datavorous/entrance-exam-dataset huggingface dataset:
The standard reduction potentials at $298 \mathrm{~K}$ for the following half-cell reactions are given on next page
$\mathrm{Zn}^{2+}(a q)+2 e^{-} \rightleftharpoons \mathrm{Zn}(s)-0.762$
$\mathrm{Cr}^{3+}(a q)+3 e^{-} \rightleftharpoons \mathrm{Cr}(s)-0.74$
$2 \mathrm{H}^{+}(a q)+2 e^{-} \rightleftharpoons \mathrm{H}_{2}(g)+0.00$
$\mathrm{Fe}^{3+}(a q)+e^{-} \rightleftharpoons \mathrm{Fe}^{2+}(a q)+0.77$
Which one of the following is the strongest reducing agent?

Options:
A $\mathrm{Zn}(s)$
B $\operatorname{Cr}(s)$
C $\mathrm{H}_{2}(s)$
D $\mathrm{Fe}^{2+}(a q)$

The correct answer is:
$\mathrm{Zn}(s)$

Solution:
The metals having higher negative value of standard reduction potential are placed above hydrogen in electrochemical series. The metals placed above hydrogen has a great tendency to donate electrons or oxidising power. The metals having great oxidising power are strongest reducing agent. Zn has higher negative value of standard reduction potential. Therefore, it is the strongest reducing agent.

Topic: Chemistry, Electrochemistry, MHT CET, MHT CET 2008
The standard reduction potentials (in V) of a few metal ion/metal electrodes are given below.
The reducing strength of the metals follows the order.

Options:
A $\mathrm{Ag}>\mathrm{Cu}>\mathrm{Pb}>\mathrm{Cr}$
B $\mathrm{Cr}>\mathrm{Pb}>\mathrm{Cu}>\mathrm{Ag}$
C $\mathrm{Pb}>\mathrm{Cr}>\mathrm{Ag}>\mathrm{Cu}$
D $\mathrm{Cr}>\mathrm{Ag}>\mathrm{Cu}>\mathrm{Pb}$

The correct answer is:
$\mathrm{Cr}>\mathrm{Pb}>\mathrm{Cu}>\mathrm{Ag}$

Solution:
Which have low value of reduction potential is a good reducing agent.
$$
\mathrm{Cr}>\mathrm{Pb}>\mathrm{Cu}>\mathrm{Ag}
$$

Topic: Chemistry, Electrochemistry, KVPY, KVPY 2015 (SB/SX)
The standard reduction potentials of $\mathrm{Zn}^{2+}\left|\mathrm{Zn}, \mathrm{Cu}^{2+}\right| \mathrm{Cu}$ and $\mathrm{Ag}^{+} \mid \mathrm{Cu}$ and $\mathrm{Ag}^{+} \mid \mathrm{Ag}$ are respectively $-0.76,0.34$ and $0.8 \mathrm{~V}$. The following cells were constructed :
(1) $\mathrm{Zn}\left|\mathrm{Zn}^{2+}\right|\left|\mathrm{Cu}^{2+}\right| \mathrm{Cu}$
(2) $\mathrm{Zn}\left|\mathrm{Zn}^{2+}\right|\left|\mathrm{Ag}^{+}\right| \mathrm{Ag}$
(3) $\mathrm{Cu}\left|\mathrm{Cu}^{2+} \| \mathrm{Ag}^{+}\right| \mathrm{Ag}$
What is the correct order of $E_{\text {cell }}^{\circ}$ of these cells?

Options:
A $2>3>1$
B $2>1>3$
C $1>2>3$
D $3>1>2$

The correct answer is:
$2>1>3$

Solution:
Given
$\mathrm{Zn}^{2+} \longrightarrow \mathrm{Zn} \quad E^{\circ}=-0.78 \mathrm{~V}$
$\mathrm{Cu}^{2+} \longrightarrow \mathrm{Cu} \quad E^{\circ}=0.34 \mathrm{~V}$
$\mathrm{Ag}^{+} \longrightarrow \mathrm{Ag} \quad E^{\circ}=0.8 \mathrm{~V}$
Cell reaction of (1) is
$\mathrm{Zn}+\mathrm{Cu}^{2+} \longrightarrow \mathrm{Zn}^{2+}+\mathrm{Cu}$
$E_{\text {cell }}^{\circ}=E_{\text {oxidation }}+E_{\text {reduction }}$
$=+0.78+0.34$
$=+1.12 \mathrm{~V}$
Cell reaction of (2) is
$\mathrm{Zn}+\mathrm{Ag}^{+} \longrightarrow \mathrm{Zn}^{2+}+\mathrm{Ag}$
$E_{\text {cell }}^{\circ}=0.78+0.8$
$=+1.58 \mathrm{~V}$
Cell reaction of (3) is :
$\mathrm{Cu}+\mathrm{Ag}^{+} \longrightarrow \mathrm{Cu}^{2+}+\mathrm{Ag}$
$E_{\text {cell }}^{\circ}=-0.34+0.8$
$=+0.46$
So, the correct order of $E_{\text {cell }}^{\circ}$ of these cell is -
$\mathrm{b}>\mathrm{a}>\mathrm{c} .$

Topic: Chemistry, Electrochemistry, AP EAMCET, AP EAMCET 2006
The standard reduction potentials of $\mathrm{Zn}^{2+}\left|\mathrm{Zn}, \mathrm{Cu}^{2+}\right| \mathrm{Cu}$ and $\mathrm{Ag}^{+} \mid \mathrm{Cu}$ and $\mathrm{Ag}^{+} \mid \mathrm{Ag}$ are respectively $-0.76,0.34$ and $0.8 \mathrm{~V}$. The following cells were constructed :
(1) $\mathrm{Zn}\left|\mathrm{Zn}^{2+}\right|\left|\mathrm{Cu}^{2+}\right| \mathrm{Cu}$
(2) $\mathrm{Zn}\left|\mathrm{Zn}^{2+}\right|\left|\mathrm{Ag}^{+}\right| \mathrm{Ag}$
(3) $\mathrm{Cu}\left|\mathrm{Cu}^{2+} \| \mathrm{Ag}^{+}\right| \mathrm{Ag}$
What is the correct order of $E_{\text {cell }}^{\circ}$ of these cells?

Options:
A $2>3>1$
B $2>1>3$
C $1>2>3$
D $3>1>2$

The correct answer is:
$2>1>3$

Solution:
Given
$\mathrm{Zn}^{2+} \longrightarrow \mathrm{Zn} \quad E^{\circ}=-0.78 \mathrm{~V}$
$\mathrm{Cu}^{2+} \longrightarrow \mathrm{Cu} \quad E^{\circ}=0.34 \mathrm{~V}$
$\mathrm{Ag}^{+} \longrightarrow \mathrm{Ag} \quad E^{\circ}=0.8 \mathrm{~V}$
Cell reaction of (1) is
$\mathrm{Zn}+\mathrm{Cu}^{2+} \longrightarrow \mathrm{Zn}^{2+}+\mathrm{Cu}$
$E_{\text {cell }}^{\circ}=E_{\text {oxidation }}+E_{\text {reduction }}$
$=+0.78+0.34$
$=+1.12 \mathrm{~V}$
Cell reaction of (2) is
$\mathrm{Zn}+\mathrm{Ag}^{+} \longrightarrow \mathrm{Zn}^{2+}+\mathrm{Ag}$
$E_{\text {cell }}^{\circ}=0.78+0.8$
$=+1.58 \mathrm{~V}$
Cell reaction of (3) is :
$\mathrm{Cu}+\mathrm{Ag}^{+} \longrightarrow \mathrm{Cu}^{2+}+\mathrm{Ag}$
$E_{\text {cell }}^{\circ}=-0.34+0.8$
$=+0.46$
So, the correct order of $E_{\text {cell }}^{\circ}$ of these cell is -
$\mathrm{b}>\mathrm{a}>\mathrm{c} .$

Topic: Chemistry, Electrochemistry, TS EAMCET, TS EAMCET 2006
The standard reduction potentials of $\mathrm{Zn}$ and $\mathrm{Ag}$ in water at $298 \mathrm{~K}$ are,
$\begin{aligned}
\mathrm{Zn}^{2+}+2 e^{-} \rightleftharpoons \mathrm{Zn}, & E^{\circ}=-0.76 \mathrm{~V} \\
\text { and } \quad \mathrm{Ag}^{+}+e^{-} \rightleftharpoons \mathrm{Ag} ; & E^{\circ}=+0.80 \mathrm{~V}.
\end{aligned}$
Which of the following reactions take place?

Options:
A $\mathrm{Zn}^{2+}(a q)+2 \mathrm{Ag}(s) \longrightarrow 2 \mathrm{Ag}^{+}(a q)+\mathrm{Zn}(s)$
B $\mathrm{Zn}(s)+2 \mathrm{Ag}^{+}(a q) \longrightarrow \mathrm{Zn}^{2+}(a q)+2 \mathrm{Ag}(s)$
C $\mathrm{Zn}^{2+}(a q)+\mathrm{Ag}^{+}(a q) \longrightarrow \mathrm{Zn}(s)+\mathrm{Ag}(s)$
D $\mathrm{Zn}(s)+\mathrm{Ag}(s) \longrightarrow \mathrm{Zn}^{2+}(a q)+\mathrm{Ag}^{+}(a q)$

The correct answer is:
$\mathrm{Zn}(s)+2 \mathrm{Ag}^{+}(a q) \longrightarrow \mathrm{Zn}^{2+}(a q)+2 \mathrm{Ag}(s)$

Solution:
For reaction to be spontaneous, $E_{\text {cell }}^{\circ}>0$
For option (b),
$$
\begin{aligned}
E_{\text {cell }}^{o} &=\left(E_{\text {red }}^{0}\right)_{\text {Cathode }}-\left(E_{\text {red }}^{0}\right)_{\text {Anode }} \\
&=0.80-(-0.76)=156 \mathrm{~V}
\end{aligned}
$$
$\therefore$ It will be spontaneous reaction
$$
\because E_{\text {cell }}^{a}>0 \text {. }
$$

Topic: Chemistry, Electrochemistry, COMEDK, COMEDK 2013
The standard temperature used in thermo chemical calculations is

Options:
A $273 \mathrm{~K}$
B $298 \mathrm{~K}$
C $297 \mathrm{~K}$
D $303 \mathrm{~K}$

The correct answer is:
$298 \mathrm{~K}$

Solution:
The standard temperature used in thermo chemical calculation is $25^{\circ} \mathrm{C}$ or $298 \mathrm{~K}$
$$
K=T+273=25+273=298 \mathrm{~K}
$$

Topic: Chemistry, Thermodynamics (C), COMEDK, COMEDK 2019
The steps involved in spermatogenesis are given below. Arrange them in proper sequence and select the correct option.

A. Primary spermatocytes undergo the first meiotic division.
B. Spermatogonia multiply by mitotic division.
C. Transformation of spermatids into sperms.
D. Secondary spermatocytes undergo the second meiotic division.
E. Release of sperms from the seminiferous tubules.

Options:
A A, C, B, E, D
B B, A, C, D, E
C B, A, D, C, E
D C, D, E, A, B

The correct answer is:
B, A, D, C, E

Solution:
The process of spermatogenesis is as follows:
1. Spermatogonia present in seminiferous tubules multiply by mitotic division and increase in numbers.
2. Few spermatogonia called 'primary spermatocytes' undergo meiotic division to form two equal 'secondary spermatocytes'.
3. Secondary spermatocytes undergo meiotic division to form four haploid spermatids.
4. Spermatids are transformed into sperms by the process of spermiogenesis.
5. Sperms are released from the seminiferous tubules by the process of spermiation.

Topic: Biology, Human Reproduction, NEET
The straight line $\frac{\mathrm{x}-3}{2}=\frac{\mathrm{y}-4}{3}=\frac{\mathrm{z}-5}{4}$ is parallel to which
one of the following ?

Options:
A $4 x+3 y-5 z=0$
B $4 x+5 y-4 z=0$
C $4 x+4 y-5 z=0$
D $5 x+4 y-5 z=0 \quad$

The correct answer is:
$4 x+4 y-5 z=0$

Solution:
A plane ax $+$ by $+\mathrm{cz}=0$ is parallel to, a straight line having direction ratios $\mathrm{a}^{\prime}, \mathrm{b}^{\prime}, \mathrm{c}^{\prime}$.
If $a a^{\prime}+b b^{\prime}+c c^{\prime}=0$
In the given problem, $\mathrm{dr}_{\mathrm{s}}$ of line is $2,3,4$. We check the equations of plane in the given choices, one by one.
(a) $4 \times 2+3 \times 3+(-5) \times 4=8+9-20 \neq 0$
(b) $4 \times 2+5 \times 3+(-4) \times 4=8+15-16 \neq 0$
(c) $4 \times 2+4 \times 3+(-5) \times 4=8+12-20=0$
Further checking is not needed

Topic: Mathematics, Three Dimensional Geometry, NDA, NDA 2008 (Phase 1)
The straight line $a x+b y+c=0$ and the coordinate axes form an isosceles triangle under which one of the following conditions?

Options:
A $|\mathrm{a}|=|\mathrm{b}|$
B $|\mathrm{a}|=|\mathrm{c}|$
C $|\mathrm{b}|=|\mathrm{c}|$
D none of these

The correct answer is:
$|\mathrm{a}|=|\mathrm{b}|$

Solution:
Co-ordinate axes and straight line ax $+$ by $+c=0$ form an isosceles triangle. This is possible when, the line makes equal intercept on both the axes.
Expressing $a x+b y+c=0$ in intercept form:
$a x+b y=-c$ or $\frac{x}{-\frac{c}{a}}+\frac{y}{-\frac{c}{b}}=1$
So, $x$ -inntercept $=-\frac{\mathrm{c}}{\mathrm{a}}$ and $\mathrm{y}$ -intercept $=-\frac{\mathrm{c}}{\mathrm{b}}$
Since, $-\frac{c}{a}=-\frac{c}{b}$
Hence, $\mathrm{a}=\mathrm{b}$
Intercepts can be on both the sides of axis. So, $|\mathrm{a}|=|\mathrm{b}|$

Topic: Mathematics, Straight Lines, NDA, NDA 2007 (Phase 1)
The straight lines $\mathrm{x}+\mathrm{y}-4=0,3 \mathrm{x}+\mathrm{y}-4=0$ and $\mathrm{x}+3 \mathrm{y}-4$
$=0$ form a triangle, which is

Options:
A isosceles
B right-angled
C equilateral
D scalene

The correct answer is:
isosceles

Solution:
Given lines: $\mathrm{L}_{1} \equiv \mathrm{x}+\mathrm{y}-4=0$
$\mathrm{L}_{2} \equiv 3 \mathrm{x}+\mathrm{y}-4=0$
$L_{3} \equiv x+3 y-4=0$
Slope of $L_{1}=m_{1}=\frac{-1}{1}=-1$
Slope of $L_{2}=m_{2}=\frac{-3}{1}=-3$
Slope of $L_{3}=m_{3}=\frac{-1}{3}$
Angle between $\mathrm{L}_{1}$ and $\mathrm{L}_{2}$ $\Rightarrow \tan \theta_{1}=\left|\frac{-1-(-3)}{1+(-1)(-3)}\right|=\frac{-1+3}{1+3}=\frac{1}{2}$
Angle between $\mathrm{L}_{2}$ and $\mathrm{L}_{3}$
$\Rightarrow \tan \theta_{3}=\left|\frac{-1-\left(\frac{-1}{3}\right)}{1+(-1)\left(\frac{-1}{3}\right)}\right|=\left|\frac{-3+1}{3+1}\right|=\frac{1}{2}$
The triangle formed is an isosceles triangle.

Topic: Mathematics, Straight Lines, NDA, NDA 2019 (Phase 1)
The strongest force among the following forces in nature is

Options:
A electromagnetic force
B weak nuclear force
C gravitational force
D strong nuclear force

The correct answer is:
strong nuclear force

Solution:
According to the fundamental forces in nature, the increasing order of strength of forces is given by
Gravitational force $ < $ Electromagnetic force $ < $ Weak nuclear force $ < $ Strong nuclear force.
Hence, the 'strong nuclear force' is the strongest force in nature.

Topic: Physics, Laws of Motion, TS EAMCET, TS EAMCET 2021 (06 Aug Shift 1)
The structure of functional grouv of secondary amide is:

Options:
A $-\mathrm{NH}-$
B

C

D

The correct answer is:

Solution:
Std.11|Ch-14|Subtopic-14.3

Topic: Chemistry, General Organic Chemistry, MHT CET, MHT CET 2023 (10 May Shift 1)
The structure of isobutyl group in an organic compound is

Options:
A CH3-CH|-CH2-CH3
B CH3-CH2-CH2-CH2-
C (CH3)3C-
D (CH3)2CH-CH2-

The correct answer is:
(CH3)2CH-CH2-

Solution:
CH3-CH|-CH2-CH3 (Secondary butyl)
CH3-CH2-CH2-CH2- (Normal butyl)
(CH3)3C- (Tertiary butyl)

Topic: Chemistry, Hydrocarbons, NEET
The structure shown below is made up of two strands of long chains of amino acids.

What does it depict?

Options:
A Primary structure with β-pleated sheet
B Tertiary structure with amino acid chains
C Secondary structure with β-conformation
D Quaternary structure with β-pleated sheet

The correct answer is:
Secondary structure with β-conformation

Solution:
The given figure depicts a secondary structure with beta confirmation. It represents the manner in which the regions of the protein threads, apart from the helix are folded onto each other.
Tertiary structure is the structure which represents how a long chain protein thread is folded upon itself.
Primary structure is the structure which represents the position of amino acids.
Quaternary structure is the structure which represents how the individual folded polypeptides and subunits are arranged.

Topic: Chemistry, Biomolecules, NEET
The structure which holds the leaf blades and helps to flutter in wind known as a __________

Options:
A Pulvinus
B Petiole
C Stipule
D Lamina

The correct answer is:
Petiole

Solution:
The petiole is a long thin flexible structure that holds leaf blades and flutter in wind. Sometimes petiole becomes tendril and leaf-like. A stipule is a small leaflike appendage to a leaf, typically borne in pairs at the base of the leaf stalk. Lamina is the leaf blade. The swollen leaf base is known as pulvinus.

Topic: Biology, Morphology of Flowering Plants, NEET
The structures present in the roots to absorb water and minerals is

Options:
A epidermal extensions
B hypodermis
C endodermis
D epidermal appendages

The correct answer is:
epidermal appendages

Solution:
Epidermal extensions are always unicellular, while epidermal appendages may be uni or multicellular. Root hairs are epidermal extensions formed by outward elongated bulging of wall of epidermal cells.

Topic: Biology, Anatomy of Flowering Plants, JIPMER, JIPMER 2013
The structures that help some bacteria to attach to rocks and/or host tissue are:

Options:
A Fimbriae
B Mesosomes
C Holdfast
D Rhizoids

The correct answer is:
Fimbriae

Solution:
Fimbriae: Hollow tubular surface appendages are present in the bacterial cells, which help in an attachment to rocks and/or host tissues.

Topic: Biology, Cell - The Unit of Life, NEET, NEET 2015 (Phase 2)
The structures that help some bacteria to attach to rocks and/or host tissue are

Options:
A Fimbriae
B Mesosomes
C Holdfast
D Rhizoids

The correct answer is:
Fimbriae

Solution:
Fimbriae: Hollow tubular surface appendages, present in the bacterial cells, which help in an attachment to rocks and/or host tissues.

Topic: Biology, Cell - The Unit of Life, NEET
The study of mutations is easy in haploids as compared to diploids, because

Options:
A Haploids are more abundant in nature than diploids
B All mutations, whether dominant or recessive are expressed in haploids
C They are reproductively more stable than diploids
D Mutagens penetrate in haploids more effectively than in diploids

The correct answer is:
All mutations, whether dominant or recessive are expressed in haploids

Solution:
Haploids are the cells which have only one set of the chromosome or can say one of the pair of the alleles of gene. Since haploids are not having the whole pair of allele, thus its allele is not influenced or affected by the counterpart. All allele types gets expressed in a haploid. It is the choice of cell for studying mutations as even the recessive allele will get expressed and not just only the dominant type since the recessive allele is singular and not in pair.

Topic: Biology, Strategies for Enhancement in Food Production, NEET
The sudden increase in which of the following hormone is directly responsible for the release of ovum from the Graafian follicle?

Options:
A LH
B FSH
C Estrogen
D Progesterone

The correct answer is:
LH

Solution:
Luteinising hormone is secreted by the anterior pituitary. It increases suddenly in the middle of the menstrual cycle. This is called LH surge. It stimulates the membrane granulosa cells of the Graafian follicle to release more liquor folliculi. This causes the pressure inside the follicle to increase causing it to rupture. The ovum in the secondary oocycte stage is released along with some follicular cells. This is called ovulation

Topic: Biology, Human Reproduction, NEET
The sum and product of matrices $\mathrm{A}$ and $\mathrm{B}$ exist. Which of the following implications are necessarily true?
$1.$ $\mathrm{A}$ and $\mathrm{B}$ are square matrices of same order.
$2.$ A and B are non-singular matrices
Select the correct answer using the code given below:

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) Since sum of matrices exist therefore $\mathrm{A}$ and $\mathrm{B}$ are square matrices of same order.
(2) Non-singularity of A and B does not depend on sum and product of $\mathrm{A}$ and $\mathrm{B}$.

Topic: Mathematics, Matrices, NDA, NDA 2012 (Phase 1)
The sum and sum of the squares corresponding to length $x$ in(cm) and weight $y$ (in gm) of 50 plant products are given below;
$\begin{aligned}
&\sum_{i=1}^{50} x_i=212, \sum_{i=1}^{50} x_i^2=902.8, \sum_{i=1}^{50} y_i=261 \\
&\sum_{i=1}^{50} y_i^2=1457.6
\end{aligned}$
Which is more varying, the length or weight?

Options:

None

Solution:
For length
$\mathrm{N}=50, \sum_{i=1}^{50} x_i=212, \sum_{i=1}^{50} x_i^2=902.8$
$\bar{x}=\frac{\sum x_i}{N}=\frac{212}{50}=4.24$
Standard deviation $=\sigma$
$=\frac{1}{N}\left[N \sum x_i^2-\left(\sum x_i\right)^2\right]$
$=\frac{1}{50} \sqrt{50 \times 902.8-(212)^2}$
$=\frac{\sqrt{196}}{50}=\frac{14}{50}=0.28$
Coefficients of variation $=\frac{\sigma}{\bar{x}} \times 100$
$=\frac{0.28}{4.24} \times 100=6.6$
For weight, $\mathrm{N}=50, \sum_{i=1}^{50} y_i=261, \sum_{i=1}^{50} y_i^2=1457.6$
Mean $\bar{x}=\frac{\sum y_i}{N}=\frac{261}{50}=5.22 \mathrm{gm}$.
Standard deviation, $\sigma$
$\begin{aligned}
&=\frac{1}{N} \sqrt{N \sum y_i^2-\left(\sum y_i\right)^2} \\
&=\frac{1}{50} \sqrt{50 \times 1457.6-(261)^2}=\frac{\sqrt{4759}}{50}=1.38 \\
&\therefore \text { Coefficients of variation }(\text { C.V })=\frac{\sigma}{\overline{\mathrm{x}}} \times 100 \\
&=\frac{1.38}{5.22} \times 100=26.4
\end{aligned}$
Coefficients of variation of weights is more than that of length
$\therefore$ Weight is more varying than length.

Topic: Mathematics, Statistics
The sum of 10 values is 12 and the sum of their squares is 16.9 , then their standard deviation $(S, D)$ is

Options:
A $0.05$
B 5
C $0.5$
D $0.005$

The correct answer is:
$0.5$

Solution:
Given $\sum x=12, \sum x^2=16.9$ and $n=10$
$$
\begin{aligned}
& \because \text { S.D. }=\sqrt{\frac{\sum x^2}{n}-\left(\frac{\sum x}{n}\right)^2}=\sqrt{\frac{16.9}{10}-\left(\frac{12}{10}\right)^2} \\
& =\sqrt{\frac{169-144}{100}}=\frac{5}{10}=0.5
\end{aligned}
$$

Topic: Mathematics, Statistics, MHT CET, MHT CET 2022 (10 Aug Shift 2)
The sum of all real roots of the equation $|x-3|^{2}+$ $|x-3|-2=0$ is

Options:
A 2
B 3
C 4
D 6

The correct answer is:
6

Solution:
$|x-3|^{2}+|x-3|-2=0$
Let $|x-3|=t$
$\therefore \mathrm{t}^{2}+\mathrm{t}-2=0 \Rightarrow \mathrm{t}^{2}+2 \mathrm{t}-\mathrm{t}-2=0$
$\Rightarrow \mathrm{t}(\mathrm{t}+2)-1(\mathrm{t}+2)=0$
$\Rightarrow(\mathrm{t}+2)(\mathrm{t}-1)=0$
$\Rightarrow \mathrm{t}=-2$ or $\mathrm{t}=1$
Since tis modulus of a number, it cannot be negative. $\therefore \mathrm{t}=1 \Rightarrow|\mathrm{x}-3|=1 \Rightarrow \mathrm{x}-3=1$ or $\mathrm{x}-3=-1$
$\Rightarrow \mathrm{x}=4$ or 2
Sum of roots $=4+2=6$

Topic: Mathematics, Quadratic Equation, NDA, NDA 2017 (Phase 2)
The sum of all the two-digit odd numbers is

Options:
A 2475
B 2530
C 4905
D 5049

The correct answer is:
2475

Solution:
Sum of odd numbers $=\mathrm{n}^{2}$
Two-digit odd numbers are from $11-99$.
Number of odd numbers from 1 to $99=50$
Sum of odd numbers from 1 to $99=50^{2}=2500$
Number of odd numbers from $1+9=5$
Sum of odd numbers from 1 to $9=5^{2}=25$
So, Sum of all two digit odd numbers
$=2500-25=2475 .$

Topic: Mathematics, Sequences and Series, NDA, NDA 2017 (Phase 1)
The sum of an infinite geometric progression is 6 , If the sum of the first two terms is $9 / 2$, then what is the first term?

Options:
A 1
B $5 / 2$
C 3 or $3 / 2$
D 9 or 3

The correct answer is:
9 or 3

Solution:
Let 'a ' be the first term and 'ar' be the second term of
G.Pwith common ratio 'r'.
Given: $\mathrm{S}_{\infty}=6$ and $a+a r=\frac{9}{2} \Rightarrow \frac{a}{1-r}=6$
$\Rightarrow \quad \mathrm{a}=6(1-\mathrm{r})$ ...(i)
and $a+a r=\frac{9}{2}$
$\Rightarrow \quad 6(1-r)+6 r(1-r)=\frac{9}{2} \quad$ [from (i)]
$\Rightarrow \quad 12-12 \mathrm{r}+12 \mathrm{r}-12 \mathrm{r}^{2}=9$
$\Rightarrow \quad r^{2}=\frac{3}{12}=\frac{1}{4} \Rightarrow r=\frac{1}{2}$ or $\frac{-1}{2} \Rightarrow a=3$ or 9

Topic: Mathematics, Sequences and Series, NDA, NDA 2010 (Phase 2)
The sum of an infinite $G P$ is $x$ and the common ratio $r$ is such that $|r| < 1$. If the first term of the $G P$ is 2, then which one of the following is correct?

Options:
A $\quad-1 < x < 1$
B $-\infty < x < 1$
C $\quad 1 < x < \infty$
D None of these

The correct answer is:
$\quad 1 < x < \infty$

Solution:
$\quad \mathrm{GP}=\mathrm{x}$
$\qquad \frac{\mathrm{a}}{1-\mathrm{r}}=\mathrm{x} \quad$ (where, $\mathrm{a}=$ lst term and $\mathrm{r}=$ common ratio $)$
$\Rightarrow \frac{2}{1-r}=x \quad$...(1) $(\because$ Given $a=2$ and $|r| < 1)$
$\Rightarrow-1 < \mathrm{r} < 1 \Rightarrow 1>-\mathrm{r}>-1$
$\Rightarrow 1+1>1-r>1-1$
$\Rightarrow 0 < 1-\mathrm{r} < 2$
$\Rightarrow \frac{1}{1-r}>\frac{1}{2}, \frac{2}{1-r}>1$
from equation (i) $x>1$
Hence, $1 < \mathrm{x} < \infty$

Topic: Mathematics, Sequences and Series, NDA, NDA 2014 (Phase 1)
The sum of first 10 terms and 20 terms of an AP are 120 and 440 respectively.
What is its first term?

Options:
A 2
B 3
C 4
D 5

The correct answer is:
3

Solution:
$\mathrm{S}_{10}=\frac{10}{2}[2 \mathrm{a}+9 \mathrm{~d}]$
$120=5(2 \mathrm{a}+9 \mathrm{~d})$
$2 \mathrm{a}+9 \mathrm{~d}=24$ ...(i)
$\mathrm{S}_{20}=\frac{20}{2}[2 \mathrm{a}+19 \mathrm{~d}]$
$440=10[2 \mathrm{a}+19 \mathrm{~d}]$
$2 \mathrm{a}+19 \mathrm{~d}=44$ ...(ii)
Solving (i) \& (11), we get
a = 3

Topic: Mathematics, Sequences and Series, NDA, NDA 2012 (Phase 2)
The sum of first 10 terms and 20 terms of an AP are 120 and 440 respectively.
What is the common difference?

Options:
A 1
B 2
C 3
D 4

The correct answer is:
2

Solution:
put a = 3 in (i) $6+9 \mathrm{~d}=24$
$9 \mathrm{~d}=18 \Rightarrow \mathrm{d}=2$

Topic: Mathematics, Sequences and Series, NDA, NDA 2012 (Phase 2)
The sum of $(p+q)^{\text {th }}$ and $(p-q)^{\text {th }}$ terms of an AP is equal to

Options:
A $(2 \mathrm{p})^{\text {th }}$ term
B $(2 q)^{\text {th }}$ term
C Twice the $\mathrm{p}^{\text {th }}$ term
D Twice the $q^{\text {th }}$ term

The correct answer is:
Twice the $\mathrm{p}^{\text {th }}$ term

Solution:
Let the first term of $\mathrm{AP}=\mathrm{a}$ Let the common difference of $\mathrm{AP}=\mathrm{d}$ $(p+q)^{\text {th }}$ term $=\mathrm{T}_{\mathrm{ptq}}=\mathrm{a}+(\mathrm{p}+\mathrm{q}-1) \mathrm{d}$
$(p-q)^{\text {th }}$ term $=T_{p-q}^{p \cdot q}=a+(p-q-1) d$
$\mathrm{T}_{\mathrm{ptq}}+\mathrm{T}_{\mathrm{p-q}}=\mathrm{a}+(\mathrm{p}+\mathrm{q}-1) \mathrm{d}+\mathrm{a}+(\mathrm{p}-\mathrm{q}-1) \mathrm{d}$
$=2 a+(p+q-1+p-q-1) d$
$=2 a+(2 p-2) d=2[a+(p-1) d]=2 T_{p}$

Topic: Mathematics, Sequences and Series, NDA, NDA 2019 (Phase 1)
The sum of the binary numbers $(11011)_{2}(10110110)_{2}$ and $(10011 \times 0 y)_{2}$ is the binary number ( 101101101$)_{2}$. What are the values of $\mathrm{x}$ and $\mathrm{y}$ ?

Options:
A $x=1, y=1$
B $x=1, y=0$
C $x=0, y=1$
D $x=0, y=0$

The correct answer is:
$x=1, y=0$

Solution:
On substraction, we get
101101101
\begin{array}{r}
-10110110 \\
\hline 10110111 \\
-11011 \\
\hline 10011100
\end{array}
$\Rightarrow x=1, y=0$

Topic: Mathematics, Sets and Relations, NDA, NDA 2018 (Phase 2)
The sum of the direction cosines of z-axis is

Options:
A 0
B $1 / 3$
C 1
D 3

The correct answer is:
1

Solution:
Direction cosines of z-axis are $0,0,1$
sum $=0+0+1=1$

Topic: Mathematics, Three Dimensional Geometry, NDA, NDA 2013 (Phase 1)