12 Mathematics Test Demo

7/30/2019 12 Mathematics Test Demo

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MATHEMATICS(Class 12)

Index

Chapters page

1. Relations & Functions 012. Inverse Trigonometric Functions 183. Matrices 504. Determinants 755. Continuity & Differentiability 1046. Applications of Derivatives 1367. Integrals 1788. Applications of Integrals 2079. Differential Equations 22610. Vectors 25811. Three Dimensional Geometry 27912. Linear Programming 30213. Probability 313


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CBSE TEST PAPER-01

CLASS - XII MATHEMATICS (Relations and Functions)

Topic: Relation and function

1. A Relation R:AA is said to be Reflexive if --------- for every a A where A is non [1]

empty set.

2. A Relation R:AA is said to be Symmetric if ---------- a,b, A [1]

3. A Relation R:AA is said to be Transitive if ------------- a,b,c A [1]

4. Define universal relation? Give example. [1]

5. What is trivial relation? [1]

6. Let T be the set of all triangles in a plane with R a relation in T given by [4]

R = {(T1, T2): T1 is congruent to T2}.

Show that R is an equivalence relation.

7. Show that the relation R in the set Z of integers given by [4]

R = {(a, b) : 2 divides a-b}. is equivalence relation.

8. Let L be the set of all lines in plane and R be the relation in L define if [4]

R = {(l1, L2 ): L1 is to L2 } .

Show that R is symmetric but neither reflexive nor transitive.

9. Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as [4]

R = {(a, b): b = a+1} is reflexive, symmetric or transitive.

10. Let L be the set of all lines in xy plane and R be the relation in L define as [4]

R = {(L1, L2): L1 || L2}

Show then R is on equivalence relation.

Find the set of all lines related to the line y=2x+4.

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CBSE TEST PAPER-01


[ANSWERS]

Ans 1. (a, a) R

Ans 2. (a, b) R, (b, a) R

Ans 3. (a, b)R, and (b, c)R (a, c) R.

Ans 4. A Relation R in a set A called universal relation if each element of A is related to every

element of A. Ex. Let = {2,3,4}

R = (A A) = {(2,2),(2,3) (2,4) (3,2) (3,3) (3,4) (4,2) (4,3) (4,4) }

Ans 5. Both the empty relation and the universal relation are some time called trivial relation.

Ans 6. R is reflexive, since every is congruent to itself.

(T1T2)R similarly (T2T1) R

since T1 T2

(T1T2) R, and (T2,T3) R

(T1T3)R Since three triangles are congruent to each other.

Ans 7. R is reflexive , as 2 divide a-a = 0

((a,b)R ,(a-b) is divide by 2

(b-a) is divide by 2 Hence (b,a) R hence symmetric.

Let a,b,c Z

If (a,b) R

And (b,c) R

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Then a-b and b-c is divided by 2

a-b +b-c is even

(a-c is even

(a,c) R

Hence it is transitive.

Ans 8. R is not reflexive , as a line L1 cannot be to itself i.e (L1,L1 ) R

L1 L2

L2 L1

(L2,L1)R

L1 L2 and L2 L3

Then L1 can never be to L3 in fact L1 || L3

i.e (L1,L2) R, (L2,L3) R.

But (L1, L3) R

Ans 9. R = {(a,b): b= a+1}

Symmetric or transitive

R = {(1,2) (2,3) (3,4) (4,5) (5,6) }

R is not reflective , because (1,1) R

R is not symmetric because (1,2) R but (2,1) R

(1,2) R and (2,3) R

But (1,3) R Hence it is not transitive

Ans 10. L1||L1 i.e (L1, L1) R Hence reflexive

L1||L2 then L2 ||L1 i.e (L1L2) R

(L2,L) R Hence symmetric

We know the

L1||L2 and L2||L3

Then L1|| L3

Hence Transitive . y = 2x+K When K is real number.

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CBSE TEST PAPER-02


Topic:- Relations and functions

1. Prove that the function f: R R, given by f(x) = 2x, is one one. [1]

2. State whether the function is one one, onto or bijective f: R R defined by f(x) = 1+ x2 [1]

3. Let S = {1, 2, 3} [1]

Determine whether the function f: S S defined as below have inverse.

f = {(1, 2), (2, 1), (3, 1)}

4. Find gof f(x) = |x|, g(x) = |5x + 1| [1]

5. Let f, g and h be function from R to R show that (f + g) oh = foh + goh [1]

6. If a * b = a + 3b2, then find 2 * 4 [1]

7. Show that the relation in the set R of real no. defined R = {(a, b) : a b3 }, is neither

reflexive nor symmetric nor transitive. [4]

8. Let A = N N and * be the binary operation on A define by (a, b) * (c, d) = (a + c, b + d) [4]

Show that * is commutative and associative.

9. Show that if f:7 3

5 5R R

is defining by f(x) =

3 4

5 7

x

x

+

and g:

3 7

5 5R R

is

define by g(x) =7 4

,5 3

x

x

+

then fog = IA and gof = IB when

3 7,

5 5A R B R

= =

; IA (x) = x,

for all xA, IB(x) = x, for all xB are called identify function on set A and B respectively. [4]

10. Let f: N N be defined by f(x) =

1, all n

2

2

nif n is odd for N

nif n is even

+

Examine whether the function f is onto, one one or bijective [4]

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CBSE TEST PAPER-02


[ANSWERS]

Topic:- Relations and functions

1. f is one one as f(x1) = f (x1)

2x1 = 2x2x1 = x2

Prove.

2. Let x1, x2 x

If f(x1) = f(x2)2 2

1 11 1x x+ = + 2 2

1 1

1 2

x x

x x

=

=

Hence not one one

( )

21

1

y x

x y

= +

=

( )1 1 (1 ) 2f y y y y = + =

3. f(2) = 1 f(3) = 1,

f is not one one, So that that f is not invertible.

4. gof (x) = g [f(x)]

= g [(x)]

= 5 2x

5. L.H.S = (f + g) oh= {(f + g) oh} (x)

= (f + g) h (x)

= f [h (x)] + g [h (x)]

= foh + goh

6. 2 * 4 = 2 + 3 (4)2

= 2 + 3 16

= 2 + 48

= 50

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7. (i) (a, a) 2R as a a Which is false R is not reflexive.

(ii)

2 2

anda b b a

Which is false R is not symmetric.(iii) 2 2, then 4a b b c a c Which is false

8. (i) (a, b) * (c, d) = (a + c, b + d)

= (c + a, d + b)

= (c, d) * (a, b)

Hence commutative

(ii) (a, b) * (c, d) * (e, f)

= (a + c, b + d) * (e, f)

= (a + c + e, b + d + f)

= (a, b) * (c + e, d + f)= (a, b) * (c, d) * (e, f)

Hence associative.

9. gof (x) =

3 47 4

3 4 5 7

3 45 75 3

5 7

x

x xg x

xx

x

+ +

+ + = =

++

+

7 43 4

3 4 5 3

( ) 7 45 35 7

5 3

x

x x

fog x f xxx

x

+ +

+ = = =

+

gof(x) = x, for all x B

fog (x) = x, for all x A

Thus

Which implies that gof = IBAnd Fog = IA

10.1 1 1

(1) 12 2

nf

+ += = =

2(2) 12 2

n

f = = =

f is not one one

1 has two pre images 1 and 2

Hence f is onto

f is not one one but onto.

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CBSE TEST PAPER-03


Topic:- Relations and Functions

1. Show that function f: N N, given by f(x) = 2x, is one one. [1]

2. State whether the function is one one, onto or bijective f: R R defined by f(x) = 3 4x [1]

3. Let S = {1, 2, 3} [1]


f = {(1, 1), (2, 2), (3, 3)}

4. Find got f(x) = |x|, g(x) = |5x -2| [1]

5. Consider f: {1, 2, 3} {a, b, c} given by f(1) = a, f(2) = b and f(3) = c find f-1 and show

that (f-1)-1 = f [1]

6. If f(x) = x + 7 and g(x) = x 7, xR find (fog) (7) [1]

7. Show that the relation R in the set of all books in a library of a collage given by

R ={(x, y) : x and y have same no of pages}, is an equivalence relation. [4]

8. Let * be a binary operation. Given by a * b = a b + ab [4]

Is * :

(a) Commutative

(B) Associative

9. Let f: R R be f (x) = 2x + 1 and g: R R be g(x) = x2 2 find (i) gof (ii) fog [4]

10. Let A = R {3} and B = R- {1}. Consider the function of f: A B defined by

f(x) =2

.3

x

x

is f one one and onto. [4]

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7. (i) (x, x) R, as x and x have the same no of pages for all xR R is reflexive.

(ii) (x, y) R

x and y have the same no. of pages

y and x have the same no. of pages

(y, x) R (x, y) = (y, x) R is symmetric.

(iii) if (x, y) R, (y, y) R

(x, z) R

R is transitive.

8. (i) a * b = a b + ab

b * a = b a + ab

a * b b * a

(ii) a * (b * c) = a * (b c + bc)

= a (b c + bc) + a. (b c + bc)= a b + c bc + ab ac + abc

(a * b) * c = (a b + ab) * c

= [ (a b + ab) c ] + ( a b + ab)

= a- b + ab c + ac bc + abc

a * (b * c) (a * b) * c.

9. (i) gof (x) = g[f(x)]

= g (2x + 1)= (2x + 1)2 2

(ii) fog (x) = f (fx)

= f (2x + 1)

= 2(2x + 1) + 1

= 4x + 2 + 1 = 4x + 3

10. Let x1 x2 A

Such that f(x1) = f(x2)

1 2

1 2

1 2

2 2

3 3

x x

x x

x x

=

=

f is one one

2

1 3

2 2

1

y xx

x

yx

y

=

=

3 2

1

yf y

y

=

Hence onto

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CBSE TEST PAPER-04



1. What is a bijective function? [1]

2. Let f: R R be define as f(x) = x4 check whether the given function is one one onto, [1]

or other.

3. Let S = {1, 2, 3} [1]


f = {(1, 3) (3, 2) (2, 1)}

4. Find gof where f(x) = 8x3, g(x) = x1/3 [1]

5. Let f, g and h be function from R + R. Show that (f.g) oh = (foh). (goh) [1]

6. Let * be a binary operation defined by a * b = 2a + b 3. find 3 * 4 [1]

7. Show that the relation R defined in the set A of all triangles as

R = { ( )1 2 1, :T T T is similar to T2 }, is an equivalence relation. Consider three right

angle triangles T1 with sides 3, 4, 5. T2 with sides 5, 12, 13 and T3 with sides 6, 8, 10.

Which triangles among T1, T2 and T3 are related? [4]

8. Determine which of the following operation on the set N are associative and

which are commutative. [4]

(a) a * b = 1 for all a, b N

(B) a * b =2

a b+for all a, b, N

9. Let A and B be two sets. Show that f: A B B A such that f(a, b) = (b, a) is

a bijective function. [4]

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CBSE TEST PAPER-04


[ANSWERS]


1. A function f: X Y is said to be one one and onto (bijective), if f is both one one and

onto.

2. Let x1, x2R

If f(x1) = f(x2)

4 41 2

2 2

1 1

1 2

x x

x x

x x

=

=

=

Not one one4

1/4

1/4( )

y x

x y

f y y

=

=

=

Not onto.1/4( )f y y =

3. f is one one and onto, Ao that f is invertible with f-1 = {(3,1) (2, 3) (1, 2)}

4. gof (x) = g[f(x)]

= g (8x3)

= ( )1

3 38x

= 2x

5. (f. g) oh

(f. g) h (x)f[h(x)]. g[h(x)]

foh. goh

6. 3 * 4 = 2 (3) + 4-3 = 7

7. (i) Each triangle is similar to at well and thus (T1, T1) R

R is reflexive.

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(ii) (T1, T2) R

T1 is similar to T2

T2 is similar to T1(T2, T1) R

R is symmetric

(iii) T1 is similar to T2 and T2 is similar to T3 T1 is similar to T3 (T1, T3) R R is transitive.

Hence R is equivalence

(II) part T1 = 3, 4, 5T2 = 5, 12, 13

T3 = 6, 8, 10

3 4 5 1

6 8 10 2= = = T1 is relative to T3.

8. (a) a * b = 1

b * a = 1

for all a, b N also

(a * b) * c = 1 * c = 1

a * (b * c) = a * (1) = 1 for all, a, b, c R N

Hence R is both associative and commutative

(b) a * b =2

a b+, b * a =

2

b a+

Hence commutative.

(a * b) * c = *2

a bc

+

=2

2 4

a b a b cc

+ + + + =

=2

*( * ) *2 2

a b

aa ba b c a

+ + +

= =

2

4

a b c+ +=

* is not associative.

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9. Let (a1 b1) and (a2, b2) A B

(i) f(a1 b1) = f(a2, b2)

b1 = b2 and a1 = a2

(a1 b1) = (a2, b2)

Then f(a1 b1) = f(a2, b2)

(a1 b1) = (a2, b2) for all

(a1 b1) = (a2, b2) A B

(ii) f is injective,

Let (b, a) be an arbitrary

Element of B A. then b B and a A

(a, b) ) (A B)Thus for all (b, a) B A their exists (a, b) ) (A B)

Hence that

f(a, b) = (b, a)

So f: A B B A

Is an onto function.

Hence bijective

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CBSE TEST PAPER-05



1. show that a one one function f: {1, 2, 3} {1, 2, 3} must be onto. [1]

2. f: R R be defined as f(x) = 3x check whether the function is one one onto or other [1]

3. Let S = {1, 2, 3} [1]


f = { (1, 2) (2, 1) (3, 1) }4. Find fog

f(x) = 8x3, g(x) = x1/3 [1]

5. If f: R R be given by f(x) = ( )1

3 33 x , find fof (x) [1]

6. If f(x) is an invertible function, find the inverse of f(x) =3 2

5

x [1]

7. Show that the relation R defined by (a, b) R (c, d) a + b = b + c on the set N N is an

equivalence relation. [4]

8. Let * be the binary operation on H given by a * b = L. C. M of a and b. find [4]

(a) 20 * 16

(b) Is * commutative

(c) Is * associative

(d) Find the identity of * in N.

9. If the function f: R R is given by f(x) =3

2

x +and g: R R is given by g(x) = 2x 3,

Find (i) fog (ii) gof. Is f-1 = g [4]

10. Let L be the set of all lines in Xy plane and R be the relation in L define as [4]

R = {(L1, L2): L1 || L2}

Show then R is on equivalence relation.

Find the set of all lines related to the line y=2x+4.

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CBSE TEST PAPER-05


[ANSWERS]


1. Since f is one one three element of {1, 2, 3} must be taken to 3 different element of the

co domain {1, 2, 3} under f. hence f has to be onto.

2. Let1 2,x x R

[ ]1 2 1 23 3 f(x ) = f(x )x x if=

1 2

is one - one

3

x x

f

y x

=

=

3

3

3 3

yx

y yf y

=

= =

3. f(2) = 1, f(3) =1f is not one one so that f is not invertible

Hence no inverse

4. fog (x) = f(gx)1

3

31

38

f x

x

=

=

= 8x

5. ( ) ( )

11 3

1 333 3f f x 3 3 x

=

= ( )1

3 33 3 x+

= x

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6. Let f(x) = y

1

3 2 5 2,

5 3

5 2( )

3

x yy x

yf y

+= =

+ =

7. (a, b) R (c, d) a + b = b + c where a, b, c, d N(a, b ) R (a, b) a + b = b + a (a, b) N NR is reflexive

(a, b) R (c, d) a + b = b + c (a, b ) (c, d) N N

d + a = c + b c + b = d + a (c, d) R (a, b) (a, b), (c, d) N NHence reflexive.

(a, b) R (c, d) a + d = b + c (1) (a, b), (c, d) N N(c, d) R (e, f) c + f = d + e (2) (c, d), (e, f) N NAdding (1) and (2)

(a + b) + [(+f)] = (b + c) + (d + e)

a + f = b + e

(a, b) R (e, f)

Hence transitive

So equivalence

8. (i) 20 * 16 = L. C.M of 20 and 16

= 80

4

( ) ( )

HCF

p x q xLCM

HCF

=

=

(ii) a * b = L.C.M of a and b

= L.C.M of b and a

= b * a

(iii) a * (b * c) = a * (L.C.M of b and c)= L.C.M of (a and L.C.M of b and c)

= L.C.M of a, b and c

Similarity

(a * b) * c = L. C.M of a, b, and c

(iv) a * 1 = L.C.M of a and 1

= a

Ans = 1

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9. (i) fog (x) = f [g(x)]

= f (2x 3)

=2 3 3

2

x +

= x

(ii) gof (x) = g [f(x)]

3

2

3

2 32

xg

x

+ =

+ =

= x

(iii) fog = gof = x

Yes,

10. L1||L1 i.e (L1, L1) R Hence reflexive

L1||L2 then L2 ||L1 i.e (L1L2) R

(L2,L) R Hence symmetric

We know the

L1||L2 and L2||L3

Then L1|| L3

Hence Transitive . y = 2x+K

When K is real no .

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CBSE TEST PAPER-01

CLASS - XII MATHEMATICS (Relations & Functions)

Topic:-Inverse Trigonometric Functions

1. Find the principal value of sin-11

2

[1]

2. Find the value of sin-13

sin5

[1]

3. Find the value of 1 1tan 3 co ( 3)t [1]

4. Find the value of sin ( )1 1sin cosa a +

[1]

5. tan-11

tanx x y

y x y

+ evaluate

[1]

6. Find the value of 1 1 11 1

tan (1) cos sin2 2

+ +

[4]

7. Show that 1 1 13 8 84

sin sin cos5 17 85

=

[4]

8. Prove that 1 1 1 11 1 1 1

tan tan tan tan5 7 3 8 4

+ + + =

[4]

9. Prove that

31 1 1

2 2

2 3

tan tan tan1 1 3

x x x

x x x

+ = [4]

10. Simplify1 1sin cos 3 4sin cos cos sin

5 52

x xor x x

+ +

[4]

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12 Mathematics Test Demo