datavorous/entrance-exam-dataset huggingface dataset:
What does the shaded region in the above diagram represent?

Options:
A $(\mathrm{A} \cap \mathrm{B}) \cap \mathrm{C}$
B $(\mathrm{A} \cup \mathrm{B}) \cap \mathrm{C}$
C $(\mathrm{A} \cup \mathrm{B})-\mathrm{C}$
D None of the above

The correct answer is:
$(\mathrm{A} \cup \mathrm{B}) \cap \mathrm{C}$

Solution:
In the given Venn diagram, shaded region shows $(\mathrm{A} \cup \mathrm{B}) \cap \mathrm{C}$

Topic: Mathematics, Sets and Relations, NDA, NDA 2008 (Phase 1)
What does the shaded region in the Venn diagram given below represent?

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

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

Solution:
In the given Venn diagram shaded region is $\mathrm{C} \cup\left(\mathrm{C}^{\prime} \cap \mathrm{A} \cap \mathrm{B}\right)$

Topic: Mathematics, Sets and Relations, NDA, NDA 2007 (Phase 2)
What does the shaded region represent in the figure given below

Options:
A $(\mathrm{P} \cup \mathrm{Q})-(\mathrm{P} \cap \mathrm{Q})$
B $\mathrm{P} \cap(\mathrm{Q} \cap \mathrm{R})$
C $(P \cap Q) \cap(P \cap R)$
D $(\mathrm{P} \cap \mathrm{Q}) \cup(\mathrm{P} \cap \mathrm{R})$

The correct answer is:
$(\mathrm{P} \cap \mathrm{Q}) \cup(\mathrm{P} \cap \mathrm{R})$

Solution:
The shaded region represents $(\mathrm{P} \cap \mathrm{Q}) \cup(\mathrm{P} \cap \mathrm{R})$ Let the intersecting sets $\mathrm{P}, \mathrm{Q}, \mathrm{R}$ divide it into $7 \mathrm{regions}$ marked, a to g as shown below.
The shaded part contains regions $\mathrm{b}, \mathrm{c}$, and $\mathrm{d}$.
(a) $(\mathrm{P} \cup \mathrm{Q})-(\mathrm{P} \cap \mathrm{Q})=$ regions $\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{f}, \mathrm{g},-\mathrm{b}, \mathrm{c}$
$=\mathrm{a}, \mathrm{d}, \mathrm{f}_{\mathrm{g}} \mathrm{g}$, notcorrect.
(b) $(\mathrm{P} \cap(\mathrm{Q} \cap \mathrm{R})=\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \cap \mathrm{c}, \mathrm{f}=\mathrm{c}$ not correct.
(c) $(P \cap Q) \cap(P \cap R)=$ regions $b, c, \cap r e g i o n, c, d=c, s o$
not correct
(d) $(P \cap Q) \cup(P \cap R)=$ regions $b, c, \cup c, d=b, c, d s o$
correct

Topic: Mathematics, Sets and Relations, NDA, NDA 2006 (Phase 2)
What does the solution of the differential equation
$x \frac{d y}{d x}=y$ represent?

Options:
A Family of straight lines through the origin
B Family of circles with their centres at the origin
C Family of parabolas with their vertices at the origin
D Family of straight lines having slope 1 and not passing through the origin

The correct answer is:
Family of straight lines through the origin

Solution:
Given differential equation is
$$
x \frac{d y}{d x}=y
$$
By separating the variables, we get
$$
\frac{\mathrm{dy}}{\mathrm{y}}=\frac{\mathrm{dx}}{\mathrm{x}}
$$
Intergrate both the sides, we get
$$
\begin{aligned}
& \int \frac{\mathrm{dy}}{\mathrm{y}}=\int \frac{\mathrm{dx}}{\mathrm{x}} \\
\Rightarrow & \log y=\log x+\log c \\
\Rightarrow & y=x c
\end{aligned}
$$
which is a family of straight lines through the origin.

Topic: Mathematics, Differential Equations, NDA, NDA 2010 (Phase 2)
What does the solution of the differential equation $x d y-y d x=0$ represent?

Options:
A Rectangular hyperbola
B Straight line passing through the origin
C Parabola whose vertex is at origin
D Circle whose centre is at origin

The correct answer is:
Straight line passing through the origin

Solution:
$x d y-y d x=0$
$\Rightarrow x d y=y d x$
$\Rightarrow \int \frac{1}{y} d y=\int \frac{1}{x} d x$
$\Rightarrow \log y=\log x+\log c$
$\Rightarrow \log y=\log c x$
$\Rightarrow y=c x$
Thus, the solution of equation $x d y-y d x=0$ represents straight line passing through the origin.

Topic: Mathematics, Differential Equations, NDA, NDA 2008 (Phase 2)
What does the solution of the differential equation $\mathrm{xdy}-\mathrm{ydx}=0$ represent?

Options:
A Rectangular hyperbola
B Straight line passing through $(0,0)$
C Parabola with vertex at $(0,0)$
D Circle with centre at $(0,0)$

The correct answer is:
Straight line passing through $(0,0)$

Solution:
Given that $x \mathrm{dy}-\mathrm{ydx}=0$ Dividing both the sides by $\mathrm{x}^{2}$
$\Rightarrow \quad \frac{\mathrm{xdy}-\mathrm{ydx}}{\mathrm{x}^{2}}=0$
$\Rightarrow \mathrm{d}\left(\frac{\mathrm{y}}{\mathrm{x}}\right)=0$
$\Rightarrow \frac{y}{x}=c \Rightarrow y=c x$, where $c$ is a constant. Thus, it a straight line passing through $(0,0)$. Aliter :
Given, $x d y-y d x=0$
$\Rightarrow \frac{d y}{y}=\frac{d x}{x}$
Integrating both the sides, $\int \frac{\mathrm{dy}}{\mathrm{y}}=\int \frac{\mathrm{dx}}{\mathrm{x}}+\log \mathrm{c}$
$\log \mathrm{y}=\log \mathrm{x}+\log \mathrm{c}$ where $\mathrm{c}$ is constant
$\Rightarrow \mathrm{y}=\mathrm{cx}$

Topic: Mathematics, Differential Equations, NDA, NDA 2006 (Phase 1)
What does the standard electrode potential of lithium metal indicate about its reactivity?

Options:
A It is highly reducing.
B It is ly reducing.
C It is ly oxidizing
D It is highly oxidizing.

The correct answer is:
It is highly reducing.

Solution:
Lithium has the lowest electrode potential indicating that lithium ion is the weakest oxidising agent while lithium metal is the most powerful reducing agent in an aqueous solution.

Topic: Chemistry, Electrochemistry, NEET
What does the symbol Λ°m represent in the context of molar conductivity?

Options:
A Molar concentration
B Molar mass
C Limiting molar conductivity
D Standard molar conductivity

The correct answer is:
Limiting molar conductivity

Solution:
When concentration approaches zero, the molar conductivity is known as limiting molar conductivity and is represented by the symbol Λ°m.

Topic: Chemistry, Electrochemistry, NEET
What does the term "secondary valence" represent in Werner's concept?

Options:
A The number of moles of AgCl formed
B The number of groups bound directly to the metal ion
C The color of the compound
D The conductivity of the compound in solution

The correct answer is:
The number of groups bound directly to the metal ion

Solution:
In Werner's concept, "secondary valence" represents the number of groups bound directly to the metal ion.

Topic: Chemistry, Coordination Compounds, NEET
What does the variable 'Q' represent in the Nernst equation for a general electrochemical reaction?

Options:
A The equilibrium constant of the reaction.
B The reaction rate constant.
C The standard electrode potential.
D The number of electrons transferred in the reaction.

The correct answer is:
The equilibrium constant of the reaction.

Solution:
'Q' represents the reaction quotient, which is the equilibrium constant of the reaction at a given point.

Topic: Chemistry, Electrochemistry, NEET
What effect is observed in emulsions due to the negatively charged droplets?

Options:
A Brownian movement
B Tyndall effect
C Electrophoresis
D Flocculation

The correct answer is:
Brownian movement

Solution:
Emulsions with negatively charged droplets often show Brownian movement.

Topic: Chemistry, Surface Chemistry, NEET
What element serves as the standard for the present system of atomic masses?

Options:
A Carbon-12 (12
C)
B Hydrogen (H)
C Oxygen (O)
D Nitrogen (N)

The correct answer is:
Carbon-12 (12
C)

Solution:
The present system of atomic masses is based on carbon-12 (12

Topic: Chemistry, Some Basic Concepts of Chemistry, NEET
What $=\frac{f(x)+1}{f(x)-1}+x$ is equal to ?
Consider the function $f(x)=\frac{x-1}{x+1}$.

Options:
A 0
B 1
C 2x
D 4x

The correct answer is:
0

Solution:
Given $\mathrm{f}(\mathrm{x})=\frac{\mathrm{x}-1}{\mathrm{x}+1}$
Applying componendo and dividendo, we get
$\frac{\mathrm{f}(\mathrm{x})+1}{\mathrm{f}(\mathrm{x})-1}=\frac{\mathrm{x}-1+\mathrm{x}+1}{\mathrm{x}-1-\mathrm{x}-1}$
$\Rightarrow \frac{\mathrm{f}(\mathrm{x})+1}{\mathrm{f}(\mathrm{x})-1}=-\mathrm{x}$
Now, $\frac{\mathrm{f}(\mathrm{x})+1}{\mathrm{f}(\mathrm{x})-1}+\mathrm{x}=-\mathrm{x}+\mathrm{x}=0$

Topic: Mathematics, Functions, NDA, NDA 2014 (Phase 2)
What factor determines the standard electrode potential of a metal?

Options:
A Metallic lustre
B Enthalpy of atomisation
C High tensile strength
D Low volatility

The correct answer is:
Enthalpy of atomisation

Solution:
The enthalpy of atomisation is an important factor in determining the standard electrode potential of a metal.

Topic: Chemistry, d and f Block Elements, NEET
What factor does the pressure of the gas increase by when it is compressed to half its original volume in a cylinder with a movable piston containing 3 moles of hydrogen at standard temperature and pressure, where the cylinder walls are heat insulators and the piston is insulated with a pile of sand?

Options:
A 2.639
B 3.223
C 4.125
D 5.335

The correct answer is:
2.639

Solution:
Correct Option is : (A)
2.639

Topic: Chemistry, Thermodynamics, NEET
What forces primarily stabilize the secondary and tertiary structures of proteins?

Options:
A Covalent bonds
B Ionic bonds
C Van der Waals and electrostatic forces
D Hydrogen bonds

The correct answer is:
Van der Waals and electrostatic forces

Solution:
The van der Waals and electrostatic forces of attraction are among the forces that stabilize the secondary and tertiary structures of proteins.

Topic: Chemistry, Biomolecules, NEET
What fraction of the total volume at Standard Temperature and Pressure (STP) is occupied by oxygen gas if the diameter of an oxygen molecule is 3 Å?

Options:
A $3.8~\times ~{{10}^{-4}}$
B $3.8~\times ~{{10}^{-3}}$
C $1.8~\times ~{{10}^{-4}}$
D $1.8~\times ~{{10}^{-3}}$

The correct answer is:
$3.8~\times ~{{10}^{-4}}$

Solution:
Correct Option is : (A)
$3.8~\times ~{{10}^{-4}}$

Topic: Chemistry, Kinetic Theory of Gases, NEET
What happens at the boundary between ice and water when ice and water are in equilibrium?

Options:
A There is no activity.
B Molecules from liquid water collide with ice and adhere to it.
C Molecules from ice escape into the liquid phase.
D The temperature increases rapidly.

The correct answer is:
Molecules from liquid water collide with ice and adhere to it.

Solution:
At the boundary between ice and water, molecules from liquid water collide with ice and adhere to it.

Topic: Chemistry, Chemical Equilibrium, NEET
What happens during vascularization in plants?

Options:
A Differentiation of procambium is immediately followed by the development of secondary xylem and phloem
B Differentiation of procambium followed by the development of xylem and phloem
C Differentiation of procambium, xylem and phloem is simultaneous
D Differentiation of procambium followed by the development of primary phloem and then by primary xylem

The correct answer is:
Differentiation of procambium followed by the development of xylem and phloem

Solution:
Vascularization of plant means the development of vascular bundle in the plant, which develops from the cambium.

Topic: Biology, Anatomy of Flowering Plants, NEET
What happens if a reaction has Qc

ChemistryChemical EquilibriumNEET

Options:

A Net reaction goes from left to right
B Net reaction goes from right to left
C Products and reactants are at equilibrium
D Cannot say

Solution:

2760 Upvotes

Verified Answer

The correct answer is:
Net reaction goes from left to right

Correct Option is : 1
Net reaction goes from left to right

Looking for more such questions to practice?

Download the MARKS App - The ultimate prep app for IIT JEE &
NEET with chapter-wise PYQs, revision notes, formula sheets,
custom tests & much more.

Use on iOS or Desktop Download from Google Play

Options:
A Net reaction goes from left to right
B Net reaction goes from right to left
C Products and reactants are at equilibrium
D Cannot say

The correct answer is:
Net reaction goes from left to right

Solution:
Correct Option is : 1
Net reaction goes from left to right

Topic: Chemistry, Chemical Equilibrium, NEET
What happens to the primary oocyte in the tertiary follicle?

Options:
A Forms the mature follicle
B Completes its first meiotic division
C Changes into the secondary follicle
D None of the above

The correct answer is:
Completes its first meiotic division

Solution:
The primary oocyte within the tertiary follicle grows in size and completes its first meiotic division, resulting in the formation of a large haploid secondary oocyte and a tiny first polar body.

Topic: Biology, Human Reproduction, NEET
What happens when a secondary alcohol reacts with Lucas reagent at room temperature in the Lucas test?

Options:
A Immediate turbidity
B Turbidity after heating
C No turbidity at any temperature
D Formation of an alkyl halide

The correct answer is:
Turbidity after heating

Solution:
Secondary alcohols may produce turbidity in the Lucas test after heating.

Topic: Chemistry, Alcohols Phenols and Ethers, NEET
What harmonic mode of a pipe 20 cm long, closed at one end, is resonantly excited by a source of 1237.5 Hz? (Assume the speed of sound in air to be 330 m/s.)

Options:
A Fundamental harmonic
B 1st harmonic
C 2nd harmonic
D 3rd harmonic

The correct answer is:
3rd harmonic

Solution:
Correct Option is : (D)
3rd harmonic

Topic: Chemistry, Waves and Sound, NEET
What is $\int_{0}^{1} \frac{e^{\tan ^{-1} x} d x}{1+x^{2}}$ equal to ?

Options:
A $e^{\frac{\pi}{4}}-1 \quad$
B $e^{\frac{\pi}{4}}+1$
C $e-1$
D $e$

The correct answer is:
$e^{\frac{\pi}{4}}-1 \quad$

Solution:
$\mathrm{I}=\int_{0}^{1} \frac{\mathrm{e}^{\mathrm{tan}^{-1} \mathrm{x}}}{1+\mathrm{x}^{2}} \mathrm{dx}$
Let $\tan ^{-1} \mathrm{x}=\mathrm{t}$
$\frac{1}{1+\mathrm{x}^{2}} \mathrm{dx}=\mathrm{dt}$
Lower limit $\rightarrow \mathrm{t}=\tan ^{-1} 0=0$
upper limit $\rightarrow \mathrm{t}=\tan ^{-1},=\pi / 4$
$\therefore \int_{0}^{\pi / 4} \mathrm{e}^{\mathrm{t}}-\mathrm{dt}=\left[\mathrm{e}^{\mathrm{t}}\right]_{0}^{\pi / 4}$
$\mathrm{e}^{\pi / 4}-\mathrm{e}^{0} \Rightarrow \mathrm{e}^{\pi / 4}-1$
$=(-\mathrm{a} \cos \theta,-\mathrm{b} \sin \theta)$

Topic: Mathematics, Definite Integration, NDA, NDA 2014 (Phase 1)
What is $\int_{0}^{1} \frac{\tan ^{-1} x}{1+x^{2}} d x$ equal to ?

Options:
A $\frac{\pi^{2}}{8}$
B $\frac{\pi^{2}}{32}$
C $\frac{\pi}{4}$
D $\frac{\pi}{8}$

The correct answer is:
$\frac{\pi^{2}}{32}$

Solution:
Let $\mathrm{I}=\int_{0}^{1} \frac{\tan ^{-1} \mathrm{x}}{1+\mathrm{x}^{2}} \mathrm{dx}$
Let, $\tan ^{-1} x=t$
$\frac{1}{1+x^{2}} d x=d t$
Also, $x=0 \rightarrow t=0$
$x=1 \rightarrow t=\frac{\pi}{4}$
$\therefore \mathrm{I}=\int_{0}^{\frac{\pi}{4}} \mathrm{t} \mathrm{dt}=\left.\frac{\mathrm{t}^{2}}{2}\right|_{0} ^{\frac{\pi}{4}}=\frac{\pi^{2}}{32}$

Topic: Mathematics, Definite Integration, NDA, NDA 2012 (Phase 2)
What is $\int_{0}^{\pi} \frac{d x}{1+2 \sin ^{2} x}$ equal to?

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

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

Solution:
$\begin{aligned} I=\int_{0}^{\pi} \frac{d x}{1+2 \sin ^{2} x}=2 \int_{0}^{\pi / 2} \frac{d x}{1+2 \sin ^{2} x} \\ &=2 \int_{0}^{\pi / 2} \frac{\sec ^{2} x d x}{\sec ^{2} x+2 \tan ^{2} x}=2 \int_{0}^{\pi / 2} \frac{\sec ^{2} x d x}{1+3 \tan ^{2} x} \\ \text { Put } \tan x=t \Rightarrow \sec ^{2} x d x=d t \\ \text { When } x=0, \mathrm{t}=0 \\ \text { When } x=\frac{\pi}{2}, t=\infty \\ \therefore \quad I=2 \int_{0}^{\infty} \frac{d t}{1+3 t^{2}}=\frac{2}{3} \int_{0}^{\infty} \frac{d t}{t^{2}+\left(\frac{1}{\sqrt{3}}\right)^{2}} \\=\frac{2}{3} \times \sqrt{3}\left[\tan ^{-1} \frac{t}{1 / \sqrt{3}}\right]_{0}^{\infty} \\=\frac{2}{\sqrt{3}}\left[\tan ^{-1} \sqrt{3} t\right]_{0}^{\infty} \\=\frac{2}{\sqrt{3}}\left[\tan ^{-1} \infty-\tan ^{-1} 0\right] \\=\frac{2}{\sqrt{3}} \times \frac{\pi}{2}=\frac{\pi}{\sqrt{3}} \end{aligned}$

Topic: Mathematics, Definite Integration, NDA, NDA 2011 (Phase 1)
What is $\int_{0}^{1} \mathrm{x}(1-\mathrm{x})^{9} \mathrm{dx}$ equal to?

Options:
A $\frac{1}{110}$
B $\frac{1}{132}$
C $\frac{1}{148}$
D $\frac{1}{240}$

The correct answer is:
$\frac{1}{110}$

Solution:
$\int_{0}^{1} \int_{0}^{1} x(1-x)^{9} \cdot d x$
$=\int_{0}^{1}(1-x)(1-(1-x))^{9} \cdot d x\left(\because \int_{0}^{a} f(x) \cdot d x=\int_{0}^{a} f(a-x) \cdot d x\right)$
$\quad=\int_{0}^{1}(1-x) \cdot x^{9} \cdot d x$
$\quad=\int_{0}^{1}\left(x^{9}-x^{10}\right) \cdot d x$
$\quad=\left(\frac{x^{10}}{10}-\frac{x^{11}}{11}\right)_{0}^{1}$
$\quad=\frac{1}{10}-\frac{1}{11}=\frac{11-10}{110}=\frac{1}{110} .$

Topic: Mathematics, Definite Integration, NDA, NDA 2018 (Phase 1)
What is $\int_{0}^{1} x(1-x)^{n} d x$ equal to? $\quad$

Options:
A $\frac{1}{n(n+1)}$
B $\frac{1}{(n+1)(n+2)}$
C 1
D 0

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

Solution:
Let $I=\int_{0}^{1} x(1-x)^{n} d x$
Put $1-x=t \Rightarrow d x=-d t$
when $x=0$ then $t=1$ when $x=1$ then $t=0$
$\begin{aligned} \therefore & I=-\int_{1}^{0}(1-t) t^{n} d t=\int_{0}^{1}\left(t^{n}-t^{n+1}\right) d t \\ &=\left[\frac{t^{n+1}}{n+1}-\frac{t^{n+2}}{n+2}\right]_{0}^{1} \\ &=\frac{1}{n+1}-\frac{1}{n+2}=\frac{1}{(n+1)(n+2)} \end{aligned}$

Topic: Mathematics, Definite Integration, NDA, NDA 2009 (Phase 1)
What is $\int_{0}^{2 \pi} \sqrt{1+\sin \frac{x}{2} \mathrm{~d} x \text { equal to? }}$

Options:
A 8
B 4
C 2
D 0

The correct answer is:
8

Solution:
$\int_{0}^{2 \pi} \sqrt{1+\sin \frac{x}{2}} \cdot d x$
$\sin \frac{x}{2}=\sin 2\left(\frac{x}{4}\right)=2 \sin \frac{x}{4} \cos \frac{x}{4}$
$\therefore \int_{0}^{2 \pi} \sqrt{\sin ^{2} \frac{x}{4}+\cos ^{2} \frac{x}{4}+2 \sin \frac{x}{4} \cos \frac{x}{4}}$
$=\int_{0}^{2 \pi} \sqrt{\left(\sin \frac{x}{4}+\cos \frac{x}{4}\right)^{2}} \cdot d x$
$=\int_{0}^{2 \pi}\left|\sin \frac{x}{4}+\cos \frac{x}{4}\right| \cdot d x$
$=4\left[-\cos \frac{x}{4}+\sin \frac{x}{4}\right]_{0}^{2 \pi}$
$=4\left[-\left(\cos \frac{2 \pi}{4}-\cos 0\right)+\sin \left(\frac{2 \pi}{4}-\sin 0\right)\right]$
$=4[-(-1)+(1)]=4 \times 2=8 .$

Topic: Mathematics, Definite Integration, NDA, NDA 2017 (Phase 2)
What is $\int_{0}^{\pi / 2} \frac{d x}{a^{2} \cos ^{2} x+b^{2} \sin ^{2} x}$ equal to?

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

The correct answer is:
$\frac{\pi}{2 a b}$

Solution:
Let $\mathrm{I}=\int_{0}^{\frac{\pi}{2}} \frac{\mathrm{d} \mathrm{x}}{\mathrm{a}^{2} \cos ^{2} \mathrm{x}+\mathrm{b}^{2} \sin ^{2} \mathrm{x}}$
$=\int_{0}^{\frac{\pi}{2}} \frac{\sec ^{2} \mathrm{x} \mathrm{dx}}{\mathrm{a}^{2}+\mathrm{b}^{2} \tan ^{2} \mathrm{x}}$
[divide numerator and denominator by $\left.\cos ^{2} \mathrm{x}\right]$ Put tan $x=t$ $\Rightarrow \sec ^{2} x d x=d t$
When $x=0$, then $t=0$ and when $x=\frac{\pi}{2}$, then $t=\infty$
$\therefore \mathrm{I}=\int_{0}^{\infty} \frac{\mathrm{dt}}{\mathrm{a}^{2}+\mathrm{b}^{2} \mathrm{t}^{2}}=\frac{1}{\mathrm{~b}^{2}} \int_{0}^{\infty} \frac{\mathrm{dt}}{\left(\frac{\mathrm{a}}{\mathrm{b}}\right)^{2}+\mathrm{t}^{2}}$
$=\frac{1}{\mathrm{~b}^{2}} \frac{1}{\left(\frac{\mathrm{a}}{\mathrm{b}}\right)}\left[\tan ^{-1}\left(\frac{\mathrm{bt}}{\mathrm{a}}\right)\right]_{0}^{\infty}$
$\quad\left[\because \int \frac{\mathrm{dx}}{\mathrm{a}^{2}+\mathrm{x}^{2}}=\frac{1}{\mathrm{a}} \tan ^{-1}\left(\frac{\mathrm{x}}{\mathrm{a}}\right)+\mathrm{C}\right]$
$=\frac{1}{\mathrm{ab}}\left[\tan ^{-1}(\infty)-\tan ^{-1}(0)\right]$
$=\frac{1}{\mathrm{ab}}\left[\frac{\pi}{2}-0\right]=\frac{\pi}{2 \mathrm{ab}}$

Topic: Mathematics, Definite Integration, NDA, NDA 2014 (Phase 2)
What is $\int_{0}^{2} e^{\ln x} d x$ equal to ?

Options:
A 1
B 2
C 4
D None of these

The correct answer is:
2

Solution:
$\int_{0}^{2} \mathrm{e}^{\ln x} \mathrm{dx}=\int_{0}^{2} x \mathrm{dx}=\left|\frac{\mathrm{x}^{2}}{2}\right|_{0}^{2}=2$

Topic: Mathematics, Definite Integration, NDA, NDA 2013 (Phase 2)
What is $\int_{0}^{\frac{\pi}{2}} \ln (\tan x) d x$ equal to ?

Options:
A $\ln 2$
B $-\ln 2$
C 0
D None of these

The correct answer is:
0

Solution:
$\quad I=\int_{0}^{\frac{\pi}{2}} \ln (\tan x) d x$
$I=\int_{0}^{\frac{\pi}{2}} \ln \left(\tan \left(\frac{\pi}{2}-x\right)\right) d x$
$=\int_{0}^{\frac{\pi}{2}} \ln \cot x d x$
Adding equations (i) and (ii)
$2 \mathrm{I}=\int_{0}^{\frac{\pi}{2}} \ln (\tan \mathrm{x} \cdot \cot \mathrm{x}) \mathrm{dx}$
$2 \mathrm{I}=0$
$\mathrm{I}=0$

Topic: Mathematics, Definite Integration, NDA, NDA 2014 (Phase 1)
What is $\int_{0}^{\pi / 2} \frac{\sin ^{3} x}{\sin ^{3} x+\cos ^{3} x} d x ? \quad$

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

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

Solution:
Let $I=\int_{0}^{\pi / 2} \frac{\sin ^{3} x}{\sin ^{3} x+\cos ^{3} x} d x$
...(i)
$=\int_{0}^{\pi / 2} \frac{\sin ^{3}(\pi / 2-x)}{\sin ^{3}(\pi / 2-x)+\cos ^{3}(\pi / 2-x)} d x$
By using the property $\int_{0}^{a} f(x) d x=\int_{0}^{a} f(a-x) d x$
$\Rightarrow I \int_{0}^{\pi / 2} \frac{\cos ^{3} x}{\sin ^{3} x+\cos ^{3} x} d x\left(\begin{array}{c}\because \sin (\pi / 2-\theta)=\cos \theta \text { and } \\ \cos (\pi / 2-\theta)=\sin \theta\end{array}\right)$
On adding Eqs. ( $i$ ) and ( $(i i)$, we get $2 I=\int_{0}^{\pi / 2} \frac{\sin ^{3} x d x}{\sin ^{3} x+\cos ^{3} x}+\int_{0}^{\pi / 2} \frac{\cos ^{3} x d x}{\sin ^{3} x+\cos ^{3} x}$
$=\int_{0}^{\pi / 2} \frac{\sin ^{3} x+\cos ^{3} x d x}{\sin ^{3} x+\cos ^{3} x}$
$2 I=\int_{0}^{\pi / 2} 1 d x \Rightarrow 2 I=[x]_{0}^{\pi / 2}=\frac{\pi}{2} \Rightarrow I=\frac{\pi}{4}$

Topic: Mathematics, Definite Integration, NDA, NDA 2010 (Phase 1)
What is $\int_{0}^{\frac{x}{2}} \frac{\mathrm{d} \theta}{1+\cos \theta}$ equal to?

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

The correct answer is:
1

Solution:
$\int_{0}^{\pi / 2} \frac{\mathrm{d} \theta}{1+\cos \theta}=\int_{0}^{\pi / 2} \frac{\mathrm{d} \theta}{2 \cos ^{2}\left(\frac{\theta}{2}\right)}=\frac{1}{2} \int_{0}^{\pi / 2} \sec ^{2} \frac{\theta}{2} \mathrm{~d} \theta$
$=\frac{1}{2}\left[\frac{\tan \theta / 2}{1 / 2}\right]_{0}^{\pi / 2}$
$=\tan \frac{\pi}{4}-\tan 0$
$=1-0=1$

Topic: Mathematics, Definite Integration, NDA, NDA 2017 (Phase 1)
What is $\int_{0}^{\sqrt{2}}\left[\mathrm{x}^{2}\right] \mathrm{dx}$ equal to (where $[.]$ is the greatest integer function)?

Options:
A $\sqrt{2-1}$
B $1-\sqrt{2}$
C $2(\sqrt{2}-1)$
D $\sqrt{3}-1$

The correct answer is:
$\sqrt{2-1}$

Solution:
$\begin{aligned} & \int_{0}^{\sqrt{2}}\left[x^{2}\right] \mathrm{d} x \\ &=\int_{0}^{1}\left[x^{2}\right] d x+\int_{1}^{\sqrt{2}}\left[x^{2}\right] d x \end{aligned}$
$=0+\int_{1}^{\sqrt{2}} 1 \mathrm{dx} \quad(\because[]$ is greatest integer function $)$
$(\mathrm{x})_{1}^{\sqrt{2}}=\sqrt{2}-1$

Topic: Mathematics, Definite Integration, NDA, NDA 2018 (Phase 1)
What is $\int_{0}^{4 \pi}|\cos x| d x$ equal to?

Options:
A 0
B 2
C 4
D 8

The correct answer is:
8

Solution:
$\begin{aligned} \int_{0}^{4 \pi}|\cos x| d x &=4 \int_{0}^{\pi}|\cos x| d x \\ &=4\left[\int_{0}^{\pi / 2} \cos x d x-\int_{\pi / 2}^{\pi} \cos x d x\right] \\ &=4\left[(\sin x)_{0}^{\pi / 2}-(\sin x)_{\pi / 2}^{\pi}\right] \\ &=4\left[\sin \frac{\pi}{2}-0-\sin \pi+\sin \frac{\pi}{2}\right] \\ &=4\left[2 \sin \frac{\pi}{2}\right]=8 \end{aligned}$

Topic: Mathematics, Definite Integration, NDA, NDA 2016 (Phase 1)
What is $\int_{0}^{\pi} \mathrm{e}^{\mathrm{x}} \sin x \mathrm{dx}$ equal to?

Options:
A $\frac{\mathrm{e}^{\pi}+1}{2}$
B $\frac{\mathrm{e}^{\pi}-1}{2}$
C $\mathrm{e}^{\pi}+1$
D $\frac{\mathrm{e}^{\pi}+1}{4}$

The correct answer is:
$\frac{\mathrm{e}^{\pi}+1}{2}$

Solution:
$\int_{0}^{\pi} \mathrm{e}^{\mathrm{x}} \sin x \cdot \mathrm{d} \mathrm{x}=\mathrm{I}$
$\mathrm{I}=\left(\sin \mathrm{x} \cdot \mathrm{e}^{\mathrm{x}}\right)_{0}^{\pi}-\int_{0}^{\pi} \cos \mathrm{x} \cdot \mathrm{e}^{\mathrm{x}} \cdot \mathrm{dx}$
$\mathrm{I}=\left(\sin \pi \cdot \mathrm{e}^{\pi}-\sin 0 . \mathrm{e}^{0}\right)-\left\{\left[\cos x \cdot e^{\mathrm{x}}\right]_{0}^{\pi}-\int_{0}^{\pi} \sin x \cdot \mathrm{e}^{\mathrm{x}} \cdot \mathrm{dx}\right\}$
$\mathrm{I}=0-\left\{\left(\cos \pi \cdot \mathrm{e}^{\pi}-\cos 0 \cdot \mathrm{e}^{0}\right)-\mathrm{I}\right\}$
$\mathrm{I}=-\left[-\mathrm{e}^{\pi}-1\right]-\mathrm{I}$
$\Rightarrow 2 \mathrm{I}=\mathrm{e}^{\pi}+1 \Rightarrow \mathrm{I}=\frac{\mathrm{e}^{\pi}+1}{2} .$

Topic: Mathematics, Definite Integration, NDA, NDA 2018 (Phase 1)