List of practice Questions

Which of the following statements is NOT correct under the IS-LM (Fixed Price) model?

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?

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 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 (-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?

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?

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?

0.25 g of an organic compound containing chlorine gave 0.40 g of silver chloride in Carius estimation. The percentage of chlorine present in the compound is ______. [in nearest integer]
(Given : Molar mass of \(Ag\) is 108 g mol^–1 and that of \(Cl\) is 35.5 g mol^–1

A ball is released from rest from point P of a smooth semi-spherical vessel as shown in figure. The ratio of the centripetal force and normal reaction on the ball at point Q is A while angular position of point Q is α with respect to point P. Which of the following graphs represent the correct relation between A and α when ball goes from Q to R?

A capacitor C₁ of capacitance 5 μF is charged to a potential of 30 V using a battery. The battery is then removed and the charged capacitor is connected to an uncharged capacitor C₂ of capacitance 10 μF as shown in figure. When the switch is closed charge flows between the capacitors. At equilibrium, the charge on the capacitor C₂ is ____ μC.

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.)

A solution containing 2.5 × 10^–3 kg of a solute dissolved in 75 × 10^–3 kg of water boils at 373.535 K. The molar mass of the solute is _____ g mol^–1. [nearest integer] (Given : K_b(H₂O) = 0.52 K kg mol^–1 and boiling point of water = 373.15 K)

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

Compound ‘P’ on nitration with dil. HNO₃ yields two isomers (A) and (B) show the intramolecular and intermolecular hydrogen bonding respectively. Compound (P) on reaction with conc. HNO₃ yields a yellow compound ‘C’, a strong acid. The number of oxygen atoms is present in compound ‘C’ _____