datavorous/entrance-exam-dataset huggingface dataset:
Consider the following statements:
1) The length $\mathrm{p}$ of the perpendicular from the origin to
the line $a x+b y=c$ satisfies the relation $p^{2}=\frac{c^{2}}{a^{2}+b^{2}}$.
2) The length $\mathrm{p}$ of the perpendicular from the origin to
the line $\frac{\mathrm{x}}{\mathrm{a}}+\frac{\mathrm{y}}{\mathrm{b}}=1$ satisfies the relation $\frac{1}{\mathrm{p}^{2}}=\frac{1}{\mathrm{a}^{2}}+\frac{1}{\mathrm{~b}^{2}}$.
3) The length $\mathrm{p}$ of the perpendicular from the origin to the line $\mathrm{y}=\mathrm{mx}+\mathrm{c}$ satisfies the relation $\frac{1}{\mathrm{p}^{2}}=\frac{1+\mathrm{m}^{2}+\mathrm{c}^{2}}{\mathrm{c}^{2}}$
Which of the above is/are correct?

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

The correct answer is:
1 and 2 only

Solution:
1: We know, the perpendicular distance from
$\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)$ to line $\mathrm{ax}+\mathrm{by}+\mathrm{c}=0$ is $\frac{\mid \mathrm{ax}_{1}+\mathrm{by}_{1}+\mathrm{c}}{\sqrt{\mathrm{a}^{2}+\mathrm{b}^{2}}}$
Here, $\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)=(0,0)$ and distance $=\mathrm{P}$.
$P=\frac{|a(0)+b(0)+(c)|}{\sqrt{a^{2}+b^{2}}}$
$\Rightarrow \mathrm{P}^{2}=\frac{\mathrm{c}^{2}}{\mathrm{a}^{2}+\mathrm{b}^{2}}$
1 is correct.
2: Line is $\frac{\mathrm{x}}{\mathrm{a}}+\frac{\mathrm{y}}{\mathrm{b}}=1 \Rightarrow\left(\frac{1}{\mathrm{a}}\right) \mathrm{x}+\left(\frac{1}{\mathrm{~b}}\right) \mathrm{y}+(-1)=0$
$P=\frac{\left|\frac{1}{a}(0)+\frac{1}{b}(0)-1\right|}{\sqrt{\frac{1}{a^{2}}+\frac{1}{b^{2}}}}=\frac{1}{\sqrt{\frac{1}{a^{2}}+\frac{1}{b^{2}}}}$
$=P^{2}=\frac{1}{\frac{1}{a^{2}}+\frac{1}{b^{2}}}=\frac{a^{2} b^{2}}{b^{2}+a^{2}}$
$\Rightarrow \frac{1}{\mathrm{P}^{2}}=\frac{\mathrm{a}^{2}+\mathrm{b}^{2}}{\mathrm{a}^{2} \mathrm{~b}^{2}}=\frac{1}{\mathrm{a}^{2}}+\frac{1}{\mathrm{~b}^{2}}$
2 is correct.
3 : Line is $\mathrm{y}=\mathrm{mx}+\mathrm{c} \Rightarrow \mathrm{mx}-\mathrm{y}+\mathrm{c}=0$
$P=\frac{|\mathrm{m}(0)-0+\mathrm{c}|}{\sqrt{\mathrm{m}^{2}+1}}$
$\Rightarrow \mathrm{P}^{2}=\frac{\mathrm{c}^{2}}{\mathrm{~m}^{2}+1} \Rightarrow \frac{1}{\mathrm{P}^{2}}=\frac{\mathrm{m}^{2}+1}{\mathrm{c}^{2}}$
3 is wrong. Only 1 and 2 are correct.

Topic: Mathematics, Straight Lines, NDA, NDA 2018 (Phase 1)
Consider the following statements:
$1.$ The matrix $\left(\begin{array}{lll}1 & 2 & 1 \\ a & 2 a & 1 \\ b & 2 b & 1\end{array}\right)$ is singular.
$2.$ The matrix $\left(\begin{array}{lll}c & 2 c & 1 \\ a & 2 a & 1 \\ b & 2 b & 1\end{array}\right)$ is non-singular.
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:
$1.$ $\left(\begin{array}{ccc}1 & 2 & 1 \\ a & 2 a & 1 \\ b & 2 b & 1\end{array}\right)=2\left(\begin{array}{lll}1 & 1 & 1 \\ a & a & 1 \\ b & b & 1\end{array}\right)=2\left|\begin{array}{lll}1 & 1 & 1 \\ a & a & 1 \\ b & b & 1\end{array}\right|=0$
Hence matrix is singular.
$2.$ $\left|\begin{array}{lll}\mathrm{c} & 2 \mathrm{c} & 1 \\ \mathrm{a} & 2 \mathrm{a} & 1 \\ \mathrm{~b} & 2 \mathrm{~b} & 1\end{array}\right|=2\left|\begin{array}{lll}\mathrm{c} & \mathrm{c} & 1 \\ \mathrm{a} & \mathrm{a} & 1 \\ \mathrm{~b} & \mathrm{~b} & 1\end{array}\right|=2\left|\begin{array}{lll}\mathrm{c} & \mathrm{c} & 1 \\ \mathrm{a} & \mathrm{a} & 1 \\ \mathrm{~b} & \mathrm{~b} & 1\end{array}\right|=0$
Hence matrix is singular.

Topic: Mathematics, Matrices, NDA, NDA 2013 (Phase 2)
Consider the following statements:
1 . The mean and median are equal in symmetric distribution.
2 . The range is the difference between the maximum value and the minimum value in the data.
3 . The sum of the areas of the rectangles in the histogra is equal to the total area bounded by the frequency polygon and the horizontal axis. 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:
Mean = Median (in symmetric distribution) Range $=$ (Max. value-Min. value) And sum of areas of rectangles in the histogram is always equal to the total area bounded by frequency polygon and the horizontal axis

Topic: Mathematics, Statistics, NDA, NDA 2016 (Phase 2)
Consider the following statements:
$1.$ The probability that there are 53 Sundays in a leap year is twice the probability that there are 53 Sundays in a non-leap year.
$2.$ The probability that there are 5 Mondays in the month of March is thrice the probability that there are 5 Mondays in the month of April. 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:
Statement 1:
Anon leap year has 365 days. i.e., 52 weeks and 1 day. 1 day can be \{Sunday\}, \{Monday $\},$ Tuesday\},
\{Wednesday $\}$, \{Thursday\}, \{Friday\}, \{Saturday\}
In total, there are 7 possibilities and 1 possibility is Sunday.
Required probability $=\frac{1}{7}$
A leap year has 366 days. i.e., 52 weeks and 2 days. 2 days can be \{Sun, Mon\}, \{Mon, Tue\}, \{Tue, Wed $\}$,
\{Wed, Thu \}, \{Thu, Fri\}, \{Fri, Sat\}, \{Sat, Sun\}.
In total, there are 7 possibilities and 2 possibilities have Sundays.
Required probability $=\frac{2}{7}$
So, statement 1 is correct. Statement 2 :
March has 31 days. i.e., 4 complete weeks and 3 days. 3 days can be $\{\mathrm{S}, \mathrm{M}, \mathrm{T}\},\{\mathrm{M}, \mathrm{T}, \mathrm{W}\},\{\mathrm{T}, \mathrm{W}, \mathrm{Th}\},\{\mathrm{W}, \mathrm{Th}, \mathrm{F}\}$
$\{\mathrm{Th}, \mathrm{F}, \mathrm{Sa}\},\{\mathrm{F}, \mathrm{Sa}, \mathrm{S}\},\{\mathrm{Sa}, \mathrm{S}, \mathrm{M}\}$
In total 7 possibilities, Monday can come in 3 possibilities
$\therefore$ Required probabilities $=\frac{3}{7}$
April has 30 days. i.e., 4 complete weeks and 2 days. 2 days can be $\{\mathrm{S}, \mathrm{M}\},\{\mathrm{M}, \mathrm{T}\},\{\mathrm{T}, \mathrm{W}\},\{\mathrm{W}, \mathrm{Th}\},\{\mathrm{Th}, \mathrm{F}\}$
$\{\mathrm{F}, \mathrm{Sa}\},\{\mathrm{Sa}, \mathrm{S}\}$
In total 7 possibilities, Monday can come in 2 possibilities.
$\therefore$ Required probability $=\frac{2}{7}$
$\therefore$ Statement 2 is wrong.

Topic: Mathematics, Probability, NDA, NDA 2010 (Phase 1)
Consider the following statements:
$1.$ The product of two non-zero matrices can never be identity matrix.
$2.$ The product of two non-zero matrices can never be zeromatrix.
Which of the above statements is/are correct?

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

The correct answer is:
Neither lnor 2

Solution:
If B $=\mathrm{A}^{-1}$, then $\mathrm{AB}=\mathrm{I}$ (identity matrix) Therefore, statement 1 is false.
$\mathrm{I}=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$ and $\mathrm{A}=\left[\begin{array}{ll}0 & 2 \\ 2 & 0\end{array}\right]$, then $\mathrm{IA}=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$
$\left[\begin{array}{ll}0 & 2 \\ 2 & 0\end{array}\right]=0$
Therefore, statement 2 is not correct.

Topic: Mathematics, Matrices, NDA, NDA 2013 (Phase 2)
Consider the following statements:
1) The sum of cubes of first 20 natural numbers is 44400 .
2) The sum of squares of first 20 natural numbers is 2870 . 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:
Consider the Ist statement $1^{3}+2^{3}+\ldots .+20^{3}$.
$=\left[\frac{\mathrm{n}(\mathrm{n}+1)}{2}\right]^{2}$ Here $\mathrm{n}=20$
$=\left[20 \times \frac{21}{2}\right]^{2}=(210)^{2}=44100$
So, statement $\mathrm{I}$ is false. Consider the Ind statement. $\begin{aligned} 1^{2}+2^{2}+3^{2}+\ldots \ldots \ldots+20^{2}=\frac{\mathrm{n}(\mathrm{n}+1)(2 \mathrm{n}+1)}{6} \\ &=\frac{20 \times 21 \times 41}{6}=2870 \end{aligned}$
So, statement II is true.

Topic: Mathematics, Sequences and Series, NDA, NDA 2012 (Phase 2)
Consider the following statements
1 . The sum of deviations from mean is always zero
2 . The sum of absolute deviations is minimum when taken around median Which of the above statements is/are correct?

Options:
A 1 only
B 2 only
C Both 1 and
D Neither $1 \mathrm{n} \alpha$

The correct answer is:
Both 1 and

Solution:
Both the given statements are correct.

Topic: Mathematics, Statistics, NDA, NDA 2018 (Phase 1)
Consider the following statements:
1. The value of $\cos 46^{\circ}-\sin 46^{\circ}$ is positive.
2. The value of $\cos 44^{\circ}-\sin 44^{\circ}$ is negative.
Which of the above statement 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:
Since $\cos \theta>\sin \theta$, in $\left[0, \frac{\pi}{4}\right]$
and $\cos \theta < \sin \theta$, in $\left[\frac{\pi}{4}, \frac{\pi}{2}\right]$
$\therefore \quad \cos 46^{\circ}-\sin 46^{\circ}=-\mathrm{ve}$
and $\cos 44^{\circ}-\sin 44^{\circ}=+\mathrm{ve}$
So, both the above statements are incorrect.

Topic: Mathematics, Trigonometric Ratios & Identities, NDA, NDA 2012 (Phase 2)
Consider the following statements:
1) There exists $\theta \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ for which $\tan ^{-1}(\tan \theta) \neq \theta$.
2) $\quad \sin ^{-1}\left(\frac{1}{3}\right)-\sin ^{-1}\left(\frac{1}{5}\right)=\sin ^{-1}\left(\frac{2 \sqrt{2}(\sqrt{3}-1)}{15}\right)$
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:
2 only

Solution:
$\theta \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ for which $\tan ^{-1}(\tan \theta)=\theta$.
Hence, statement (1) is incorrect.
if $x \leq 1 ; y \leq 1 \& x^{2}+y^{2} \leq 1$
$\therefore \sin ^{-1}(x)-\sin ^{-1} y=\sin ^{-1}\left[x \sqrt{1-y^{2}}-y \sqrt{1-x^{2}}\right]$
$\Rightarrow \sin ^{-1} \frac{1}{3}-\sin ^{-1} \frac{1}{5}$
$\quad=\sin ^{-1}\left[\frac{1}{3} \sqrt{1-\frac{1}{25}}-\frac{1}{5} \sqrt{1-\frac{1}{9}}\right]$
$=\sin ^{-1}\left[\frac{1}{3 \times 5} \sqrt{24}-\frac{1}{5 \times 3} \sqrt{8}\right]$
$=\sin ^{-1}\left[\frac{\sqrt{8 \times 3}-\sqrt{8}}{15}\right]=\sin ^{-1}\left[\frac{2 \sqrt{2}(\sqrt{3}-1)}{15}\right]$
Hence, statement ( 2 ) is correct.

Topic: Mathematics, Inverse Trigonometric Functions, NDA, NDA 2016 (Phase 1)
Consider the following statements:
1) There exists no triangle $A B C$ for which $\sin A+\sin B=$ $\sin C$
2) If the angle of a triangle are in the ratio $1: 2: 3$, then its sides will be in the ratio $1: \sqrt{3}: 2$.
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:
1) Given, $\sin A+\sin B=\sin C$
$a+b=c \quad\left(\because B y\right.$ sine law,$\left.\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}=K\right)$
Here, the sum of two sides of $\Delta \mathrm{ABC}$ is equal to the third side, but it is not possible (Because by triangle inequality, the sum of the length of two sides of a triangle is always greater than the length of the third side) $\mathrm{a}+\mathrm{b}>\mathrm{c}$
2) Ratio of angles of a triangle $A: B: C=1: 2: 3$
$\mathrm{A}+\mathrm{B}+\mathrm{C}=180^{\circ}$
$\begin{aligned} \therefore \mathrm{A} &=30^{\circ} \\ \mathrm{B} &=60^{\circ} \end{aligned}$
$\mathrm{C}=90^{\circ}$
the ratio in sides according to sine rule $a: b: c=\sin A: \sin B: \sin C$
$=\sin 30^{\circ}: \sin 60^{\circ}: \sin 90^{\circ}$
$=\frac{1}{2}, \frac{\sqrt{3}}{2}, 1=\frac{1}{2}: \frac{\sqrt{3}}{2}: 1 \Rightarrow 1: \sqrt{3}: 2$

Topic: Mathematics, Properties of Triangles, NDA, NDA 2014 (Phase 1)
Consider the following statements:
$1$ Two events are mutually exclusive if the occurrence of one event prevents the occurrence of the other.
$2$ The probability of the union of two mutually exclusive events is the sum of their individual probabilities. 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:
No Solution

Topic: Mathematics, Probability, NDA, NDA 2017 (Phase 1)
Consider the following statements
1 . Two independent variables are always uncorrelated.
2 . The coefficient of correlation between two variables $\mathrm{x}$ and $\mathrm{Y}$ is positive when $\mathrm{X}$ decreases then $\mathrm{Y}$ decreases 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:
It is a property.

Topic: Mathematics, Statistics, NDA, NDA 2012 (Phase 1)
Consider the following statements:
1. Value of $\sin \theta$ oscillates between $-1$ and 1 .
2 Value of $\cos \theta$ oscillates between 0 and 1 . 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:
$\sin \theta \in[-1,1] ; \mathrm{Q} \in \mathrm{R}$, the value of $\sin \theta$ lies between
$-1$ to 1 . $\cos \theta \in[-1,1] ; \mathrm{Q} \in \mathrm{R}$, the value of $\sin \theta$ lies between $-$
1 to 1 .

Topic: Mathematics, Trigonometric Ratios & Identities, NDA, NDA 2014 (Phase 1)
Consider the following statements: $\quad$
1 . Variance is unaffected by change of origin and change of scale.
2 . Coefficient of variance is independent of the unit of observations. 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:
Only statement 2 is correct

Topic: Mathematics, Statistics, NDA, NDA 2017 (Phase 2)
Consider the following statements: $\quad$
1.$\quad y=\frac{e^{x}+e^{-x}}{2}$ is an increasing function on $[0, \infty)$.
2.$y=\frac{e^{x}-e^{-x}}{2}$ is an increasing function on $(-\infty, \infty)$. 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:
From statement $-1$
$y=\frac{e^{x}+e^{-x}}{2}$
$\therefore \quad \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{2}\left[\mathrm{e}^{\mathrm{x}}-\mathrm{e}^{-\mathrm{x}}\right]$
Now equating $\frac{\mathrm{dy}}{\mathrm{dx}}$ to zero we get, $\mathrm{e}^{\mathrm{x}}-\mathrm{e}^{-\mathrm{x}}=0$
$\therefore \quad \mathrm{e}^{\mathrm{x}}=\mathrm{e}^{-\mathrm{x}}$
$\mathrm{e}^{x}-\mathrm{e}^{-\mathrm{x}}=0$
$\mathrm{e}^{\mathrm{x}}=\mathrm{e}^{-\mathrm{x}}$
$\frac{\mathrm{e}^{\mathrm{x}}}{\mathrm{e}^{-\mathrm{x}}}=1$
$\mathrm{e}^{2 \mathrm{x}}=1$
$2 \mathrm{x}=0$
$\mathrm{x}=0$
The given function is increasing on interval $[0, \infty]$ From Statement-2
$\mathrm{y}=\frac{\mathrm{e}^{\mathrm{x}}-\mathrm{e}^{-\mathrm{x}}}{2}$
$\frac{d y}{d x}=\frac{1}{2}\left[e^{x}-e^{x}\right]$
Now equating $\frac{\mathrm{dy}}{\mathrm{dx}}$ to zero, we get. $\mathrm{e}^{\mathrm{x}}+\mathrm{e}^{-\mathrm{x}}=0$
$\therefore \quad \mathrm{e}^{\mathrm{x}}+\frac{1}{\mathrm{e}^{\mathrm{x}}}=0$
$\mathrm{e}^{2 \mathrm{x}}+1=0$
$\mathrm{e}^{2 \mathrm{x}}=-1$
Hence, the given function is increacing from $[-\infty, \infty]$
- Both statement are correct Option (c) is correct.

Topic: Mathematics, Application of Derivatives, NDA, NDA 2015 (Phase 1)
Consider the following statements:
1.Parallelism of lines is an equivalence relation.
2.$\mathrm{xRy}$,ifx is a father of y, is an equivalence relation. 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:
Statement 1: Let $l, \mathrm{~m}, \mathrm{n}$ areparallel line and $\mathrm{R}$ is a relation. $I \| l$, then $\mathrm{R}$ is reflexive and $I \| \mathrm{m}$ and $\mathrm{m} \| l$, the $\mathrm{R}$ is symmetric. also $l\|\mathrm{~m}, \mathrm{~m}\| \mathrm{n} \Rightarrow I \| \mathrm{n}$, then $\mathrm{R}$ is transitive
Hence, Ris an equivalence relation Statement $2: \mathrm{x}$ is father of $\mathrm{y}$ then $\mathrm{x}$ is not the father of $\mathrm{x}$, so relation is not reflexive. Also, $x$ is father of $y$ but yis not father of $x$, so it is not symmetric. And $x$ is father of $y$ and $y$ is father of $z$ does not imply that $\mathrm{x}$ is father of $\mathrm{z}$ so, it is not transitive too. So, this is not an equivalence relation. so, only statement 1 is
correct

Topic: Mathematics, Sets and Relations, NDA, NDA 2007 (Phase 1)
Consider the following statements:
1.The function $f(x)=\sin x$ decreases on the interval $(0, \pi / 2)$
2.The function $f(x)=\cos x$ increases on the interval $(0, \pi / 2)$
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:
Neither 1 nor 2

Solution:
$\sin x$ increases on the interval $\left(0, \frac{\pi}{2}\right)$
$\cos x$ decreases on the interval $\left(0, \frac{\pi}{2}\right)$

Topic: Mathematics, Functions, NDA, NDA 2014 (Phase 1)
Consider the following statements:
1.The general solution of $\frac{d y}{d x}=f(x)+x$ is of the form $y$ $=g(x)+c$, where $c$ is an arbitrary constant.
2.The degree of $\left(\frac{d y}{d x}\right)^{2}=f(x)$ 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:
Statement 1:
The general solution of $\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{f}(\mathrm{x})+\mathrm{x}$
Now integrating on both side
$\int \frac{\mathrm{dy}}{\mathrm{dx}}=\int[\mathrm{f}(\mathrm{x})+\mathrm{x}] \mathrm{dx}$
$\mathrm{y}=\mathrm{f}(\mathrm{x})+\mathrm{C}$
Statement 1 is correct. Statement 2:
$\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{f}(\mathrm{x})+\mathrm{c}$
Squaring both sides,
$\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^{2}=(\mathrm{f}(\mathrm{x})+\mathrm{c})^{2}$
$\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^{2}=\left[\mathrm{f}(\mathrm{x})^{2}+2 \mathrm{c} \mathrm{f}(\mathrm{x})+\mathrm{c}^{2}\right]$
Hence, the differential equation is of order $-1$ and degree $-2$.
- both Statements 1 and 2 are correct.
$\therefore$ Option (c) is correct.

Topic: Mathematics, Differential Equations, NDA, NDA 2015 (Phase 1)
Consider the following statements: $\quad[2009-I]$
1) The equation to a straight line parallel to the axis of $x$ is $y=d$, where $d$ is a constant.
2) The equation to the axis of $x$ is $x=0$. Which of the statement (s) 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:
We know that, the equation of $\mathrm{x}$ -axis is $y=0$ Statement 1 says that the equation to a straight line parallel to the axis of $x$ is $y=d$. Since, $d$ is constant therefore it can be zero. Thus, only statement 1 is correct.

Topic: Mathematics, Straight Lines, NDA, NDA 2009 (Phase 1)
Consider the following statements: $\quad[2016-I]$
1) If $\mathrm{ABC}$ is an equilateral triangle, then $3 \tan (A+B) \tan C=1$
2) If $\mathrm{ABC}$ is a triangle in which $\mathrm{A}=78^{\circ}, \mathrm{B}=66^{\circ}$, then $\tan \left(\frac{A}{2}+C\right) < \tan A$
3) If $\mathrm{ABC}$ is any triangle, then
$$
\tan \left(\frac{\mathrm{A}+\mathrm{B}}{2}\right) \sin \left(\frac{\mathrm{C}}{2}\right) < \cos \left(\frac{\mathrm{C}}{2}\right)
$$
Which of the above statements is/are correct?

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

The correct answer is:
2 only

Solution:
$\because A B C$ is an equilateral triangle. $\therefore A=B=C=60^{\circ}$
L.H.S. $=3 \tan (A+B) \tan C$
$=3 \tan 120^{\circ} \tan 60^{\circ}$
$\quad=3(-\sqrt{3})(\sqrt{3})$
$\quad=-9 \neq 1$
Hence statement $(1)$ is incorrect. Statement-2 $A B C$ is a triangle such that $A=78^{\circ}$ and $B=66^{\circ}$ $C=180-(78+66)=180-144=36^{\circ}$
$\frac{A}{2}+C=\frac{78}{2}+36^{\circ}$
$=39+36=75^{\circ}$
$\tan \left(\frac{A}{2}+C\right) < \tan A$
$\Rightarrow \tan 75^{\circ} < \tan 78^{\circ}$
Hence statement $(2)$ is correct. Statement (3) In a triangle $\mathrm{ABC}$ $A+B+C=180^{\circ}$
$A+B=180^{\circ}-C$
$\frac{A+B}{2}=\frac{180^{\circ}-C}{2}$
$\Rightarrow \frac{A+B}{2}=90-\frac{C}{2}$
$\Rightarrow \tan \left(\frac{A+B}{2}\right)=\tan \left(90-\frac{C}{2}\right)$
$\Rightarrow \tan \left(\frac{A+B}{2}\right)=\cot \frac{C}{2}$
$\tan \left(\frac{A+B}{2}\right) \cdot \sin \frac{C}{2}=\cot \frac{C}{2} \cdot \sin \frac{C}{2}=\cos \frac{C}{2}$
$\Rightarrow \tan \left(\frac{A+B}{2}\right) \cdot \sin \frac{C}{2}=\cos \frac{C}{2}$
We can see that statement $(3)$ is not correct.
Hence only 2 nd statement is correct.

Topic: Mathematics, Properties of Triangles, NDA, NDA 2016 (Phase 1)
Consider the following statements: $\quad[2016-I]$
1) $\quad \tan ^{-1} x+\tan ^{-1}\left(\frac{1}{x}\right)=\pi$
2) There exist $\mathrm{x}, \mathrm{y} \in[-1,1]$, where $\mathrm{x} \neq \mathrm{y}$ such that
$$
\sin ^{-1} x+\cos ^{-1} y=\frac{\pi}{2}
$$
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:
Neither 1 nor 2

Solution:
Statement-1
$\because\left(\tan ^{-1} x+\tan ^{-1} y=\tan ^{-1} \frac{x+y}{1-x y}\right)$
$\tan ^{-1} x+\tan ^{-1} \frac{1}{x}=\tan ^{-1} \frac{x+\frac{1}{x}}{1-x \cdot \frac{1}{x}}$
$$
\begin{array}{l}
=\tan ^{-1} \frac{x^{2}+1}{0} \\
=\tan ^{-1} \infty=\tan ^{-1} \tan \frac{\pi}{2}
\end{array}
$$
$\tan ^{-1} x+\tan ^{-1} \frac{1}{x}=\frac{\pi}{2}$
Statement (1) is wrong. Statement 2,
$$
\sin ^{-1} x+\cos ^{-1} y=\frac{\pi}{2}
$$
Only when $x=y$ Here $x \neq y$.
Statement $(2)$ is also wrong.

Topic: Mathematics, Inverse Trigonometric Functions, NDA, NDA 2016 (Phase 1)
Consider the following statements: $\quad[2017-I]$
1- If $\lim _{x \rightarrow a} f(x)$ and $\lim _{x \rightarrow a} g(x)$ both exist, then $\lim _{x \rightarrow a}\{f(x) g$
(x) $\}$ exists.
2- If $\lim _{x \rightarrow a}\{f(x) g(x)\}$ exists, then both $\lim _{x \rightarrow a} f(x)$ and
$\lim _{\mathrm{x} \rightarrow \mathrm{a}} \mathrm{g}(\mathrm{x})$ must exist.
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:
If $\lim _{x \rightarrow a} f(x)$ and $\lim _{x \rightarrow a} g(x)$ both exists, then
$\lim _{\mathrm{x} \rightarrow \mathrm{a}} \mathrm{f}(\mathrm{x}) \cdot \mathrm{g}(\mathrm{x})$ exists. But if $\lim _{\mathrm{x} \rightarrow \mathrm{a}} \mathrm{f}(\mathrm{x}) \cdot \mathrm{g}(\mathrm{x})$ exists, then
it is not necesary that both $\lim _{\mathrm{x} \rightarrow \mathrm{a}} \mathrm{f}(\mathrm{x})$ and $\lim _{\mathrm{x} \rightarrow \mathrm{a}} \mathrm{g}(\mathrm{x})$ exists.

Topic: Mathematics, Limits, NDA, NDA 2017 (Phase 1)
Consider the following statements:
$[2018-I I]$
1 . The angle between the planes
$2 x-y+z=1$ and $x+y+2 z=3$ is $\frac{\pi}{3}$
2 . The distance between the plancs
$6 x-3 y+6 z+2=0$ and $2 x-y+2 z+4=0$ is $\frac{10}{9}$
Which of the above statements is/are correct

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

The correct answer is:
Both and 2

Solution:
Angle between planes
$=\cos ^{-1}\left(\frac{2-1+2}{6}\right)$
$=\cos ^{-1}\left(\frac{1}{2}\right)=\frac{\pi}{3}$
Distance between planes
$=\frac{\left|\frac{2}{3}-4\right|}{\sqrt{9}}=\frac{10}{3 \times 3}=\frac{10}{9}$

Topic: Mathematics, Three Dimensional Geometry, NDA, NDA 2018 (Phase 2)
Consider the following statements: $\quad[2019-I]$
1) For an equation of a line, $x \cos q+y \sin q=p$, in normal form, the length of the perpendicular from the point $(a, b)$ to the line is $a \cos q+b \sin q+p \mid$
2) The length of the perpendicular from the point $(a, b)$ to
the line $\frac{x}{a}+\frac{y}{b}=1$ is $\left|\frac{a \alpha+b \beta-a b}{\sqrt{a^{2}+b^{2}}}\right|$.
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:
Neither 1 nor 2

Solution:
The length of perpendicular from $(\alpha, \beta)$ to line $x \cos \theta+y \sin \theta-p=0$
$|\alpha \cos \theta+\beta \sin \theta-p|$
Statement 1 is false. The length of perpendicular from $(\alpha, \beta)$ to line
$\frac{x}{a}+\frac{y}{b}=1$
$\frac{x}{a}+\frac{y}{b}=1 \Rightarrow b x+a y-a b=0$
So, perpendicular is $\left|\frac{b \alpha+a \beta-a b}{\sqrt{a^{2}+b^{2}}}\right|$
Statement 2 is also false.

Topic: Mathematics, Straight Lines, NDA, NDA 2019 (Phase 1)