List of practice Questions

The number of $3$-digit numbers, that are divisible by either $2$ or $3$ but not divisible by $7$ , is_________

$A(2,6,2), B(-4,0, \lambda), C(2,3,-1)$ and $D(4,5,0),|\lambda| \leq 5$ are the vertices of a quadrilateral $A B C D$ If its area is 18 square units, then $5-6 \lambda$ is equal to________

Let $f(x)=2 x+\tan ^{-1} x$ and $g(x)=\log _e\left(\sqrt{1+x^2}+x\right), x \in[0,3]$ Then

If $y=y(x)$ is the solution curve of the differential equation $\frac{d y}{d x}+y \tan x=x \sec x, 0 \leq x \leq \frac{\pi}{3}, y(0)=1$ then $y\left(\frac{\pi}{6}\right)$ is equal to

If the center and radius of the circle $\left|\frac{z-2}{z-3}\right|=2$ are respectively $(\alpha, \beta)$ and $\gamma$, then $3(\alpha+\beta+\gamma)$ is equal to

Given below are two statements:
Statement I : Acceleration due to gravity is different at different places on the surface of earth
Statement II : Acceleration due to gravity increases as we go down below the earth's surface
In the light of the above statements, choose the correct answer from the options given below

The shortest distance between the lines $\frac{x-5}{1}=\frac{y-2}{2}=\frac{z-4}{-3}$ and $\frac{x+3}{1}=\frac{y+5}{4}=\frac{z-1}{-5}$ is

For a triangle $A B C$, the value of $\cos 2 A+\cos 2 B+\cos 2 C$ is least If its inradius is $3$ and incentre is $M$, then which of the following is NOT correct?

Let $a, b, c>, a^3, b^3$ and $c^3$ be in AP, and $\log _a b, \log _c a$ and $\log _b c$ be in GP If the sum of first 20 terms of an AP, whose first term is $\frac{a+4 b+c}{3}$ and the common difference is $\frac{a-8 b+c}{10}$ is $-444$, then $a b c$ is equal to:

If $P$ is a $3 \times 3$ real matrix such that $P^T=a P+(a-1) I$, where $a>$, then

The number of ways of selecting two numbers $a$ and $b, a \in\{2,4,6, \ldots , 100\}$ and $b \in\{1,3,5, \ldots , 99\}$ such that 2 is the remainder when $a+b$ is divided by 23 is

For $\alpha, \beta \in R$, suppose the system of linear equations $ x-y+z=5 $ $2 x+2 y+\alpha z=8 $ $3 x-y+4 z=\beta$ has infinitely many solutions Then $\alpha$ and $\beta$ are the roots of

The solution of the differential equation $\frac{d y}{d x}=-\left(\frac{x^2+3 y^2}{3 x^2+y^2}\right), y(1)=0$ is

If a plane passes through the points $(-1, k, 0),(2, k,-1),(1,1,2)$ and is parallel to the line $\frac{x-1}{1}=\frac{2 y+1}{2}=\frac{z+1}{-1}$, then the value of $\frac{k^2+1}{(k-1)(k-2)}$ is

See the following chemical reaction: \[ \text{Cr}_2\text{O}_7^{2-} + \text{XH}^+ + \text{6Fe}^{2+} \rightarrow \text{YCr}^{3+} + \text{6Fe}^{3+} + \text{ZH}_2\text{O}. \] The sum of \(X\), \(Y\), and \(Z\) is _____.

Select the correct graph showing the difference (d) between total energy and potential energy of a particle in linear SHM with position x of the particle (x = 0 is the mean position)

The output from a NAND gate having inputs A and B given below will be.

The rms speed of oxygen molecule in a vessel at particular temperature is \((1+\frac{5}{x})^{\frac{1}{2}} v\), where \(v\) is the average speed of the molecule. The value of x will be : (Take \(π = \frac{22}{7}\))

The figure shows a liquid of given density flowing steadily in horizontal tube of varying cross - section. Cross sectional areas at A is 1.5 cm², and B is is 25 mm², if the speed of liquid at B is 60 cm/s then \((P_A-P_B)\) is: 
(Given \(P_A\) and \(P_B\) are liquid pressures at A and B points.
density \(ρ = 1000 \space kg \space m^{-3}\)
A and B are on the axis of tube)

A planet having mass \(9M_e\) and radius \(4R_e\), where \(M_e\) and \(R_e\) are mass and radius of earth respectively, has escape velocity in km/s given by: (Given escape velocity on earth \(V_e= 11.2 × 10^3 m/s\))

A bullet of 10 g leaves the barrel of gun with a velocity of 600 m/s. If the barrel of gun is 50 cm long and mass of gun is 3 kg, then value of impulse supplied to the gun will be :

$KMnO _4$ oxidises $I ^{-}$in acidic and neutral/faintly alkaline solution, respectively, to

Given below are two statements:
Statement I : During Electrolytic refining, the pure metal is made to act as anode and its impure metallic form is used as cathode
Statement II : During the Hall-Heroult electrolysis process, purified $Al _2 O _3$ is mixed with $Na _3 AlF _6$ to lower the melting point of the mixture
In the light of the above statements, choose the most appropriate answer from the options given below:

Maximum number of electrons that can be accommodated in shell with $n=4$ are:

Two bodies having the same linear momentum have a ratio of kinetic energy as 16:9. Find the ratio of masses of these bodies.