List of practice Questions

Let P and Q be two distinct points on a circle which has center at C(2, 3) and which passes through origin O. If OC is perpendicular to both the line segments CP and CQ, then the set\{P, Q\} is equal to :

Two tangents are drawn from the point P(-1, 1) to the circle $x^2 + y^2 - 2x - 6y + 6 = 0$. If these tangents touch the circle at points A and B, and if D is a point on the circle such that length of the segments AB and AD are equal, then the area of the triangle ABD is equal to :

Let
A = $\{(x, y) \in R \times R | 2x^2 + 2y^2 - 2x - 2y = 1\}$,
B = $\{(x, y) \in R \times R | 4x^2 + 4y^2 - 16y + 7 = 0\}$ and
C = $\{(x, y) \in R \times R | x^2 + y^2 - 4x - 2y + 5 \le r^2\}$.
Then the minimum value of $|r|$ such that $A \cup B \subseteq C$ is

The image of the point (3, 5) in the line x - y + 1 = 0, lies on :

If one of the diameters of the circle $x^2 + y^2 - 2x - 6y + 6 = 0$ is a chord of another circle 'C', whose center is at (2, 1), then its radius is ________

The point P (a, b) undergoes the following three transformations successively :
(a) reflection about the line y=x.
(b) translation through 2 units along the positive direction of x-axis.
(c) rotation through angle $\frac{\pi}{4}$ about the origin in the anti-clockwise direction.
If the co-ordinates of the final position of the point P are $(-\frac{1}{\sqrt{2}}, \frac{7}{\sqrt{2}})$, then the value of 2a+b is equal to :

If α, β are natural numbers such that 100$^α - 199 β = (100)(100) + (99)(101) + (98)(102) + ⋯ + (1)(199)$, then the slope of the line passing through (α, β) and origin is :

Choose the correct statement about two circles whose equations are given below : x² + y² - 10x - 10y + 41 = 0 ; x² + y² - 22x - 10y + 137 = 0

A square ABCD has all its vertices on the curve x²y² = 1. The midpoints of its sides also lie on the same curve. Then, the square of area of ABCD is __________.

Let a, b, c be in arithmetic progression. Let the centroid of the triangle with vertices (a, c), (2, b) and (a, b) be (10/3, 7/3). If α, β are the roots of the equation ax² + bx + 1 = 0, then the value of α² + β² - αβ is :

If the area of the triangle formed by the positive x-axis, the normal and the tangent to the circle $ (x-2)^2 + (y-3)^2 = 25 $ at the point $ (5, 7) $ is $ A $, then $ 24A $ is equal to _______

Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): Synthesis of ethyl phenyl ether may be achieved by Williamson synthesis.
Reason (R): Reaction of bromobenzene with sodium ethoxide yields ethyl phenyl ether.
In the light of the above statements, choose the most appropriate answer from the options given below :

Match List - I with List - II.

Consider the given reaction, Identify 'X' and 'Y' :

The major products formed in the following reaction sequence A and B are :

The correct sequential addition of reagents in the preparation of 3-nitrobenzoic acid from benzene is :

The structure of the starting compound P used in the reaction given below is:

Which one of the following reactions will not yield propionic acid ?

Which one of the following is the major product of the given reaction ?

The structure of product C, formed by the following sequence of reactions is :
$ CH_3COOH + SOCl_2 \to A \xrightarrow[\text{AlCl}_3]{\text{Benzene}} B \xrightarrow[\text{}^-OH]{\text{KCN}} C $

The major products A and B in the following set of reactions are :

For the reaction given below :
The compound which is not formed as a product in the reaction is a :

The structures of A and B formed in the following reaction are : [Ph = $-C_6H_5$]

The major product of the following reaction is :
CH$_3$CH$_2$CH=CH$_2$ + H$_2$/CO $\xrightarrow{\text{Rh catalyst}}$

Compound(s) which will liberate carbon dioxide with sodium bicarbonate ($NaHCO_3$) solution is/are : A = Urea, B = Benzoic acid, C = p-Nitrophenol