datavorous/entrance-exam-dataset huggingface dataset:
Consider the following statements $A$ is relative error in the area of a square when the relative error in its side is 0.4 $B$ is relative error in the volume of a sphere when the relative error in its radius is 0.3 $C$ is relative error in the surface area of a closed cylinder whose height is equal to its radius, when the relative error in its height is 0.2 $D$ is approximate error in $y=x^2+x-3$ when $x=2$ and $\delta x=0.1$
The ascending order of the values of errors in these statements is

Options:
A $B, C, A D$
B $A, C, B, D$
C $C, D, A, B$
D $D, A, C, B$

The correct answer is:
$C, D, A, B$

Solution:
Calculation of $A$
Let $S$ be the area of square of side $a$
$$
\begin{aligned}
& \therefore \quad S=a^2 \\
& \Rightarrow \quad \Delta S=\frac{d s}{d a} \times \Delta a \Rightarrow \Delta S=2 a \times \Delta a \\
& \Rightarrow \quad \frac{\Delta S}{S}=\frac{2 a \times \Delta a}{S} \Rightarrow \frac{\Delta S}{S}=\frac{2 a \times \Delta a}{a^2} \\
& =\quad \frac{\Delta a}{a}=2 \times 0.4=0.8 \quad\left[\therefore \frac{\Delta a}{a}=0.4\right] \\
& \therefore \quad A=0.8 \\
&
\end{aligned}
$$
Calculation of $B$
Let $V$ be the volume of the sphere of radius $r$.
$$
\therefore \quad V=\frac{4}{3} \pi r^3
$$
$$
\begin{aligned}
& \Rightarrow \quad \Delta V=\frac{d V}{d r} \times \Delta r \Rightarrow \Delta V=4 \pi r^2 \times \Delta r \\
& \Rightarrow \quad \frac{\Delta V}{V}=\frac{4 \pi r^2 \times \Delta r}{V} \\
& \begin{array}{r}
\Rightarrow \quad \frac{\Delta V}{V}=\frac{4 \pi r^2 \times \Delta r}{\frac{4}{3} \pi r^3}=\frac{3 \Delta r}{r}=3 \times 0.3=0.9 \\
{[\because \Delta r}
\end{array} \\
& {\left[\because \frac{\Delta r}{r}=0.3\right]} \\
& \therefore \quad B=0.9 \\
&
\end{aligned}
$$
Calculation of $C$
Let $S^{\prime}$ be the surface area of closed cylinder with radius $r$ and height $h$
$$
\begin{aligned}
& \therefore \quad S^{\prime}=2 \pi r h+2 \pi r^2 \\
& \Rightarrow \quad S^{\prime}=2 \pi h^2+2 \pi h^2 \\
& {[\because r=h]} \\
& \Rightarrow \quad S^{\prime}=4 \pi h^2 \\
& \Rightarrow \quad \Delta S^{\prime}=\frac{d S^{\prime}}{d h} \cdot \Delta h \\
& \Rightarrow \quad \Delta S^{\prime}=8 \pi h \times \Delta h \\
& \Rightarrow \quad \frac{\Delta S^{\prime}}{S}=\frac{8 \pi h \times \Delta h}{S} \\
& \Rightarrow \quad \frac{\Delta S^{\prime}}{S}=\frac{8 \pi h \times \Delta h}{4 \pi h^2} \\
& =\frac{2 \Delta h}{h}=2 \times 0.2=0.4 \quad\left[\because \frac{\Delta h}{h}=0.2\right] \\
& \therefore \quad c=0.4 \\
&
\end{aligned}
$$
Calculation of $D$
We have, $y=x^2+x-3 \Rightarrow \frac{d y}{d x}=2 x+1$
Now, $\quad \Delta y=\frac{d y}{d x} \Delta x \quad \Rightarrow \Delta y=(2 x+1) \Delta x$
$$
\begin{array}{lcc}
\Rightarrow & \Delta y=(2 \times 2+1) \times 01 & {[\because x=2 \Delta x=01]} \\
\Rightarrow & \Delta y=5 \times 01=0.5 \\
\therefore & \quad D=0.5 \\
\therefore & C < D < A < B .
\end{array}
$$

Topic: Mathematics, Application of Derivatives, JEE Main
Consider the following statements:
A: Rishi is a judge.
B: Rishi is honest.
C: Rishi is not arrogant.

The negation of the statement "if Rishi is a judge and he is not arrogant, then he is honest" is

Options:
A B→A∨C
B ~B∧A∧C
C B→~A∨~C
D B→A∧C

The correct answer is:
~B∧A∧C

Solution:
if Rishi is a judge and he is not arrogant, then he is honest can be written as A∧C→B

Now, Negation of A∧C→B is ~A∧C→B

=~~A∧C∨B {Using De-Morgan's law}

=A∧C∧~B

Topic: Mathematics, Mathematical Reasoning, JEE Main, JEE Main 2022 (24 Jun Shift 2)
Consider the following statements :

(A) The principal quantum number 'n' is a positive integer with values of 'n' =1,2,3,…

(B) The azimuthal quantum number 'l' for a given 'n' (principal quantum number) can have values as 'l' =0,1,2,….n has 2 n+1 values.

(C) Magnetic orbital quantum number '$\mathrm{m}$' for a particular '$\mathrm{l}$' (azimuthal quantum number) has $(2\mathrm{l}+1)$ values.

(D) ±12 are the two possible orientations of electron spin.

(E) For l=5, there will be a total of 9 orbital

Which of the above statements are correct?

Options:
A A,B & C
B A,C,D& E
C A,C & D
D A,B,C & D

The correct answer is:
A,C & D

Solution:
A) Number of values of -n=1,2,3....∞

B) Number of values of l=0 to (n-1)

C) Number of values of m=-l to +l

Total values=2l+1

D)The two possible orientations of electronic spin, which are spin quantum numbers are +12 and -12

E) For l=5 total number of orbital is 2l+1=2x5+1=11

Topic: Chemistry, Structure of Atom, JEE Main, JEE Main 2022 (28 Jun Shift 2)
Consider the following statements.

A. The productivity and distribution of plants have nothing to do with water.
B. Next to temperature, water is the most important factor influencing the life of organisms.
C. Mammals from colder climates generally have longer ears and limbs to minimize heat loss.
D. Life on Earth originated in water and is unsustainable without water.

Of the above statements:

Options:
A A and B are correct
B A and D are correct
C B and C are correct
D B and D are correct

The correct answer is:
B and D are correct

Solution:
Water is indispensable for every living organism on earth. It is used for transportation of nutrients, for photosynthesis, for maintaining shape & structure and also used as natural habitat by certain species. It plays a critical role in all the vital processes carried out by organisms. Mammals from colder climates generally have shorter ears and limbs to minimise heat loss. This is called the Allen’s rule.

Topic: Biology, Organisms and Populations, NEET
Consider the following statements about biomedical technologies
(a) During open heart surgery blood is circulated in the heart-lung machine
(b) Blockage in coronary arteries is removed by angiography
(c) Computerised Axial Tomography (CAT) shows detailed internal structure as seen in a section of body
(d) X-ray provides clear and detailed images or organs like prostate glands and lungs Which two of the above statements are correct?

Options:
A a and b
B $b$ and $d$
C c and $d$
D a and c

The correct answer is:
a and c

Solution:
Angioplasty not angiography is unblocking of blocked arteries. Radiograph produced by $\mathrm{x}$-rays is just a shadow of the dense body parts.

Topic: Biology, Human Health and Diseases, NEET, NEET 2008 (Mains)
Consider the following statements about Cleistogamy:

(i) Reported in Viola, Oxalis, Commelina.
(ii) Flowers are invariably autogamous.
(iii) Pollinators are required for seed set.
(iv) Anthesis is absent.

How many statements are incorrect?

Options:
A Four
B Three
C Two
D One

The correct answer is:
One

Solution:
Cleistogamy is the phenomenon in which the flower never opens up and always remains closed. Anthesis or floral bud is absent. Such flowers are always or invariably self-pollinated, thus pollinators are not required. Plants like Viola, Oxalis and Commelina shows cleistogamous flower.

Topic: Biology, Sexual Reproduction in Flowering Plants, NEET
Consider the following statements about the blocks shown in the diagram that are being pushed by a constant force on a frictionless table
A. All blocks move with the same acceleration
B. The net force on each block is the same Which of these statements are/is correct

Options:
A A only
B B only
C Both A and B
D Neither A nor B

The correct answer is:
A only

Solution:
The correct option is $\mathbf{A}$
(i) Only
As we can see from the free body diagram, forces acting on each block is different. But, they will all move together as a system as same net force acts on them, and therefore they will have the same acceleration.

Topic: Physics, Laws of Motion, JEE Main
Consider the following statements and choose the correct option.

1. The thread-like cytoplasmic strands, running from one cell to other is known as plasmodesmata.
2. Xylem and phloem constitute the vascular bundle of the stem.
3. The first formed xylem elements are described as metaxylem.
4. Radial vascular bundles are mainly found in the leaves.

Options:
A II is true, but I, III and IV are false
B III is true, but I, II and IV are false
C IV is true, but I, II and III are false
D I and II are true, but III and IV are false

The correct answer is:
I and II are true, but III and IV are false

Solution:
Plasmodesmata are thread-like cytoplasmic strands running from one cell to other and these make connections between adjacent cells. Each vascular bundle is made up necessarily of xylem and phloem elements and the cambium may or may not be present. The first differentiated xylem is called protoxylem, whereas those differentiated afterward are called metaxylem. Radial vascular bundles are mainly found in the roots. In these, there are separate and alternate strands of phloem and xylem present on different radii.

Topic: Biology, Anatomy of Flowering Plants, NEET
Consider the following statements concerning food chains
(a) Removal of $80 \%$ tigers from an area resulted in greatly increased growth of vegetation
(b) Removal of most of the carnivores resulted in an increased population of deers
(c) The length of food chains is generally limited to 3-4 trophic levels due to energy loss
(d) The length of food chains may vary from 2 to 8 trophic levels.
Which two of the above statements are correct?

Options:
A $\mathrm{a}, \mathrm{b}$
B $b, c$
C $\mathrm{c}, \mathrm{d}$
D a, d

The correct answer is:
$b, c$

Solution:
Removal of tigers from an area will lead to an increase in number of herbivores and hence there will be decreased growth of vegetation in that particular area. The number of trophic levels in a food chain does not reach 8 rather there are 3 or 4 trophic levels.

Topic: Biology, Ecosystem, NEET, NEET 2008 (Mains)
Consider the following statements for air molecules in an air tight container.
(I) the average speed of molecules is larger than root mean square speed
(II) mean free path of molecules is larger than the mean distance between molecules
(III) mean free path of molecules increases with temperature
(IV) the rms speed of nitrogen molecule is smaller than oxygen molecule The true statements are :

Options:
A only II
B II \& III
C II \& IV
D I, II \& IV

The correct answer is:
only II

Solution:
For ideal gas mean free path of molecules is larger than mean distance between molecules

Topic: Physics, Kinetic Theory of Gases, JEE Main
Consider the following statements. For any integer $\mathrm{n}$,
I. $n^{2}+3$ is never divisible by 17 .
II. $n^{2}+4$ is never divisible by 17 .
Then

Options:
A both I and II are true
B both I and II are fal se
C I is false and II is true
D I is true and II is fal se

The correct answer is:
I is true and II is fal se

Solution:
$n^{2}+4$ is divisible by 17 check at $n=9$ $\because \frac{n^{2}+3}{17}=\frac{n^{2}+4}{17}-\frac{1}{17}$ not divisible by 17

Topic: Mathematics, Binomial Theorem, JEE Main
Consider the following statements
I. If a piece of bar magnet is broken into two equally long pieces, then the pieces will not lose the magnetic properties.
II. Magnetic properties of a substance lie at the atomic level.
Which of the statement(s) given above is/are correct?

Options:
A Only I
B Only II
C Both I and II
D Neither I nor II

The correct answer is:
Both I and II

Solution:
A magnetic monopole does not exist, i.e., even if we break a magnet it will behave as two magnets, and we cannot isolate a magnetic monopole.

If we keep on dividing the magnet till the atomic level the magnetic property will still be there, thus the magnetic property of a substance lie in the atomic level.

Topic: Physics, Magnetic Properties of Matter, JEE Main
Consider the following statements.

(i). In man, the vertebral column has 33 bones organized as 28 bones.
(ii). The pelvic girdle is made up of two fused bones only.
(iii). Osteoporosis is characterized by micro-architectural deterioration of the bone.

Which of the following is a correct statement?

Options:
A I is correct
B II is correct
C III is correct
D Both (B) and (C)

The correct answer is:
III is correct

Solution:
Osteoporosis is a disease, in which bones loses minerals and fibres from its matrix. Imbalance of hormones like calcitonin, parathyroid and sex hormones, deficiencies of calcium and vitamin-D are the main causes of osteoporosis.
In man, the vertebral column has 33 bones organized as 26 bones. The pelvic girdle is made up of three fused bones only.

Topic: Biology, Locomotion and Movement, NEET
Consider the following statements:

(I) LaOH3 is the least basic among hydroxides of lanthanides.

(II) Zr4+ and Hf4+ possess almost the same ionic radii.

(III) Ce4+ can act as an oxidizing agent.

Which of the above is/are true?

Options:
A (I) and (III)
B (II) only
C (II) and (III)
D (I) only

The correct answer is:
(II) and (III)

Solution:
The basic character of hydroxides decreases from LaOH3 to LuOH3. Due to smaller size of Lu, the Lu−OH bond attains more covalent character.
Also, due to Lanthanide contraction Zr 4+ and Hf 4+ possess almost the same ionic radii.
Ce 4+ is a strong oxidant.

Topic: Chemistry, d and f Block Elements, JEE Main
Consider the following statements
I. limn→∞2n+(-2)n2n does not exist
II. limn→∞3n+(-3)n4n does not exist
Then

Options:
A I is true and II is false
B I is false and II is true
C I and II are true
D neither Inor II is true

The correct answer is:
I is true and II is false

Solution:
I. limn→∞2n2n+-22n
=limn→∞1+(-1)n does not exist

II. II.limn→∞34n+-34n=0+0=0

Topic: Mathematics, Limits, JEE Main
Consider the following statements
i. Number of ways of placing 'n' objects in k bins k≤n ) such that no bin is empty is Ck-1(n-1)

ii. Number of ways of writing a positive integer " n ' into a sum of k positive integers is Ck-1(n-1)

iii. Number of ways of placing ' n ' objects in k bins such that at least one bin is non-empty is Ck-1(n-1)

iv. Ckn-Ckn-1=Ck-1(n-1)

Options:
A all the four statements
B (iii) and (iv) only
C all except (iii)
D all except i

The correct answer is:
all except (iii)

Solution:
i. Number of ways of placing 'n' objects in k bins k≤n ) such that no bin is empty is equal to number of integral solution of

x1+x2+x3+...+xk=n

which is

=Ck-1n-1

ii Let

x1+x2+x3+...+xk=n

Then, number of positive integral solutions are

=Ck-1n-1

iii. Number of ways of placing ' n ' objects in k bins such that at least one bin is non-empty is equal to the number of non-negative integrals solution of

x1+x2+x3+...+xk=n

which is =Ck-1n+k-1

iv

Ckn-Ckn-1=Ck-1(n-1)

∵Ckn=n!(n-r)!r!

LHS

=n!(n-k)!k!-(n-1)!(n-k-1)!k!

=n!n-k·n-k-1!·k!-n-1!n-k-1!·k!

=n!-n-kn-1!n-k·n-k-1!·k!

=n·n-1!-n-kn-1!n-k·n-k-1!·k!

=n-1!n-n-kn-k·n-k-1!·k!

=k·n-1!n-k·n-k-1!·k!

=k·n-1!n-k·n-k-1!·k·k-1!

=(n-1)!(n-k)!(k-1)!=Ck-1n-1

Topic: Mathematics, Permutation Combination, JEE Main
Consider the following statements.
I. $\sin ^{-1}\left(y^2-4 y+6\right)+\cos ^{-1}\left(y^2-4 y+6\right) =\frac{\pi}{2}, \forall y \in R$
II. $\sec ^{-1}\left(y^2-4 y+6\right)+\operatorname{cosec}^{-1}\left(y^2-4 y+6\right) =\frac{\pi}{2}, \forall y \in R$
Which of the above statement(s) is/are true?

Options:
A Only I
B Only II
C Both I and II
D Neither I nor II

The correct answer is:
Only II

Solution:
Given statements,
$\sin ^{-1}\left(y^2-4 y+6\right)+\cos ^{-1}\left(y^2-4 y+6\right)=\frac{\pi}{2}$
$\because y^2-4 y+6=(y-2)^2+2 \geq 2, \forall y \in R \text {. }$
So, statement (i) is false and statement (ii) is true.

Topic: Mathematics, Inverse Trigonometric Functions, JEE Main
Consider the following statement(s):
I. The focal length of the objective of a microscope is less than the focal length of the eyepiece.
II. The minimum distance between an object and its real image formed by a convex lens of focal length \( f \) is equal to \( 4 f \).
Which of the statement(s) given above is/are correct?

Options:
A Only I.
B Only II.
C Both I and II.
D Neither I nor II.

The correct answer is:
Both I and II.

Solution:
When we see an image through a compound microscope, our image is usually very close to the objective lens. So, in order to get a real image with objective lens (convex in nature), the focus is kept small.

From the above ray diagram,
object distance, u=d-x,
image distance, v=x,
distance between object and image=d,
focal length=f.

Using lens formula,

1v-1u=1f
1x-1-d-x=1fx2-xd-fd=0

Real image must be real means roots of the equation should be real.

∴d2≥4fd∴dmin=4d

Topic: Physics, Ray Optics, JEE Main
Consider the following statements:
If $A$ and $B$ are independent events, then
$1.$ $A$ and $\bar{B}$ are independent.
$2.$ $\bar{A}$ and $\mathrm{B}$ are independent.
$3.$ $\bar{A}$ and $\bar{B}$ are independent. Which of the above statements is/are correct?

Options:
A 3 only
B 1 and 2 only
C 1,2 and 3
D None of these

The correct answer is:
1,2 and 3

Solution:
Let $\mathrm{A}$ and $\mathrm{B}$ are independent events. $\Rightarrow \quad P(A \cap B)=P(A) \cdot P(B)$
$$
\text { 1. } \begin{aligned}
\text { Consider } \mathrm{P}(\mathrm{A} \cap \overline{\mathrm{B}}) &=\mathrm{P}(\mathrm{A})-\mathrm{P}(\mathrm{A} \cap \mathrm{B}) \\
&=\mathrm{P}(\mathrm{A})-\mathrm{P}(\mathrm{A}) \mathrm{P}(\mathrm{B})
\end{aligned}
$$
(from A)
$$
=\mathrm{P}(\mathrm{A})[1-\mathrm{P}(\mathrm{B})]=\mathrm{P}(\mathrm{A}) \mathrm{P}(\overline{\mathrm{B}})
$$
Hence, $\mathrm{A}$ and $\overline{\mathrm{B}}$ are independent.
$2.$ Similarly, $\overline{\mathrm{A}}$ and $\mathrm{B}$ are independent.
$3.$
$$
\begin{aligned}
\text { Consider } \mathrm{P}(\overline{\mathrm{A}} \cap \overline{\mathrm{B}}) &=\mathrm{P}(\overline{\mathrm{A} \cup \mathrm{B}}) \\
&=1-\mathrm{P}(\mathrm{A} \cap \mathrm{B}) \\
=& 1-[\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cup \mathrm{B})] \\
=& 1-\mathrm{P}(\mathrm{A})-\mathrm{P}(\mathrm{B})+\mathrm{P}(\mathrm{A}) \cdot \mathrm{P}(\mathrm{B}) \\
=&[1-\mathrm{P}(\mathrm{A})][1-\mathrm{P}(\mathrm{B})] \\
=& \mathrm{P}(\overline{\mathrm{A}}) . \mathrm{P}(\overline{\mathrm{B}})
\end{aligned}
$$
Hence, $\overline{\mathrm{A}}$ and $\overline{\mathrm{B}}$ are independent.

Topic: Mathematics, Probability, JEE Main
Consider the following statements (I-IV) about organic farming.
I. Utilizes genetically modified crops like $B t$ cotton
II. Uses only naturally produced inputs like compost
III. Does not use pesticides and urea
IV. Produces vegetables rich in vitamins and minerals
Which of the above statements are correct?

Options:
A (II), (III) and (IV)
B (III) and (IV) only
C (II) and (III) only
D (I) and (II) only

The correct answer is:
(II) and (III) only

Solution:
Organic forming is the form of agriculture that relies on techniques such as crop rotation, green manure, compost and biological pest control to maintain soil productivity. It excludes the use of manufactured fertilizers, pesticides, plant growth regulators, genetically modified organisms, live stock antibiotics, etc. It uses only naturally produced inputs like composts.

Topic: Biology, Biotechnology and Its Applications, NEET, NEET 2011 (Mains)
Consider the following statements (I)-(IV) each with one or two blanks.
I. Bears go into (1) during winter to ......(2) ...... cold weather.
II. A conical age pyramid with a broad base represents......(3)..... human population.
III. A wasp pollinating a fig flower is an example of (4) ......
IV. An area with high levels of species richness is known as ......(5)......
Which one of the following options give the correct fill up the respective blanks numbers from (1) to (5) in the statements.

Options:
A (2) stable, (4) commensalism, (5) marsh
B (1) aestivation, (5) escape, (3) stable, (4) mutualism
C (3) expanding, (4) commensalism, (5) biodiversity park
D (1) hibernation, (2) escape, (3) expanding, (5) hot spot

The correct answer is:
(1) hibernation, (2) escape, (3) expanding, (5) hot spot

Solution:
Bears go into hibernation during winter to escape cold weather.
A conical age pyramid with a broad base represents expanding human population.
A wasp pollinating a fig flower is an example of mutualism.
An area with high levels of species richness is known as hot spots.

Topic: Biology, Organisms and Populations, NEET, NEET 2011 (Mains)
Consider the following statements in respect of symmetric matrices $A$ and $B$
$1. A B$ is symmetric.
$2. A^{2}+B^{2}$ is symmetric.
Which of the above statement(s) is/are correct?

Options:
A 1 only
B 2 only
C Both 1 and 2
D Neither 1 nor 2

The correct answer is:
2 only

Solution:
We know, a matrix $\mathrm{A}$ is said to be symmetric matrix if $A^{\prime}=A$ where ' ' ' represents the transpose.
Consider $(A B)^{\prime}=B^{\prime} A^{\prime}$
Since, $(A B)^{\prime} \neq A B$
$\therefore \mathrm{AB}$ is not symmetric.
and consider $\left(A^{2}+B^{2}\right)^{\prime}=\left(A^{\prime}\right)^{2}+\left(B^{\prime}\right)^{2}=A^{2}+B^{2}$
$\therefore \mathrm{A}^{2}+\mathrm{B}^{2}$ is symmetric.

Topic: Mathematics, Matrices, JEE Main
Consider the following statements in respect of the determinant $\left|\begin{array}{ll}\cos ^{2} \frac{\alpha}{2} & \sin ^{2} \frac{\alpha}{2} \\ \sin ^{2} \frac{\beta}{2} & \cos ^{2} \frac{\beta}{2}\end{array}\right|$
where $\alpha, \beta$ are complementary angles
$1.$ The value of the determinant is $\frac{1}{\sqrt{2}} \cos \left(\frac{\alpha-\beta}{2}\right)$.
$2.$ The maximum value of the determinant is $\frac{1}{\sqrt{2}}$.
Which of the above statements is/ are correct?

Options:
A 1 only
B 2 only
C Both 1 and 2
D Neither 1 nor 2

The correct answer is:
Both 1 and 2

Solution:
$\alpha+\beta=90^{\circ}$
$\left|\begin{array}{rr}\cos ^{2} \frac{\alpha}{2} & \sin ^{2} \frac{\alpha}{2} \\ \sin ^{2} \frac{\beta}{2} & \cos ^{2} \frac{\beta}{2}\end{array}\right|$
$=\cos ^{2} \frac{\alpha}{2} \cos ^{2} \frac{\beta}{2}-\sin ^{2} \frac{\alpha}{2} \sin ^{2} \frac{\beta}{2}$
$=\left(\cos \frac{\alpha}{2} \cos \frac{\beta}{2}+\sin \frac{\alpha}{2} \sin \frac{\beta}{2}\right)\left(\cos \frac{\alpha}{2} \cos \frac{\beta}{2}-\sin \frac{\alpha}{2} \sin \frac{\beta}{2}\right)$
$=\cos \frac{(\alpha-\beta)}{2} \times \cos \frac{(\alpha+\beta)}{2}$
$=\cos \frac{(\alpha-\beta)}{2} \times \cos \frac{\left(90^{\circ}\right)}{2}$
$=\cos \frac{(\alpha-\beta)}{2} \times \frac{1}{\sqrt{2}}$
Maximum value of $\cos \left(\frac{\alpha-\beta}{2}\right)$ is $1 .$ Somaximum value
of determinent is $\left(\frac{1}{\sqrt{2}}\right)$
So both 1 and 2 are correct.

Topic: Mathematics, Determinants, JEE Main
Consider the following statements
L. If $\theta=1200^{\circ}$, then $(\sec \theta+\tan \theta)^{-1}$ is positive.
II. If $\theta=1200^{\circ}$, then $(\operatorname{cosec} \theta-\cot \theta)$ is negative.
Which of the statements given above is/are correct?

Options:
A I only
B II only
C Both I and II
D Neither Inor II

The correct answer is:
Neither Inor II

Solution:
Statement-I: Let $\theta=1200^{\circ}$
Consider $(\sec \theta+\tan \theta)^{-1}=\frac{1}{\sec \theta+\tan \theta}$
$=\frac{\sec \theta-\tan \theta}{(\sec \theta+\tan \theta)(\sec \theta-\tan \theta)}$
$=\frac{\sec \theta-\tan \theta}{\left(\sec ^{2} \theta-\tan ^{2} \theta\right)}=\sec \theta-\tan \theta$
When $1200 \div 360$
we get remainder as $120^{\circ}$ Now, put $\theta=120^{\circ}$
$\Rightarrow(\sec \theta+\tan \theta)^{-1}=\sec 120^{\circ}-\tan 120^{\circ}$
$=\frac{1}{\cos 120^{\circ}}-\tan \left(90^{\circ}+30^{\circ}\right)$
$=\frac{1}{\cos (90+30)}-\cot 30^{\circ}$
$=\frac{1}{-\sin 30^{\circ}}+\sqrt{3}$
$=-2+\sqrt{3} \quad$ which is negative
Now, Statement-II :
Consider $\operatorname{cosec} \theta-\cot \theta$
$$
\begin{array}{l}
=\operatorname{cosec} 120^{\circ}-\cot 120^{\circ} \\
=\frac{1}{\sin (90+30)}-\cot (90+30) \\
=\frac{1}{\cos 30^{\circ}}+\tan 30^{\circ} \\
=\frac{2}{\sqrt{3}}+\frac{1}{\sqrt{3}}=\frac{3}{\sqrt{3}}
\end{array}
$$
which is positive. Hence, both statements are incorrect.

Topic: Mathematics, Trigonometric Ratios & Identities, NDA, NDA 2009 (Phase 2)
Consider the following statements:
Mode can be computed from histogram
Median is not independent of change of scale
Variance is independent of change of origin and scale.

Options:
A
only (a)

B
only (b)

C
only (a) and (b)

D
(a), (b) and (c)

The correct answer is:
only (a) and (b)

Solution:
Mode can be computed from histogram and median is dependent on the scale. Hence statement (a) and (b) are correct

Topic: Mathematics, Statistics, JEE Main, JEE Main 2004