List of practice Questions

In a market, two firms F₁ and F₂ are producing homogenous products. The inverse demand function is given by p=120-0.5(q₁+q₂), where p is the unit price of the product, and q₁ and q₂ are the outputs from F₁ and F₂, respectively. Suppose the cost functions of F₁ and F₂ are C₁ = 20q₁ and C₂ = 10 +0.5q₂², respectively. Then the total profit earned by both the firms assuming a competitive situation is (in integer).

Let (-c,c) be the largest open interval in \(\R\) (where c is either a positive real number or c = ∞) on which the solution y(x) of the differential equation \(\frac{dy}{dx}=x^2+y^2+1\) with initial condition y(0) = 0 exists and is unique. Then which of the following is/are true?

Let S be the set of all continuous functions f: [-1,1]→\(\R\) satisfying the following three conditions:
(i) f is infinitely differentiable on the open interval (-1,1),
(ii) the Taylor series
\(f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+...\) of f at 0 converges to f(x) for each x ∈ (-1,1),
(iii) \(f(\frac{1}{n})=0\ \text{for all}\ n\isin\N\)
Then which of the following is/are true?

Define f: [0,1] → [0,1] by
\(f(x) = \begin{cases} 1 & \text{if } x =0 \\ \frac{1}{n} & \text{if } x=\frac{m}{n}\ \text{for some}\ m, n \isin \N\ \text{with}\ m\le n\ \text{and gcd(m, n)} = 1, \\ 0 & \text{if} x\isin[0,1]\ \text{is irrational}\end{cases}\)
and define g: [0,1]→ [0,1] by
\(g(n) = \begin{cases} 0 & \text{if } x=0 \\ 1 & \text{if } x\isin(0,1]. \end{cases}\)
Then which of the following is/are true?

Let S be the set of all functions f: \(\R\rightarrow\R\) satisfying \(|f(x)-f(y) |^2 \le |x - y|^3\ for\ all\ x, y \isin \R.\) Then which of the following is/are true?

A real-valued function y(x) defined on \(\R\) is said to be periodic if there exists a real number \(T\gt0\) such that y(x + T) = y(x) for all \(x\isin\R\). Consider the differential equation
\(\frac{d^2y}{dx^2}+4y=\sin(ax), x\isin\R, \quad (*)\)
where \(a\isin\R\) is a constant. 
Then which of the following is/are true?

Let be a finite group of order at least two and let e denote the identity element of G. Let σ: G →g be a bijective group homomorphism that satisfies the following two conditions:
(i): if σ(g)=g for some gεG, then g=e, 
(ii) (σ o σ) (g)=g for all g σ G. 
The n which of the following is/are correct ?

Let (x_n) be a sequence of real numbers. Consider the set P = {n\(\isin\N:x_n\gt x_m\) for all \(m\isin\N\) with \(m\gt n\)}. Then which of the following is/are true?

A solid cylinder and a solid sphere, having same mass M and radius R, roll down the same inclined plane from top without slipping They start from rest The ratio of velocity of the solid cylinder to that of the solid sphere, with which they reach the ground, will be :

For $x \in R$, let the function $y ( x )$ be the solution of the differential equation $\frac{d y}{d x}+12 y=\cos \left(\frac{\pi}{12} x\right), y(0)=0 \text { }$.
Then, which of the following statements is/are TRUE?

The number of distinct integer values of n satisfying \(4−\log\frac{2n}{3}−\log4n\lt0\), is

Let S = {4, 6, 9} and T = {9, 10, 11, …,1000}. If A = {a1 + a2 + … +ak :k∈N, a1, a2, a3, …, ak∈S}, then the sum of all the elements in the set T – A is equal to ____________.

A small toy starts moving from the position of rest under a constant acceleration. If it travels a distance of 10 m in t s, the distance travelled by the toy in the next t s will be :

Since their inception in the 1970s, video games have often been criticized by some for violent content. Politicians, parents, and other activists have claimed that violence in video games can be tied to violent behavior, particularly in children, and have sought ways to regulate the sale of video games. However, numerous studies have failed to find any connection between video games and violent behavior. Based on the information above, which conclusion necessarily follows ?
I. Video games have no relationship with violence.
II. Better studies need to be conducted to find a link between video games and violence.
Choose the most appropriate answer.

\(\alpha = \sin 36^\circ\)  is a root of which of the following equation?

For positive integer $n$, define $f(n)=n+\frac{16+5 n-3 n^2}{4 n+3 n^2}+\frac{32+n-3 n^2}{8 n+3 n^2}+\frac{48-3 n-3 n^2}{12 n+3 n^2}+\ldots+\frac{25 n-7 n^2}{7 n^2}$.
Then, the value of $\displaystyle\lim _{n \rightarrow \infty} f(n)$ is equal to

In a college 200 students are randomly chosen. 140 like tea and 120 like Coffee and 70 like both tea and coffee.How many like neither tea nor coffee?

Water is being filled at the rate of 1 cm³/sec in a right circular conical vessel (vertex downwards) of height 35 cm and diameter 14 cm. When the height of the water level is 10 cm, the rate (in cm²/sec) at which the wet conical surface area of the vessel increase, is

The cell potential for \(\text{Zn}|\text{Zn}^{2+}\text{(aq)}||\text{Sn}^{x+}|\text{Sn}\) is 0.801 V at 298 K.The reaction quotient for the above reaction is 10^–2. The number of electrons involved in the given electrochemical cell reaction is _____. (Given: \(Eº_Zn^{2+}\)|Zn=-0.763V,Eºsn^x+|sn=+0.008V and 2.303RT/F=0.06V)

What will be the effect on the root mean square velocity of oxygen molecules if the temperature is doubled and oxygen molecule dissociates into atomic oxygen?

A coil is placed in a time varying magnetic field. If the number of turns in the coil were to be halved and the radius of wire doubled, the electrical power dissipated due to the current induced in the coil would be: (Assume the coil to be short circuited.)

L-isomer of a compound ‘A’ \((C_4H_8O_4)\) gives a positive test with \([Ag(NH_3)^2]^+\). Treatment of ‘A’ with acetic anhydride yields triacetate derivative. Compound ‘A’ produces an optically active compound (B) and an optically inactive compound (C) on treatment with bromine water and \(HNO_3\) respectively. Compound (A) is:

In isolation of which one of the following metals from their ores, the use of cyanide salt is not commonly involved?

Which of the following compound does not contain sulphur atom?

A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is