| Class XII |
Mathematics |
Mathematics Part-II |
5 |
But point B is common to both AB and BC. Therefore, A, B, C are collinearpoints. EXERCISE11.1 1. Ifalinemakesangles90°,135°,45°withthe x,yandz-axes... |
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| Class XII |
Mathematics |
Mathematics Part-II |
5 |
the parallel to the line. Here, b should not be confused with | | . Derivation of cartesian form from vector form Let the coordinates of the given ... |
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| Class XII |
Mathematics |
Mathematics Part-II |
5 |
Now, is the position vector of any point P(x, y, z) on the line. Therefore, xi + y j + z k ˆ = 5i +2 j−4k+ λ ( 3i + 2 j − 8 k) ˆ = (5+3λ)i +(2+ 2λ)... |
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| Class XII |
Mathematics |
Mathematics Part-II |
5 |
2 2 2 2 ... (2) a1+ b + 1 1 a2+ b + 2 2 Reprint 2026-27 384 MATHEMATICS Note In case the lines L and L do not pass through the origin, we may take ... |
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| Class XII |
Mathematics |
Mathematics Part-II |
5 |
= ... (2) 2 b 2 c 2 where,a ,b c anda b ,c arethedirectionratiosofthelines(1)and(2),respectively, 1 1, 1 2, 2 2 then a a +bb +c c cos θ = 1 2 1 2 1... |
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| Class XII |
Mathematics |
Mathematics Part-II |
5 |
then the shortest distance between them will be the perpendicular distance, i.e. the length of the perpendicular drawn from a pointononelineontothe... |
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| Class XII |
Mathematics |
Mathematics Part-II |
5 |
| r r d n⋅( a 2 − a1) uur r r = (since ST = a 2− a1) d ST r r r r = (b1 × b2r⋅(ar2 − a1) [From (3)] ST b × b 1 2 Reprint 2026-27 THREE DIMENSIONAL ... |
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| Class XII |
Mathematics |
Mathematics Part-II |
5 |
r r i.e., |b × (a2 − a1)| = |b |PT⋅1 (as |n | = 1) Hence, the distance between the given parallel lines is d = Example 9 Find the shortest distance... |
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| Class XII |
Mathematics |
Mathematics Part-II |
5 |
+ 3 j − 5 k + µ ( 2 i +3 j + 6 k )ˆ Reprint 2026-27 THREE DIMENSIONAL GEOMETRY 389 Solution The two lines are parallel (Why? ) We have r a1= i + 2 ... |
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| Class XII |
Mathematics |
Mathematics Part-II |
5 |
line is = = . Write its vector form. 3 7 2 8. Findtheanglebetweenthefollowingpairsoflines: r (i) r = 2i − 5 j + k + λ(3i + 2 j + 6k) and r = 7i − 6... |
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| Class XII |
Mathematics |
Mathematics Part-II |
5 |
µ (2 i + 3 j + k) 15. Find the shortest distance between the lines whose vector equations are r r = (1− t) i + (t − 2) j + (3− 2 t) k r ˆ ˆ ˆ r = (... |
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| Class XII |
Mathematics |
Mathematics Part-II |
5 |
z 2 1, 2 1, 2 1 PQ PQ PQ where PQ = (x − x ) + (y − y ) + 2 (z − z )2 2 1 2 1 2 1 ® Direction ratios of a line are the numbers which are proportion... |
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| Class XII |
Mathematics |
Mathematics Part-II |
5 |
z − z 1= 1 = 1 l m n Thevectorequationofalinewhichpassesthroughtwopointswhoseposition ® r vectors are and isr = a + λ (b − a)r . r r r r r r ® If θ... |
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| Class XII |
Mathematics |
Mathematics Part-II |
5 |
r r b × (a 2− a1) r |b | — — ❖ Reprint 2026-27 |
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| Class XII |
Mathematics |
Mathematics Part-II |
6 |
394 MATHEMATICS Chapter 12 LINEARPROGRAMMING ❖ The mathematical experience of the student is incomplete if he never had the opportunity to solve a ... |
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| Class XII |
Mathematics |
Mathematics Part-II |
6 |
Rs50,000, he can buy 50000 ÷ 500, i.e. 100 chairs. But he can store only 60 pieces. Therefore, he is forced to buy only 60 chairs which will give h... |
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| Class XII |
Mathematics |
Mathematics Part-II |
6 |
function subject to certain conditions determinedbyasetoflinearinequalitieswithvariablesasnon-negative.Suchproblems are called Linear Programming P... |
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| Class XII |
Mathematics |
Mathematics Part-II |
6 |
y ≤ 60 ... (2) x ≥ 0 ... (3) y ≥ 0 ... (4) Thegraphofthissystem(shadedregion)consistsofthepointscommontoallhalf planes determined by the inequaliti... |
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| Class XII |
Mathematics |
Mathematics Part-II |
6 |
When Z has an optimal value (maximumorminimum),wherethevariablesxandyaresubjecttoconstraintsdescribed by linear inequalities, this optimal value mu... |
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| Class XII |
Mathematics |
Mathematics Part-II |
6 |
points. 3. (i) When the feasible region is bounded, M and m are the maximum and minimumvaluesofZ. (ii) In case, the feasible region is unbounded, w... |
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| Class XII |
Mathematics |
Mathematics Part-II |
6 |
3000 Fig 12.3 Reprint 2026-27 LINEAR PROGRAMMING 401 A, B and C are (0,5), (4,3) and (0,6) respectively. Now we evaluate Z = 200x + 500y at these p... |
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| Class XII |
Mathematics |
Mathematics Part-II |
6 |
feasible region is unbounded. We now evaluate Z at the corner points. Corner Point Z = – 50x + 20y (0, 5) 100 (0, 3) 60 (1, 0) –50 (6, 0) –300 ← sm... |
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| Class XII |
Mathematics |
Mathematics Part-II |
6 |
you can see that there is no point satisfying all the constraints simultaneously. Thus, the problemishavingnofeasibleregionand hencenofeasiblesolut... |
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| Class XII |
Mathematics |
Mathematics Part-II |
6 |
y ≤ 0, 2x + y ≤ 200; x, y ≥ 0. 9. Maximise Z = – x + 2y, subject to the constraints: x ≥ 3, x + y ≥ 5, x + 2y ≥ 6, y ≥ 0. 10. Maximise Z = x + y, s... |
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| Class XII |
Mathematics |
Mathematics Part-II |
7 |
406 MATHEMATICS Chapter 13 PROBABILITY ❖ The theory of probabilities is simply the Science of logic quantitatively treated. – C.S. PEIRCE ❖ 13.1 In... |
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