datavorous/entrance-exam-dataset huggingface dataset:
In which one of the following preparations are you likely to come across cell junctions most frequently?

Options:
A thrombocytes
B tendon
C hyaline cartilage
D ciliated epithelium.

The correct answer is:
ciliated epithelium.

Solution:
Cell junctions are especially abundant in epithelial tissues. They consist of protein complexes and provide contact between neighbouring cells, between a cell and the extracellular matrix, or they built up the paracellular barrier of epithelia and control the paracellular transport. The other three (thrombocytes, tendon and hyaline cartilage) belongs to connective tissue. Thrombocytes or platelets aid the formation of blood clots by releasing various protein substances. Tendon attaches a muscle to a bone and hyaline cartilage gives flexibility and support at the joints.
Related Theory
Ciliated epithelium is a region of epithelium consisting of columnar or cuboidal cells bearing hairlike appendages that are capable of beating rapidly. Ciliated epithelium performs the function of moving particles or fluid over the epithelial surface in such structures as the trachea, bronchial tubes, and nasal cavities. It ofien occurs in the vicinity of mucus-secreting goblet cells.

Topic: Biology, Structural Organisation in Animals, NEET, NEET 2007
In which one of the following the genus name, its two characters and its phylum are incorrectly matched?

Options:
A

Genus name	Two characters

a)	Pila	Body Segmented Mouth with radula	Mollusca

B

Genus name	Two characters

b)	Asterias	Spiny Skinned Water vascular system	Echinoermata

C

Genus name	Two characters

c)	Sycon	Pore bearing Canal system	Porifera

D

Genus name	Two characters

d)	Periplaneta	Jointed appendages Chitinous exoskeleton	Arthropoda

The correct answer is:

Genus name	Two characters

a)	Pila	Body Segmented Mouth with radula	Mollusca

Solution:
Molluscs are soft-bodied animals, bilaterally symmetric, triploblastic and coelomate. Their soft body is covered by a calcareous shell and is unsegmented with a distinct head, muscular foot and visceral hump.
e.g. Pila (apple snail), sepia (cuttle fish), pinctada (pearl oyster), etc.

Topic: Biology, Biological Classification, NEET
In which part of the plant is Potassium required in more abundant quantities?

Options:
A Meristematic tissues
B Roots
C Flowers
D Fruits

The correct answer is:
Meristematic tissues

Solution:
In plants, Potassium is required in more abundant quantities in the meristematic tissues, buds, leaves and root tips

Topic: Biology, Mineral Nutrition, NEET
In which region of pBR322, 5' GAATTC 3' sequence will be present?

Options:
A rop
B tetR
C ampR
D none of these

The correct answer is:
none of these

Solution:
Bolivar and Rodriguez inserted genes for antibiotic resistance in a plasmid from E.coli to form a new plasmid called pBR 322. pBR322 is one of the standard cloning vectors widely used in gene cloning experiments. pBR322 has the following regions ori: Region for origin for replication. rop: This region codes for the proteins involved in the replication of plasmid. This region also has recognition sequence for restriction endonuclease Pvu II (Proteus vulgaris). tetR - This region provides resistance against the antibiotic tetracycline. This region also has recognition sequence for restriction endonucleases: Bam H I from Bacillus amyloliquefaciens and Sal I from Streptomyces albus. ampR - This region provides resistance against the antibiotic ampicillin. This region also has recognition sequence for restriction endonucleases: Pvu I from Proteus vulgaris and Pst I from Providencia stuarti. Apart from these, pBR 322 also shows recognition sequence for restriction endonucleases: EcoR I (its recognition site is 5' GAATTC 3') from Escherichia coli, Hind III from Haemophilus influenza and Cla I from Caryophanon latum.

Topic: Biology, Biotechnology - Principles and Processes, NEET
$\int(\operatorname{In} x)^{-1} \mathrm{~d} x-\int(\operatorname{In} x)^{-2} \mathrm{~d} x$ is equal to

Options:
A $x(\operatorname{In} x)^{-1}+c$
B $x(\operatorname{In} x)^{-2}+c$
C $x(\operatorname{In} x)+c$
D $x(\ln x)^{2}+c$

The correct answer is:
$x(\operatorname{In} x)^{-1}+c$

Solution:
$\int(\ln x)^{-1} \mathrm{dx}-\int(\ln x)^{-2} \mathrm{dx}$
$=\int\left[\frac{1}{\ell^{n} x}-\frac{1}{(\ell n x)^{2}}\right] \cdot d x$
Put $\ell \mathrm{n} \mathrm{x}=\mathrm{t} \Rightarrow \mathrm{x}=\mathrm{e}^{\mathrm{t}}$
$\mathrm{dx}=\mathrm{e}^{\mathrm{l}} \cdot \mathrm{dt}$
$\therefore \int\left[\frac{1}{\ln x}-\frac{1}{(\ell n x)^{2}}\right] \mathrm{dx}=\int\left(\frac{1}{t}-\frac{1}{t^{2}}\right) \cdot \mathrm{e}^{\mathrm{t}} \cdot \mathrm{dt}$
$=\int \mathrm{e}^{\mathrm{t}} \cdot\left(\frac{1}{\mathrm{t}}-\frac{1}{\mathrm{t}^{2}}\right) \mathrm{dt}$
$=\frac{\mathrm{e}^{\mathrm{t}}}{\mathrm{t}}+\mathrm{c}=\frac{\mathrm{x}}{\ln \mathrm{x}}+\mathrm{c}$
$=x(\ln x)^{-1}+c$

Topic: Mathematics, Indefinite Integration, NDA, NDA 2017 (Phase 2)
Industrial melanism is an example of:

Options:
A Natural selection
B Mutation
C Neo Lamarckism
D Neo Darwinism

The correct answer is:
Natural selection

Solution:
One of the most striking examples, which demonstrates the action of natural selection, is the industrial melanism in England. The peppered moth Biston betularia, with a dull grey colour, was abundant in England before the industrial revolution. A black coloured form of the same moth (melanic, a dominant mutant differing in a single gene), Biston carbonaria, was very rare. After industrial revolution, the proportion of Biston carbonaria, increased to almost 90 percent.

Topic: Biology, Evolution, NEET, NEET 2015 (Phase 2)
Initiation of lateral roots and vascular cambium during secondary growth takes place in cells of :

Options:
A Epiblema
B Cortex
C Endodermis
D Pericycle

The correct answer is:
Pericycle

Solution:
In Dicot root, the vascular bundles are radial, in which vascular cambium is purely secondary and arise from pericycle with lateral roots as well. Lateral roots thus have endogenous origin.

Topic: Biology, Anatomy of Flowering Plants, NEET, NEET 2022 (Phase 2)
Inner lining of blood vessels is formed by

Options:
A ciliated epithelium
B squamous epithelium
C cubical epithelium
D columnar epithelium

The correct answer is:
squamous epithelium

Solution:
Inner lining of blood vessels is formed by squamous epithelium. Spuamous epithelium consists of a layer of thin, flat, scale-like cells with prominent nuclei. The cell have irregular boundaries that fit closely into those of neighbouring cells.

Topic: Biology, Structural Organisation in Animals, JIPMER, JIPMER 2005
Interfascicular cambium and cork cambium are formed due to

Options:
A
cell division

B
cell differentiation

C
cell dedifferentiation

D
redifferentiation

The correct answer is:
cell dedifferentiation

Solution:
Dedifferentiation is the phenomenon of regeneration of permanent tissue to become meristematic. Cork cambium, wound cambium and interfascicular vascular cambium are the examples of secondary meristems which are always produced through dedifferentiation.

Topic: Biology, Anatomy of Flowering Plants
Interfascicular cambium is present between :

Options:
A Primary xylem and primary phloem
B Pericycle and endodermis
C Two vascular bundles
D Secondary xylem and secondary phloem

The correct answer is:
Two vascular bundles

Solution:
Inter fascicular cambium is present between two vascular bundles and formed from de differentiation of permanent tissues in medullary rays. Interfascicular cambium is present in between primary xylem and primary phloem.

Topic: Biology, Anatomy of Flowering Plants, NEET, NEET 2022 (Phase 2)
______ is a couple of layers thick, made of narrow, thin walled and nearly rectangular cells. It cuts off cells on both sides. The outer cells differentiate into __________ or __________ while the inner differentiate into _________ or ____________. The three together is called ____________.

Select the right sequence of words to complete the above.

Options:
A Phellogen, cork, phellem, secondary cortex, phelloderm, periderm
B Phellem, phelloderm, cork, secondary cortex, phellogen, periderm
C Phellogen, cork, periderm, secondary cortex, phellem, phellem
D Phellem, secondary cortex, periderm, cork, phellogen, phelloderm

The correct answer is:
Phellogen, cork, phellem, secondary cortex, phelloderm, periderm

Solution:
Phellogen is thick with a couple of layers. It is made of narrow, thin-welled and nearly rectangular cells. Phellogen cuts off cells on both sides. The outer cells differentiate into cork or phellem, whereas the inner cells differentiate into secondary cortex or phelloderm. The cork is impervious to water due to suberin deposition in the cell wall. The cells of the secondary cortex are parenchymatous. Phellogen, phellem and phelloderm are collectively known as periderm.

Topic: Biology, Anatomy of Flowering Plants, NEET
_________ is a taxonomic category and identified on the basis of aggregates of characters.

Options:
A Species
B Genus
C Family
D Order

The correct answer is:
Order

Solution:
Order and above taxonomic categories based on aggregate of characters.

Order	Group of closely related families.
Family	Group of closely related genera.
Genus	Group of closely related species.
Species	Group of individuals with fundamental similarities

Topic: Biology, The Living World, NEET
Is adding a scalar to a vector of the same dimensions a meaningful algebraic operation?

Options:
A Yes, because it results in a new vector with the same dimensions.
B Yes, because it results in a scalar quantity.
C No, because scalar and vector quantities cannot be combined algebraically.
D No, because the dimensions of scalar and vector quantities are fundamentally different.

The correct answer is:
No, because scalar and vector quantities cannot be combined algebraically.

Solution:
Correct Option is : (C)
No, because scalar and vector quantities cannot be combined algebraically.

Topic: Chemistry, Motion In Two Dimensions, NEET
Is competition always based on limiting resources?

Options:
A No
B Yes
C Sometimes
D Only in laboratory experiments

The correct answer is:
No

Solution:
Competition need not be based on limiting resources as in interference competition, the feeding efficiency of one species might be reduced due to the interfering and inhibitory presence of the other species, even if resources (food and space) are abundant.

Topic: Biology, Organisms and Populations, NEET
_________ is defined as an ecological state of a species being unique to a specific geographic location.

Options:
A Exotic species
B Endemic species
C Ecosystem
D Endagered species

The correct answer is:
Endemic species

Solution:
A species which is located to a specific geographic area or location is called as endemic species. Species which is brought form other geographical is called as endemic species.

Topic: Biology, Biodiversity and its Conservation, NEET
Is rubber a primary metabolite or a secondary metabolite? Write four sentences about rubber.

Options:

None

Solution:
Secondary metabolites are chemicals produced by plants which do not play any [role] in growth, photosynthesis reproduction or other primary functions of the plant. Rubber (cis 1, 4-polyisopyrene) is a secondary metabolite.
(i) Rubber is extracted from Hevea brasiliensis (rubber tree)
(ii) It is a byproduct of the lactiferous tissue of the vessels that are in the form of latex.
(iii) It contains over 400 isoprene units and thus is the largest of the terpenoids.
(iv) It is elastic, water proof and a good conductor of electricity.

Topic: Biology, Biomolecules
Iso-propyl amine is a

Options:
A primary amine
B secondary amine
C tertiary amine
D quaternary amine

The correct answer is:
primary amine

Solution:
The structure of iso-propyl amine is
It is a primary amine.

Topic: Chemistry, General Organic Chemistry, MHT CET, MHT CET 2009
It has been found that if $A$ and $B$ play a game 12 times, $A$ wins 6 times, $B$ wins 4 times and they draw twice. $A$ and $B$ take part in a series of 3 games. The probability that they win alternately, is :

Options:
A $5 / 12$
B $5 / 36$
C $19 / 27$
D $5 / 27$

The correct answer is:
$5 / 36$

Solution:
$\mathrm{P}(\mathrm{A})=\frac{6}{12}=\frac{1}{2}, \mathrm{P}(\mathrm{B})=\frac{4}{12}=\frac{1}{3}$
Req. probability $=\frac{1}{2} \cdot \frac{1}{3} \cdot \frac{1}{2}+\frac{1}{3} \cdot \frac{1}{2} \cdot \frac{1}{3}=\frac{1}{12}+\frac{1}{18}=\frac{5}{36}$

Topic: Mathematics, Probability, NDA, NDA 2014 (Phase 1)
It is claimed that two cesium clocks, if allowed to run for 100 years, free from any disturbance, may differ by only about $0.025 \mathrm{~s}$. What does this imply for the accuracy of the standard cesium clock in measuring a time interval of $1 \mathbf{s}$ ?

Options:

None

Solution:
Error in 100 years $=0.02 \mathrm{~s}$
$\therefore \quad$ Error in $1 \mathrm{sec}$
$$
\begin{aligned}
& \frac{\Delta t}{t}=\frac{0.02}{100 \times 365 \times 2.5 \times 24 \times 60 \times 60}=\frac{0.02}{3.5576 \times 10^9} \\
=& 0.0063 \times 10^{-9}=0.63 \times 10^{-11}
\end{aligned}
$$
So, there is an accuracy of $1 \mathrm{~s}$ is $10^{-11} \mathrm{~s}$.

Topic: Physics, Units and Dimensions
It is given that $\cos (\theta-u)-a, \cos (\theta-\beta)-\mathrm{b}$
What is $\cos (\alpha-\beta)$ equal to ?

Options:
A $a b+\sqrt{1-a^{2}} \sqrt{1-b^{2}}$
B $a b-\sqrt{1-a^{2}} \sqrt{1-b^{2}}$
C $a \sqrt{1-b^{2}}-b \sqrt{1-a^{2}}$
D $a \sqrt{1-b^{2}}+b \sqrt{1-a^{2}}$

The correct answer is:
$a b+\sqrt{1-a^{2}} \sqrt{1-b^{2}}$

Solution:
Given, $\cos (\theta-\alpha)=\mathrm{a} \Rightarrow \sin (\theta-\alpha)=\sqrt{1-a^{2}}$
$$
\begin{array}{l}
\cos (\theta-\beta)=\mathrm{b} \Rightarrow \sin (\theta-\beta)=\sqrt{1-b^{2}} \\
\therefore \cos (\alpha-\beta)=\cos (\theta-\beta-(\theta-\alpha)) \\
=\cos (\theta-\beta) \cos (\theta-\alpha)+\sin (\theta-\beta) \sin (\theta-\alpha) \\
=(b)(a)+\sqrt{1-b^{2}} \sqrt{1-a^{2}} \\
=a b+\sqrt{1-a^{2}} \sqrt{1-b^{2}}
\end{array}
$$

Topic: Mathematics, Trigonometric Ratios & Identities, NDA, NDA 2019 (Phase 1)
It is given that $\cos (\theta-u)-a, \cos (\theta-\beta)-\mathrm{b}$
What is $\sin ^{2}(\alpha-\beta)+2 a b \cos (\alpha-\beta)$ equal to ?

Options:
A $a^{2}+b^{2}$
B $a^{2}-b^{2}$
C $b^{2}-a^{2}$
D $-\left(a^{2}+b^{2}\right)$

The correct answer is:
$a^{2}+b^{2}$

Solution:
$\begin{aligned} & \sin ^{2}(\alpha-\beta)+2 a b \cos (\alpha-\beta)=1-\cos ^{2}(\alpha-\beta)+2 a b \cos \\ &(\alpha-\beta)=1-\cos (\alpha-\beta)[\cos (\alpha-\beta)-2 a b] \\ &=1-\left(a b+\sqrt{1-a^{2}} \sqrt{1-b^{2}}\right) \\ & \quad\left[a b+\sqrt{1-a^{2}} \sqrt{1-b^{2}}-2 a b\right] \\ &=1-\left(\sqrt{1-a^{2}} \sqrt{1-b^{2}}+a b\right)\left(\sqrt{1-a^{2}} \sqrt{1-b^{2}}-a b\right) \\ &=1-\left[\left(\sqrt{1-a^{2}} \sqrt{1-b^{2}}\right)^{2}-(a b)^{2}\right] \\ &=1-\left[\left(1-a^{2}\right)\left(1-b^{2}\right)-a^{2} b^{2}\right) \\ &=1-\left(1-b^{2}-a^{2}+a^{2} b^{2}-a^{2} b^{2}\right) \\ &=1-1+b^{2}+a^{2}=a^{2}+b^{2}

Topic: Mathematics, Trigonometric Ratios & Identities, NDA, NDA 2019 (Phase 1)
It is given that the roots of the equation $x^{2}-4 x-\log _{3} P=$ are real. For this, the minimum value of $\mathrm{P}$ is

Options:
A $\frac{1}{27}$
B $\frac{1}{64}$
C $\frac{1}{81}$
D 1

The correct answer is:
$\frac{1}{81}$

Solution:
$x^{2}-4 x-\log _{3} P=0$
We know, roots are real if discriminant is greater than or equal to 0 . i.e., $b^{2}-4 a c \geq 0 \Rightarrow b^{2} \geq 4 a c$
In the given equation, $\mathrm{a}=1, \mathrm{~b}=-4, \mathrm{c}=-\log _{3} \mathrm{P}$.
$\therefore b^{2} \geq 4 a c \Rightarrow(-4)^{2} \geq 4(1)\left(-\log _{3} P\right)$
$\Rightarrow 16 \geq-4 \log _{3} \mathrm{P}$
$\Rightarrow 4 \geq-\log _{3} P$
$\Rightarrow 4 \geq \log _{3}\left(\frac{1}{\mathrm{P}}\right) \quad\left(\because \sin \theta-\log a=\log \frac{1}{\mathrm{a}}\right)$
$\Rightarrow 3^{4} \geq \frac{1}{\mathrm{P}}$
$\Rightarrow 81 \geq \frac{1}{\mathrm{P}} \Rightarrow \mathrm{P} \geq \frac{1}{81}$
$\therefore$ The minimum value of $\mathrm{P}$ is $\frac{1}{81}$

Topic: Mathematics, Quadratic Equation, NDA, NDA 2017 (Phase 2)
It is given that $\overline{\mathrm{X}}=10, \overline{\mathrm{Y}}=90, \sigma_{\mathrm{X}}=3, \sigma_{\mathrm{Y}}=12$ and
$\mathrm{I}_{\mathrm{XY}}=0.8$. The regression equation of $\mathrm{X}$ on $\mathrm{Y}$ is

Options:
A $\quad Y=3.2 X+58$
B $\mathrm{X}=3.2 \mathrm{Y}+58$
C $\quad X=-8+0.2 Y$
D $Y=-8+0.2 X$

The correct answer is:
$\quad X=-8+0.2 Y$

Solution:
$\overline{\mathrm{X}}=10, \overline{\mathrm{Y}}=90, \sigma_{\mathrm{x}}=3, \sigma_{\mathrm{y}}=12, \mathrm{r}_{\mathrm{xy}}=0.8$
Regression equation $\mathrm{x}$ on $\mathrm{y}$ is
$\mathrm{x}-10=\mathrm{r} \cdot \frac{\sigma_{\mathrm{x}}}{\sigma_{\mathrm{y}}}(\mathrm{y}-90)$
$\Rightarrow x-10=0.8 \times \frac{3}{12}(y-90)$
$\Rightarrow x-10=\frac{2.4}{12}(y-90)$
$10=0.2(\mathrm{y}-90)$
$\Rightarrow x-10=0.2 y-18$
$\Rightarrow x=0.2 y-8$

Topic: Mathematics, Statistics, NDA, NDA 2017 (Phase 2)
It the sum of first 10 terms of an arithmetic progression with first term $p$ and common difference $q$, is 4 times the sum of the first 5 terms, then what is the ratio $p: q$ ?

Options:
A $1: 2$
B $1: 4$
C $2: 1$
D $4: 1$

The correct answer is:
$1: 2$

Solution:
Since first term $=\mathrm{p}$ and common difference $=\mathrm{q}$. Sum of first 10 terms $=\frac{10}{2}[2 \mathrm{p}+(10-1) \mathrm{q}]$ and
Sum of first 5 terms $=\frac{5}{2}[2 \mathrm{p}+(5-1) \mathrm{q}]$ According to question, $\frac{10}{2}[2 p+9 q]=4 \times \frac{5}{2}[2 p+4 q]$
$\Rightarrow 2 \mathrm{p}+9 \mathrm{q}=4 \mathrm{p}+8 \mathrm{q}$
$\Rightarrow 2 \mathrm{p}=\mathrm{q}$
$\Rightarrow \mathrm{p}: \mathrm{q}=1: 2$

Topic: Mathematics, Sequences and Series, NDA, NDA 2006 (Phase 1)
Jute fibres are obtained from the:

Options:
A Secondary phloem
B Pith
C Xylem
D Endodermis

The correct answer is:
Secondary phloem

Solution:
Jute fibers are bast fibers or phloem fibers. Secondary phloem forms the fibre which are long and are commercially important. Other fibers like flax and hemp are also phloem fibers.

Topic: Biology, Anatomy of Flowering Plants, NEET
Jute is a

Options:
A bast fibre from secondary xylem
B bast fibre from primary xylem
C bast fibre from secondary phloem
D bast fibre from primary phloem

The correct answer is:
bast fibre from secondary phloem

Solution:
Fibres are elongated cells with usually lignified walls. Bast fibres from phloem is called jute. There are secondary phloem cells.

Topic: Biology, Anatomy of Flowering Plants, JIPMER, JIPMER 2015
Ketones
where, $R=R^{\prime}=$ alkyl group can be obtained in one step by

Options:
A Hydrolysis of esters
B Oxidation of primary alcohols
C Oxidation of secondary alcohols
D Reaction of acid halide with alcohols

The correct answer is:
Oxidation of secondary alcohols

Solution:
Oxidation of Ketones, yield secondary alcohol

Topic: Chemistry, Aldehydes and Ketones, VITEEE, VITEEE 2013
\(K_p\) for the conversion of oxygen to ozone at \(400 \mathrm{~K}\) is \(1.0 \times 10^{-30}\), its standard Gibbs energy change in \(\mathrm{kJ} \mathrm{mol}^{-1}\) is approximately

Options:
A 229.8
B 114.9
C -229.8
D -114.9

The correct answer is:
229.8

Solution:
Given, \(K_p\) for the conversion of oxygen to ozone at \(400 \mathrm{~K}\) is \(1.0 \times 10^{-30}\).
\(\left(\Delta G^{\circ}\right)\) standard Gibbs energy change \(=\) ?
\(\Delta G^{\circ}=-R T \ln K_p=2.303 R T \log _{10} K_p\)
Here,
\(R=8.314 \mathrm{~J} / \mathrm{mol}\)
\(\begin{aligned}
T & =400 \mathrm{~K} \\
\Delta G^{\circ} & =-R T \log K_p \\
& =-8.314 \times 400 \times 2.303 \times \log 10^{-30} \\
& =22976570 \mathrm{~J} \mathrm{~mol}^{-1} \\
& =229.765 \mathrm{~kJ} \mathrm{~mol}^{-1}
\end{aligned}\)

Topic: Chemistry, Chemical Equilibrium, AP EAMCET, AP EAMCET 2019 (22 Apr Shift 1)
LAB stands for

Options:
A Lactic acid bacteria
B Lactobacillus basophilus bacteria
C Lactose acetaldehyde bacteria
D Laboratory

The correct answer is:
Lactic acid bacteria

Solution:
Although many genera of bacteria produce lactic acid as a primary or secondary end-product of fermentation, the term Lactic acid bacteria is conventionally reserved for genera in the order Lactobacillales, which includes Lactobacillus, Leuconostoc, Pediococcus, etc. Lactic acid bacteria (LAB) are responsible for a great diversification in the flavor and texture of food products due to their fermentation of food raw materials. However, in some circumstances, they can be responsible for food spoilage.

Topic: Biology, Microbes in Human Welfare, NEET
Lateral roots and vascular cambium originates due to the activity of which tissue in dicot root?

Options:
A Endodermis
B Pericycle
C Casparian strip
D Periderm

The correct answer is:
Pericycle

Solution:
In dicot root, lateral root originates endogenously due to the activity of cells of pericycle, which undergoes dedifferentiation and becomes meristematic. During secondary growth cells of pericycle and the parenchyma cells of conjuctive tissue becomes meristematic and forms vascular cambium.

Topic: Biology, Anatomy of Flowering Plants, NEET
\(\left|\begin{array}{lll}a-b & b-c & c-a \\ b-c & c-a & a-b \\ c-a & a-b & b-c\end{array}\right|=\)

Options:
A 0
B -1
C 1
D 2

The correct answer is:
0

Solution:
Hints : \(\mathrm{c}_1 \rightarrow \mathrm{c}_1+\mathrm{c}_2+\mathrm{c}_3\)

Topic: Mathematics, Determinants, WBJEE, WBJEE 2011
Let $0^{\circ} < \theta < 45^{\circ}$. Which one of the following is correct?

Options:
A $\sin ^{2} \theta+\cos ^{6} \theta=\sin ^{6} \theta+\cos ^{2} \theta$
B $\operatorname{cosec}^{2} \theta+\cot ^{6} \theta=\operatorname{cosec}^{6} \theta+\cot ^{2} \theta$
C $\sin ^{2} \theta-\cos ^{4} \theta=\sin ^{4} \theta+\cos ^{2} \theta$
D $\operatorname{cosec}^{2} \theta+\cot ^{4} \theta=\operatorname{cosec}^{4} \theta+\cot ^{2} \theta

The correct answer is:
$\operatorname{cosec}^{2} \theta+\cot ^{4} \theta=\operatorname{cosec}^{4} \theta+\cot ^{2} \theta

Solution:
(a) In such a problem, we have to check option, oneby-one $\begin{aligned} & \sin ^{2} \theta+\cos ^{6} \theta=\sin ^{6} \theta+\cos ^{2} \theta \\ \Rightarrow & \sin ^{6} \theta-\cos ^{6} \theta=\sin ^{2} \theta-\cos ^{2} \theta \end{aligned}$
Taking L.H.S, $\sin ^{6} \theta-\cos ^{6} \theta=\left(\sin ^{2} \theta\right)^{3}-\left(\cos ^{2} \theta\right)^{3}$
Which is not equal to $\mathrm{R}$.H.S., $\sin ^{2} \theta-\cos ^{2} \theta$
Option (a) is not correct.
(b) $\because \operatorname{cosec}^{6} \theta-\cot ^{6} \theta$
$\left(\operatorname{cosec}^{2} \theta-\cot ^{2} \theta\right)\left[\left(\operatorname{cosec}^{2} \theta-\cot ^{2} \theta\right)^{2}+(\operatorname{cosec} \theta \cot \theta)\right]$
Which is not equal to $\operatorname{cosec}^{2} \theta-\cot ^{2} \theta$.
$\therefore$ Option (b) is also not correct.
(c) $\sin ^{4} \theta+\cos ^{4} \theta=\left(\sin ^{2} \theta+\cos ^{2} \theta\right)^{2}-2 \sin ^{2} \theta \cos ^{2} \theta$
$=1-2 \sin ^{2} \theta \cos ^{2} \theta .$
Which is not equal to $\sin ^{2} \theta-\cos ^{2} \theta$.
Hence, option (c) is also not correct. (d) $\begin{aligned} &\left(\operatorname{cosec}^{2} \theta+\cot ^{4} \theta\right) \\=& \operatorname{cosec}^{2} \theta+\left(\operatorname{cosec}^{2} \theta-1\right)^{2} \\ &=\operatorname{cosec}^{2} \theta+\operatorname{cosec}^{4} \theta+1-2 \operatorname{cosec}^{2} \theta \\ &=\operatorname{cosec}^{4} \theta+1-\operatorname{cosec}^{2} \theta \\ &=\operatorname{cosec}^{4} \theta+\cot ^{2} \theta \end{aligned}$
Thus option (d) is correct.

Topic: Mathematics, Trigonometric Equations, NDA, NDA 2007 (Phase 2)
Let $\sigma_1, \sigma_2$ be the standard deviations of two distributions $D_1$ and $D_2$ respectively and $D_1$ be more consistent than $D_2$. If the means of $D_1$ and $D_2$ are same, then the percentage increase in the standard deviation of $D_2$ over the standard deviation of $D_1$ is

Options:
A $\frac{\sigma_1-\sigma_2}{\sigma_2} \times 100$
B $\frac{\sigma_1-\sigma_2}{\sigma_1} \times 100$
C $\frac{\sigma_2-\sigma_1}{\sigma_2} \times 100$
D $\frac{\sigma_2-\sigma_1}{\sigma_1} \times 100$

The correct answer is:
$\frac{\sigma_2-\sigma_1}{\sigma_1} \times 100$

Solution:
Since mean of both the distribution are same, therefore
Percentage increase in the standard deviation of $D_2$ over the standard deviation of $D_1$
$$
=\frac{\sigma_2-\sigma_1}{\sigma_1} \times 100
$$

Topic: Mathematics, Statistics, TS EAMCET, TS EAMCET 2018 (07 May Shift 1)