| Class XI |
Mathematics |
Mathematics |
3 |
3. 2 cos x 10 1 2 2 π 2 π 3 . Example 22 Prove that cos x + cos x + +cos x − = 3 3 2 Solution We have 2π 2π 1+cos 2... |
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| Class XI |
Mathematics |
Mathematics |
14 |
CDAB, CDBA DABC, DACB, DBCA, DBAC, DCAB, DCBA} (i) Let the event ‘she visits Abefore B’ be denoted by E Therefore,E = {ABCD, CABD, DABC, ABDC, CADB... |
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| Class XI |
Mathematics |
Mathematics |
14 |
= P( )+ P ( )P (A∩E ) ... (1) Now P( )= P (∪C ) = P( )+ P ( )P B∩C( ) ... (2) Also A∩E = A∩ B∪C ( ) = (A∩B ∪ ) ( )[using distribution property of i... |
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| Class XI |
Mathematics |
Mathematics |
14 |
In a certain lottery 10,000 tickets are sold and ten equal prizes are awarded. What is the probability of not getting a prize if you buy (a) one ti... |
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| Class XI |
Mathematics |
Mathematics |
14 |
set fi Sure event: The whole sample space fi Complementary event or ‘not event’ : The set A′ or S – A fi Event A or B: The set A ∪ B fi Event A and B: ... |
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| Class XI |
Mathematics |
Mathematics |
14 |
Aleae). It was published in 1663 after his death. In 1654, a gambler Chevalier de Metre approached the well known French Philosopher and Mathematic... |
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| Class XI |
Mathematics |
Mathematics |
2 |
Chapter 2 RELATIONSANDFUNCTIONS v Mathematics is the indispensable instrument of all physical research. – BERTHELOT v 2.1 Introduction Much of math... |
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| Class XI |
Mathematics |
Mathematics |
2 |
pairs of elements from P and Q, i.e., P × Q = { (p,q) : p ∈ P, q ∈ Q } If either P or Q is the null set, then × Q will also be empty set, i.e., P ×... |
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| Class XI |
Mathematics |
Mathematics |
2 |
the position (b , a ). 2 1 Remarks (i) Twoorderedpairsareequal,ifandonlyif thecorrespondingfirstelements are equal and the second elements are also... |
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| Class XI |
Mathematics |
Mathematics |
2 |
= {(1,3), (1,4), (2,3), (2,4), (3,3), (3,4)} and (A × C) = {(1,4), (1,5), (1,6), (2,4), (2,5), (2,6), (3,4), (3,5), (3,6)} Therefore, (A × B) ∩ (A ... |
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| Class XI |
Mathematics |
Mathematics |
2 |
= {5, 4, 2}, fin× H and H × G. 4. State whether each of the following statements are true or false. If the statement is false, rewrite the given st... |
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| Class XI |
Mathematics |
Mathematics |
2 |
a subset of P × Q by introducing a relation R between the first element x and the second element y of each ordered pair Fig 2.4 (x, y) as R= { (x,y... |
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| Class XI |
Mathematics |
Mathematics |
2 |
6} and the codomain = {1, 2, 3, 4, 5, 6}. Fig 2.5 Example 8 The Fig 2.6 shows a relation between the sets Pand Q.Write this relation (i) in set-bui... |
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| Class XI |
Mathematics |
Mathematics |
2 |
A}. Write down its domain, codomain and range. Reprint 2026-27 30 MATHEMATICS 2. Define a relation R on the set N of natural numbers by R = {(x, y)... |
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| Class XI |
Mathematics |
Mathematics |
2 |
many terms such as ‘map’ or ‘mapping’ used to denote a function. Definition 5 A relation f from a set A to a set B is said to be a function if ever... |
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| Class XI |
Mathematics |
Mathematics |
2 |
each of the following relations given below and state in each case, giving reasons whether it is a function or not? (i) R={(2,1),(3,1),(4,2)}, (ii)... |
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| Class XI |
Mathematics |
Mathematics |
2 |
Fig 2.8 (ii) Constant function Define the function f: R → R by y = f (x) = c, x ∈ R where c is a constant and each x ∈ R. Here domain of f is R and... |
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| Class XI |
Mathematics |
Mathematics |
2 |
f(–2) = –8, f(3) = 27; f(–3) = –27, etc. Therefore, f = {(x,x ): x∈R}. The graph of f is given in Fig 2.11. Fig 2.11 f (x) (iv) Rational functions ... |
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| Class XI |
Mathematics |
Mathematics |
2 |
by the Fig 2.14. Fig 2.14 (vii) Greatest integer function The function f: R → R defined by f(x) = [x], x∈ R assumes the value of the greatest integ... |
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| Class XI |
Mathematics |
Mathematics |
2 |
(iv) MultiplicationoftworealfunctionsTheproduct(ormultiplication)oftworeal functions f:X → R and g:X → R is a function fg:X → R defined by (fg) (x)... |
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| Class XI |
Mathematics |
Mathematics |
2 |
= – x (ii) f(x) = 9− x . 3. A function f is defined by f(x) = 2x –5. Write down the values of (i) f (0), (ii) f (7), (iii) f (–3). 4. The function ... |
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| Class XI |
Mathematics |
Mathematics |
2 |
follows that (a, a) ∈ R. (ii) (a,b) ∈ R implies that a – b ∈ Z. So, b – a ∈ Z. Therefore, (b, a) ∈ R (iii) (a, b) and (, c) ∈ R implies that a – b ... |
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| Class XI |
Mathematics |
Mathematics |
2 |
, 0≤ x ≤ 2 The relation g is defined by 3x , 2≤ x ≤10 Show that f is a function and g is not a function. 2. If f (x) = x , find f (1.1) – f (.) ... |
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| Class XI |
Mathematics |
Mathematics |
2 |
= {(ab, a + b) : a, b ∈ Z}. Is f a function from Z to Z? Justify your answer. 12. LetA= {9,10,11,12,13} and let f :A → N be defined by f (n) = the ... |
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| Class XI |
Mathematics |
Mathematics |
2 |
image y in set B. We write f: A → B, where f(x) = y. fi A is the domain and B is the codomain of f. Reprint 2026-27 42 MATHEMATICS fi The range of th... |
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