datavorous/entrance-exam-dataset huggingface dataset:
Consider the following statements:
A continuous random variable can take all values in an
interval.
2 . A random variable which takes a finite number of values is necessarily discrete.
3. Construction of a frequency distribution is based on data which are discrete Which of the above statements are correct? $\quad$

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:
Statement 2 and 3

Topic: Mathematics, Statistics, NDA, NDA 2012 (Phase 1)
Consider the following statements : Both the regression coefficients have same sign 2 . If one of the regression coefficients is greater thar unity, the other must be less than unity. Which of the above statements is/are correct

Options:
A 1 only
B 2 only $y$
C Both 1 and
D Neither $[\mathrm{m}$

The correct answer is:
Both 1 and

Solution:
No Solution

Topic: Mathematics, Statistics, NDA, NDA 2013 (Phase 2)
Consider the following statements:
For non empty sets A, B and C
1.$\quad \mathrm{A}-(\mathrm{B}-\mathrm{C})=(\mathrm{A}-\mathrm{B}) \cup \mathrm{C}$
2.$\quad$ A- $(B \cup C)=(A-B)-C$
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:
Let there be three non empty, non overlapping sets; inside a universal set U. This creates 8 regionsmarked as: $a, b, c, d, e, f, g, h$
Statement 1: $\mathrm{A}-(\mathrm{B}-\mathrm{C})=(\mathrm{A}-\mathrm{B}) \cup \mathrm{C}$
LHS represent region a, RHS represents a, $\mathrm{d}, \mathrm{g}$. Hence, this is not correct. Statement $2: \mathrm{A}-(\mathrm{B} \cup \mathrm{C})=(\mathrm{A}-\mathrm{B})-\mathrm{C}$
LHS represents, region 'a' RHS also represents a. Hence, only statement 2 is correct.

Topic: Mathematics, Sets and Relations, NDA, NDA 2007 (Phase 1)
Consider the following statements for the two non-empty sets A and B:
(1) $(\mathrm{A} \cap \mathrm{B}) \cup(\mathrm{A} \cap \overline{\mathrm{B}}) \cup(\overline{\mathrm{A}} \cap \mathrm{B})=\mathrm{AUB}$
(2) $\quad(\mathrm{A} \cup(\overline{\mathrm{A}} \cap \overline{\mathrm{B}}))=\mathrm{AUB}$
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. $(A \cap B) \cup(A \cap \bar{B}) \cup(\bar{A} \cap B)=A \cup B$
Let us draw Venn diagram
$-1$ is correct.
2.$A \cup(\bar{A} \cap \bar{B})=A$
$\therefore 2$ is not correct

Topic: Mathematics, Sets and Relations, NDA, NDA 2019 (Phase 1)
Consider the following statements:
I. $\quad f(x)=|x-3|$ is continuous at $x=0$.
II. $\quad f(x)=|x-3|$ is differentiable at $x=0$. 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:
Both I and II

Solution:
Let $f(x)=|x-3|$ $\begin{aligned} \text { L.H.L } &=\lim _{h \rightarrow 0} f(0-h)=\lim _{h \rightarrow 0}|0-h-3| \\ &=\lim _{h \rightarrow 0}(h+3)=3 \\ \text { R.H.L } &=\lim _{h \rightarrow 0} f(0+h)=\lim _{h \rightarrow 0}|0+h-3| \\ &=\lim _{h \rightarrow 0}|h-3|=3 \end{aligned}$
Since. L.H.L $=$ R.H.L at $x=0$ $\therefore f(x)=|x-3|$ is continuous at $x=0$. Now, LHD $=f^{\prime}\left(0^{-}\right)=\lim _{h \rightarrow 0} \frac{f(0)-f(0-h)}{h}$
$=\lim _{h \rightarrow 0} \frac{3-(3-h)}{h}=1$
and $\mathrm{RHD}=f^{\prime}\left(0^{+}\right)=\lim _{h \rightarrow 0} \frac{f(0+h)-f(0)}{h}$
$=\lim _{h \rightarrow 0} \frac{3+h-3}{h}=1$
$\Rightarrow \mathrm{LHD}=\mathrm{RHD}$
$\therefore \quad f(x)$ is differentiable Hence, both statements (I) and (II) are correct.

Topic: Mathematics, Continuity and Differentiability, NDA, NDA 2010 (Phase 2)
Consider the following statements
I. The coefficient of the middle term in the expansion of $(1+x)^{8}$ is equal to the middle term of $\left(x+\frac{1}{x}\right)^{8}$.
II. The coefficient of the middle term in the expansion of $(1+x)^{8}$ is less than the coefficient of the fifth term in the expansion of $(1+x)^{7}$. Which of the above statements is/ are correct?

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

The correct answer is:
I only

Solution:
Statement I : Given expansion is $(1+x)^{8}$ Since, $n=8$ is even
$\therefore\left(\frac{n}{2}+1\right)$ th term is the middle term.
ie- $\left(\frac{8}{4}+1\right)^{\text {th }}=5^{\text {th }}$ term $=$ middle term
Now, $5^{\text {th }}$ term $={ }^{8} C_{4} x^{4} 1^{4}={ }^{8} C_{4} x^{4}$
Coeff of $5^{\text {th }}$ term (middle term) $={ }^{8} C_{4}$
Now, consider the expansion $\left(x+\frac{1}{x}\right)^{8}$
It's middle term $=5^{\text {th }}$ term
and $5^{\text {th }}$ term $={ }^{8} C_{4} x^{4}\left(\frac{1}{x}\right)^{4}={ }^{8} C_{4}$
Hence, statement I is correct.
Statement-II : Coeff. of middle term in $(x+1)^{8}$ is
${ }^{8} C_{4}=\frac{8 !}{4 ! 4 !}=70$
Coeff of $5^{\text {th }}$ term in $(1+x)^{7}={ }^{7} C_{4}=\frac{7 !}{4 ! 3 !}=35$
Hence, statement II is incorrect.

Topic: Mathematics, Binomial Theorem, NDA, NDA 2009 (Phase 2)
Consider the following statements
I. The data, which are collected from the unit or individual respondents directly for the purpose of certain study or information are known as primary data.
II. The data obtained in a census study are primary data. Which of the above statements is/are correct?

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

The correct answer is:
Both I and II

Solution:
Both the given statements $\mathrm{I}$ and $\mathrm{II}$ are true.

Topic: Mathematics, Statistics, NDA, NDA 2009 (Phase 2)
Consider the following statements.
I. The inverse of a square matrix, if it exists, is unique.
II. If $A$ and $B$ are singular matrices of order $n$, then $A B$ is also a singular matrix of order $n$. Which of the statements 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:
Only I

Solution:
The inverse of a square matrix if it exists, is unique but if $A$ and $B$ are singular matrices of order $n$, then $A B$ is not a singular matrices of order $n$. Hence, only statement I is correct.

Topic: Mathematics, Matrices, NDA, NDA 2011 (Phase 1)
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, NDA, NDA 2010 (Phase 2)
Consider the following statements in respect of a function $\mathrm{f}(\mathrm{x}):$
1- $\mathrm{f}(\mathrm{x})$ is continuous at $\mathrm{x}=$ a iff $\lim _{\mathrm{x} \rightarrow \mathrm{a}} \mathrm{f}(\mathrm{x})$ exists.
2- If $\mathrm{f}(\mathrm{x})$ is continuous at a point, then $\frac{1}{\mathrm{f}(\mathrm{x})}$ is also continuous at that point.
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:
$\mathrm{f}(\mathrm{x})$ is continuous if $\lim _{\mathrm{x} \rightarrow \mathrm{a}} \mathrm{f}(\mathrm{x})$ exists and equals $\mathrm{f}(\mathrm{a})$
So, statement $-1$ is not correct. Also, statement $-2$ incorrect.

Topic: Mathematics, Continuity and Differentiability, NDA, NDA 2013 (Phase 1)
Consider the following statements in respect of a histogram:
1 . The total area of the rectangles in a histogram is equal to the total area bounded by the corresponding frequnecy polygon and the $x$ -axis.
2 . When class intervals are unequal in a frequency distribution, the area of the rectangle is proportional to the frequency. 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) is correct because total area of the rectangles in a histogram is equal to the total area bounded by the corresponding frequency polygon and $x$ -axis. Statement $(2)$ is also correct

Topic: Mathematics, Statistics, NDA, NDA 2016 (Phase 1)
Consider the following statements in respect of a square matrix $A$ and its transpose $A^{\mathrm{T}} .$
$1.$ $A+A^{\mathrm{T}}$ is always symmetric.
$2.$ $A-A^{\mathrm{T}}$ is always anti-symmetric 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:
We know, that A matrix 'A' is said to be symmetric $\mathrm{A}=\mathrm{A}^{\mathrm{T}}$ and anti- symmetric if $\mathrm{A}=-\mathrm{A}^{\mathrm{T}}$
Now, consider $\left(\mathrm{A}+\mathrm{A}^{\mathrm{T}}\right)^{\mathrm{T}}=\mathrm{A}^{\mathrm{T}}+\left(\mathrm{A}^{\mathrm{T}}\right)^{\mathrm{T}}=\mathrm{A}^{\mathrm{T}}+\mathrm{A}$
$\Rightarrow \mathrm{A}+\mathrm{A}^{\mathrm{T}}$ is always symmetric
Now, consider $\left(\mathrm{A}-\mathrm{A}^{\mathrm{T}}\right)^{\mathrm{T}}=\mathrm{A}^{\mathrm{T}}-\left(\mathrm{A}^{\mathrm{T}}\right)^{\mathrm{T}}=\mathrm{A}^{\mathrm{T}}-\mathrm{A}$
$=-\left(\mathrm{A}-\mathrm{A}^{\mathrm{T}}\right)$
$\Rightarrow \mathrm{A}-\mathrm{A}^{\mathrm{T}}$ is always anti- symmetric.

Topic: Mathematics, Matrices, NDA, NDA 2010 (Phase 2)
Consider the following statements in respect of circles $x^{2}+y^{2}-2 x-2 y=0$ and $x^{2}+y^{2}=1
1. The radius of the first circle is twice that of the second circle.
2. Both the circles pass through the origin. 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:
The equation of first circle is $x^{2}+y^{2}-2 x-2 y=0$.
Radius $=\sqrt{(-1)^{2}+(-1)^{2}}=\sqrt{2}$
and equation of second circle is $x^{2}+y^{2}=1$.
Radius $=1$
From above it is clear that the radius of first circle is not twice that of second circle.
$\therefore$ Statement 1 is not correct.
Also, first circle passes through the origin while second circle does not pass through the origin. Hence, neither 1 nor 2 statement is correct.

Topic: Mathematics, Circle, NDA, NDA 2010 (Phase 1)
Consider the following statements in respect of class intervals of grouped frequency distribution:
1 . Class intervals need not be mutually exclusive.
2 . Class intervals should be exhaustive
3. Class intervals need not be of equal width Which of the above statements are correct?

Options:
A 1 only 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:
No Solution

Topic: Mathematics, Statistics, NDA, NDA 2016 (Phase 2)
Consider the following statements in respect of histogram
1 . The histogram is a suitable representation of frequency distribution of a continuous variable
2 . The area included under the whole histogram is the total frequency. Which of the above statements is/are correct?

Options:
A 1 only
B 2 only
C Both 1 and 2
D Neithe

The correct answer is:
1 only

Solution:
1 . It is true that, the histogram is a suitable representation of a frequency distribution of continuous variable Hence, Statement 1 is correct 2 . We know that, the area of histogram is proportional to the frequency, so it is not true statement.

Topic: Mathematics, Statistics, NDA, NDA 2014 (Phase 2)
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, NDA, NDA 2009 (Phase 1)
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, NDA, NDA 2015 (Phase 2)
Consider the following statements in respect of the differential equation $\quad$
$\frac{d^{2} y}{d x^{2}}+\cos \left(\frac{d y}{d x}\right)=0$
1.The degree of the differential equation is not defined.
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:
Statement 1: Differential equation is not a polynomial equation in its derivatives. So, its degree is not defined. Statement 2 : The highest order derivative in the given polynomial is 2 .

Topic: Mathematics, Differential Equations, NDA, NDA 2014 (Phase 1)
Consider the following statements in respect of the function $f(x)=x^{3}-1, x \in[-1,1] \quad$
I $f(x)$ is increasing in $[-1,1]$
II $f(x)$ has noroot in $(-1,1)$. Which of the statements given above is/are correct?

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

The correct answer is:
OnlyI

Solution:
Since $f(x)$ is an increasing function in $[-1,1]$ and $i t$ has a root in $(-1,1)$. $\therefore \quad$ Only statement I is correct.

Topic: Mathematics, Application of Derivatives, NDA, NDA 2011 (Phase 1)
Consider the following statements in respect of the given
equation
$\left(x^{2}+2\right)^{2}+8 x^{2}=6 x\left(x^{2}+2\right)$
1.All the roots of the equation are complex
2.The sum of all the roots of the equation is 6 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:
$\quad\left(x^{2}+2\right)^{2}+8 x^{2}=$2
$+8 x^{2}$
$y^{2}-6 x y+8 x$
$y=\frac{6 x \pm \sqrt{36 x^{2}-32 x^{2}}}{2}$
$y=4 x$
At
$x^{2}-4 x+2=0$
Discriminant, $\mathrm{D}=$
Roots are real Sum of roots $=-(-4)$
At
$\mathrm{D}=4-8=-$
Roots are complex Sum of roots $=$ Sum of all roots $=4+2=6$ only statement 2 is (b)

Topic: Mathematics, Quadratic Equation, NDA, NDA 2015 (Phase 1)
Consider the following statements in respect of the matrix
$A=\left[\begin{array}{ccc}0 & 1 & 2 \\ -1 & 0 & -3 \\ -2 & 3 & 0\end{array}\right] \quad[2014-I]$
$1.$ The matrix $A$ is skew-symmetric.
$2.$ The matrix $A$ is symmetric.
$3.$ The matrix $A$ is invertible.
Which of the above statements is/are correct?

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

The correct answer is:
1 only

Solution:
$A=\left[\begin{array}{ccc}0 & 1 & 2 \\ -1 & 0 & -3 \\ -2 & 3 & 0\end{array}\right]$
$\mathrm{A}^{\mathrm{T}}=\left[\begin{array}{ccc}0 & -1 & -2 \\ 1 & 0 & 3 \\ 2 & -3 & 0\end{array}\right]=-\left[\begin{array}{ccc}0 & 1 & 2 \\ -1 & 0 & -3 \\ -2 & 3 & 0\end{array}\right]=-\mathrm{A}$
Hence, $\mathrm{A}$ is skew symmetric matrix
$|A|=\left|\begin{array}{ccc}0 & 1 & 2 \\ -1 & 0 & -3 \\ -2 & 3 & 0\end{array}\right|=1(-6)-2(-3)=-6+6=0$
Therefore A is non-invertible.

Topic: Mathematics, Matrices, NDA, NDA 2014 (Phase 1)
Consider the following statements in respect of the square matrices $A$ and $B$ of same order:
$1.$ $A$ and $B$ are non-zero and $A B=0 \Rightarrow$
either $|A|=0$ or $|B|=0$
$2.$ $A B=0 \Rightarrow A=0$ or $B=0$
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:
1 only

Solution:
If the matrix $A B$ is zero then it is not necessary that either $A=0$ or $B=0$ therefore statement 2 is incorrect. Let $A B=0$
$\Rightarrow|A B|=0$
$\Rightarrow|A||B|=0 \Rightarrow$ either $|A|=0$ or $|B|=0$

Topic: Mathematics, Matrices, NDA, NDA 2011 (Phase 2)
Consider the following statements in respect of the vectors
$\overrightarrow{\mathbf{u}_{1}}=(1,2,3), \overrightarrow{\mathbf{u}_{2}}=(2,3,1), \overrightarrow{\mathbf{u}_{3}}=(1,3,2)$ and $\overrightarrow{\mathbf{u}_{4}}=(4,6,2)$
I. $\overrightarrow{\mathbf{u}}_{1}$ is parallel to $\overrightarrow{\mathbf{u}_{4}}$.
$\overrightarrow{\mathrm{ll}}$
$\rightarrow$
II. $\mathbf{u}_{2}$ is parallel to $\mathbf{u}_{4}$.
\Pi. $\overrightarrow{\mathbf{u}_{2}}$ is parallel to $\overrightarrow{\mathbf{u}_{3}}$.
Which of the statements given above is/are correct?

Options:
A Only I
B Only II
C OnlyIII
D Both I and III

The correct answer is:
Only II

Solution:
Since, $\vec{u}_{4}=2 \vec{u}_{2}$
$\therefore \vec{u}_{2}$ is parallel to $\vec{u}_{4}$
Hence, only statement II is correct.

Topic: Mathematics, Vector Algebra, NDA, NDA 2011 (Phase 1)
Consider the following statements related to the nature of Bayes' theorem
$1.$ Bayes' theorem is a formula for computation of a conditional probability.
$2.$ Bayes' theorem modifies an assumed probability of an event in the light of a related event which is observed. 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:
Baye's theorem says If $A_{1}, A_{2}, \ldots, A_{n}$ are $(n)$ mutuallyexclusive and exhaustive events in sample space $S$ and $E$ is any event is $S$ intersecting events $A_{i}($ iz $1,2, \ldots, n)$ such that $\mathrm{P}(\mathrm{E}) \neq 0$ then $P\left(A_{i} / E\right)=\frac{P\left(E / A_{i}\right)}{P(E)}$
$=\frac{P\left(A_{i}\right) P\left(E / A_{i}\right)}{\sum_{|2|}^{n} P\left(A_{i}\right) P\left(E / A_{i}\right)}$
Thus, both statement 1 and 2 are correct.

Topic: Mathematics, Probability, NDA, NDA 2008 (Phase 2)
Consider the following statements:
Statement $1:$ Range is not a good measure of dispersion. Statement 2 : Range is highly affected by the existence of extreme values Which one of the following is correct in respect of the above statements?

Options:
A Both Statement 1 and Statement 2 are correct and Statement 2 is the correct explanation of Statement
B Both Statement 1 and Statement 2 are correct bui Statement 2 is not the correct explanation of Statement
C Statement 1 is correct but Statement 2 is not correct
D Statement 2 is correct but Statement 1 is not correct

The correct answer is:
Both Statement 1 and Statement 2 are correct and Statement 2 is the correct explanation of Statement

Solution:
No Solution

Topic: Mathematics, Statistics, NDA, NDA 2017 (Phase 1)
Consider the following statements :
Statement $1:$ The function $\mathrm{f}: \mathrm{IR} \rightarrow \mathrm{IR}$ such that $\mathrm{f}(\mathrm{x})=\mathrm{x}^{3}$ for all $\mathrm{x} \in \mathrm{IR}$ is one-one.
Statement $2: \mathrm{f}(\mathrm{a}) \Rightarrow \mathrm{f}(\mathrm{b})$ for all $\mathrm{a}, \mathrm{b} \in \mathrm{IR}$ if the function $\mathrm{f}$ is
one-one. Which one of the following is correct in respect of the above statements?

Options:
A Both the statements are true and Statement 2 is the correct explanation of Statement 1 .
B Both the statements are true and Statement 2 is not the correct explanation of Statement 1 .
C Statement 1 is true but Statement 2 is false.
D Statement 1 is false but Statement 2 is true.

The correct answer is:
Both the statements are true and Statement 2 is the correct explanation of Statement 1 .

Solution:
$\mathrm{f}(\mathrm{x})=\mathrm{x}^{3}$
$\mathrm{f}\left(\mathrm{x}_{1}\right)=\mathrm{x}_{1}^{3}$
$\mathrm{f}\left(\mathrm{x}_{2}\right)=\mathrm{x}_{2}^{3}$
if $\mathrm{x}_{1}=\mathrm{x}_{2}$
then $\mathrm{f}\left(\mathrm{x}_{1}\right)=\mathrm{f}\left(\mathrm{x}_{2}\right)$
So it is one-one function
Hence option (a) is correct.

Topic: Mathematics, Functions, NDA, NDA 2015 (Phase 2)
Consider the following statements:
Statement $\mathbf{I}:$ If the line segment joining the points $\mathrm{P}(\mathrm{m}, \mathrm{n})$ and $\mathrm{Q}(\mathrm{r}, \mathrm{s})$ subtends an angle $\alpha$ at the
$$
\text { origin, then } \cos \alpha=\frac{\mathrm{ms}-\mathrm{nr}}{\sqrt{\left(\mathrm{m}^{2}+\mathrm{n}^{2}\right)\left(\mathrm{r}^{2}+\mathrm{s}^{2}\right)}} \text { . }
$$
Statements II : In any triangle $\mathrm{ABC}$, it is true that
$$
a^{2}=b^{2}+c^{2}-2 b c \cos A
$$
What of the following is correct in respect of the above two statements?

Options:
A Both Statement I and Statement II are true and Statement II is the correct explanation of Statement I
B Both Statement I and Statement II are true, but Statement II is not the correct explanation of Statement I
C Statement I is true, but Statement II is false
D Statement I is false, but Statement II is true

The correct answer is:
Statement I is false, but Statement II is true

Solution:
$\cos \alpha=\frac{m r+n s}{\sqrt{m^{2}+n^{2}} \sqrt{r^{2}+s^{2}}}$
Statement 1 is false, statement 2 is true.

Topic: Mathematics, Straight Lines, NDA, NDA 2018 (Phase 2)
Consider the following statements:
Statement I: $x>\sin x$ for all $x>0$
Statement II: $f(\mathrm{x})=\mathrm{x}-\sin \mathrm{x}$ is an increasing function for all $\mathrm{x}>0$
Which one of the following is correct in respect of the above statements?

Options:
A Both Statement I and Statement II are true and Statement II is the correct explanation of Statement II.
B Both Statement I and Statement II are true and Statement II is not the correct explanation of Statement $\mathrm{I}$.
C Statement I is true but Statement II is false
D Statement I is false but Statement II is true

The correct answer is:
Both Statement I and Statement II are true and Statement II is the correct explanation of Statement II.

Solution:
No Solution

Topic: Mathematics, Application of Derivatives, NDA, NDA 2017 (Phase 2)