Q88.If S = {a ∈R : |2a −1| = 3[a] + 2{a}} , where [t] denotes the greatest integer less than or equal to t and {t} represents the fractional part of t , then 72 ∑a∈S a is equal to ______
What This Question Tests
This problem requires a strong understanding of the properties of the greatest integer function and fractional part function, substituting t = [t] + {t} to simplify the equation and solve for 'a' in different integer intervals.
Concepts Tested
Formulas Used
t = [t] + {t}
📚 NCERT Sections This Tests
8.17 — Complete Each Synthesis By Giving Missing Starting Material, Reagent Or Products
Chemistry Class 12 · Chapter 8
8.17 Complete each synthesis by giving missing starting material, reagent or products
8.3 — Structural Representations Similarly, Ch3Ch2Ch2Ch2Ch2Ch2Ch2Ch3
Chemistry Class 12 · Chapter 8
8.3 STRUCTURAL RepresenTATIONS Similarly, CH3CH2CH2CH2CH2CH2CH2CH3 OF organic COMPOUNDs can be further condensed to CH3(CH2)6CH3. 8.3.1 Complete, Condensed and Bond-line For further simplification, organic chemists Structural Formulas use another way of representing the structures, in which only lines are used.Structures of organic compounds are In this bond-line structural representationrepresented in several ways. The Lewis of organic compounds, carbon andstructure or dot structure, dash structure, hydrogen atoms are not shown and thecondensed structure and bond line structural lines representing carbon-carbon bonds areformulas are some of the specific types. The drawn in a zig-zag fashion. The only atomsLewis structures, however, can be simplified specifically written are oxygen, chlorine,by representing the two-electron covalent nitrogen etc. The terminals denote methylbond by a dash (–). Such a structural formula (–CH3) groups (unless indicated otherwise byfocuses on the electrons involved in bond a functional group), while the line junctionsformation. A single dash represents a single denote carbon atoms bonded to appropriatebond, double dash is used for double bond number of hydrogens required to satisfy theand a triple dash represents triple bond. Lone- valency of the carbon atoms. Some of thepairs of electrons on heteroatoms (e.g., oxygen, examples are represented as follows:nitrogen, sulphur, halogens etc.) may or may (i) 3-Methyloctane can be represented innot be shown. Thus, ethane (C2H6), ethene various forms as:(C2H4), ethyne (C2H2) and methanol (CH3OH) can be represented by the following structural (a) CH3CH2CHCH2CH2CH2CH2CH3 formulas. Such structural representations are | called complete structural formulas. CH3 (b) Ethane Ethene (c) Ethyne Methanol These structural formulas can be further abbreviated by omitting some or all of the dashes representing covalent bonds and by (ii) Various ways of representing 2-bromo indicating the number of identical groups butane are: attached to an atom by a subscript. The resulting expression of the compound is called a condensed structural formula. Thus, (a) CH3CHBrCH2CH3 (b)ethane, ethene, ethyne and methanol can be written as: CH3CH3 H2C=CH2 HC≡CH CH3OH (c) Ethane Ethene Ethyne Methanol Reprint 2025-26 organic chemistry – some basic principles and techniques 259 In cyclic compounds, the bond-line formulas may be given as follows: (b) Solution Condensed formula: Cyclopropane (a) HO(CH2)3CH(CH3)CH(CH3)2 (b) HOCH(CN)2 Bond-line formula: (a) Cyclopentane (b) chlorocyclohexane Problem 8.6 Problem 8.4 Expand each of the following bond-line Expand each of the following condensed formulas to show all the atoms including formulas into their complete structural carbon and hydrogen formulas. (a) (a) CH3CH2COCH2CH3 (b) CH3CH=CH(CH2)3CH3 Solution (b) (a) (c) (b) (d) Solution Problem 8.5 For each of the following compounds, write a condensed formula and also their bond-line formula. (a) HOCH2CH2CH2CH(CH3)CH(CH3)CH3 Reprint 2025-26 260 chemistry Molecular Models Molecular models are physical devices that are used for a better visualisation and perception of three-dimensional shapes of organic molecules. These are made of wood, plastic or metal and are commercially available. Commonly three types of molecular models are used: (1) Framework model, (2) Ball-and-stick model, and (3) Space filling model. In the framework model only the bonds connecting the atoms of a molecule and not the atoms themselves are shown. This model emphasizes the pattern of bonds of a molecule while ignoring the size of atoms. In the ball-and-stick model, both the atoms and the bonds are shown. Balls represent atoms and the stick denotes a bond. Compounds containing C=C (e.g., ethene) can best be represented by using8.3.2 Three-Dimensional springs in place of sticks. These models are Representation of Organic referred to as ball-and-spring model. The Molecules space-filling model emphasises the relative The three-dimensional (3-D) structure of size of each atom based on its van der Waals organic molecules can be represented on radius. Bonds are not shown in this model. paper by using certain conventions. For It conveys the volume occupied by each atom in the molecule. In addition to these models,example, by using solid ( ) and dashed computer graphics can also be used for( ) wedge formula, the 3-D image of a molecular modelling. molecule from a two-dimensional picture can be perceived. In these formulas the solid-wedge is used to indicate a bond projecting out of the plane of paper, towards the observer. The dashed-wedge is used to depict the bond projecting out of the plane of the paper and away from the observer. Wedges are shown in such a way that the broad end of the wedge is towards the observer. The bonds Framework model Ball and stick model lying in plane of the paper are depicted by using a normal line (—). 3-D representation of methane molecule on paper has been shown in Fig. 8.1 Space filling model Fig. 8.2 Fig. 8.1 Wedge-and-dash representation of CH4 Reprint 2025-26 organic chemistry – some basic principles and techniques 261
12.5 — A Hydrogen Atom Initially In The Ground Level Absorbs A Photon,
Physics Class 12 · Chapter 12
12.5 A hydrogen atom initially in the ground level absorbs a photon, which excites it to the n = 4 level. Determine the wavelength and frequency of photon.
📋 Question Details
- Chapter
- Sets Relations Functions
- Topic
- Greatest Integer and Fractional Part Functions
- Year
- 2024
- Shift
- 05 Apr Shift 1
- Q Number
- Q88
- Type
- Numerical
- NCERT Ref
- Class 11 Mathematics Ch 2: Relations and Functions
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