List of practice Questions

A vector $\vec{v}$ in the first octant is inclined to the $x$-axis at $60^{\prime}$, to the $y$-axis at $45$ and to the $z$-axis at an acute angle If a plane passing through the points $(\sqrt{2},-1,1)$ and $(a, b, c)$, is normal to $\vec{v}$, then

A vector $\vec{v}$ in the first octant is inclined to the $x$-axis at $60^{\prime}$, to the $y$-axis at $45$ and to the $z$-axis at an acute angle If a plane passing through the points $(\sqrt{2},-1,1)$ and $(a, b, c)$, is normal to $\vec{v}$, then

Let $f, g$ and $h$ be the real valued functions defined on $R$ as $f(x)=\begin{cases} \frac{x}{|x|}, & x \neq 0 \\1, & x=0\end{cases}, g(x)=\begin{cases} \frac{\sin (x+1)}{(x+1)}, & x \neq-1 \\1, & x=-1\end{cases}$ and $h(x)=2[x]-f(x)$, where $[x]$ is the greatest integer $\leq x$ Then the value of $\displaystyle\lim _{x \rightarrow 1} g(h(x-1))$ is

Let $a_1=1, a_2, a_3, a_4, \ldots $ be consecutive natural numbersThen $\tan ^{-1}\left(\frac{1}{1+a_1 a_2}\right)+\tan ^{-1}\left(\frac{1}{1+a_2 a_3}\right)+\ldots +\tan ^{-1}\left(\frac{1}{1+a_{2021} a_{2022}}\right)$ is equal to

If the functions $f(x)=\frac{x^3}{3}+2 b x+\frac{a x^2}{2}$ and $g(x)=\frac{x^3}{3}+a x+b x^2, a \neq 2 b$ have a common extreme point, then $a+2 b+7$ is equal to:

Let $\vec{a}$ and $\vec{b}$ be two vectors, Let $|\vec{a}|=1,|\vec{b}|=4$ and $\vec{a} \cdot \vec{b}=2$ If $\vec{c}=(2 \vec{a} \times \vec{b})-3 \vec{b}$, then the value of $\vec{b} \cdot \vec{c}$ is

Let \(a,b,c>1,a^3,b^3 \text{ and } c^3\) be in A.P., and \(log_a​b, log_c​a \text{ and } log_b​c\) be in G.P. If the sum of first 20 terms of an A.P., whose first term is \(\frac{a+4b+c}{3}\)​ and the common difference is \(\frac{a−8b+c}{10}\)​ is −444, then abc is equal to:

The range of the function $f(x)=\sqrt{3-x}+\sqrt{2+x}$ is:

Let $S$ be the set of all values of $a_1$ for which the mean deviation about the mean of $100$ consecutive positive integers $a_1, a_2, a_3, \ldots , a_{100}$ is $25$ Then $S$ is

Let $A$ be a point on the $x$-axis Common tangents are drawn from $A$ to the curves $x^2+y^2=8$ and $y^2=16 x$ If one of these tangents touches the two curves at $Q$ and $R$, then $(Q R)^2$ is equal to

Consider the following statements: 
P : I have fever 
Q: I will not take medicine 
R : I will take rest 
The statement "If I have fever, then I will take medicine and I will take rest" is equivalent to:

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

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

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

In Young's double slits experiment, the position of $5^{\text {th }}$ bright fringe from the central maximum is $5 \,cm$ The distance between slits and screen is $1 \,m$ and wavelength of used monochromatic light is $600 \,mm$ The separation between the slits is:

A car is moving with a constant speed of $20\, m / s$ in a circular horizontal track of radius $40\, m$ A bob is suspended from the roof of the car by a massless string The angle made by the string with the vertical will be : (Take $g=10 \, m / s ^2$ )

An electromagnetic wave is transporting energy in the negative $z$ direction At a certain point and certain time the direction of electric field of the wave is along positive $y$ direction What will be the direction of the magnetic field of the wave at that point and instant?

The root mean square velocity of molecules of gas is

The ratio of the density of oxygen nucleus $\left({ }_8^{16} O \right)$ and helium nucleus $\left({ }_2^4 He \right)$ is

An object of mass $8 kg$ is hanging from one end of a uniform rod CD of mass $2 kg$ and length $1 m$ pivoted at its end $C$ on a vertical wall as shown in figure It is supported by a cable $AB$ such that the system is in equilibrium The tension in the cable is : (Take $g =10 m / s ^2$ )

A uniform metallic wire carries a current $2 A$, when $34 V$ battery is connected across it The mass of uniform metallic wire is $892 \times 10^{-3} kg$, density is $892 \times 10^3 kg / m ^3$ and resistivity is $17 \times 10^{-8} \Omega- m$ The length of wire is :

A message signal of frequency $5 \,kHz$ is used to modulate a carrier signal of frequency $2\, MHz$ The bandwidth for amplitude modulation is:

Which of the following represents the correct order of metallic character of the given elements?

Given below are two statements, one is labelled as Assertion $A$ and the other is labelled as Reason R 
Assertion A : Butylated hydroxy anisole when added to butter increases its shelf life
Reason R : Butylated hydroxy anisole is more reactive towards oxygen than food
In the light of the above statements,choose the most appropriate answer from the options given below

Let A,B,C be 3×3 matrices such that A is symmetric and B and C are skew symmetric. Consider the statements:
(S1) A¹³B²⁶ − B²⁶A¹³ is symmetric.
(S2) A²⁶C¹³ − C¹³A²⁶ is symmetric.Then: