| 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 |
= {(1,4), (1,5), (1,6), (2,4), (2,5), (2,6), (3,4), (3,5), (3,6)} Therefore, (A × B) ∩ (A × C) = {(1, 4), (2, 4), (3, 4)}. (iii)Since, (B ∪ C) = {3... |
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| Class XI |
Mathematics |
Mathematics |
2 |
State whether each of the following statements are true or false. If the statement is false, rewrite the given statement correctly. (i) If P = {m, ... |
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| Class XI |
Mathematics |
Mathematics |
2 |
between the first element x and the second element y of each ordered pair Fig 2.4 (x, y) as R= { (x,y): x is the first letter of the name y, x ∈ P,... |
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| Class XI |
Mathematics |
Mathematics |
2 |
Fig 2.5 Example 8 The Fig 2.6 shows a relation between the sets Pand Q.Write this relation (i) in set-builder form, (ii) in roster form. Whatisitsd... |
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| Class XI |
Mathematics |
Mathematics |
2 |
2. Define a relation R on the set N of natural numbers by R = {(x, y) : y = x + 5, x is a natural number less than 4; x, y ∈N}. Depict this relatio... |
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| Class XI |
Mathematics |
Mathematics |
2 |
Definition 5 A relation f from a set A to a set B is said to be a function if every element of set Ahas one and only one image in set B. In other w... |
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| Class XI |
Mathematics |
Mathematics |
2 |
giving reasons whether it is a function or not? (i) R={(2,1),(3,1),(4,2)}, (ii)R={(2,2),(2,4),(3,3),(4,4)} (iii) R={(1,2),(2,3),(3,4),(4,5),(5,6),(... |
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| Class XI |
Mathematics |
Mathematics |
2 |
by y = f (x) = c, x ∈ R where c is a constant and each x ∈ R. Here domain of f is R and its range is {c}. Fig 2.9 Reprint 2026-27 RELATIONS AND FUN... |
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| Class XI |
Mathematics |
Mathematics |
2 |
= {(x,x ): x∈R}. The graph of f is given in Fig 2.11. Fig 2.11 f (x) (iv) Rational functions are functions of the type , where f(x) and g(x) are g(... |
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| Class XI |
Mathematics |
Mathematics |
2 |
f: R → R defined by f(x) = [x], x∈ R assumes the value of the greatest integer, less thanorequaltox.Suchafunction is called the greatest integer fu... |
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| Class XI |
Mathematics |
Mathematics |
2 |
function fg:X → R defined by (fg) (x) = f(x) g(x), for all x ∈ X. This is also called pointwise multiplication. (v) Quotient of two real functions ... |
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| Class XI |
Mathematics |
Mathematics |
2 |
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 ‘t’ which maps temperature in ... |
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| Class XI |
Mathematics |
Mathematics |
2 |
that a – b ∈ Z. So, b – a ∈ Z. Therefore, (b, a) ∈ R (iii) (a, b) and (, c) ∈ R implies that a – b ∈ Z. b – c ∈ Z. So, a – c = (a – b) + (b – c) ∈ ... |
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| Class XI |
Mathematics |
Mathematics |
2 |
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 (.) (1.1–1) x + 2x+1 3. Find the domain of t... |
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| Class XI |
Mathematics |
Mathematics |
2 |
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 highest prime factor of n. Find... |
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| Class XI |
Mathematics |
Mathematics |
3 |
Chapter 3 TRIGONOMETRICFUNCTIONS v A mathematician knows how to solve a problem, he can not solve it. – MILNE v 3.1 Introduction The word ‘trigonom... |
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| Class XI |
Mathematics |
Mathematics |
3 |
of measurement of an angle which are most commonly used, viz. degree measure and radian measure. 1 th 3.2.1 Degree measure Ifarotationfromtheinitia... |
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| Class XI |
Mathematics |
Mathematics |
3 |
as initial side of an angle. Thenthelength ofanarcofthecirclewillgivetheradian 1 measure of the angle which the arc will subtend at the centre of t... |
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| Class XI |
Mathematics |
Mathematics |
3 |
π Radian measure = 180 × Degree measure Degree measure = π ×Radian measure Example 1 Convert 40° 20′ into radian measure. Solution We know that 180... |
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| Class XI |
Mathematics |
Mathematics |
3 |
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 + 1+co... |
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| Class XI |
Mathematics |
Mathematics |
3 |
radian π 22π and θ 2 110° = ×110 = radian 180 36 Let l be the length of each of the arc. Then l = r θ = r θ , which gives 1 1 2 2 13π 22π 1 22 × r1... |
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| Class XI |
Mathematics |
Mathematics |
3 |
2 2 OM + MP = OP or a + b = 1 Thus, for every point on the unit circle, we have a + b = 1 or cos x + sin x = 1 Since one complete revolution subten... |
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| Class XI |
Mathematics |
Mathematics |
3 |
n is any integer. sinx 1 π sec x = cosx , x ≠ (2n + 12 , where n is any integer. sinx π tan x = , x ≠ (2n +12 , where n is any integer. cosx cosx c... |
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| Class XI |
Mathematics |
Mathematics |
3 |
xb are both positive, in the π second quadrant ( < x <π) a is negative and b is positive, in the third quadrant 3π 3π (π < x < ) a and b are both n... |
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