A natural number $N$ is such that it can be expressed as $\mathbf{N}=\mathbf{p}+\mathbf{q}+\mathbf{r}$, where $\mathbf{p}, \mathbf{q}$ and $\mathbf{r}$ are distinct factors of $\mathbf{N}$. How many numbers below 50 have this property?

Three prime numbers $p, q$ and $r$, each less than 20 , are such that $p-q=q-r$. How many distinct possible values can we get for $(p+q+r)$ ?

How many possible values of $(p+q+r)$ are there satisfying $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=1$, where $p, q$ and $r$ are natural numbers (not necessarily distinct)?

Team $X$ scored a total of $N$ runs in 20 overs. Team $Y$ tied the score in $10 \%$ less overs. Had team Y's average run rate (runs per over) been $50 \%$ higher, the scores would have been tied in 12 overs. How many runs were scored by team X?

The price (p) of a commodity is first increased by $\mathbf{k} \boldsymbol{\%}$; then decreased by $\mathrm{k} \%$; again increased by $\mathrm{k} \%$; and again decreased by $\mathrm{k} \%$. If the new price is q , then what is the relation between $p$ and $q$ ?

The petrol price shot up by $10 \%$ as a result of the hike in crude oil prices. The price of petrol before the hike was ₹$ 90$ per litre. A person travels 2200 km every month and his car gives a mileage of 16 km per litre. By how many km. should he reduce his travel if he wants to maintain his expenditure at the previous level?

A 4-digit number $N$ is such that when divided by $3,5,6$, 9 leaves a remainder $1,3,4,7$ respectively. What is the smallest value of $\mathbf{N}$ ?

What is the unit digit in the multiplication of $1 \times 3 \times 5 \times 7 \times 9 \times \ldots \times 999$ ?

Consider the first 100 natural numbers. How many of them are not divisible by any one of $2,3,5,7$ and 9 ?

If $4 \leq x \leq 8$ and $2 \leq y \leq 7$, then what is the ratio of maximum value of $(x+y)$ to minimum value of $(x-y)$ ?

Let both $p$ and $k$ be prime numbers such that $\left(p^2+k\right)$ is also a prime number less than 30 . What is the number of possible values of $k$ ?

There are $n$ sets of numbers each having only three positive integers with LCM equal to 1001 and HCF equal to 1 . What is the value of $n$ ?

Let PQR be a 3-digit number, PPT be a 3-digit number and $\mathbf{P S}$ be a 2-digit number, where $\mathbf{P}, \mathbf{Q}, \mathbf{R}, \mathbf{S}, \mathbf{T}$ are distinct non-zero digits. Further, $\mathrm{PQR}-\mathrm{PS}=\mathrm{PPT}$. If $Q=3$ and $T<6$, then what is the number of possible values of $(\mathbf{R}, \mathbf{S})$ ?

What is the maximum value of $n$ such that $7 \times 343 \times 385 \times 1000 \times 2401 \times 77777$ is divisible by $35^{\mathrm{n}}$ ?

$P$ and $Q$ walk along a circular track. They start at 5:00 a.m. from the same point in opposite directions. $P$ walks at an average speed of 5 rounds per hour and Q walks at an average speed of 3 rounds per hour. How many times will they cross each other between 5:20 a.m. and 7:00 a.m.?

A tram overtakes 2 persons $X$ and $Y$ walking at an average speed of $3 \mathrm{~km} / \mathrm{hr}$ and $4 \mathrm{~km} / \mathrm{hr}$ in the same direction and completely passes them in 8 seconds and 9 seconds respectively. What is the length of the tram?

If $N^2=12345678987654321$, then how many digits does the number N have?

A set $(X)$ of 20 pipes can fill $70 \%$ of a tank in 14 minutes. Another set ( $\mathbf{Y}$ ) of 10 pipes fills 3/8th of the tank in $\mathbf{6}$ minutes. A third set (Z) of 16 pipes can empty half of the tank in 20 minutes. If half of the pipes of set $\mathbf{X}$ are closed and only half of the pipes of set $\mathbf{Y}$ are open, and all pipes of the set (Z) are open, then how long will it take to fill 50\% of the tank?

If $n$ is a natural number, then what is the number of distinct remainders of $\left(1^n+2^n\right)$ when divided by 4 ?

Let $P=Q Q Q$ be a 3-digit number. What is the HCF of P and 481 ?

What is the 489 th digit in the number $123456789101112 \ldots$ ?

The 5-digit number PQRST (all distinct digits) is such that $\mathbf{T} \neq \mathbf{0} . \mathbf{P}$ is thrice $\mathbf{T} . \mathbf{S}$ is greater than $\mathbf{Q}$ by 4, while $Q$ is greater than $R$ by 3 . How many such 5-digit numbers are possible?

$\mathbf{X}$ can complete one-third of a certain work in $\mathbf{6}$ days, $\mathbf{Y}$ can complete one-third of the same work in $\mathbf{8}$ days and $\mathbf{Z}$ can complete three-fourth of the same work in 12 days. All of them work together for $n$ days and then $X$ and $Z$ quit and $Y$ alone finishes the remaining work in $8 \frac{2}{3}$ days. What is $n$ equal to?

Consider the following statements:

I. There exists a natural number which when increased by $50 \%$ can have its number of factors unchanged.

II. There exists a natural number which when increased by $150 \%$ can have its number of factors unchanged.

Which of the statements given above is/are correct?

What is the remainder when $9^3+9^4+9^5+9^6+\ldots+9^{100}$ is divided by 6 ?

In a T20 cricket match, three players X, Y and Z scored a total of 37 runs. The ratio of number of runs scored by $\mathbf{X}$ to the number of runs scored by $\mathbf{Y}$ is equal to ratio of number of runs scored by $Y$ to number of runs scored by $\mathbf{Z}$.

$$ \begin{aligned} & \text { Value-I = Runs scored by X } \\ & \text { Value-II = Runs scored by Y } \end{aligned} $$

Value-III $=$ Runs scored by $\mathbf{Z}$

Which one of the following is correct?

Let $p+q=10$, where $p, q$ are integers.

Value-I $=$ Maximum value of $\mathbf{p} \times \mathbf{q}$ when $\mathbf{p}, \mathbf{q}$ are positive integers.

Value-II $=$ Maximum value of $p \times q$ when $p \geq-6, q \geq-4$.

Which one of the following is correct?

Consider a set of 11 numbers:

Value-I = Minimum value of the average of the numbers of the set when they are consecutive integers $\geq-5$.

Value-II = Minimum value of the product of the numbers of the set when they are consecutive non-negative integers.

Which one of the following is correct?

The average of three numbers $p, q$ and $r$ is $k . p$ is as much more than the average as $q$ is less than the average. What is the value of $r$ ?

Let $x$ be a real number between 0 and 1 . Which of the following statements is/are correct?

I. $x^2>x^3$.

II. $x>\sqrt{x}$.

Select the correct answer using the code given below:

The difference between any two natural numbers is 10 . What can be said about the natural numbers which are divisible by 5 and lie between these two numbers?

Which one of the following statements best reflects the critical message conveyed by the author of the passage?

With reference to above passage, the following assumptions have been made:

I. No country needs ecosystems to depend boost national income.

II. Resource-rich countries need to share their resources with those of scant the resources so as to prevent degradation of ecosystems.

Which of the above assumptions is/are valid?

Which one of the following statements best reflects the central idea of the passage?

With reference to the above passage, the following assumptions have been made:

I. Path-dependent green investments will eventually most likely benefit growth as well as public finances in a country like India.

II. If other green technologies follow the same pattern as that of solar energy, there will most likely be an easy green transition.

Which of the above assumptions is/are valid?

Which one of the following statements best and reflects the most logical, rational pragmatic message conveyed by the author of the passage?

With reference to the passage, the following assumptions have been made:

I. Green energy production can be linked to/integrated with the climate change mitigation and adaptation strategies.

II. Effects of climate change are much more severe in coastal and mountainous regions.

Which of the above assumptions is/are valid?

Which one of the following statements best reflects the critical message conveyed by the passage?

With reference to the above passage, the following assumptions have been made:

I. People above the poverty line also are prone to suffer from anxiety about economic insecurity.

II. Eradication of poverty can result in peace and social equality in the country.

Which of the above assumptions is/are valid?

Which one of the following statements best reflects the corollary to the above passage?

With reference to the passage, the following assumptions have been made:

The growing divergence between the fortunes of the agricultural and non-agricultural economy in India could have been economy reduced/contained by:

I. adapting large-scale cultivation of commercial crops and viable corporate farming.

II. providing free insurance for all crops and heavily subsidising seeds, fertilisers, electricity and farm machinery at par with developed countries.

Which of the above assumptions is/are valid?

Which one of the following statements best reflects the most logical and rational message conveyed by the author of the passage?

With reference to the above passage, the following assumptions have been made:

I. There is no need for the State to be involved in any manner in the handloom sector.

II. Handloom products are no longer appealing and attractive in the rapidly changing modern world.

Which of the above assumptions is/are valid?

Which one of the following statements best reflects the central idea conveyed by the passage?

With reference to the above passage, the following assumptions have been made:

I. Higher education is a constantly evolving subject that needs to align towards new developments in all spheres of society.

II. In our country, sufficient funds are not allocated for promoting higher education.

Which of the above assumptions is/are valid?

Which one of the following statements best reflects the crux of the passage?

With reference to the above passage, the following assumptions have been made:

I. The food manufacturing and processing industries in every country should align their objectives and processes in accordance with the changing needs of the societies.

II. Wealthier societies tend to incur great loss of calories of food materials due to indirect utilisation of their agricultural produce.

Which of the above assumptions is/are valid?

Which one of the following statements best reflects the central idea conveyed by the passage?

With reference to the above passage, the following assumptions have been made:

I. Supply-side investments in companies can result in low-carbon behaviour in people.

II. People are not capable of adapting low-carbon behaviour without the involvement of Government and Companies.

Which of the above assumptions is/are valid?

Which one of the following statements best reflects the most logical, rational and pragmatic message conveyed by the passage?

With reference to the above passage, the following assumptions have been made:

I. Automobile factories are examples of the circular economy.

II. Economic growth is compatible with circular use of mineral resources.

Which of the above assumptions is/are valid?

Which one of the following statements best reflects the central idea conveyed by the passage?

With reference to the above passage, the following assumptions have been made:

I. Global climate change can result in the migration of several plant diseases to new areas.

II. Scientific understanding of the wild relatives of our present crops would enable us to strengthen food security.

Which of the above assumptions is/are valid?

Which one of the following statements best reflects the critical message conveyed by the author of the passage?

With reference to the above passage, the following assumptions have been made:

I. In a democracy, a strong opposition is required only if the Head of Government is indifferent.

II. The more aggressive the opposition, the better is the governance in a democracy.

Which of the above assumptions is/are valid?

Which one of the following statements best reflects the critical message conveyed by the author of the passage?

With reference to the above passage, the following assumptions have been made:

I. Corporate capitalism promotes the growth of labour force and provides more employment opportunities.

II. Poor and marginalised sections of population are benefited by corporate capitalism due to trickle-down effect.

Which of the above assumptions is/are valid?

Which one of the following statements reflects the best explanation of the above passage?

Which one of the following statements best reflects the most logical, rational and crucial message conveyed by the passage?

Which one of the following statements best reflects the crux of the passage?

A question is given followed by Statements I and II. Consider the Question and the Statements and mark the correct option.

Question:

What is the smallest 1 -digit number having exactly 4 distinct factors?

Statement I: 2 is one of the factors.

Statement II: $\mathbf{3}$ is one of the factors.

Which one of the following is correct in respect of the above Question and the Statements?

A question is given followed by two Statements I and II. Consider the Question and the Statements and mark the correct option.

Question:

Let $\mathrm{P}, \mathrm{Q}, \mathrm{R}, \mathrm{S}$ be distinct non-zero digits. If $\mathrm{PP} \times \mathrm{PQ}=$ RRSS, where $\mathrm{P} \leq 3$ and $\mathrm{Q} \leq 4$, then what is Q equal to ?

Statement I: $\mathbf{R = 1}$.

Statement II: $\mathrm{S}=2$.

Which one of the following is correct in respect of the above Question and the Statements?

A question is given followed by two Statements I and

II. Consider the Question and the Statements and mark the correct option.

Question:

How is Q related to P?

Statement I: P has two sisters, R and S.

Statement II: R's father is the brother of Q.

Which one of the following is correct in respect of the above Question and the Statements?

A question is given followed by two Statements I and

II. Consider the Question and the Statements and mark the correct option.

Question: In a football match, team P playing against Q was behind by 3 goals with 10 minutes remaining. Does team P win the match?

Statement I: Team P scored 4 goals in the last 10 minutes.

Statement II: Team Q scored a total of 4 goals in the match.

Which one of the following is correct in respect of the above Question and the Statements?

A question is given followed by two Statements I and

II. Consider the Question and the Statements and mark the correct option.

Question:

Is $(p+q)^2-4 p q$, where $p, q$ are natural numbers, positive?

Statement I: $\mathrm{p}<\mathrm{q}$.

Statement II: p>q.

Which one of the following is correct in respect of the above Question and the Statements?

What comes at $X$ and $Y$ respectively in the following sequence?

January, January, December, October, X, March, October, Y, September

A solid cube is painted yellow on all its faces. The cube is then cut into 60 smaller but equal pieces by making the minimum number of cuts. Which of the following statements is/are correct?

I. The minimum number of cuts is 9 .

II. The number of smaller pieces which are not painted on any face is 6 .

Select the correct answer using the code given below:

If $7 * 24=25$ and $12 * 16=20$, then what is $16^* 63$ equal to?

Consider the following statements:

I. If $\mathrm{A} \leq \mathrm{B}>\mathrm{C}<\mathrm{D}>\mathrm{E}>\mathrm{F} \geq \mathrm{G}=\mathrm{H}$; then B is always greater than E .

II. If $\mathrm{P}>\mathrm{Q}=\mathrm{R} \geq \mathrm{S}=\mathrm{T} \leq \mathrm{U}=\mathrm{V}>\mathrm{W}$; then S is always less than V .

Which of the statements given above is/are correct?

Consider the sequence

AB_CC_A_BCCC_BBC_C that follows a certain pattern. Which one of the following completes the sequence?

What is $\mathbf{X}$ in the sequence

$$ 24, X, 12,18,36,90 ? $$

If $\mathbf{P}=+, \mathbf{Q}=-, \mathbf{R}=\times, \mathbf{S}=\div$, then insert the proper notations between the successive numbers in the equation $60 \_15 \_3 \_20 \_4=20$ :

A mobile phone has been stolen. There are 3 suspects $P$, $\mathbf{Q}$ and $\mathbf{R}$. They were questioned knowing that only one of them is guilty. Their responses are as follows:

P: I did not steal. Q stole it.

Q: R did not steal. I did not steal.

R: I did not steal. I do not know who did it.

Who stole the mobile phone?

Three teams $\mathbf{P}, \mathbf{Q}, \mathbf{R}$ participated in a tournament in which the teams play with one another exactly once. A win fetches a team 2 points and a draw 1 point. A team gets no point for a loss. Each team scored exactly one goal in the tournament. The team $\mathbf{P}$ got $\mathbf{3}$ points, $\mathbf{Q}$ got 2 points and R got 1 point. Which of the following statements is/are correct?

I. The result of the match between $P$ and $Q$ is a draw with the score 0-0.

II. The number of goals scored by R against Q is 1 .

Which of the statements given above is/are correct?

$P$ is the brother of $Q$ and R. S is R's mother. T is P's father. How many of the following statements are definitely true?

I. S and T are a couple.

II. Q is T's son.

III. T is Q's father.

IV. S is P's mother.

V. R is T's daughter.

VI. P is S's son.

Select the correct answer using the code given below:

If NO is coded as 210,NOT is coded as 4200 and NOTE is coded as 21000 , then how is NOTES coded?

If FRANCE is coded as 654321 and GERMANY is coded as 9158437 , then how is YEMEN coded?

What is $\mathbf{X}$ in the sequence $1,3,6,11,18, X, 42$ ?

There are 7 places $\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E}, \mathrm{F}$ and G in a city connected by various roads $\mathrm{AB}, \mathrm{AC}, \mathrm{CD}, \mathrm{DE}, \mathrm{BF}, \mathrm{EG}$ and FG. A is 6 km south of B. A is 10 km west of C. D is 5 km east of $E$. $C$ is 6 km north of $D$. $F$ is 9 km west of $B . F$ is 12 km north of $G$. A person travels from $D$ to F through these roads. What is the distance covered by the person?

In a certain code if 64 is written as 343 and 216 is written as $\mathbf{7 2 9}$, then how is $\mathbf{5 1 2}$ written in that code?