List of practice Questions

In the balanced condition, the values of the resistances of the four arms of a Wheatstone bridge are shown in the figure below. The resistance $R_{3}$ has temperature coefficient $00004^{\circ} C ^{-1}$. If the temperature of $R_{3}$ is increased by $100{ }^{\circ} C$, the voltage developed between $S$ and $T$ will be ______ volt

Let $a$ and $b$ be positive real numbers such that \(a >1\) and \(b < a\). Let $P$ be a point in the first quadrant that lies on the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$. Suppose the tangent to the hyperbola at P passes through the point $(1,0)$, and suppose the normal to the hyperbola at $P$ cuts off equal intercepts on the coordinate axes. Let $\Delta$ denote the area of the triangle formed by the tangent at $P$, the normal at $P$ and the $x$-axis. If e denotes the eccentricity of the hyperbola, then which of the following statements is/are TRUE?

Let $\alpha, \beta, \gamma, \delta$ be real numbers such that $\alpha^{2}+\beta^{2}+\gamma^{2} \neq 0$ and $\alpha+\gamma=1$. Suppose the point $(3,2,-1)$ is the mirror image of the point $(1,0,-1)$ with respect to the plane $\alpha x+\beta y+\gamma z=\delta$. Then which of the following statements is/are TRUE?

Let $f : R \rightarrow R$ and $g : R \rightarrow R$ be functions satisfying
$f(x+y)=f(x)+f(y)+f(x) f(y)$ and $f(x)=x g(x)$
for all $x, y \in R $. If $\displaystyle\lim _{x \rightarrow 0} g(x)=1$, then which of the following statements is/are TRUE?

Let $O$ be the centre of the circle $x ^{2}+ y ^{2}= r ^{2}$, where \(r >\frac{\sqrt{5}}{2}\). Suppose $PQ$ is a chord of this circle and the equation of the line passing through $P$ and $Q$ is $2 x +4 y =5$. If the centre of the circumcircle of the triangle OPQ lies on the line $x+2 y=4$, then the value of $r$ is _____

Let the function $f :(0, \pi) \rightarrow R$ be defined by 
$f (\theta)=(\sin \theta+\cos \theta)^{2}+(\sin \theta-\cos \theta)^{4} \text { }$. 
Suppose the function f has a local minimum at $\theta$ precisely when $\theta \in\left\{\lambda_{1} \pi, \ldots, \lambda_{r} \pi\right\}$, where \(0 < \lambda_{1} < \ldots < \lambda_{r} < 1\) Then the value of $\lambda_{1}+\ldots+\lambda_{r}$ is ______

Starting at time $t =0$ from the origin with speed $1\, ms ^{-1}$, a particle follows a two-dimensional trajectory in the $x-y$ plane so that its coordinates are related by the equation $y=\frac{x^{2}}{2}$. The $x$ and $y$ components of its acceleration are denoted by $a _{ x }$ and $a _{y}$, respectively Then

The probability that a missile hits a target successfully is $0.75$. In order to destroy the target completely, at least three successful hits are required. Then the minimum number of missiles that have to be fired so that the probability of completely destroying the target is NOT less than $0.95$, is______

Two capacitors with capacitance values $C_{1}=2000 \pm 10\, pF$ and $C_{2}=3000 \pm 15\, pF$ are connected in series. The voltage applied across this combination is $V =5.00 \pm 002 \,V$. The percentage error in the calculation of the energy stored in this combination of capacitors is _____

Two fair dice, each with faces numbered $1,2,3,4,5$ and $6$, are rolled together and the sum of the numbers on the faces is observed. This process is repeated till the sum is either a prime number or a perfect square. Suppose the sum turns out to be a perfect square before it turns out to be a prime number. If $p$ is the probability that this perfect square is an odd number, then the value of $14\, p$ is _____

Consider a $70 \%$ efficient hydrogen-oxygen fuel cell working under standard conditions at $1$ bar and $298 \,K$ Its cell reaction is
$H _{2}( g )+\frac{1}{2} O _{2}( g ) \rightarrow H _{2} O ( l )$
The work derived from the cell on the consumption of $1.0 \times 10^{-3} mol$ of $H _{2}( g )$ is used to compress $100\, mol$ of a monoatomic ideal gas in a thermally insulated container. What is the change in the temperature (in $K$ ) of the ideal gas?
The standard reduction potentials for the two half-cells are given below
$O _{2}(g)+4 H ^{+}(a q)+4 e^{-} \rightarrow 2 H _{2} O ( I ), E^{\circ}=1.23 \,V$
$2 H ^{+}(a q)+2 e^{-} \rightarrow H _{2}(g), E^{\circ}=0.00 \,V$
Use $F =96500 \,C\,mol ^{-1}, R=8.314\, J\,mol ^{-1} K ^{-1}$

Let e denote the base of the natural logarithm. The value of the real number a for which the right hand limit
$\displaystyle\lim_{x\rightarrow 0^+} \frac{(1-x)^{\frac{1}{x}} - e^{-1}}{x^a}$
is equal to a nonzero real number, is_____

Flippers of Penguins and Dolphins are examples of:

Biolistics method is suitable for gene transfer into_________

A positive hole in a semiconductor is

In the following circuit what are $P$ and $Q$:

If the parabola $x^2=4ay$ passes through the point $(2, 1)$, then the length of the latus rectum is

The probability of solving a problem by three persons $A, B$ and $C$ independently is $\frac{1}{2}$, $\frac{1}{4}$ and $\frac{1}{3}$ respectively. Then the probability of the problem is solved by any two of them is

If A = {1,2,3,4,5,6}, then the number of subsets of A which contain at least two elements is

Which of the following features of plants is not helpful in adapting to desert life?

In the following equation of Verhulst - Pearl logistic growth, the letter '$r$' denotes________ $\frac{dN}{dt} =rN(\frac{K-N}{K})$

Which one of the following is not included under in-situ conservation?

The percentage of $s$-character in the hybrid orbitals of nitrogen in $NO^+_2, NO^-_3$ and $NH_4^+$ respectively are

The passages given here are followed by some questions that have four answer choices; read the passage carefully and pick the option whose answer best aligns with the passage
In the late 1960s, while studying the northern-elephant-seal population along the coasts of Mexico and California, Burney Le Boeuf and his colleagues couldn’t help but notice that the threat calls of males at some sites sounded different from those of males at other sites. . .. That was the first time dialects were documented in a nonhuman mammal. . ..
All the northern elephant seals that exist today are descendants of the small herd that survived on Isla Guadalupe [after the near extinction of the species in the nineteenth century]. As that tiny population grew, northern elephant seals started to recolonize former breeding locations. It was precisely on the more recently colonized islands where Le Boeuf found that the tempos of the male vocal displays showed stronger differences to the ones from Isla Guadalupe, the founder colony.
In order to test the reliability of these dialects over time, Le Boeuf and other researchers visited Año Nuevo Island in California—the island where males showed the slowest pulse rates in their calls—every winter from 1968 to 1972. “What we found is that the pulse rate increased, but it still remained relatively slow compared to the other colonies we had measured in the past” Le Boeuf told me.
At the individual level, the pulse of the calls stayed the same: A male would maintain his vocal signature throughout his lifetime. But the average pulse rate was changing. Immigration could have been responsible for this increase, as in the early 1970s, 43 percent of the males on Año Nuevo had come from southern rookeries that had a faster pulse rate. This led Le Boeuf and his collaborator, Lewis Petrinovich, to deduce that the dialects were, perhaps, a result of isolation over time, after the breeding sites had been recolonized. For instance, the first settlers of Año Nuevo could have had, by chance, calls with low pulse rates. At other sites, where the scientists found faster pulse rates, the opposite would have happened—seals with faster rates would have happened to arrive first.
As the population continued to expand and the islands kept on receiving immigrants from the original population, the calls in all locations would have eventually regressed to the average pulse rate of the founder colony. In the decades that followed, scientists noticed that the geographical variations reported in 1969 were not obvious anymore. . . . In the early 2010s, while studying northern elephant seals on Año Nuevo Island, [researcher Caroline] Casey noticed, too, that what Le Boeuf had heard decades ago was not what she heard now. . . . By performing more sophisticated statistical analyses on both sets of data, [Casey and Le Boeuf] confirmed that dialects existed back then but had vanished. Yet there are other differences between the males from the late 1960s and their great-greatgrandsons: Modern males exhibit more individual diversity, and their calls are more complex. While 50 years ago the drumming pattern was quite simple and the dialects denoted just a change in tempo, Casey explained, the calls recorded today have more complex structures, sometimes featuring doublets or triplets. . . .

Direction for Reading Comprehension: The passages given here are followed by some questions that have four answer choices; read the passage carefully and pick the option whose answer best aligns with the passage
Vocabulary used in speech or writing organizes itself in seven parts of speech (eight, if you count interjections such as Oh! and Gosh! and Fuhgeddaboudit!). Communication composed of these parts of speech must be organized by rules of grammar upon which we agree. When these rules break down, confusion and misunderstanding result. Bad grammar produces bad sentences. My favorite example from Strunk and White is this one: “As a mother of five, with another one on the way, my ironing board is always up.”
Nouns and verbs are the two indispensable parts of writing. Without one of each, no group of words can be a sentence, since a sentence is, by definition, a group of words containing a subject (noun) and a predicate (verb); these strings of words begin with a capital letter, end with a period, and combine to make a complete thought which starts in the writer’s head and then leaps to the reader’s.
Must you write complete sentences each time, every time? Perish the thought. If your work consists only of fragments and floating clauses, the Grammar Police aren’t going to come and take you away. Even William Strunk, that Mussolini of rhetoric, recognized the delicious pliability of language. “It is an old observation,” he writes, “that the best writers sometimes disregard the rules of rhetoric.” Yet he goes on to add this thought, which I urge you to consider: “Unless he is certain of doing well, [the writer] will probably do best to follow the rules.”
The telling clause here is Unless he is certain of doing well. If you don’t have a rudimentary grasp of how the parts of speech translate into coherent sentences, how can you be certain that you are doing well? How will you know if you’re doing ill, for that matter? The answer, of course, is that you can’t, you won’t. One who does grasp the rudiments of grammar finds a comforting simplicity at its heart, where there need be only nouns, the words that name, and verbs, the words that act.
Take any noun, put it with any verb, and you have a sentence. It never fails. Rocks explode. Jane transmits. Mountains float. These are all perfect sentences. Many such thoughts make little rational sense, but even the stranger ones (Plums deify!) have a kind of poetic weight that’s nice. The simplicity of noun-verb construction is useful—at the very least it can provide a safety net for your writing. Strunk and White caution against too many simple sentences in a row, but simple sentences provide a path you can follow when you fear getting lost in the tangles of rhetoric—all those restrictive and nonrestrictive clauses, those modifying phrases, those appositives and compound-complex sentences. If you start to freak out at the sight of such unmapped territory (unmapped by you, at least), just remind yourself that rocks explode, Jane transmits, mountains float, and plums deify. Grammar is . . . the pole you grab to get your thoughts up on their feet and walking.