(1+\cos 112 \frac{1^{\circ}}{2}\right)$ is

Options:
A an irrational number and is greater than 1
B a rational number but not an integer
C an integer
D an irrational number and is less than $1$

The correct answer is:
an irrational number and is less than $1$

Solution:
The given expression $\left(1+\cos 67 \frac{1^{\circ}}{2}\right)\left(1+\cos 112 \frac{1^{\circ}}{2}\right)$
Can also be writters as :
$\left(1+\cos 67 \frac{1^{\circ}}{2}\right)\left\{1+\cos \left(180^{\circ}-67 \frac{1^{\circ}}{2}\right)\right\}$
$=\left(1+\cos 67 \frac{1^{\circ}}{2}\right)\left(1-\cos 67 \frac{1^{\circ}}{2}\right)$
$=1-\cos ^{2} 67^{\circ} \frac{1^{\circ}}{2}=\sin ^{2} 67 \frac{1^{\circ}}{2}$
$=\frac{1-\cos 135^{\circ}}{2}=\frac{\sqrt{2}+1}{2 \sqrt{2}} \quad\left(\because \sin ^{2} \mathrm{~A}=\frac{1-\cos 2 \mathrm{~A}}{2}\right)$
Which is an irrational number and is less than 1

Topic: Mathematics, Trigonometric Ratios & Identities, NDA, NDA 2008 (Phase 1)
Which one of the following is correct?

Options:
A $A \times(B-C)=(A-B) \times(A-C)$
B $A \times(B-C)=(A \times B)-(A \times C)$
C $A \cap(B \cup C)=(A \cap B) \cup C$
D $A \cup(B \cap C)=(A \cup B) \cap C$

The correct answer is:
$A \times(B-C)=(A \times B)-(A \times C)$

Solution:
We know that $A \times(B-C)=(A \times B)-(A \times C)$

Topic: Mathematics, Sets and Relations, NDA, NDA 2009 (Phase 1)
Which one of the following is correct?

Options:
A $\sin 1^{\circ}>\sin 1$
B $\sin 1^{\circ} < \sin 1$
C $\sin 1^{\circ}=\sin 1$
D $\sin 1^{\circ}=\frac{\pi}{180} \sin 1$

The correct answer is:
$\sin 1^{\circ} < \sin 1$

Solution:
We know that, $1^{\circ} < 1$ radian $\quad\left(\because 1^{\circ}=\frac{\pi}{180}\right.$ radian $)$ $\Rightarrow \sin 1^{\circ} < \sin 1$

Topic: Mathematics, Trigonometric Ratios & Identities, NDA, NDA 2010 (Phase 2)
Which one of the following is correct?

Options:
A $\mathrm{A} \cup \mathrm{P}(\mathrm{A})=\mathrm{P}(\mathrm{A})$
B $A \cap P(A)=A$
C $\mathrm{A}-\mathrm{P}(\mathrm{A})=\mathrm{A}$
D $\mathrm{P}(\mathrm{A})-\{\mathrm{A}\}=\mathrm{P}(\mathrm{A})$
Here $\mathrm{P}(\mathrm{A})$ denotes the power set of a set $\mathrm{A}$.

The correct answer is:
$\mathrm{A} \cup \mathrm{P}(\mathrm{A})=\mathrm{P}(\mathrm{A})$

Solution:
$\mathrm{A} \cup \mathrm{P}(\mathrm{A})=\mathrm{P}(\mathrm{A})$ is correct
Since $\mathrm{A}$ is a subset of its power set.

Topic: Mathematics, Sets and Relations, NDA, NDA 2008 (Phase 1)
Which one of the following is correct ?

Options:
A An event having no sample point is called an elementary event.
B An event having one sample point is called an elementary event.
C An event having two sample points is called an elementary event.
D An event having many sample points is called an elementary event.

The correct answer is:
An event having one sample point is called an elementary event.

Solution:
An event having one sample point is called an elementary event.

Topic: Mathematics, Probability, NDA, NDA 2012 (Phase 2)
Which one of the following is correct?

Options:
A $\mathrm{A} \cup(\mathrm{B}-\mathrm{C})=\mathrm{A} \cap(\mathrm{B} \cap \mathrm{C})$
B $\mathrm{A}-(\mathrm{B} \cup \mathrm{C})=\left(\mathrm{A} \cap \mathrm{B}^{\prime}\right) \cap \mathrm{C}^{\prime}$
C $\mathrm{A}-(\mathrm{B} \cap \mathrm{C})=\left(\mathrm{A} \cap \mathrm{B}^{\prime}\right) \cap \mathrm{C}$
D $\mathrm{A} \cap(\mathrm{B}-\mathrm{C})=(\mathrm{A} \cap \mathrm{B}) \cap \mathrm{C}$

The correct answer is:
$\mathrm{A}-(\mathrm{B} \cup \mathrm{C})=\left(\mathrm{A} \cap \mathrm{B}^{\prime}\right) \cap \mathrm{C}^{\prime}$

Solution:
Leta Venn-diagram be drawn taking three intersecting sets $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ under a universal set $\mathrm{U}$. This makes 8 regions a to $h$ as shown
A has regions $a, b, d, e$ B has regions $b, c, e, f$
regions $\mathrm{e}, \mathrm{t}, \mathrm{g}$
$\mathrm{C}^{\prime}$ has regions $\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{h}$
$\mathrm{B}^{\prime}$ has regions $\mathrm{a}, \mathrm{d}, \mathrm{g}, \mathrm{h}$
Statement $(\mathrm{a}): \mathrm{A} \cup(\mathrm{B}-\mathrm{C})=\mathrm{A} \cap(\mathrm{B} \cap \mathrm{C})$
$\mathrm{LHS}=(\mathrm{a}, \mathrm{b}, \mathrm{e}, \mathrm{d}) \cup \mathrm{b}, \mathrm{c}=\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}$
$\mathrm{RHS}=\mathrm{a}, \mathrm{b}, \mathrm{d}, \mathrm{e} \cap \mathrm{e}, \mathrm{f}=\mathrm{e}$
So, statement (a) is not correct. Statement(b): $A-(B \cup C)=\left(A \cap B^{\prime}\right) \cap C^{\prime}$
$\mathrm{LHS}=(\mathrm{a}, \mathrm{b}, \mathrm{d}, \mathrm{e})-(\mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}, \mathrm{g})=\mathrm{a}$
$\mathrm{RHS}=(\mathrm{a}, \mathrm{b}, \mathrm{d}, \mathrm{e} \cap \mathrm{a}, \mathrm{d}, \mathrm{g}, \mathrm{h}) \cap(\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{h})=\mathrm{a}$,
So, statement (b) is correct. Correct statement is $A-(B \cup C)=\left(A \cap B^{\prime}\right) \cap C$

Topic: Mathematics, Sets and Relations, NDA, NDA 2006 (Phase 1)
Which one of the following is correct?

Options:
A $\mathrm{As}_2 \mathrm{~S}_3$ is a positively charged sol.
B The refractive indices of the dispersed phase and the dispersion medium differ greatly in magnitude to observe Tyndall effect.
C Ultra microscope provides information about the size and shape of colloidal particles.
D Finest gold sol is green in colour.

The correct answer is:
The refractive indices of the dispersed phase and the dispersion medium differ greatly in magnitude to observe Tyndall effect.

Solution:
No solution. Refer to answer key.

Topic: Chemistry, Surface Chemistry, AP EAMCET, AP EAMCET 2017 (26 Apr Shift 2)
Which one of the following is correct? If $4 < x^{2} < 9$, then

Options:
A $2 < x < 3$ only
B $-3 < x < -2$ only
C $2 < x < 3,-3 < x < -2$
D None of these

The correct answer is:
$2 < x < 3,-3 < x < -2$

Solution:
In the given inequality $\because 4 < x^{2} < 9$
We consider $x^{2}>4$ and $x^{2} < 9$
We first consider, $x^{2}>4 \Rightarrow x^{2}-4>0$
$\Rightarrow x>-2$ and $x < 2$
Next; $\mathrm{x}^{2} < 9$
$\Rightarrow x^{2}-9 < 0 \quad \Rightarrow-3 < x < 3$
Combining both we get $-3 < x < -2,2 < x < 3$ We represent this on number line
$\Rightarrow-3 < x < -2$ and $2 < x < 3$

Topic: Mathematics, Basic of Mathematics, NDA, NDA 2007 (Phase 2)