datavorous/entrance-exam-dataset huggingface dataset:
Consider the following in respect of the differential equation:
$\frac{d^{2} y}{d x^{2}}+2\left(\frac{d y}{d x}\right)^{2}+9 y=x \quad$
1.The degree of the differential equation is $1 .$
2.The order of the differential equation is 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:
$\frac{d^{2} y}{d x^{2}}+2\left(\frac{d y}{d x}\right)^{2}+9 y=x$
Order $=2$
Degree $=1$

Topic: Mathematics, Differential Equations, NDA, NDA 2019 (Phase 1)
Consider the following in respect of the function [2016-II]
$f(x)=\left\{\begin{array}{l}2+x, \quad x \geq 0 \\ 2-x, x < 0\end{array}\right.$
1- $\operatorname{limf}_{x \rightarrow 1}(x)$ does not exist.
2- $\mathrm{f}(\mathrm{x})$ is differentiable at $\mathrm{x}=0$
3- $\mathrm{f}(\mathrm{x})$ is continuous at $\mathrm{x}=0$ Which of the above statements is/are correct?

Options:
A 1 only
B 3 only
C 2 and 3 only
D 1 and 3 only

The correct answer is:
3 only

Solution:
For $x \geq 1$
$\lim _{x \rightarrow 1} f(x)=\lim _{x \rightarrow 1} 2+x=2+1=3$
For $x < 1$
$\lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1^{-}} 2+x=2+1=3$
So, $\lim _{x \rightarrow 1} f(x)$ exist.
At $x=0$
RHL: $\lim _{h \rightarrow 0^{+}} f(0+h)=\lim _{h \rightarrow 0} 2+h=2$
LHL $\lim _{h \rightarrow 0^{-}} f(0-h)=\lim _{h \rightarrow 0} 2-h=2$
$f(0)=2+0=2 .$
So, $\mathrm{RHL}=\mathrm{LHL}=f(0)$
$\Rightarrow f(x)$ is continuous at $x=0$ Differentiability at $x=0$
LHD: $\lim _{h \rightarrow 0^{-}} \frac{f(0-h)-f(0)}{-h}=\lim _{h \rightarrow 0^{-}} \frac{2+h-2}{-h}$
$=\frac{-h}{h}=-1$
RHD : $\lim _{h \rightarrow 0^{+}} \frac{f(0+h)-f(0)}{h}=\lim _{h \rightarrow 0^{+}} \frac{2+h-2}{h}=1$
Since $\mathrm{LHD} \neq \mathrm{RHD}$
So, $f(x)$ is not differentiable at $x=0$.

Topic: Mathematics, Continuity and Differentiability, NDA, NDA 2016 (Phase 2)
Consider the following in respect of the function $\mathrm{f}(\mathrm{x})=|\mathrm{x}-3|:$
1- $\mathrm{f}(\mathrm{x})$ is continuous at $\mathrm{x}=3$
2- $\mathrm{f}(\mathrm{x})$ is differentiable at $\mathrm{x}=0$. 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:
2 only

Solution:
Given $\mathrm{f}(\mathrm{x})=|\mathrm{x}-3|$ is not continuous at $\mathrm{x}=3$ but it is differentiable at $\mathrm{x}=0$.

Topic: Mathematics, Continuity and Differentiability, NDA, NDA 2012 (Phase 1)
Consider the following in respect of the matrix $\mathrm{A}=\left(\begin{array}{cc}-1 & 1 \\ 1 & -1\end{array}\right):$
$1.$ $\mathrm{A}^{2}=-\mathrm{A}$
$2.$ $A^{3}=4 A$
Which of the above 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:
$A=\left[\begin{array}{cc}-1 & 1 \\ 1 & -1\end{array}\right]$
$\begin{aligned} A \cdot A &=\left[\begin{array}{cc}-1 & 1 \\ 1 & -1\end{array}\right]\left[\begin{array}{cc}-1 & 1 \\ 1 & -1\end{array}\right] \\ &=\left[\begin{array}{cc}2 & -2 \\ -2 & 2\end{array}\right]=-2\left[\begin{array}{cc}-1 & 1 \\ 1 & -1\end{array}\right] \\ A^{2} &=-2 A \end{aligned}$
$A^{2}, A=-2\left[\begin{array}{cc}-1 & 1 \\ 1 & -1\end{array}\right]\left[\begin{array}{cc}-1 & 1 \\ 1 & -1\end{array}\right]$
$\quad=-2\left[\begin{array}{cc}2 & -2 \\ -2 & 2\end{array}\right]=4\left[\begin{array}{cc}-1 & 1 \\ 1 & -1\end{array}\right]$
$A^{3}=4 A$
Hence $A^{2} \neq-A, A^{3}=4 A$

Topic: Mathematics, Matrices, NDA, NDA 2016 (Phase 1)
Consider the following in respect of two events $\mathrm{A}$ and $\mathrm{B}$ :
$1.$ $\quad \mathrm{P}($ A occurs but not $\mathrm{B})=\mathrm{P}(\mathrm{A})-\mathrm{P}(\mathrm{B})$ if $\mathrm{B} \subset \mathrm{A}$
$2.$ $\quad \mathrm{P}(\mathrm{A}$ alone or $\mathrm{B}$ alone occurs $)=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})$
$3.$ $\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})$ if $\mathrm{A}$ and $\mathrm{B}$ are mutually exclusive
Which of the above is/are correct?

Options:
A 1 only
B 1 and 3 only
C 2 and 3 only
D 1 and 2 only

The correct answer is:
1 and 3 only

Solution:
Statement $1:$ If $B \subset A$, then $\mathrm{P}(\mathrm{A}-\mathrm{B})=\mathrm{P}(\mathrm{A})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})$
$=\mathrm{P}(\mathrm{A})-\mathrm{P}(\mathrm{B})$
$\therefore$ It is correct.
Statement $2: \mathrm{P}(\mathrm{A}$ alone or $\mathrm{B}$ alone $)$ $=\mathrm{P}(\mathrm{A})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})$
$=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-2 \mathrm{P}(-\mathrm{A} \cap \mathrm{B})$
$\therefore$ It is false. Statement $3:$ If $A, B$ are mutually exclusive events, then $\mathrm{P}(\mathrm{A} \cap \mathrm{B})=0$
$\Rightarrow \mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})$
It is correct.

Topic: Mathematics, Probability, NDA, NDA 2018 (Phase 1)
Consider the following in respect of two non-singular matrices $A$ and $B$ of same order: $\quad[2015-I]$
$1.$ $\operatorname{det}(A+B)=\operatorname{det} A+\operatorname{det} B$
$2.$ $(A+B)^{-1}=A^{-1}+B^{-1}$
Which of the above is/are correct?

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

The correct answer is:
Neither 1 nor 2

Solution:
Non-singular matrix is a matrix whose determinate Value is non-zero.
Let $\mathrm{A}=\left[\begin{array}{ll}1 & 1 \\ 1 & 2\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right]$
Here, $\mathrm{A}$ and $\mathrm{B}$ are non-singular matrix Now from Statement 1 :
$\mathrm{A}+\mathrm{B}=\left[\begin{array}{ll}1 & 1 \\ 1 & 2\end{array}\right]+\left[\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right]=\left[\begin{array}{ll}3 & 2 \\ 2 & 4\end{array}\right]$
$\operatorname{det}(\mathrm{A}+\mathrm{B})=12-4=8$
Now det $(\mathrm{A})=\left|\begin{array}{ll}1 & 1 \\ 1 & 2\end{array}\right|=1$
and det $(\mathrm{B})=\left|\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right|=3$
Now, $\operatorname{det}(\mathrm{A}+\mathrm{B})=4 \neq \operatorname{det}(\mathrm{A}+\mathrm{B})$
Statement 1 is wrong. Now from Statement 2 :
$(\mathrm{A}+\mathrm{B})^{-1}=\frac{1}{|\mathrm{~A}+\mathrm{B}|} \operatorname{adj}(\mathrm{A}+\mathrm{B})$
$=\frac{1}{8}\left[\begin{array}{cc}4 & -2 \\ -2 & 3\end{array}\right]$
and $\mathrm{A}^{-1}=\frac{1}{|\mathrm{~A}|} \operatorname{adj}(\mathrm{A})=\frac{1}{1}\left[\begin{array}{cc}2 & -1 \\ -1 & 1\end{array}\right]$
and $\mathrm{B}^{-1}=\frac{1}{|\mathrm{~B}|} \operatorname{adj}(\mathrm{B})=\frac{1}{3}\left[\begin{array}{cc}2 & -1 \\ -1 & 2\end{array}\right]$
Now, $A^{-1}+B^{-1}=\left[\begin{array}{cc}2 & -1 \\ -1 & 1\end{array}\right]+\left[\begin{array}{cc}\frac{2}{3} & \frac{-1}{3} \\ \frac{-1}{3} & \frac{2}{3}\end{array}\right]$
$=\left[\begin{array}{cc}\frac{8}{3} & \frac{-4}{3} \\ \frac{-4}{3} & \frac{5}{3}\end{array}\right] \neq(\mathrm{A}+\mathrm{B})^{-1}$
$\therefore$ Statement 2 is wrong
$\therefore$ Option (d) is correct.

Topic: Mathematics, Matrices, NDA, NDA 2015 (Phase 1)
Consider the following inequalities in respect of vectors
$\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}:$
$1.$ $\quad|\vec{a}+\vec{b}|_{Â£}|\vec{a}|+|\vec{b}|$
$2.$ $\quad|\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}|^{3}|\overrightarrow{\mathrm{a}}|-|\overrightarrow{\mathrm{b}}|$
Which of the above 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:
$\begin{aligned} \text { } &|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}| \leq|\overrightarrow{\mathrm{a}}|+|\overrightarrow{\mathrm{b}}| \\ &|\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}| \geq|\overrightarrow{\mathrm{a}}|-|\overrightarrow{\mathrm{b}}| \end{aligned}$
both are correct.

Topic: Mathematics, Vector Algebra, NDA, NDA 2015 (Phase 2)
Consider the following points:
1) $(0,5)$
2) $(2,-1)$
3) $(3,-4)$
Which of the above lie on the line $3 x+y=5$ and at a distance
$\sqrt{10}$ from $(1,2) ?$

Options:
A 1 only
B 2 only
C 1 and 2 only
D 1,2 and 3

The correct answer is:
1 and 2 only

Solution:
All three points $(0,5),(2,-1)$ and $(3,-4)$ lie on $3 x+y=5$
$\sqrt{(0-1)^{2}+(5-2)^{2}}=\sqrt{10}$
$\sqrt{(2-1)^{2}+(-1-2)^{2}}=\sqrt{10}$
$\sqrt{(3-1)^{2}+(-4-2)^{2}}=\sqrt{40}=2 \sqrt{10}$

Topic: Mathematics, Straight Lines, NDA, NDA 2014 (Phase 1)
Consider the following relations among the angles $\alpha, \beta$ and $\gamma$ made by a vector with the coordinate axes
I. $\quad \cos 2 \alpha+\cos 2 \beta+\cos 2 \gamma=-1$
II. $\quad \sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma=1$
Which of the 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:
Only I

Solution:
We have, $\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1$
$\Rightarrow 2 \cos ^{2} \alpha+2 \cos ^{2} \beta+2 \cos ^{2} \gamma=2$
$\Rightarrow 2 \cos ^{2} \alpha-1+2 \cos ^{2} \beta-1+2 \cos ^{2} \gamma-1=2-3$
$\Rightarrow \cos 2 \alpha+\cos 2 \beta+\cos 2 \gamma=-1$
Hence statement $-\mathrm{I}$ is correct.
and now from (i), $1-\sin ^{2} \alpha+1-\sin ^{2} \beta+1-\sin ^{2} \gamma=1$
$\Rightarrow \sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma=2$
Hence, only statement I is correct.

Topic: Mathematics, Three Dimensional Geometry, NDA, NDA 2011 (Phase 1)
Consider the following relations from $A$ to $B$ where $A=\{u, v, w, x, y, z\}$ and $B=\{p, q, r, s\}$
1.$\{(u, p),(v, p),(w, p),(x, q),(y, q),(z, q)\}$
2.\{(u, p),(v, q),(w, r),(z, s)\}
3.\{(u, s),(v, r),(w, q),(u, p),(v, q),(z, q)
4.$\{(u, q),(v, p),(w, s),(x, r),(y, q),(z, s)$
Which of the above relations are not functions ?

Options:
A 1 and 2
B 1 and 4
C 2 and 3
D 3 and 4

The correct answer is:
2 and 3

Solution:
Given that, $A=\{u, v, x, y, z\} ; B=\{p, q, r, s\}$ As we know, a mapping $\mathrm{f}: \mathrm{x} \rightarrow \mathrm{y}$ is said to be a function if each element in the set $\mathrm{x}$ has its image in set $\mathrm{y} .$ It $\mathrm{k}$
also possible that these are few elements in set y which are not the image of any element in set $\mathrm{x}$. Every element
in set $x$ should have one and only one image.
(i)
(ii)
(iii)
(iv)
(ii) and (iii) are not function

Topic: Mathematics, Sets and Relations, NDA, NDA 2014 (Phase 1)
Consider the following standard electrode potentials (Eo in volts) in aqueous solution:

Element_M3+/M_M+/M_Al-1.66+0.55Tl+1.26-0.34

Based on these data, which of the following statements is correct?

Options:
A Tl+ is more stable than Al3+
B Al+ is more stable than Al3+
C Tl3+ is more stable than Al3+
D Tl+ is more stable than Al+

The correct answer is:
Tl+ is more stable than Al+

Solution:
âGo=-nFEo
if Eo= positive; then âGo=negative,
For those conversions whose E o values are more positive are more favourable.

Topic: Chemistry, Electrochemistry, NEET
Consider the following statement:
"The mean of a binomial distribution is 3 and variance is $4 . "$ Which of the following is correct regarding this statement?

Options:
A It is always true
B It is sometimes true
C It is never true
D No conclusion can be drawn

The correct answer is:
It is never true

Solution:
Given, that $\mathrm{np}=3$ and $\mathrm{npq}=4$ where $\mathrm{p}$ is the probability of success in one trial and $\mathrm{q}$ is the probability of failure and $\mathrm{n}$ is the number of trials
$\Rightarrow q=\frac{4}{3}$
and this is not possible. Thus, the given statement is never true.

Topic: Mathematics, Probability, NDA, NDA 2006 (Phase 2)
Consider the following statements:
1. $1^{\circ}$ in radian measure is less than $0.02$ radians.
2. 1 radian in degree measure is greater than $45^{\circ}$. 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:
(1) $1^{\circ}=\left(\frac{\pi}{180}\right)$ radian $=0.01746$ radian which is less than $0.02$ radian.
(2) 1 radian $=\left(\frac{180}{\pi}\right)^{\circ}=57^{\circ} 16^{\prime} 22^{\prime \prime}$ approx. which is
greater than $45^{\circ}$. Hence, both statements are true.

Topic: Mathematics, Trigonometric Ratios & Identities, NDA, NDA 2012 (Phase 1)
Consider the following statements
1.$\phi \in\{\phi\}$
2.$\quad\{\phi\} \subseteq \phi$
Which of the statements given above is/are correct?

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

The correct answer is:
Neither 1 nor $2$

Solution:
Both statements are incorrect

Topic: Mathematics, Sets and Relations, NDA, NDA 2008 (Phase 1)
Consider the following statements :
1- $\left(\omega^{10}+1\right)^{7}+\omega=0$
2- $\left(\omega^{105}+1\right)^{10}=\mathrm{p}^{10}$ for some prime number p where $\omega \neq 1$ is a cube root of unity. 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:
2 only

Solution:
Statement-1 $\left(\omega^{10}+1\right)^{7}+\omega=0$
L.H.S $=\left[\left(\omega^{3}\right)^{3} \omega+1\right]^{7}+\omega$
$\Rightarrow(\omega+1)^{7}+\omega\left(\because \omega^{3}=1\right)$
$\Rightarrow-\left[\omega^{2}\right]^{7}+\omega=-\left(\omega^{3}\right)^{4} \cdot \omega^{2} \cdot+\omega_{.}=-\omega^{2}+\omega \neq 0$
Hence, LHS $\neq$ RHS
Statement $-2$
L.H.S $=\left(\omega^{105}+1\right)^{10}=\left[\left(\omega^{3}\right)^{35}+1\right]^{10}=2^{10}$
where 2 is a prime number. Hence, statement $-2$ is correct.

Topic: Mathematics, Complex Number, NDA, NDA 2012 (Phase 2)
Consider the following statements:
$1.$ A matrix is not a number
$2.$ Two determinants of different order may have the same value.
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:
Both the statements are correct.

Topic: Mathematics, Matrices, NDA, NDA 2013 (Phase 1)
Consider the following statements:
1 . Both variance and standard deviation are measures of variability in the population.
2 . Standard deviation is the square of the variance. 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:
Neither 1 nor 2

Solution:
Both statements are wrong. Also, $\mathrm{S} . \mathrm{D}=\sqrt{\text { variance }}$

Topic: Mathematics, Statistics, NDA, NDA 2013 (Phase 1)
Consider the following statements:
1 . Coefficient of variation depends on the unit of measurement of the variable.
2 . Range is a measure of dispersion.
3 . Mean deviation is least when measured about median. Which of the above statements are correct?

Options:
A 1 and 2 only
B 2 and 3 only
C 1 and 3 only
D 1,2 and 3

The correct answer is:
2 and 3 only

Solution:
Only statements 2 and 3 are correct

Topic: Mathematics, Statistics, NDA, NDA 2017 (Phase 2)
Consider the following statements:
1- Derivative of $f(\mathrm{x})$ may not exist at some point.
2- Derivative of $f(\mathrm{x})$ may exist finitely at some point.
3- Derivative of $f(\mathrm{x})$ may be infinite (geometrically) at some point. Which of the above statements are correct?

Options:
A 1 and 2 only
B 2 and 3 only
C 1 and 3 only
D 1,2 and 3

The correct answer is:
1,2 and 3

Solution:
All statements are correct.

Topic: Mathematics, Continuity and Differentiability, NDA, NDA 2017 (Phase 2)
Consider the following statements:
$1.$ Determinant is a square matrix.
$2.$ Determinant is a number associated with a square matrix. 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:
2 only

Solution:
$1.$ We know that, determinant is not a square matrix, so it is not a true statement.
$2.$ It is true that, determinant is a number associated with a square matrix.
Hence, Statement 2 is correct

Topic: Mathematics, Determinants, NDA, NDA 2014 (Phase 2)
Consider the following statements:
1.$\frac{\mathrm{dy}}{\mathrm{dx}}$ at a point on the curve given slope of the tangent
at that point.
2.If a(t) denotes acceleration of a particle, then $\int \mathrm{a}(\mathrm{t}) \mathrm{dt}+\mathrm{c}$ gives velocity of the particle.
3.If $\mathrm{s}(\mathrm{t})$ gives displacement of a particle at time $\mathrm{t}$, then
$\frac{\mathrm{ds}}{\mathrm{dt}}$ gives its acceleration at that instant. Which of the above statements is/are correct?

Options:
A 1 and 2 only
B 2 only
C 1 only
D 1,2 and 3

The correct answer is:
1 and 2 only

Solution:
Statements (1), (2) are correct.

Topic: Mathematics, Application of Derivatives, NDA, NDA 2017 (Phase 2)
Consider the following statements:
1 . Equations ax $+b y+c z+d=0, a^{\prime} x+b^{\prime} y+c^{\prime} z+d^{\prime}=0$
represent a straight line.
2 . Equation of the form
$\frac{\mathrm{x}-\alpha}{\ell}=\frac{\mathrm{y}-\beta}{\mathrm{m}}=\frac{\mathrm{z}-\gamma}{\mathrm{n}}$
represent a straight line passing through the point $(\alpha, \beta, \gamma)$ and having direction ratio proportional to $l, \mathrm{~m}, \mathrm{n}$. Which of the statements given above 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:
Equations $a x+b y+c z+d=0, a^{\prime} x+b^{\prime} y+c^{\prime} z+d^{\prime}=0$
represent a straight line and equation of the form $\frac{x-\alpha}{\ell}=\frac{y-\beta}{m}=\frac{z-y}{n}$
represent a straight line passing through the point $(\alpha, \beta, \gamma)$ and having direction ratios proportional to $l, \mathrm{~m}$
n. Thus, both statements are correct.

Topic: Mathematics, Three Dimensional Geometry, NDA, NDA 2006 (Phase 2)
Consider the following statements
1- Every function has a primitive.
2- A primitive of a function is unique. Which of the statements given above 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 every function does not has a primitive but a primitive of a function is unique.

Topic: Mathematics, Functions, NDA, NDA 2010 (Phase 1)
Consider the following statements:
$1.$ every zero matrix is a square matrix.
$2.$ A matrix has a numerical value.
$3.$ A unit matrix is a diagonal matrix.
Which of the above statements is/are correct?

Options:
A 2 only
B 3 only
C 2 and 3
D 1 and 3

The correct answer is:
3 only

Solution:
Only statement $-3$ is correct
$\text { Unit matrix }=\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right]$

Topic: Mathematics, Matrices, NDA, NDA 2012 (Phase 1)
Consider the following statements :
1- $f^{\prime}(2+0)$ does not exist.
2- $f^{\prime}(2-0)$ does not exist.
Let $f(x)$ be a function defined in $1 \leq x < \infty$ by $\quad[2014-I]$
$f(x)=\left\{\begin{array}{cc}2-x & \text { for } 1 \leq x \leq 2 \\ 3 x-x^{2} & \text { for } x>2\end{array}\right.$
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:
1 only

Solution:
$\mathrm{f}^{\prime}(2+0)=\lim _{\mathrm{h} \rightarrow 0} \mathrm{f}^{\prime}(2+\mathrm{h})$
$=\lim _{\mathrm{h} \rightarrow 0} 3-4-2 \mathrm{~h}=-1$
$\mathrm{f}^{\prime}(2-0)=\lim _{\mathrm{h} \rightarrow 0} \mathrm{f}^{\prime}(2-\mathrm{h})=(3-4-2 \mathrm{~h})=-1$
So, $\mathrm{f}^{\prime}(\mathrm{x})$ exist at $\mathrm{x}=2$

Topic: Mathematics, Continuity and Differentiability, NDA, NDA 2014 (Phase 1)
Consider the following statements:
1- $f(x)=[x]$, where $[.]$ is the greatest integer function, is discontinuous at $x=n$, where $n \in Z$.
2- $f(x)=\cot x$ is discontinuous at $x=n \pi$, where $n \in Z$.
which of the above statements is/are correct?

Options:
A 1 only
B 2 only
C Both 1 and 2
D Neither 1 or 2

The correct answer is:
Both 1 and 2

Solution:
From statement-1 From the definition of greatest integer function $\mathrm{f}(\mathrm{x})=[\mathrm{x}]$ is discontinuous at $\mathrm{x}=\mathrm{n}$ for any value of $\mathrm{n} \in \mathrm{Z}$
$\therefore \quad$ Statement 1 is correct From statement- 2 $f(x)=\cot x$
for $x=\pi=\cot \pi=-\infty$
for $\mathrm{x}=2 \pi$
$f(2 \pi)=\cot 2 \pi=-\infty$
Hence the function $\mathrm{f}(\mathrm{x})=\cot \mathrm{x}$ is discontinuous at $\mathrm{x}=\mathrm{n} \pi$ where $\mathrm{n} \in \mathrm{Z}$.
$\therefore$ Option (c) is correct.

Topic: Mathematics, Continuity and Differentiability, NDA, NDA 2015 (Phase 1)
Consider the following statements
$1.$ For any three vectors $\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}, \overrightarrow{\mathbf{c}}$;
$\overrightarrow{\mathbf{a}} \cdot\{(\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}) \times(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}})\}=0$
$2.$ For any three coplanar unit vectors
$\overrightarrow{\mathbf{d}}, \overrightarrow{\mathbf{e}}, \overrightarrow{\mathbf{f}} ;(\overrightarrow{\mathbf{d}} \times \overrightarrow{\mathbf{e}}) \cdot \overrightarrow{\mathbf{f}}=1$
Which of the statements given above is/are correct?

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

The correct answer is:
1 only

Solution:
Consider statement 1
$\vec{a} \cdot\{(\vec{b}+\vec{c}) \times(\vec{a}+\vec{b}+\vec{c})\}=0$
$=\vec{a} \cdot\{\vec{b} \times \vec{a}+\vec{b} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}+\vec{c} \times \vec{b}+\vec{c} \times \vec{c}\}$
$=0+0+\vec{a} \cdot(\vec{b} \times \vec{c})+0+\vec{a} \cdot(\vec{c} \times \vec{b})+0 \quad(\because \vec{a} \times \vec{a}=0)$
$=\vec{a} \cdot(\vec{b} \times \vec{c})-\vec{a} \cdot(\vec{b} \times \vec{c})=0$
and for any three coplanar vectors $\overrightarrow{\mathrm{d}}, \overrightarrow{\mathrm{e}}, \overrightarrow{\mathrm{f}}$
$(\vec{d} \times \vec{e}) \cdot \vec{f}=0$
Hence, statement (1) is correct and statement-2 is incorrect.

Topic: Mathematics, Vector Algebra, NDA, NDA 2010 (Phase 1)
Consider the following statements:
1. If $3 \theta$ is an acute angle such that $\sin 3 \theta=\cos 2 \theta$, then
the mesurement of $\theta$ in radian equals to $\frac{\pi}{10}$.
2 One radian is the angle subtended at the centre of a circle by an arc of the same circle whose length is equal to the diameter of that circle. 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:
1 only

Solution:
Statement: 1 $\sin 3 \theta=\cos 2 \theta$
$\sin 3 \theta=\sin \left(\frac{\pi}{2}-2 \theta\right)$
$3 \theta=\frac{\pi}{2}-2 \theta$
$5 \theta=\frac{\pi}{2} \Rightarrow \theta=\frac{\pi}{10}$
Statement : 2 One radian is the angle subtended at the centre of a circle by an arc of the same circle whose length is equal to radius of that circle. Hence, statement 1 is correct.

Topic: Mathematics, Trigonometric Ratios & Identities, NDA, NDA 2014 (Phase 1)
Consider the following statements: $\quad$
1 . If 10 is added to each entry on a list then the average increase by $10 .$
2 . If 10 is added to each entry on a list, then the standard deviation increase by 10 .
3 . If each entry on a list is doubled, then the average doubles
Which of the above statement are correct?

Options:
A 1,2 and 3
B 1 and 2 only
C 1 and 3 only
D 2 and 3 only

The correct answer is:
1 and 3 only

Solution:
No Solution

Topic: Mathematics, Statistics, NDA, NDA 2018 (Phase 2)
Consider the following statements
$1.$ If $\mathrm{A}^{\prime}=\mathrm{A}$; then $A$ is a singular matrix, where $\mathrm{A}^{\prime}$ is the transpose of $A$.
$2.$ If $A$ is a square matrix such that $A^{3}=I$, then $A$ is non-singular. Which of the statements given above 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:
If $A^{\prime}=\mathrm{A}$ where $A$ ' is transpose of matrix then $|A|=\left|A^{\prime}\right|$
But it is not neccessary that $|A|=0$
i.e. $A$ is singular matrix Hence, statement 1 is wrong. Given $A^{3}=I$
$\left|A^{3}\right|=|I|=1$
$\Rightarrow|A|=1$
Thus, $A$ is non-singular matrix. Hence, only statement 2 is correct.

Topic: Mathematics, Matrices, NDA, NDA 2010 (Phase 1)
Consider the following statements :
$1$ If $A$ and $B$ are exhaustive events, then their union is the sample space.
$2$ If $A$ and $B$ are exhaustive events, then their intersection must be an empty event. 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:
1 only

Solution:
No Solution

Topic: Mathematics, Probability, NDA, NDA 2013 (Phase 2)