# Recent questions tagged cat2018-2 1
Each visitor to an amusement park needs to buy a ticket. Tickets can be Platinum, Gold, or Economy. Visitors are classified as Old, Middle-aged, or Young. The following facts are known about visitors and ticket sales on a particular day: $140$ tickets ... of Middle-aged and Young visitors buying Gold tickets were equal The numbers of Gold and Platinum tickets bought by Young visitors were equal
2
The smallest integer $n$ such that $n^{3} - 11n^{2} + 32n - 28 >0$ is
3
The value of the sum $7 \times 11 + 11 \times 15 + 15 \times 19 + \dots$ + $95 \times 99$ is $80707$ $80773$ $80730$ $80751$
4
How many two-digit numbers, with a non-zero digit in the units place, are there which are more than thrice the number formed by interchanging the positions of its digits? $5$ $6$ $8$ $7$
5
A $20\%$ ethanol solution is mixed with another ethanol solution, say, $\text{S}$ of unknown concentration in the proportion $1:3$ by volume. This mixture is then mixed with an equal volume of $20\%$ ethanol solution. If the resultant mixture is a $31.25\%$ ethanol solution, then the unknown concentration of $\text{S}$ is $52\%$ $50\%$ $55\%$ $48\%$
6
A chord of length $5$ cm subtends an angle of $60^\circ$ at the centre of a circle. The length, in cm, of a chord that subtends an angle of $120^\circ$ at the centre of the same circle is $8$ $6\sqrt2$ $5\sqrt3$ $2\pi$
7
On a triangle $\text{ABC}$, a circle with diameter $\text{BC}$ is drawn, intersecting $\text{AB}$ and $\text{AC}$ at points $\text{P}$ and $\text{Q}$, respectively. If the lengths of $\text{AB, AC}$, and $\text{CP}$ are $30$ cm, $25$ cm, and $20$ cm respectively, then the length of $\text{BQ}$, in cm, is __________
8
Gopal borrows Rs. $\text{X}$ from Ankit at $8\%$ annual interest. He then adds Rs. $\text{Y}$ of his own money and lends Rs. $\text{X+Y}$ to Ishan at $10\%$ annual interest. At the end of the year, after returning Ankit's dues, the net interest ... the net interest retained by him would have increased by Rs. $150$. If all interests are compounded annually, then find the value of $\text{X+Y}$.
9
Let $f\left (x \right ) = \max\left \{5x, 52 – 2x^{2}\right \}$ , where $x$ is any positive real numbers. Then the minimum possible value of $f(x)$ is ________
10
On a long stretch of east-west road, $\text{A}$ and $\text{B}$ are two points such that $\text{B}$ is $350$ km west of $\text{A}$. One car starts from $\text{A}$ and another from $\text{B}$ at the same time. If they move towards each other, then they meet after $1$ hour. If they both move towards east, then they meet in $7$ hrs. The difference between their speeds, in km per hour, is _______
11
A water tank has inlets of two types $\text{A}$ and $\text{B}$. All inlets of type $\text{A}$ when open, bring in water at the same rate. All inlets of type $\text{B}$, when open, bring in water at the same rate. The empty tank is completely filled in $30$ ... many minutes will the empty tank get completely filled if $7$ inlets of type $\text{A}$ and $27$ inlets of type $\text{B}$ are open?
12
If $a$ and $b$ are integers such that $2x^{2}- ax + 2 > 0$ and $x^{2}-bx+8 \geq 0$ for all real numbers $x$, then the largest possible value of $2a-6b$ is _________
13
A tank is emptied everyday at a fixed time point. Immediately thereafter, either pump $\text{A}$ or pump $\text{B}$ or both start working until the tank is full. On Monday, $\text{A}$ alone completed filling the tank at $8$ pm. On Tuesday, $\text{B}$ alone completed filling ... was the tank filled on Thursday if both pumps were used simultaneously all along? $4:36$ pm $4:12$ pm $4:24$ pm $4:48$ pm
14
A jar contains a mixture of $175$ ml water and $700$ ml alcohol. Gopal takes out $10\%$ of the mixture and substitutes it by water of the same amount. The process is repeated once again. The percentage of water in the mixture is now $35.2$ $30.3$ $20.5$ $25.4$
15
In a tournament, there are $43$ junior level and $51$ senior level participants. Each pair of juniors play one match. Each pair of seniors play one match. There is no junior versus senior match. The number of girl versus girl matches in junior level is $153$, while the number of boy versus boy matches in senior level is $276$. The number of matches a boy plays against a girl is _________
16
Ramesh and Ganesh can together complete a work in $16$ days. After seven days of working together, Ramesh got sick and his efficiency fell by $30\%$. As a result, they completed the work in $17$ days instead of $16$ days. If Ganesh had worked alone after Ramesh got sick, in how many days would he have completed the remaining work? $13.5$ $11$ $12$ $14.5$
17
If $p^{3}=q^{4}=r^{5}=s^{6}$, then the value of $\log_{s}\left ( pqr \right )$ is equal to $16/5$ $1$ $24/5$ $47/10$
18
Let $t_{1}, t_{2},\dots$ be a real numbers such that $t_{1}+t_{2}+\dots+t_{n}=2n^{2}+9n+13$, for every positive integers $n\geq2$.If $t_{k}=103$ , then $k$ equals
19
If $\text{N}$ and $x$ are positive integers such that $\text{N}^{\text{N}}=2^{160}$ and $\text{N}^{2} + 2^{\text{N}}$ is an integral multiple of $2^{x}$, then the largest possible $x$ is _______
20
If the sum of squares of two numbers is $97$, then which one of the following cannot be their product? $-32$ $48$ $64$ $16$
21
The arithmetic mean of $x,y$ and $z$ is $80$, and that of $x,y,z,u$ and $v$ is $75$, where $u=\left (x+y \right)/2$ and $v=\left (y+z \right)/2$. If $x\geq z$, then the minimum possible value of $x$ is ____________
22
Points $\text{A}$ and $\text{B}$ are $150$ km apart. Cars $1$ and $2$ travel from $\text{A}$ to $\text{B}$, but car $2$ starts from $\text{A}$ when car $1$ is already $20$ km away from $\text{A}$. Each car travels at a speed of $100$ kmph for the first $50$ ... , and at $25$ kmph for the last $50$ km. The distance, in km, between car $2$ and $\text{B}$ when car $1$ reaches $\text{B}$ is ________
23
Let $a_{1},a_{2},\dots , a_{52}$ be a positive integers such that $a_{1}<a_{2}<\dots < a_{52}$. Suppose, their arithmetic mean is one less than the arithmetic mean of $a_{2},a_{3},\dots , a_{52}$. If $a_{52}=100$ , then the largest possible value of $a_{1}$ is ________ $20$ $23$ $48$ $45$
24
A parallelogram $\text{ABCD}$ has area $48$ sqcm. If the length of $\text{CD}$ is $8$ cm and that of $\text{AD}$ is $s$ cm, then which one of the following is necessarily true? $s\geq6$ $s\neq6$ $s\leq6$ $5\leq s\leq7$
1 vote
25
If $\text{A}=\left \{6^{2n} - 35n - 1: n=1,2,3 \dots \right \}$ and $\text{B}= \left \{35\left (n - 1 \right ) : n=1,2,3\dots \right \}$ then which of the following is true? Neither every member of $\text{A}$ is in $\text{B}$ nor every member of $\text{B}$ ... $\text{B}$ is not in $\text{A}$ Every member of $\text{B}$ is in $\text{A}$ At least one member of $\text{A}$ is not in $\text{B}$
26
From a rectangle $\text{ABCD}$ of area $768$ sq cm, a semicircular part with diameter $\text{AB}$ and area $72\pi$ sq cm is removed. The perimeter of the leftover portion, in cm, is $80 + 16\pi$ $86+8\pi$ $82+24\pi$ $88+12\pi$
27
The scores of Amal and Bimal in an examination are in the ratio $11: 14$. After an appeal, their scores increase by the same amount and their new scores are in the ratio $47 : 56$. The ratio of Bimal's new score to that of his original score is $5:4$ $8:5$ $4:3$ $3:2$
28
The area of a rectangle and the square of its perimeter are in the ratio $1:25$. Then the lengths of the shorter and longer sides of the rectangle are in the ratio $1:4$ $2:9$ $1:3$ $3:8$
29
The strength of a salt solution is $p\%$ if $100$ ml of the solution contains $p$ grams of salt. If three salt solutions $\text{A, B, C}$ are mixed in the proportion $1:2:3$, then the resulting solution has strength $20\%$. If instead the proportion is $3:2:1$, then the resulting solution has ... ratio $2:7$. The ratio of the strength of $\text{D}$ to that of $\text{A}$ is $2:5$ $1:3$ $1:4$ $3:10$
30
For two sets $\text{A}$ and $\text{B}$, let $\text{A} \triangle \text{B}$ denote the set of elements which belong to $\text{A}$ or $\text{B}$ but not both. If $\text{P} = \{1,2,3,4\}, \text{Q} = \{2,3,5,6\}, \text{R} = \{1,3,7,8,9\}, \text{S} = \{2,4,9,10\},$ then the number of elements in $(\text{P} \triangle \text{Q}) \triangle (\text{R}\triangle \text{S})$ is $9$ $7$ $6$ $8$
31
The smallest integer $n$ for which $4^{n}>17^{19}$ holds, is closest to $33$ $37$ $39$ $35$
32
Points $\text{A, P, Q}$ and $\text{B}$ lie on the same line such that $\text{P, Q}$ and $\text{B}$ are, respectively, $100$ km, $200$ km and $300$ km away from $\text{A}$. Cars $1$ and $2$ leave $\text{A}$ at the same time and move towards $\text{B}$. Simultaneously, ... . If each car is moving in uniform speed then the ratio of the speed of car $2$ to that of car $1$ is $1:2$ $2:9$ $1:4$ $2:7$
1 vote
33
There are two drums, each containing a mixture of paints $\text{A}$ and $\text{B}$. In drum $1, \text{A}$ and $\text{B}$ are in the ratio $18: 7$. The mixtures from drums $1$ and $2$ are mixed in the ratio $3: 4$ and in this final mixture, $\text{A}$ and $\text{B}$ are in the ratio $13 :7$. In drum $2$, then $\text{A}$ and $\text{B}$ were in the ratio $229:141$ $220:149$ $239: 161$ $251: 163$
34
A triangle $\text{ABC}$ has area $32$ sq units and its side $\text{BC}$, of length $8$ units, lies on the line $x =4$. Then the shortest possible distance between $\text{A}$ and the point $(0,0)$ is $4$ units $8$ units $4\sqrt2$ units $2\sqrt2$ units
35
$\frac{1}{\log_{2}100} – \frac{1}{\log_{4}100} + \frac{1}{\log_{5}100} – \frac{1}{\log_{10}100} + \frac{1}{\log_{20}100} – \frac{1}{\log_{25}100} + \frac{1}{\log_{50}100}=?$ $1/2$ $0$ $10$ $-4$
36
An agency entrusted to accredit colleges looks at four parameters: faculty quality $\text{(F)}$, reputation $\text{(R)},$ placement quality $\text{(P)}$, and infrastructure $\text{(I)}.$ The four parameters are used to arrive at an overall score, which the agency uses to give an ... than $\text{A}$-one. How many colleges have overall scores between $31$ and $40$, both exclusive? $1$ $3$ $0$ $2$
37
An agency entrusted to accredit colleges looks at four parameters: faculty quality $\text{(F)}$, reputation $\text{(R)},$ placement quality $\text{(P)}$, and infrastructure $\text{(I)}.$ The four parameters are used to arrive at an overall score, which the ... better than Cosmopolitan; and Education Aid is better than $\text{A}$-one. How many colleges receive the accreditation of $\text{AAA}?$
An agency entrusted to accredit colleges looks at four parameters: faculty quality $\text{(F)}$, reputation $\text{(R)},$ placement quality $\text{(P)}$, and infrastructure $\text{(I)}.$ The four parameters are used to arrive at an overall score, which the ... is better than Cosmopolitan; and Education Aid is better than $\text{A}$-one. What is the highest overall score among the eight colleges?
Each visitor to an amusement park needs to buy a ticket. Tickets can be Platinum, Gold, or Economy. Visitors are classified as Old, Middle-aged, or Young. The following facts are known about visitors and ticket sales on a particular day: $140$ tickets were sold. The ... buying Platinum tickets, then which among the following could be the total number of Platinum tickets sold? $34$ $38$ $32$ $36$
According to a coding scheme the sentence Peacock is designated as the national bird of India is coded as $5688999\;35\;1135556678\;56\;458\;13666689\;1334\;79\;13366$ This coding scheme has the following rules: The scheme is case-insensitive (does not distinguish between upper case and lower case letters). Each letter ... digit? $\text{S, U, V}$ $\text{I, B, M}$ $\text{X, Y, Z}$ $\text{S, E, Z}$