# Recent activity 1
The ratio of boys to girls in a class is $7$ to $3$. Among the options below, an acceptable value for the total number of students in the class is $21$ $37$ $50$ $73$
2
Some people suggest anti-obesity measures $\text{(AOM)}$ such as displaying calorie information in restaurant menus. Such measures sidestep addressing the core problems that cause obesity: poverty and income inequality. Which one of the following statements summarizes the ... $\text{AOM}$ are addressing the problem superficially
3
If the area of a triangle with the vertices $(k,0), (2,0)$ and $(0, -2)$ is $2$ square units, the value of $k$ is _________
4
The value of $\underset{x \to 0} \lim \left [ \dfrac{x- \sin 2x}{x-\sin 5x} \right]$ (rounded off to two decimal places) is __________
5
The sum of the infinite geometric series $1 + 1/3 + 1/3^2+ 1/3^3+ \dots$ (rounded off to one decimal place) is ___________
6
$\dfrac{d}{dx} \left [ \ln (2x) \right ]$ is equal to $1/2x \\$ $1/x \\$ $1 /2 \\$ $x$
7
The Cartesian coordinates $(x,y)$ of a point $A$ with polar coordinates $\left ( 4, \pi/4 \right)$ is $\left( \sqrt{3}, 2 \sqrt{2} \right ) \\$ $\left( 2, 2 \sqrt{3} \right ) \\$ $\left( 2\sqrt{2}, \sqrt{3} \right ) \\$ $\left( 2 \sqrt{2}, 2 \sqrt{2} \right )$
8
Given below are two statements $1$ and $2$, and two conclusions $I$ and $II$. $\text{Statement 1}$: All bacteria are microorganisms. $\text{Statement 2}$: All pathogens are microorganisms. $\text{Conclusion I}$: Some pathogens are bacteria. $\text{Conclusion II}$: All ... is correct Either conclusion $\text{I}$ or $\text{II}$ is correct Neither conclusion $\text{I}$ nor $\text{II}$ is correct
9
A circular sheet of paper is folded along the lines in the direction shown. The paper, after being punched in the final folded state as shown and unfolded in the reverse order of folding, will look like ________.
10
There are five bags each containing identical sets of ten distinct chocolates. One chocolate is picked from each bag. The probability that at least two chocolates are identical is _________ $0.3024$ $0.4235$ $0.6976$ $0.8125$
11
Consider the following sentences: Everybody in the class is prepared for the exam. Babu invited Danish to his home because he enjoys playing chess. Which of the following is the $\text{CORRECT}$ observation about the above two sentences? $\text{(i)}$ is grammatically correct and ... correct and $\text{(ii)}$ is ambiguous $\text{(i)}$ is grammatically incorrect and $\text{(ii)}$ is ambiguous
12
A polygon is convex if, for every pair of points, $P$ and $Q$ belonging to the polygon, the line segment $PQ$ lies completely inside or on the polygon. Which one of the following is $\underline {\text{NOT}}$ a convex polygon?
13
____________ is to $\textit{surgery}$ as $\textit{writer}$ is to __________ Which one of the following options maintains a similar logical relation in the above sentence? Plan, outline Hospital, library Doctor, book Medicine, grammar
14
We have $2$ rectangular sheets of paper, $M$ and $N$, of dimensions $6$ cm $\times$ $1$ cm each. Sheet $M$ is rolled to form an open cylinder by bringing the short edges of the sheet together. Sheet $N$ is cut into equal square patches and assembled to form the largest possible closed cube. ... to that of the cube is _____________ $\frac{\pi}{2}\\$ $\frac{3}{\pi} \\$ $\frac{9}{\pi} \\$ $3 \pi$
15
$\begin{array}{|c|c|c|c|} \hline \textbf{Items} & \textbf{Cost} & \textbf{Profit} \% & \textbf{Marked Price} \\ & (₹) && (₹) \\ \hline P & 5,400 & \dots & 5,860 \\ \hline Q & \dots & 25 & 10,000 \\ \hline \end{array}$ ... $Q$, as a percentage of its marked price, is ___________ $25$ $12.5$ $10$ $5$
16
If the values of two random variables $(X,Y)$ are $(121, 360)$, $(242, 364)$ and $(363, 362)$, the value of correlation coefficient between $X$ and $Y$ (rounded off to one decimal place) is ________.
17
The determinant of matrix $A = \begin{pmatrix} 1 & 1& 1 & 1 \\ -1 & 1 & 1 & 1 \\ -1 & -1 & 1 & 1 \\ 1 & 1 & 1 & 3 \end{pmatrix}$ is __________.
18
Calculate the following integral $\int \limits_0^{\pi^2/4} \sin \sqrt{x} dx =$____________.
19
A system consists of two reactors, connected by a valve. The first reactor $(R1)$ contains an ideal gas $A$ of volume $5\: L$ and the second reactor $(R2)$ has an ideal gas $B$ of volume $10\: L$. Initially, the valve is closed and ... equilibrium. If the temperature $T$ of both the reactors is maintained constant, the final equilibrium pressure in $\textit{atm}$ of the system is _____________
20
Under standard temperature $(T)$ and pressure $(P)$ conditions, $128 \: g$ of an ideal has molecule $A$ occupies a volume of $1 \: L$. The gas molecule $A$ obeys the relationship $\text{RT=0.25 PV}$. $R$ and $V$ are universal gas constant and ideal gas volume, respectively. The molecule $A$ is $\text{CO}_2$ $\text{H}_2$ $\text{N}_2$ $\text{O}_2$
21
The largest eigenvalue of the matrix $\begin{bmatrix} 4 &1 \\ -2& 1 \end{bmatrix}$ is ______________.
22
A normal random variable has mean equal to $0$, and standard deviation equal to $3$. The probability that on a random draw the value of this random variable is greater than $0$ is __________ (round off to $2$ decimal places).
23
The system of linear equations $cx+y=5$ $3x+3y=6$ has no solution when $c$ is equal to _______________________.
24
A function $f$ is given as : $f(X)=4X-X^{2}$ The function $f$ is maximized when $X$ is equal to _________________.
25
An infinite series $S$ is given as: $S=1+2/3+3/9+4/27+5/81+\:.\dots$ (to infinity) The value of $S$ is ______________________ (round off to $2$ decimal places).
26
A function $f$ is as follows: $f(x) = \begin{cases} 15 & \text{if }x<1 \\ cx& \text{if } x\geq 1 \end{cases}$ The function $f$ is a continuous function when $c$ is equal to ____________________ (answer in an integer).
27
A variable $Y$ is a function of $t$. Given that $Y\left ( t=0 \right )=1$ and $Y\left ( t=1 \right )=2,\dfrac{dY}{dt}$ in the interval $t=\left [ 0,1 \right ]$ can be approximated as _________________.
28
If $1+r + r^2+ r^3 +\dots \infty = 1.5$, then, $1 + 2r + 3r^2 + 4r^3 + \dots \infty =$ (up to two dcimal places) ________
29
Consider the following infinite series: $1+ r+r^2 +r^3+ \dots \dots \infty$ If $r = 0.3$, then the sum of this infinite series is ____________
30
Which of the following are geometric series? $1,\:6,\:11,\:16,\:21,\:26,\:\dots$ $9,\:6,\:3,\:0,\:-3,\:-6,\:\dots$ $1,\:3,\:9,\:27,\:81,\:\dots$ $4,\:-8,\:16,\:-32,\:64,\:\dots$ $P$ and $Q$ only $R$ and $S$ only $Q$ and $S$ only $P, Q$ and $R$ only
31
Given that $Z=X^{2}+Y^{2}$, the value of $\dfrac{\partial Z}{\partial X}$ for $X=1$ and $Y=0$ is ________________ (answer is an integer).
32
The solution of $\underset{x\rightarrow 8 }\lim\left ( \dfrac{x^{2}-64}{x-8} \right )$ is _____________.
33
The median value for the dataset $\left ( 12, 10,16,8,90,50,30,24 \right )$ is ______________.
34
$\int _{-1}^{1}\:f\left ( x \right )dx$ calculated using trapezoidal rule for the values given in the table is ______ (rounded off to $2$ ...
35
Matrix $A=\begin{bmatrix} 0&6 \\ p&0 \end{bmatrix}$ will be skew-symmetric when $p$= ____.
36
The Laplace transform of the function $f\left ( t \right )=t^{2}+2t+1$ is $\dfrac{1}{s^{3}}+\dfrac{3}{s^{2}}+\dfrac{1}{s} \\$ $\dfrac{4}{s^{3}}+\dfrac{4}{s^{2}}+\dfrac{1}{s} \\$ $\dfrac{1}{s^{3}}+\dfrac{2}{s^{2}}+\dfrac{1}{s} \\$ $\dfrac{2}{s^{3}}+\dfrac{2}{s^{2}}+\dfrac{1}{s}$
37
What is the solution of the differential equation $\dfrac{\mathrm{dy} }{\mathrm{d} x}=\dfrac{x}{y}$, with the initial condition, at $x=0, y=1?$ $x^{2}=y^{2}+1$ $y^{2}=x^{2}+1$ $y^{2}=2x^{2}+1$ $x^{2}-y^{2}=0$
Which one of the following equations represents a one-dimensional wave equation? $\dfrac{\partial u}{\partial t}=C^{2}\dfrac{\partial ^{2}u}{\partial x^{2}} \\$ $\dfrac{\partial ^{2}u}{\partial t^{2}}=C^{2}\dfrac{\partial ^{2}u}{\partial x^{2}} \\$ ... $\dfrac{\partial ^{2}u}{\partial t^{2}}+\dfrac{\partial ^{2}u}{\partial x^{2}}=0$
The variable $z$ has a standard normal distribution. If $P(0 \leq z \leq 1)= 0.34$ then $P(z^2 > 1)$ is equal to (up to two decimal places) ____________
The determinant of the matrix $\begin{pmatrix} 4 & -6 \\ -3 & 2 \end{pmatrix}$ is _________