List of practice Questions

Let $[t]$ denote the greatest integer less than or equal to t Then the value of the integral $\int\limits_{-3}^{101}\left([\sin (\pi x)]+e^{[\cos (2 \pi x)]}\right) d x$ is equal to

Let the point $P(\alpha, \beta)$ be at a unit distance from each of the two lines $L _1: 3 x -4 y +12=0$, and $L _2: 8 x +6 y +11=0$ If $P$ lies below $L _1$ and above $L _2$, then $100(\alpha+\beta)$ is equal to

The equations of two sides of a variable triangle are \(x-0\) and \(y=3\), and its third side is a tangent to the parabola \(y^2=6 x\). The locus of its circumcentre is:

Let a smooth curve $y=f(x)$ be such that the slope of the tangent at any point $(x, y)$ on it is directly proportional to $\left(\frac{-y}{x}\right)$ If the curve passes through the point $(1,2)$ and $(8,1)$, then $\left|y\left(\frac{1}{8}\right)\right|$ is equal to

If the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ meets the line $\frac{x}{7}+\frac{y}{2 \sqrt{6}}=1$ on the $x$-axis and the line $\frac{x}{7}-\frac{y}{2 \sqrt{6}}=1$ on the $y$-axis, then the eccentricity of the ellipse is

Let $T$ and $C$ respectively be the transverse and conjugate axes of the hyperbola $16 x^2-y^2+64 x+4 y$ $+44=0$. Then the area of the region above the parabola $x^2=y+4$, below the transverse axis $T$ and on the right of the coujugate axis $C$ is:

The tangents at the point $A (1,3)$ and $B (1,-1)$ on the parabola $y^2-2 x-2 y=1$ meet at the point $P$ Then the area (in unit $^2$ ) of $\triangle PAB$ is :-

Let the foci of the ellipse $\frac{x^2}{16}+\frac{y^2}{7}=1$ and the hyperbola $\frac{x^2}{144}-\frac{y^2}{\alpha}=\frac{1}{25}$ coincide Then the length of the latus rectum of the hyperbola is:-

The number of numbers, strictly between 5000 and 10000 can be formed using the digits 1,3,5,7,9 without repetition, is

A plane $E$ is perpendicular to the two planes $2 x-2 y+z=0$ and $x-y+2 z=4$, and passes through the point $P (1,-1,1)$ If the distance of the plane $E$ from the point $Q(a, a, 2)$ is $3 \sqrt{2}$, then $( PQ )^2$ is equal to

The shortest distance between the lines $\frac{x+7}{-6}=\frac{y-6}{7}=z$ and $\frac{7-x}{2}=y-2=z-6$ is

The foot of perpendicular of the point $(2,0,5)$ on the line $\frac{x+1}{2}=\frac{y-1}{5}-\frac{z+1}{-1}$ is $(a, \beta, \gamma)$. Then which of the following is NOT correct?

Let $\vec{a}=\hat{i}-\hat{j}+2 \hat{k}$ and $\vec{b}$ be a vector such that $\vec{a} \times \vec{b}=2 \hat{i}-\hat{k}$ and $\vec{a} \cdot \vec{b}=3$ Then the projection of $\vec{b}$ on the vector $\vec{a}-\vec{b}$ is :-

If the mean deviation about median for the number $3,5,7,2 k , 12,16,21,24$ arranged in the ascending order, is 6 then the median is

Consider the following statements : $P$ : Ramu is intelligent $Q: $ Ramu is rich $R$ : Ramu is not honest The negation of the statement "Ramu is intelligent and honest if and only if Ramu is not rich" can be expressed as :

As per given figure, a weightless pulley $P$ is attached on a double inclined frictionless surfaces. The tension in the string (massless) will be (if $g =10 \,/ s ^2$ )

The isomeric deuterated bromide with molecular formula $C _4 H _8 DBr$ having two chiral carbon atoms is

Match List I with List II :

LIST I Isomeric pairs		LIST II Type of isomers
A	Propanamine and N-Methylethanamine	I	Metamers
B	Hexan-2-one and Hexan-3-one	II	Positional isomers
C	Ethanamide and Hydroxyethanimine	II	Functional isomers
D	o-nitrophenol and p-nitrophenol	IV	Tautomers

Choose the correct answer from the options given below:

Potassium dichromate acts as a strong oxidizing agent in acidic solution During this process, the oxidation state changes from

Match List I with List II

LIST I		LIST II
A	Cobalt catalyst	I	\((H_2​+Cl_2​)\) production
B	Syngas	II	Water gas production
C	Nickel catalyst	III	Coal gasification
D	Brine solution	IV	Methanol production

Choose the correct answer from the options given below:

Find out the major product from the following reaction

Match List I with List II

LIST I (Amines)		LIST II (pKb​)
A	Aniline	I	3.25
B	Ethanamine	II	3.00
C	N-Ethylethanamine	III	9.38
D	N, N-Diethylethanamine	IV	3.29

 Choose the correct answer from the options given below:

Given below are two statements :
Statement I : If the Brewster's angle for the light propagating from air to glass is $\theta_{ B }$, then the Brewster's angle for the light propagating from glass to air is $\frac{\pi}{2}-\theta_m$
Statement II : The Brewster's angle for the light propagating from glass to air is $\tan ^{-1}\left(\mu_{ g }\right)$ where $\mu_{ g }$ is the refractive index of glass
In the light of the above statements choose the correct answer from the options given below:

'R formed in the following sequence of reactions is :

Compound $( X )$ undergoes following sequence of reactions to give the Lactone $( Y )$