Q83.If the sum of squares of all real values of α, for which the lines 2x - y + 3 = 0, 6x + 3y + 1 = 0 and αx + 2y - 2 = 0 do not form a triangle is p, then the greatest integer less than or equal to p is ________.
What This Question Tests
This question requires identifying the conditions under which three given lines do not form a triangle (i.e., they are parallel or concurrent) and solving for the possible values of 'α'.
Concepts Tested
Formulas Used
Slope m1 = m2 for parallel lines
Condition for collinearity of 3 lines by determinant: |x1 y1 1; x2 y2 1; x3 y3 1| = 0 (using intersection points) or det of coeff matrix
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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. 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📋 Question Details
- Chapter
- Straight Lines
- Topic
- Conditions for lines not forming a triangle
- Year
- 2024
- Shift
- 27 Jan Shift 2
- Q Number
- Q83
- Type
- Numerical
- NCERT Ref
- Class 11 Mathematics Ch 10: Straight Lines
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