List of practice Questions

For a natural number \(n\), let \(α_n = 19n – 12n\). Then, the value of \(\frac {31α_9−α_{10}}{57α_8}\) is _____.

Let \(f :R→R\) be a function defined by \(f(x)=(2(1−\frac {x^{25}}{2})(2+x^{25}))^{\frac {1}{50}}\). If the function \(g(x) = f (f (f (x))) + f (f (x))\), then the greatest integer less than or equal to \(g(1)\) is _________.

Let the lines
\(L_1:\vec r=λ(\hat i+2\hat j+3 \hat k), \ λ∈R\)
\(L_2:\vec r=(\hat i+3\hat j+\hat  k)+μ(\hat i+\hat j+\hat 5k), \ μ∈R\)
intersect at the point \(S\). If a plane \(ax + by – z + d = 0\) passes through \(S\) and is parallel to both the lines \(L_1\) and \(L_2\) then the value of \(a + b + d\) is equal to _______.

Let A be a 3 × 3 matrix having entries from the set {–1, 0, 1}. The number of all such matrices A having sum of all the entries equal to 5, is ______.

The greatest integer less than or equal to the sum of first \(100\) terms of the sequence \(\frac 13, \frac 59, \frac {19}{27}, \frac {65}{81},…...\) is equal to ______.

If the initial velocity in horizontal direction of a projectile is unit vector \(\hat i\) and the equation of trajectory is \(y = 5x(1 – x)\). The y component vector of the initial velocity is ______ \(\hat j\).
(Take \(g=10\ m/s^2\))

A disc of mass 1 kg and radius R is free to rotate about a horizontal axis passing through its centre and perpendicular to the plane of disc. A body of same mass as that of disc of fixed at the highest point of the disc. Now the system is released, when the body comes to the lowest position, it angular speed will be \(4\sqrt {\frac {x}{3R}}\) rad s⁻¹ where x = ______ .(g = 10 ms^–2)

In an experiment of determine the Young’s modulus of wire of a length exactly 1 m, the extension in the length of the wire is measured as 0.4 mm with an uncertainty of ±0.02 mm when a load of 1 kg is applied. The diameter of the wire is measured as 0.4 mm with an uncertainty of ±0.02 mm when a load of 1 kg is applied. The diameter of the wire is measured as 0.4 mm with an uncertainty of ± 0.01 mm. The error in the measurement of Young’s modulus (ΔY) is found to be x × 10¹⁰ Nm^–2. The value of x is ___.
(Take g=10 m/s²)

When a car is approaching the observer, the frequency of horn is 100 Hz. After passing the observer, it is 50 Hz. If the observer moves with the car, the frequency will be \(\frac x3\) Hz, where x = _____ .

A composite parallel plate capacitor is made up of two different dielectric materials with different thickness (t₁ and t₂) as shown in figure. The two different dielectric materials are separated by a conducting foil F. The voltage of the conducting foil is _____V.

0.056 kg of Nitrogen is enclosed in a vessel at a temperature of 127°C. The amount of heat required to double the speed of its molecules is ___ k cal.(Take R = 2 cal mol^–1 K^–1)

Chlorophyll extracted from the crushed green leaves was dissolved in water to make 2 L solution of Mg of concentration 48 ppm. The number of atoms of Mg in this solution is \(x × 10^{20}\) atoms. The value of \(x\) is ________. (Nearest integer)
(Given : Atomic mass of Mg is 24 g mol^–1; N_A = 6.02 × 10²³ mol^–1)

A ball is projected vertically upward with an initial velocity of 50 ms^–1 at t = 0 s. At t = 2 s, another ball is projected vertically upward with same velocity. 
At t = _____s, second ball will meet the first ball. (g = 10 ms^–2)

Let \(ƒ : N→R\) be a function such that \(ƒ(x + y) = 2ƒ(x) ƒ(y)\) for natural numbers x and y. If \(ƒ(1) = 2\), then the value of α for which \(\displaystyle\sum_{k=1}^{10} f(α+k)=\frac {512}{3} (2^{20}−1)\) holds, is:

A batsman hits back a ball of mass 0.4 kg straight in the direction of the bowler without changing its initial speed of 15 ms^–1. The impulse imparted to the ball is _________ Ns.

A curved in a level road has a radius 75 m. The maximum speed of a car turning this curved road can be 30 m/s without skidding. If radius of curved road is changed to 48 m and the coefficient of friction between the tyres and the road remains same, then maximum allowed speed would be ______ m/s.

A geyser heats water flowing at a rate of 2.0 kg per minute from 30°C to 70°C. If geyser operates on a gas burner, the rate of combustion of fuel will be __________ g min^–1.
[Heat of combustion = 8 × 10³ Jg^–1, Specific heat of water = 4.2 Jg^–1 °C^–1]

Let A be a \(3×3\) real matrix such that \(A\begin{pmatrix}   1 \\1  \\ 0  \end{pmatrix}\)=\(\begin{pmatrix}   1 \\1  \\ 0  \end{pmatrix}\); \(A\begin{pmatrix}   1 \\0  \\ 1  \end{pmatrix}\)=\(A\begin{pmatrix}   -1 \\0  \\ 1  \end{pmatrix}\) and \(A\begin{pmatrix}   0 \\0  \\ 1  \end{pmatrix}\)=\(\begin{pmatrix}   1 \\1  \\ 2  \end{pmatrix}\)
If \(X = [x_1, x_2, x_3]^T \)and \(I\) is an identity matrix of order \(3\), then the system \([A−2I]X \)= \(\begin{pmatrix}   4 \\1  \\ 1 \end{pmatrix}\) has:

A block of mass 200 g is kept stationary on a smooth inclined plane by applying a minimum horizontal force  \(F=\sqrt xN\) as shown in figure. 

The value of x = ______.

A heat engine operates with the cold reservoir at temperature 324 K. The minimum temperature of the hot reservoir, if the heat engine takes 300 J heat from the hot reservoir and delivers 180 J heat to the cold reservoir per cycle, is _____ K.

A set of 20 tuning forks is arranged in a series of increasing frequencies. If each fork gives 4 beats with respect to the preceding fork and the frequency of the last fork is twice the frequency of the first, then the frequency of last fork is _______ Hz

Two 10 cm long, straight wires, each carrying a current of 5 A are kept parallel to each other. If each wire experienced a force of 10^–5 N, then separation between the wires is _____ cm.

A small bulb is placed at the bottom of a tank containing water to a depth of √7 m. The refractive index of water is 4/3. The area of the surface of water through which light from the bulb can emerge out is xπ m². The value of x is ________.