datavorous/entrance-exam-dataset huggingface dataset:
The foci of the hyperbola $4 x^{2}-9 y^{2}-1=0$ are

Options:
A $(\pm \sqrt{13,0})$
B $\left(\pm \frac{\sqrt{13}}{6}, 0\right)$
C $\left(0, \pm \frac{\sqrt{13}}{6}\right)$
D None of these

The correct answer is:
$\left(\pm \frac{\sqrt{13}}{6}, 0\right)$

Solution:
$4 x^{2}-9 y^{2}=1$
$\frac{x^{2}}{\left(\frac{1}{2}\right)^{2}}-\frac{y^{2}}{\left(\frac{1}{3}\right)^{2}}=1$
eccentricity, $\mathrm{e}=\sqrt{1+\frac{\left(\frac{1}{3}\right)^{2}}{\left(\frac{1}{2}\right)^{2}}}=\frac{\sqrt{13}}{3}$
foci $=\left(\pm \frac{1}{2} \times \frac{\sqrt{13}}{3}, 0\right)=\left(\pm \frac{\sqrt{13}}{6}, 0\right)$

Topic: Mathematics, Hyperbola, NDA, NDA 2013 (Phase 2)
The following anatomical features belong to

A. Hypodermis made by collenchyma.
B. Pericycle in patches of parenchyma and sclerenchyma
C. Bundle cap is present.
D. Vascular bundle open.

Options:
A Sunflower stem - Dicot
B Sunflower stem - Monocot
C Maize stem - Dicot
D Maize stem - Monocot

The correct answer is:
Sunflower stem - Dicot

Solution:
Features	Dicot stem

1. Epidermis	Single layered with trichomes (hair)

2. Hypodermis	Collenchymatous

3. Cortex	Parenchymatous

4. Pericycle	Sclerenchymatous.

5. Medullary rays	Parenchymatous

6. Vascular bundles	Vascular bundles are conjoint, open and with endarch protoxylem. These are arranged in a ring-like manner.

7. Pith	Parenchymatous

Topic: Biology, Anatomy of Flowering Plants, NEET
The following are some statements about leucocytes. Which of these is incorrect?

I. The most abundant leucocyte is a granulocyte, whereas the least abundant leucocyte is an agranulocyte.
II. The largest leucocyte is a granulocyte, whereas the smallest leucocyte is an agranulocyte.
III. Phagocytic leucocytes may be agranulocyte or granulocyte.
IV. Polymorphonuclear leucocytes are the largest granulocyte, whereas the largest agranulocyte has a kidney-shaped nucleus.

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

The correct answer is:
I and II

Solution:
The most abundant leucocyte (neutrophil) is a granulocyte, whereas the least abundant leucocyte (basophil) is also a granulocyte. The largest leucocyte (monocyte) is an agranulocyte, whereas the smallest leucocyte (lymphocyte) is also an agranulocyte. Phagocytic leucocytes are neutrophils (granulocyte) and monocyte (agranulocyte). Polymorphonuclear leucocytes (neutrophils) are the largest granulocytes, whereas the largest agranulocyte (monocyte) has a kidney shaped nucleus.

Topic: Biology, Body Fluids and Circulation, NEET
The following is the record of goals scored by team A in a football session.
For the team $B$, mean number of goals scored per match was 2 with a standard deviation $1.25$ goals.
Find which team may be considered more consistent?

Options:

None

Solution:
For Team A:
$\bar{x}=\frac{50}{25}=2$
S.D. $=\sqrt{\frac{\Sigma f_i x_i^2}{n}-\left(\frac{\Sigma f_i x_i}{n}\right)^2}=\sqrt{\frac{130}{25}-\left(\frac{50}{25}\right)^2}$
$=\sqrt{5.2-4}=\sqrt{1.2}=1.09$
For team $B, \bar{x}=2$, S.D. $=1.25$
Since their means are same, $\therefore \sigma_{\mathrm{A}}$ is less than that $\sigma_B$, therefore team $A$ is more consistent than team $B$.

Topic: Mathematics, Statistics
The following item consists of two statements, one labelled the Assertion (A) and the other labelled the Reason (R). You are to examine these two statements carefully and decide if the Assertion (A) and Reason (R) are individually true and if so, whether the reason is a correct explanation of the Assertion. Select your answer using the codes given below. Assertion (A) : The work done when the force and displacement are perpendicular to each other is zero.
Reason (R) : the dot product $\vec{A} \cdot \vec{B}$ vanishes, if the vector
$\vec{A}$ and $\vec{B}$ are perpendicular.

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

The correct answer is:
Both A and $\mathrm{R}$ are true but $\mathrm{R}$ is not the correct explanation of A

Solution:
(A) We know that
Work done $=\overrightarrow{\mathbf{F}} \cdot \overrightarrow{\mathbf{d}}=|\overrightarrow{\mathbf{F}}| \cdot|\overrightarrow{\mathbf{d}}| \cos \theta$
Since, $\theta=90^{\circ}$
$\Rightarrow$ work done $=|\overrightarrow{\mathbf{F}}| \cdot|\overrightarrow{\mathbf{d}}| \cos 90^{\circ}=0$
(R) $\overrightarrow{\mathbf{A}} \cdot \overrightarrow{\mathbf{B}}=0$
$\Rightarrow \overrightarrow{\mathbf{A}}$ and $\overrightarrow{\mathbf{B}}$ are perpendicular. Both $\mathrm{A}$ and $\mathrm{R}$ are true but $\mathrm{R}$ is not correct explanation of
A.

Topic: Mathematics, Vector Algebra, NDA, NDA 2009 (Phase 1)
The following item consists of two statements, one labelled the Assertion $(A)$ and the other labelled the Reason $(R) .$ You are to examine these two statements carefully and decide if the Assertion $(A)$ and Reason $(R)$ are individually true and if so, whether the reason is a correct explanation of the Assertion. Select your answer using the codes given below.
Assertion (A): $M=\left[\begin{array}{cc}5 & 10 \\ 4 & 8\end{array}\right]$ is invertible.
Reason (R): M is singular.

Options:
A Both A and $\mathrm{R}$ are true and $\mathrm{R}$ is the correct explanation of $\mathrm{A}$
B Both Aand Rare truebut $\mathrm{R}$ is not the correct explanation of $\mathrm{A}$
C A is true but $\mathrm{R}$ is false
D $\mathrm{A}$ is false but $\mathrm{R}$ is true

The correct answer is:
$\mathrm{A}$ is false but $\mathrm{R}$ is true

Solution:
(A) Consider $M=\left[\begin{array}{rr}5 & 10 \\ 4 & 8\end{array}\right]$
Now, $|M|=\left|\begin{array}{rr}5 & 10 \\ 4 & 8\end{array}\right|=40-40=0$
Since, $|M|=0$
$\therefore M$ is not invertible.
(R) Since, determinant of $M$ is zero therefore $M$ is singular matrix. Therefore, $\mathrm{A}$ is false and $\mathrm{R}$ is true.

Topic: Mathematics, Matrices, NDA, NDA 2009 (Phase 1)
The following item consists of two statements, one labelled the Assertion $(A)$ and the other labelled the Reason $(R) .$ You are to examine these two statements carefully and decide if the Assertion (A) and Reason $(R)$ are individually true and if so, whether the reason is a correct explanation of the Assertion. Select your answer using the codes given below.
$Assertion (A)$: $\int \frac{e^{x}}{x}(1+x \log x) d x+c=e^{x} \log x$
$Reason$ $(\mathbf{R}): \int e^{x}\left[f(x)+f^{\prime}(x)\right] d x=e^{x} f(x)+c$

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

The correct answer is:
Both A and $\mathrm{R}$ are true but $\mathrm{R}$ is not the correct explanation of $\mathrm{A}$

Solution:
(A) Consider $\int \frac{e^{x}}{x}(1+x \log x) d x$
$=\int \frac{e^{x}}{x} d x+\int e^{x} \log x d x$
$=e^{x} \log x-\int e^{x} \log x d x+\int e^{x} \log x d x=e^{x} \log x$
(R) $\int e^{x}\left[f(x)+f^{\prime}(x)\right] d x$
$=\int e^{x} f(x) d x+\int e^{x} f^{\prime}(x) d x$
$=e^{x} f(x)-\int e^{x} f^{\prime}(x) d x+\int e^{x} f^{\prime}(x) d x=e^{x} f(x)+c$
Both Aand $\mathrm{R}$ are true but $\mathrm{R}$ is not the correct explanation of $\mathrm{A}$.

Topic: Mathematics, Indefinite Integration, NDA, NDA 2009 (Phase 1)
The following organic compound in living tissue is

Options:
A carbohydrate
B protein
C lipid
D nucleic acid

The correct answer is:
lipid

Solution:
The structure is of cholesterol. This is a derived lipid as it does not contain any fatty acids but are lipids as these have fat-like properties. This is composed of four fused carbon rings. Cholesterol is the most abundant steroid in the animal tissues whose precursor is acetic acid. It is essential component of animal plasma membrane too.

Topic: Chemistry, Biomolecules, NEET
The following pie chart gives the distribution of funds in a five year plan under the major heads of development expenditures:
Agriculture(A), Industry (B), Education (C), Employment (D) and Miscellaneous (E) The total allocation is 36,000 (in crores of rupees).
How much excess money (in crores) is allocated to Miscellaneous over Education? $\quad$

Options:
A 3600
B 4200
C 4500
D 4800

The correct answer is:
4500

Solution:
Required $=7500-3000=4500$

Topic: Mathematics, Statistics, NDA, NDA 2011 (Phase 2)
The following pie chart gives the distribution of funds in a five year plan under the major heads of development expenditures:
Agriculture(A), Industry (B), Education (C), Employment (D) and Miscellaneous (E) The total allocation is 36,000 (in crores of rupees).
$\quad$ Which head is allocated maximum funds? $\quad$

Options:
A Agriculture
B Industry
C Employment
D Miscellaneous

The correct answer is:
Employment

Solution:
Agriculture : $\frac{90}{360} \times 36000=9000$ Miscellaneous : $\frac{75}{360} \times 36000=7500$
Industry: $\frac{45}{360} \times 36000=4500$
Education:3000 $\quad$ Employment : 12000 Hence, funds.

Topic: Mathematics, Statistics, NDA, NDA 2011 (Phase 2)
The following pie chart gives the distribution of funds in a five year plan under the major heads of development expenditures:
Agriculture(A), Industry (B), Education (C), Employment (D) and Miscellaneous (E) The total allocation is 36,000 (in crores of rupees).
How much money (in crores) is allocated to $\begin{array}{ll}\text { Education? } & {}\end{array}$

Options:
A 3000
B 6000
C 9000
D 10800

The correct answer is:
3000

Solution:
Education: 3000

Topic: Mathematics, Statistics, NDA, NDA 2011 (Phase 2)
The following pie chart gives the distribution of funds in a five year plan under the major heads of development expenditures:
Agriculture(A), Industry (B), Education (C), Employment (D) and Miscellaneous (E) The total allocation is 36,000 (in crores of rupees).
How much money (in crores) is allocated to both Agriculture and Employment? $\quad$

Options:
A 20000
B 21000
C 24000
D 27000

The correct answer is:
21000

Solution:
Required $=9000+12000=21000$

Topic: Mathematics, Statistics, NDA, NDA 2011 (Phase 2)
The following question consist of two statements, one labelled as the 'Assertion $(A)^{\prime}$ and the other as 'Reason $(R)^{\prime}$. You are to examine these two statements carefully and select the answer.
Assertion (A): We cannot find out the regression of $x$ on $y$ from that of y on $\mathrm{x}$. Reason (R): In one equation $x$ is dependent variable and $y$ is independent whereas in other equation $\mathrm{y}$ is dependent variable and $\mathrm{x}$ is independent.

Options:
A Both $\mathbf{A}$ and $\mathbf{R}$ are individually true, and $\mathbf{R}$ is the correct explanation of $\mathbf{A}$.
B Both $\mathbf{A}$ and $\mathbf{R}$ are individually true but $\mathbf{R}$ is not the correct explanation of $\mathbf{A}$
C $\mathbf{A}$ is true but $\mathbf{R}$ is false.
D $\quad A$ is false but $\mathbf{R}$ is true. $\quad$

The correct answer is:
Both $\mathbf{A}$ and $\mathbf{R}$ are individually true, and $\mathbf{R}$ is the correct explanation of $\mathbf{A}$.

Solution:
All are correct statement and $\mathrm{R}$ is correct explanation
of $\mathrm{A}$.

Topic: Mathematics, Statistics, NDA, NDA 2006 (Phase 1)
The following question consist of two statements, one labelled as the 'Assertion $(A)^{\prime}$ and the other as 'Reason $(R):$ You are to examine these two statements carefully and select the answer.
Let $X=\{\theta \in[0,2 \pi] \cdot \sin \theta=\cos \theta\}$
Assertion (A): The number of elements in $X$ is 2. Reason (R) : $\sin \theta$ and $\cos \theta$ are both negative both in second and fourth quadrants.

Options:
A Both $\mathbf{A}$ and $\mathbf{R}$ are individually true, and $\mathbf{R}$ is the correct explanation of $\mathbf{A}$.
B Both $\mathbf{A}$ and $\mathbf{R}$ are individually true but $\mathbf{R}$ is not the correct explanation of $\mathbf{A}$.
C $\mathbf{A}$ is true but $\mathbf{R}$ is false.
D $\mathbf{A}$ is false but $\mathbf{R}$ is true.

The correct answer is:
$\mathbf{A}$ is true but $\mathbf{R}$ is false.

Solution:
$\quad(\mathrm{A}): \mathrm{X}=\{\theta \in[0,2 \pi]: \sin \theta=\cos \theta\}$
Number of elements in $\mathrm{X}$ is 2 . Since, $\sin \theta=\cos \theta$ is possible at $\theta=45^{\circ}$ and $225^{\circ}$ Since, $\cos \theta$ is negative in IInd quadrant but $\sin \theta$ is positive, $R$ is false.

Topic: Mathematics, Trigonometric Equations, NDA, NDA 2007 (Phase 2)
The following question consist of two statements, one labelled as the 'Assertion (A)' and the other as 'Reason $(R)^{\prime} .$ You are to examine these two statements carefully and select the answer.
Assertion (A): $\left\{x \in R \mid x^{2} < 0\right\}$ is not a set. Here $R$ is the set ofreal numbers. Reason (R): For every real number $\mathrm{x}, \mathrm{x}^{2}>0 .$

Options:
A Both A and $\mathbf{R}$ are individually true, and $\mathbf{R}$ is the correct explanation of $\mathbf{A}$.
B Both $\mathbf{A}$ and $\mathbf{R}$ are individually true but $\mathbf{R}$ is not the correct explanation of $\mathbf{A}$.
C $\mathbf{A}$ is true but $\mathbf{R}$ is false.
D $\mathbf{A}$ is false but $\mathbf{R}$ is true.

The correct answer is:
Both A and $\mathbf{R}$ are individually true, and $\mathbf{R}$ is the correct explanation of $\mathbf{A}$.

Solution:
Since $x^{2} < 0$ is not possible for real numbers. A is true Since $x^{2}>0$ for $\forall x \in R$. Both $($ A $)$ and $(R)$ are true and
(R) is the correct explanation of ( $\mathrm{A}$ ).

Topic: Mathematics, Sets and Relations, NDA, NDA 2008 (Phase 1)
The following question consist of two statements, one labelled as the 'Assertion (A)' and the other as 'Reason (R)'. You are to examine these two statements carefully and select the answer.
Assertion (A): $1 / 8, \log _{a^{2}} a \rightarrow$ exponent should be to the base, $\log ^{2}$ are in GP but not in AP
Reason: $(\mathbf{R}): \mathrm{x}, \mathrm{y}, \mathrm{z}$ are in APas well as in GPif $\mathrm{x}=\mathrm{y}=\mathrm{z}$.

Options:
A Both A and $\mathrm{R}$ are individually true and $\mathrm{R}$ is the correct explanation of $\mathrm{A}$
B Both A and $\mathrm{R}$ are individually true but $\mathrm{R}$ is not the correct explanation $\mathrm{A}$
C $\mathrm{A}$ is true but $\mathrm{R}$ is false
D $A$ is false but $R$ is true

The correct answer is:
Both A and $\mathrm{R}$ are individually true and $\mathrm{R}$ is the correct explanation of $\mathrm{A}$

Solution:
If $x, y, z$ are in GP then $x \cdot z=y^{2}$
Here $x=\frac{1}{8}, z=\log _{a} a^{2}, y=\left(\log _{a}^{2} a\right)$
$\frac{1}{8} \times \log _{a} a^{2}=\frac{1}{8} \cdot 2=\frac{1}{4}$
and $\left(\log _{\mathrm{a} 2} \mathrm{a}\right)^{2}=\left(\frac{1}{2}\right)^{2}=\frac{1}{4}$
Hence, $\frac{1}{8}, \log _{\mathrm{a} 2} \mathrm{a}, \log _{\mathrm{a}} \mathrm{a}^{2}$ arein GP.
Thus, both A and Rare individuallytrue and Ris correct explanation of $\mathrm{A}$

Topic: Mathematics, Sequences and Series, NDA, NDA 2006 (Phase 2)
The following question consist of two statements, one labelled as the 'Assertion (A)' and the other as 'Reason (R)'. You are to examine these two statements carefully and select the answer.

Options:
A Both $\mathbf{A}$ and $\mathbf{R}$ are individually true, and $\mathbf{R}$ is the correct explanation of $\mathbf{A}$.
B Both $\mathbf{A}$ and $\mathbf{R}$ are individually true but $\mathbf{R}$ is not the correct explanation of $\mathbf{A}$.
C $\mathbf{A}$ is true but $\mathbf{R}$ is false.
D A is false but $\mathbf{R} \text { is true. }$

The correct answer is:
Both $\mathbf{A}$ and $\mathbf{R}$ are individually true, and $\mathbf{R}$ is the correct explanation of $\mathbf{A}$.

Solution:
$\frac{z_{1}}{z_{2}}=\frac{3+4 i}{3+5 i}=\frac{(3+4 i)(3-5 i)}{3^{2}-5^{2}}=\frac{9-3 i+20}{9-25}=\frac{29-3 i}{-16}$
which is complex number. Both A and R are individually true and $\mathrm{R}$ is correct explanation of $\mathrm{A}$.

Topic: Mathematics, Complex Number, NDA, NDA 2006 (Phase 2)
The following question consist of two statements, one labelled as the 'Assertion (A)' and the other as 'Reason (R)' You are to examine these two statements carefully and select the answer.
Assertion $(\mathbf{A}):$ If $\overrightarrow{\mathrm{a}}=2 \overrightarrow{\mathrm{i}}+\overrightarrow{\mathrm{j}}-2 \overrightarrow{\mathrm{k}}, \overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{i}}+\overrightarrow{\mathrm{j}}-\overrightarrow{\mathrm{k}}$
then $|\overrightarrow{\mathrm{a}}| \neq|\overrightarrow{\mathrm{b}}|$
Reason (R) : Two unequal vectors can never have same magnitude.

Options:
A Both $\mathbf{A}$ and $\mathbf{R}$ are individually true, and $\mathbf{R}$ is the correct explanation of $\mathbf{A}$.
B Both $\mathbf{A}$ and $\mathbf{R}$ are individually true but $\mathbf{R}$ is not the correct explanation of $\mathbf{A}$.
C $\mathbf{A}$ is true but $\mathbf{R}$ is false.
D A is false but R is true.

The correct answer is:
A is false but R is true.

Solution:
A is true, but R is false.

Topic: Mathematics, Vector Algebra, NDA, NDA 2006 (Phase 1)
The following question consist of two statements, one labelled as the 'Assertion (A)' and the other as 'Reason (R)'. You are to examine these two statements carefully and select the answer.
Assertion $(\mathbf{A}): \int_{0}^{\pi} \sin ^{7} x d x=2 \int_{0}^{\pi / 2} \sin ^{7} x d x$
Reason(R) $: \sin ^{7} x$ is an odd function

Options:
A Both A and $\mathrm{R}$ are individually true and $\mathrm{R}$ is the correct explanation of $\mathrm{A}$
B Both $\mathrm{A}$ and $\mathrm{R}$ are individually true but $\mathrm{R}$ is not the conect explamation A
C $\mathrm{A}$ is true but $\mathrm{R}$ is false
D $\mathrm{A}$ is false but $\overline{\mathrm{R}}$ is true

The correct answer is:
Both $\mathrm{A}$ and $\mathrm{R}$ are individually true but $\mathrm{R}$ is not the conect explamation A

Solution:
$\int_{0}^{\pi} \sin ^{7} x d x=2 \int_{0}^{\pi / 2} \sin ^{7} x d x$
$\sin x$ is an odd function and for an odd function
$\int_{0}^{a} f(x) d x=2 \int_{0}^{a / 2} f(x) d x$
Hence, $\int_{0}^{\pi} \sin ^{7} x d x=2 \int_{0}^{\pi / 2} \sin ^{7} x d x$ is true.
So, A and $\mathrm{R}$ both are individually true but $\mathrm{R}$ is not the correct explanation of A.

Topic: Mathematics, Definite Integration, NDA, NDA 2007 (Phase 1)
The following question consist of two statements, one labelled as the 'Assertion (A)' and the other as 'Reason $(R)^{\prime} .$ You are to examine these two statements carefully and select the answer.
Assertion $(\mathbf{A}): 0.3+0.03+0.003+\ldots=\frac{1}{3}$
Reason $(\mathbf{R}):$ For each $(+)$ ve integer $\mathrm{n}$, let $\mathrm{a}_{\mathrm{n}}=\mathrm{a}+\mathrm{nd}, \mathrm{a}$ and
dare real numbers. Then, $\mathrm{a}_{1}+\ldots .+\mathrm{a}_{\mathrm{n}}=\frac{\mathrm{n}}{2}[2 \mathrm{a}+(\mathrm{n}+1) \mathrm{d}] .$

Options:
A Both A and $\mathrm{R}$ are individually true and $\mathrm{R}$ is the correct explanation of A
B Both $\mathrm{A}$ and $\mathrm{R}$ are individually true but $\mathrm{R}$ is not the correct explanation of $\mathrm{A}$
C $\mathrm{A}$ is true but $\mathrm{R}$ is false
D $\mathrm{A}$ is false but $\mathrm{R}$ is true

The correct answer is:
Both $\mathrm{A}$ and $\mathrm{R}$ are individually true but $\mathrm{R}$ is not the correct explanation of $\mathrm{A}$

Solution:
(A): $0.3+0.03+0.003+\ldots=\frac{1}{3}$
Let $\mathrm{S}=0.3+0.03+0.003+\ldots$
This is geometric series with
first term, $\mathrm{a}=0.3$ and common ration, $\mathrm{r}=\frac{1}{10}$.
So, $S=\frac{a}{1-r}=\frac{0.3}{1-\frac{1}{10}}=\frac{0.3 \times 10}{9}$
$$
\begin{array}{l}
\quad=\frac{3}{9}=\frac{1}{3} \cdot \text { So },(\mathrm{A}) \text { is true. } \\
(\mathrm{R}): \text { As given }: \mathrm{a}_{\mathrm{n}}=\mathrm{a}+\mathrm{nd} \\
\Rightarrow \mathrm{a}_{1}+\mathrm{a}_{2}+\ldots+\mathrm{a}_{\mathrm{n}} \\
=\mathrm{a}+\mathrm{d}+\mathrm{a}+2 \mathrm{~d}+\ldots+\mathrm{a}+\mathrm{nd} \\
=\mathrm{n} \mathrm{a}+(1+2+\ldots+\mathrm{n}) \mathrm{d} \\
=\mathrm{na}+\frac{\mathrm{n}(\mathrm{n}+1) \mathrm{d}}{2}=\frac{\mathrm{n}}{2}[2 \mathrm{a}+(\mathrm{n}+1) \mathrm{d}]
\end{array}
$$
So, $(\mathrm{R})$ is also true Hence, both (A) and (R) are true. but $\mathrm{R}$ is not the correct explanation of (A)

Topic: Mathematics, Sequences and Series, NDA, NDA 2007 (Phase 2)
The following question consist of two statements, one labelled as the 'Assertion (A)' and the other as 'Reason (R)' You are to examine these two statements carefully and select the answer. Assertion (A) : If P (A) $=\frac{3}{4}$ and $P(B)=\frac{3}{8}$, then $\mathrm{P}(\mathrm{A} \cup \mathrm{B}) \geq \frac{3}{4}$
Reason $(\mathrm{R}): \mathrm{P}(\mathrm{A}) \leq \mathrm{P}(\mathrm{A} \cup \mathrm{B})$ and $\mathrm{P}(\mathrm{B}) \leq \mathrm{P}(\mathrm{A} \cup \mathrm{B})$; hence
$\mathrm{P}(\mathrm{A} \cup \mathrm{B}) \geq \max .\{\mathrm{P}(\mathrm{A}), \mathrm{P}(\mathrm{B})\}$

Options:
A Both $A$ and $R$ are individually true, and $R$ is the correct explanation of $\mathbf{A}$
B Both $\mathrm{A}$ and $\mathrm{R}$ are individually true but $\mathrm{R}$ is not the correct explanation of $\mathbf{A}$
C $\mathrm{A}$ is true but $\mathrm{R}$ is false.
D $\mathrm{A}$ is false but $\mathrm{R}$ is true.

The correct answer is:
Both $\mathrm{A}$ and $\mathrm{R}$ are individually true but $\mathrm{R}$ is not the correct explanation of $\mathbf{A}$

Solution:
A and R true but $\mathrm{R}$ is not correct explanation of $\mathrm{A}$.

Topic: Mathematics, Probability, NDA, NDA 2006 (Phase 2)
The following question consist of two statements, one labelled as the 'Assertion (A)' and the other as 'Reason (R)'. You are to examine these two statements carefully and select the answer. Assertion (A): If two triangles with vertices $\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)$, $\left(\mathrm{x}_{2}, \mathrm{y}_{2}\right),\left(\mathrm{x}_{3}, \mathrm{y}_{3}\right)$ and $\left(\mathrm{a}_{1}, \mathrm{~b}_{1}\right),\left(\mathrm{a}_{2}, \mathrm{~b}_{2}\right),\left(\mathrm{a}_{3}, \mathrm{~b}_{3}\right)$ satisfy the
relation
$\left|\begin{array}{lll}\mathrm{x}_{1} & \mathrm{y}_{1} & 1 \\ \mathrm{x}_{2} & \mathrm{y}_{2} & 1 \\ \mathrm{x}_{3} & \mathrm{y}_{3} & 1\end{array}\right|=\left|\begin{array}{lll}\mathrm{a}_{1} & \mathrm{~b}_{1} & 1 \\ \mathrm{a}_{2} & \mathrm{~b}_{2} & 1 \\ \mathrm{a}_{3} & \mathrm{~b}_{3} & 1\end{array}\right|$, then the triangles are
congruent.
Reason (R): For the given triangles satisfying the above relation implies that the triangles have equal area.

Options:
A Both $A$ and $R$ are individually true, and $R$ is the correct explanation of $\mathrm{A}$.
B Both $A$ and $R$ are individually true but $R$ is not the correct explanation of $\mathbf{A}$.
C $\mathrm{A}$ is true but $\mathrm{R}$ is false.
D $\quad$ A is false but $\mathbf{R}$ is true.

The correct answer is:
Both $A$ and $R$ are individually true, and $R$ is the correct explanation of $\mathrm{A}$.

Solution:
(A) and (R) are true and (R) is correct explanation of
A.

Topic: Mathematics, Straight Lines, NDA, NDA 2007 (Phase 2)