Matric 10 Yrs

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Compiled by: Faizan Ahmed math.pgseducation.com

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X MATHEMATICS

10 YEARS PAST PAPERS

S E T S

• If U={x|x�ℕ, x≤10}, A = {2, 4, 6, 8, 10 } and B = {3, 6, 9, 10}, then prove that (AUB)’=A’∩B’.

(2012)

• If A = {1, 2, 3, 4} and B = {2, 4, 6, 8}, then show that (AUB)−(A∩B) = A∆∆∆∆B. (2011)

• If A = {1, 2, 3, 5, 6} and B = {2, 4, 6, 7}, find A∆∆∆∆B. (2010)

• If {{{{ }}}} {{{{ }}}} {{{{ }}}}/ ^ 10 1, 3,5,7 1,5,6,8U x x N x A and B= ∈ ≤ = == ∈ ≤ = == ∈ ≤ = == ∈ ≤ = = Prove that (((( )))) ' ' 'A B A B====I UI UI UI U (2009)

• If {{{{ }}}} {{{{ }}}} {{{{ }}}}/ ^ 20 , 2,4,6, .....20 1, 3,5, ....,19U x x N x A and B= ∈ ≤ = == ∈ ≤ = == ∈ ≤ = == ∈ ≤ = = verify that

(((( )))) ' ' 'A B A B====I UI UI UI U (2008)

• If {{{{ }}}} {{{{ }}}} {{{{ }}}}/ ^ 12 2,4,6,8,10,12 3,6,9,12U x x N x A and B= ∈ ≤ = == ∈ ≤ = == ∈ ≤ = == ∈ ≤ = = Prove that

(((( )))) ' ' 'A B A B====I UI UI UI U (2007, 2004)

• If {{{{ }}}} {{{{ }}}} {{{{ }}}}/ ^1 12 1,2,5,9 2,3,8,9,10U x x N x A B= ∈ ≤ ≤ = == ∈ ≤ ≤ = == ∈ ≤ ≤ = == ∈ ≤ ≤ = = Prove that ' 'A BIIII (2005)

• If {{{{ }}}} {{{{ }}}} {{{{ }}}}/ ^ 10 2,3,5,7 2,4,6,8,10U x x N x A B= ∈ ≤ = == ∈ ≤ = == ∈ ≤ = == ∈ ≤ = = verify that (((( )))) ' ' 'A B A B====U IU IU IU I

(2006, 2002)

• If {{{{ }}}} {{{{ }}}} {{{{ }}}}1,2,3,4,5,6 , 1,2 2,4,6U A and B= = == = == = == = = Prove that (((( )))) ' ' 'A B A B====I UI UI UI U (2003)

• If {{{{ }}}} {{{{ }}}}/ ^1 12 , 2,4,6,8,10,12U x x N x A= ∈ ≤ ≤ == ∈ ≤ ≤ == ∈ ≤ ≤ == ∈ ≤ ≤ = and {{{{ }}}}2,3,5,7,11B ==== then find ' 'A BUUUU

(2001)

• {{{{ }}}} {{{{ }}}}/ ^ 10 2,4,6,8U x x N x A= ∈ ≤ == ∈ ≤ == ∈ ≤ == ∈ ≤ = and {{{{ }}}}2,3,5,7B ==== Verify that (((( )))) ' ' 'A B A B====U IU IU IU I (2000)

• If {{{{ }}}}0,1A ==== and {{{{ }}}}1,2B ==== prove that A B B A× ≠ ×× ≠ ×× ≠ ×× ≠ × (2003)

• If {{{{ }}}}2, 3,4A ==== and {{{{ }}}},B a b==== find A B×××× (2009)

• If (((( )))), ,A a b c==== find (((( ))))P A . (2007, 2004)

• Find P (B) when {{{{ }}}}, ,B x y z==== (2008, 2009)

• If {{{{ }}}} {{{{ }}}}, 2,3A a b B= == == == = and {{{{ }}}}3,4C ==== Find the value of (((( ))))A B C×××× UUUU (2006)

• If {{{{ }}}} {{{{ }}}}, 8,9A a b B= == == == = and {{{{ }}}}9,10C ==== then find (((( ))))B C A××××UUUU (2001)

• If (((( )))) (((( ))))10, 4 2 3 ,2 2x y x y+ − − = − ++ − − = − ++ − − = − ++ − − = − + Find the values of x and y (2005)

• If {{{{ }}}}1,2,3S ==== then find P(S). (2002)

B L A N K S

• A�B= ________: (AUB, A∩B, (A∩B) - (AUB), (AUB) - (A∩B) )

• (A’)’ = _______: (A, A’, Ø, U)

• If R={(1,2), (2,3), (3,4)} Domain R = ______: ({1}, {1,2}, {1,2,3}, {2,3})

• {0, 1, 2, . . .} is the set of ______: (Prime number, Even number, whole number, odd number)

• The null set is considered to be a SUBSET of every set.

• {{{{ }}}}0,1,2, 3.... is a set of WHOLE NUMBERS.

• If ‘a’ is a real number then the point (0, a) lies on Y-AXIS.

• If a relation is given by (((( )))) (((( )))) (((( )))){{{{ }}}}0,1 , 1, 2 , 3,4R ==== then Range is {{{{ }}}}1,2,4 .

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• The Y coordinate of every point on x axis−−−− is zero.

• If {{{{ }}}}, , ,A a b c d==== then P(A) has 16 elements.

• If a set has 3 elements the number of its possible subsets is 8.

• If set A has 2 elements and set B has 3 elements then A B×××× has 6 elements.

• The ordered pair (((( ))))3, 4−−−− lies in IV quadrant.

• If the elements of P(A) are 16 then the number of elements of set 4A ==== .

S Y S T E M O F R E A L N U M B E R S ,

E X P O N E N T S A N D R A D I C A L S

• Simplify:

l m m n n ll m n

m n l

x x x

x x x

+ + ++ + ++ + ++ + +

× ×× ×× ×× ×

(2012)

• If P=3+2√, find the value of � + � (2012)

• Simplify: � ��

��

�� (2011)

• Simplify:

l m m n n ll m n

m n l

x x x

x x x

+ + ++ + ++ + ++ + +

× ×× ×× ×× ×

(2010)

• Simplify:

2 2 2a b c

a b b c c a

x x x

x x x+ + ++ + ++ + ++ + +

(2008, 2007)

• If 2 3x = += += += + , find the value of 1

xx

++++ (2007, 2003)

• Simplify:

(((( ))))4 .

a b b ca b

a cc a

b c

x xx x

x x

+ ++ ++ ++ +−−−−

÷÷÷÷

(2006, 2009)

• (((( ))))(((( ))))

12 3

2

125 8

64

××××

(2005, 2004)

• 2 2 2

p q q r r pp q r

p q q r r p

x x x

x x x

+ + ++ + ++ + ++ + +

+ + ++ + ++ + ++ + +

(2003)

• (((( )))) (((( ))))

(((( ))))

2 /3 /6

/2

27 8

18

n n

n

−−−−

−−−−

×××× (2002)

• 2 2 2

a b b c c aa b c

a b b c c a

x x x

x x x

− − −− − −− − −− − −

− − −− − −− − −− − −

(2001)

•

2 2 2 2 2 2a ab b b bc c c ca aa b c

b c a

x x x

x x x

+ + + + + ++ + + + + ++ + + + + ++ + + + + +

(2000)

B L A N K S

• [-1(-1)5]2 = _________ (-1, 1, 0,2)

• Multiplicative inverse of (� − �) is: (a+b, -a+b, ��,

��)

• If 22 3x then x= + == + == + == + =

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• The additive inverse of 1

a b− −− −− −− − is

1

a b++++.

• The additive inverse of a b−−−− is b a−−−− .

• (((( )))) (((( ))))7 2 2 7 2 2− + =− + =− + =− + =

• (((( )))) (((( ))))2 2 7 2 2 7+ − =+ − =+ − =+ − =

• 9 85 5 5÷ =÷ =÷ =÷ =

• ππππ is an irrational number.

• (((( ))))1/249 121 77× =× =× =× =

• 1 1

3 28 36 8× =× =× =× =

• 1 1

15 15o

x====

• Degree of polynomial 2 3 2 5 2 7

8 5x y x y x y− −− −− −− − is

• 1a a a

−−−−× =× =× =× =

L O G A R I T H M S

• With the help of log tables, find the value of �.��

(�.�)×(�.�� ). (2012)

• With the help of log tables, find the value of !.�" ×#�.�( !. �) . (2011)

• With the help of log tables, find the value of 85.7 2.47

8.89

××××. (2010)

• Find the value of (((( )))) (((( ))))286.2 37.37

591

×××× with the help of log tables. (2009, 2002)

• (((( )))) (((( ))))

0.87

28.9 0.785 (2008)

• 82.2 88.6

2.25

×××× (2007)

• (((( ))))2780.6 3.0

4.0 (2006)

• (((( )))) (((( ))))

13

3

9310

1.08 62.4

(2005)

• (((( )))) (((( ))))

48.7

83.8 3.14 (2004)

• Solve (((( )))) (((( ))))20.96 87.5

4850 (2003, 2001)

• Solve (((( )))) (((( ))))20.96 87.5

4850 (2000)

B L A N K S

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• $%&# � = − " , ()*+� = ________ ( � , �,

! ,

�)

• $%& � �� = ,, ()*+, = ________ ( �, ", , �) • $%&� �",

- = ________ (.). $%&� �" + $%&� , − $%&� - (/)." $%&� � + $%&� , − $%&� -

(C). " $%&� � − $%&� , + $%&� - (D). " $%&� �� $%&� -

$%&� ,

• If 81

log 9 x then x= == == == = • If log 32 5x

then x= == == == =

• If 3481

log x then x= − == − == − == − = • If log 81 4x

then x====

• If 238

log x then x= == == == = • If log 49 2x

then x= == == == =

• The characteristic of 0.00396 is –3. • log 3 + log 6 – log 2 = log 9

• If log 36 2x

then x= == == == =

A L G E B R A I C E X P R E S S I O N S

• None. (2012)

• Find the value of �" − �" when � −

� = #. (2011)

• If a + b = 5 and a – b = 3, find the value of a2 + b

2. (2010)

• If 9, 20a b ab+ = =+ = =+ = =+ = = then find the value of 2 2a b++++ . (2009, 2004)

• If 9a b c+ + =+ + =+ + =+ + = and 2 2 2 29a b c+ + =+ + =+ + =+ + = then find the value of ab bc ca+ ++ ++ ++ + . (2008)

• Find 2 2a b++++ when 7a b+ =+ =+ =+ = and 12ab ==== . (2007, 2003)

• If 8a b+ =+ =+ =+ = when 2a b− =− =− =− = find the values of 2 2a b and ab++++ . (2005)

• Find a b++++ when 5 21a b and ab− = =− = =− = =− = = . (2001)

• Find 33

1xx

==== when 1 4xx

− = −− = −− = −− = − . (2006, 2002)

• Find the value of 2

2

1 16x when x

x x+ + =+ + =+ + =+ + = (2000)

B L A N K S

• Degree of the polynomial x2+xy

2+y is: (2,3,4,1)

• x2+64 can be made a perfect square by adding _________.

(a) 4x2 (b) 8x

2 (c) 2x

2 (d) 16x

2

• (x-4)(x-6) = __________

(a) x2+10x-24 (b) x

2-10x-24 (c) x

2+10x+24 (d) x

2-10x+24

• If a+b=2 and a−b=2 then value of a2+b

2 is:

(a) -1 (b) 2 (c) 4 (d) "

• (((( )))) (((( ))))2 24a b a b ab+ − − =+ − − =+ − − =+ − − = . • (((( )))) (((( )))) (((( ))))2 2 2 22a b a b a b+ + − = ++ + − = ++ + − = ++ + − = + .

F A C T O R I Z A T I O N

• Resolve the following into factors: r2(s-t)+s

2(t-r)+t

2(r-s) (2012)

x6-64, (ab+cd)

2−(ac-bd)2, x

2+15x−100 (2012)

• 18x2+9x−20, 27x

3-1+8y

6+18xy

2: (2011)

• Resolve the following into factors: (2010)

(((( )))) (((( )))) (((( ))))x y z y z x z x y2 2 2− + − + −− + − + −− + − + −− + − + −

• 210 17 6x x− +− +− +− + . (2009)

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• 2ab amx bx mx+ − −+ − −+ − −+ − − . (2009, 2005, 2001, 2000)

• 2 216 64x y xy− +− +− +− + . (2009)

• 6 6a b−−−− . (2009)

• 4 211 1x x− +− +− +− + . (2009)

• 2 214 15 9x xy y− −− −− −− − . (2003)

• 3 22x x− +− +− +− + . (2011, 2009, 2007, 2006, 2004)

• 2x yz xy xz− + −− + −− + −− + − . (2008, 2007, 2004)

• 24 5 21x x+ −+ −+ −+ − . (2008)

• 44a ++++ . (2011, 2008)

• (((( ))))2 21 2ab a b+ − ++ − ++ − ++ − + . (2008, 2007, 2004)

• 3 32 8x x y y− − +− − +− − +− − + . (2008, 2001)

• 3 3 327 9a b c abc− − −− − −− − −− − − . (2008)

• 212 5 2x x+ −+ −+ −+ − . (2007, 2004)

• 4 64x ++++ . ( 2007, 2006, 2004, 2002)

• 3 3

8 1 6x y xy− + +− + +− + +− + + . (2007, 2004)

• 3 2 2ax ax bx bx cx c+ − − − −+ − − − −+ − − − −+ − − − − . (2006)

• 2 216 64x y xy− +− +− +− + . (2006)

• 2 32 8x x y y− − +− − +− − +− − + . (2006)

• 3 6 227 64 1 36a b ab− − −− − −− − −− − − . (2006)

• 2 212 17 5a ab b− −− −− −− − . (2005)

• 264 25 80m m+ ++ ++ ++ + . (2005)

• 7 5 3m n n m−−−− . (2005)

• 8 4 1x x+ ++ ++ ++ + . (2005, 2012)

• 3 327 8 1 18a b ab− + +− + +− + +− + + . (2005)

• 2 216 40 25a ab b− +− +− +− + . (2003)

• 3 327 64x y++++ . (2003)

• 44 1y ++++ . (2003)

• 2 2

4 4x x y+ + −+ + −+ + −+ + − . (2003)

• 3 3 327 8 18a b c abc+ + −+ + −+ + −+ + − . (2003)

• 2 2 2 2 2 2 2 26 12 9 8a x b y b x a y+ − −+ − −+ − −+ − − . (2002)

• 3 3 38 6x y z xyz− − −− − −− − −− − − . (2002)

• 22 15x x+ −+ −+ −+ − . (2002)

• 2 225 9 30a b ab+ −+ −+ −+ − . (2002)

• 2 7x y xy−−−− . (2002)

• 2 2

25 4 20a b ab+ −+ −+ −+ − . (2001)

• 3 38 1 6a b ab+ + −+ + −+ + −+ + − . (2001)

• 23 7 6a a− −− −− −− − . (2001)

• 4

4 81y ++++ . (2001)

• 2 216 40 25a ab b− +− +− +− + . (2000)

• 3 327 8x y++++ . (2000)

• 2 22 1x x y+ + −+ + −+ + −+ + − . (2000)

• 3 3 38 6a b c abc+ + −+ + −+ + −+ + − . (2000)

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• 2 213 30x xy y− +− +− +− + . (2000)

• For what values of a and b, x4+4x

3+10x

2+ax+b will be a perfect square? (2012)

• For what values of ‘q’, 4x4+12x

3+25x

2+24x+q will be a perfect square? (2011)

• What should be added to 4 3 24 10 14 5x x x x+ + + ++ + + ++ + + ++ + + + to make it a perfect square? (2009)

• What should be subtracted from 4 3 22 3 2x x x x+ + + −+ + + −+ + + −+ + + − so that it becomes a perfect square?

Find the value of x also. (2008, 2006, 2002)

• If 4 3 24 12 21 9x x x ax+ + + ++ + + ++ + + ++ + + + is a perfect square. Find the value of ' 'a ? (2007, 2003)

• For what values of a and b will the expression 4 3 24 12 21x x x ax b+ + + ++ + + ++ + + ++ + + + be a perfect square?

(2005)

• For what values of a and b will the expression 4 3 24 12 25x x x ax b− + − +− + − +− + − +− + − + be a perfect square?

(2004)

• If 4 2

2 4

8 44x x q

x x+ + + ++ + + ++ + + ++ + + + is a perfect square then find the value of q . (2001)

• For what value of p will the expression 4 3 24 16 24 16x x x x p− + − +− + − +− + − +− + − + become a perfect square.

(2000)

REMAINDER THEOREM

• x3-x

2-14x+24 (2012)

• x3+3x2+4x-28 (2011)

• x x x3 211 36 36− + −− + −− + −− + − (2010)

• 3 25 2 24x x x+ − −+ − −+ − −+ − − . (2009, 2004, 2001)

• 2 2 14 24x x x− − +− − +− − +− − + . (2008)

• 3 26 11 6x x x− + −− + −− + −− + − . (2007, 2003)

• 3 26 32x x− +− +− +− + . (2006)

• 3 22 5 4 3x x x+ − −+ − −+ − −+ − − . (2005)

• 3 17 26x x− +− +− +− + . (2002)

SIMPLIFY

• 2

41

2 2

a b a

a b a ab

++++ − ÷− ÷− ÷− ÷ − −− −− −− −

. (2009)

• 2 2

4 3 1

9 4 3 2

x y

x y x y

−−−−−−−−

− −− −− −− −. (2008)

• 2 2 2 2

1 b a

a b a b a b− −− −− −− −

− − +− − +− − +− − +. (2007, 2004)

• 2 2 2 4

1 1 2

1 1 1

a

a a a a a a− −− −− −− −

− + + + + +− + + + + +− + + + + +− + + + + +. (2006)

• 2

41

2 2

x y xy

x y x xy

+++++ ÷+ ÷+ ÷+ ÷ − −− −− −− −

. (2005)

• 2 2

1 1 1

4 2 2x y x y x y+ ++ ++ ++ +

− − −− − −− − −− − −. (2000)

• 2

2

1 1 14

2 2b

a b a b a+ + −+ + −+ + −+ + −

+ −+ −+ −+ −. (2002)

• Find the second polynomial when one polynomial is 25 14,x x− −− −− −− − G.C.D is 7x −−−− and

L.C.M. = 3 210 11 70x x x− + +− + +− + +− + + . (2001)

B L A N K S

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• G.C.D of 5 2 3 418 12x y and x y is

• 4 64x ++++ can be made perfect square by adding 216 .x

• 9

4 should be added to 2 3a a−−−− to make it a perfect square.

• L.C.M. of 24 5x and x is

• 22x should be added to 4 4x ++++ to make it a perfect square.

• 2b should be added to 2

16 8a ab++++ to make it a perfect square.

• 2 y should be added to 21y ++++ to make it a perfect square.

• 24b should be added to 4 4a ab−−−− to make it a perfect square.

• To make 28x x−−−− a perfect square 16 should be added.

M A T R I C E S

• By using Cramer’s Rule, solve the equations: (2012)

2x+5y=9, 4x-2y=1

• If A=0" �1; prove that: A.A

-1 =I (2011)

• Apply Cramer’s Rule to solve the given equations:

x y4 2+ =+ =+ =+ =

x y7 2 3+ =+ =+ =+ =

• If 3 2

5 6A

−−−− ====

find 1A−−−− and PROVE THAT 1AA I−−−− ==== . (2009, 2008)

• If 2 1

3 2A

−−−− ==== −−−−

find 1A−−−− . (2007, 2003)

• If 6 2

4 3A

====

find 1A−−−− and PROVE THAT

1AA I−−−− ==== . (2006, 2002)

• If 5 3

2 1A

−−−− ==== −−−−

then find 1A−−−− and PROVE THAT

1AA I−−−− ==== . (2001)

Solve the following equations with the help of matrices. • 2 3 1x y− =− =− =− = (2004)

4 6x y+ =+ =+ =+ =

• 2 2x y− = −− = −− = −− = − (2005)

2 3x y+ =+ =+ =+ =

• 2 5x y+ =+ =+ =+ = (2000)

3 7x y+ =+ =+ =+ =

B L A N K S

• If A=2 " 34 is a singular matrix, the value of p is: (5,6,1,-1)

• Scalar matrix is :

(a) 0" �� "1 (b) 0 �

� 1 (c) 0� "" "1 (d) 0�

1 • If number of rows and columns is not equal then matrix is called rectangular matrix.

• If 1 2

3A then p

p

= == == == =

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• If 2 3

4 x

is a singular matrix then x =

• If 1 2 1 2

3 4 3 4A and B

− −− −− −− − = == == == = − −− −− −− −

then A + B is a 0 matrix.

• If 1 1

2 2

a aA and B then A B

b b

− −− −− −− − = = − == = − == = − == = − = −−−−

.

A L G E B R A I C S E N T E N C E S

Solve graphically the following sets of equations. (Find four ordered pairs)

(1) x y

x y

5 7 13

7 6 3

+ =+ =+ =+ =

+ =+ =+ =+ = (2010)

(2) 4 5

5 17

x y

x y

− =− =− =− =

+ =+ =+ =+ = (2009)

(3) 3 7 2

5 3

x y

x y

= += += += +

+ =+ =+ =+ = (2008, 2006, 2001)

(4) 3 5

2 7 3

x y

x y

− = −− = −− = −− = −

+ =+ =+ =+ = (2007, 2004)

(5) 3 5 21

4 11

x y

x y

+ =+ =+ =+ =

+ =+ =+ =+ = (2005)

(6) 8 29

2 11

x y

x y

− =− =− =− =

+ =+ =+ =+ = (2003)

(7) 2 12

3 2 4

x y

x y

+ =+ =+ =+ =

− = −− = −− = −− = − (2002)

(8) 7 3

3 13

x y

x y

− =− =− =− =

+ =+ =+ =+ = (2002)

(9) x−2y=−3, 2x+y=14 (2011)

(10) 4x−y−10=0, 3x+5y−19=0 (2012)

Solve the following equations:

• 5�� ! 5 − " = , (2011)

• 8 5

1 32

x ++++− =− =− =− = (2009)

• 25 6 4 3x x− = +− = +− = +− = + (2011, 2009, 2007, 2006, 2004)

• 2 10 24 0x x+ − =+ − =+ − =+ − = (2009)

• 2 1

2 03

x −−−−− =− =− =− = (2008, 2002)

• 3 5

2 102

x −−−−+ =+ =+ =+ = (2008)

• 26 40 0x x+ − =+ − =+ − =+ − = (2008, 2007, 2004)

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• 2 3

2 83

x ++++− =− =− =− = (2007, 2006, 2004)

• (((( )))) (((( ))))5 1 2 2 17x x+ − − =+ − − =+ − − =+ − − = (2006, 2003)

• 5 3 6 3x − − =− − =− − =− − = (2005, 2001)

• (((( ))))4 3 1 2 8x x− = +− = +− = +− = + (2005, 2001)

• 22 7 15 0x x− − =− − =− − =− − = (2005)

• 28 15 0x x+ + =+ + =+ + =+ + = (2003)

• x x22 7 6 0− + =− + =− + =− + = (2010)

• x x4 5 3 7− = +− = +− = +− = + . (2010)

• 3 5 1 8x − + =− + =− + =− + = (2003, 2000)

• 2 3 6 ,x x x N− > + ∈− > + ∈− > + ∈− > + ∈ (2002)

• 23 10 2 0x x− + =− + =− + =− + = (2002)

• 23 10 6 0x x− + =− + =− + =− + = (2001)

• 26 5 1 0x x+ + =+ + =+ + =+ + = (2000)

• 2 1

2 53

a −−−−− =− =− =− = (2000)

B L A N K S

• Sol set of {{{{ }}}}4 2x is= −= −= −= − • Sol set of {{{{ }}}}2 2 3x is+ = −+ = −+ = −+ = −

E L I M I N A T I O N

• Find a relation independent of ‘t’ from the following equation: (2012)

� = �( �() �( , , = �( �()

(

• Find a relation independent of ‘x’ from the following equation: (2011)

� + � = �, �" +

�" = �"

• Eliminate ‘ x ’ from the following equations and find the new relationship: (2010)

x px

x qx

12

12 1

+ =+ =+ =+ =

− = +− = +− = +− = +

• Find relation free of x (2009)

2x a

ba x+ =+ =+ =+ =

2x a

ca x− =− =− =− =

• Eliminate t

2 3,x at y bt= == == == = (2008)

• If 2

2

12x p

x+ = =+ = =+ = =+ = = and

2

13x q

x− + =− + =− + =− + = find relation free from x . (2006)

• Eliminate t (2005)

5 7 2x t and y t= == == == =

• Find relation independent of x . (2004)

1

x tx

+ =+ =+ =+ =

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1

2

tx

x− =− =− =− =

• If 1 1

2 2x t and x tx x

+ = + − = −+ = + − = −+ = + − = −+ = + − = − find the relation free from x . (2003)

• Eliminate 1

: 2x x px

+ =+ =+ =+ = . (2002)

1

2 1x qx

− = +− = +− = +− = +

• Eliminate :x (2001)

1

2x ax

− =− =− =− =

2 2

2

1x b

x+ =+ =+ =+ =

• Eliminate :x (2000)

2 2

2

1 12x p and x q

x x− = + =− = + =− = + =− = + =

R A T I O A N D P R O P O R R T I O N

• If �

�� =�

�� =�

�� and a+b+c≠ �, prove that a=b=c (2012)

• If a c e

b d f= == == == = , prove that (� + � + *)(� + 7 + 8) = (�� + �7 + *8) (2011)

• If a c e

b d f= == == == = , prove that

a b a e e f a

b b f f b

4 2 2 2 4 4

6 2 2 5 4

+ −+ −+ −+ −====

+ −+ −+ −+ −. (2010)

I N F O R M A T I O N H A N D L I N G

• Marks obtained by some students in a computer science exam are given below. Find the

median of their marks: (2012)

Marks Obtained 20 – 24 25 – 29 30 – 34 35 – 39 40 – 44 45 – 49

No. of Students 25 28 32 25 13 12

• The marks obtained by 84 students in an examination are given below. Find mean. (2011)

Marks Obtained 25 - 29 30- 34 35 – 39 40 – 44 45 – 49

No. of Students 9 18 35 17 5

• The marks obtained by some students in a Chemistry examination are given below. Find the

median of their marks: (2010)

Marks Obtained 25 – 29 30 – 34 35 – 39 40 – 44 45 – 49


• Find variance from the given information. (2009)

212.5 , 6666x x= ∑ == ∑ == ∑ == ∑ =

• The marks obtained by some students in a subject are below. Find mean. (2009)

Marks Obtained 15-19 20-24 25-29 30-34 35-39


• Find the variance of the following set of numbers. (2008, 2000)

3,5,7,9,11,13X ====

• The following are the marks obtained by 10 students in Mathematics: (2008)

23,15, 35,48,41,5,8,9,11,51X ==== . Find the median of the marks of the students.

• The marks obtained by 60 students in an examination are given below. Find their mode. (2007)

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Marks 40-42 43-45 46-48 49-51 52-54


• A hospital is six storied. The number of rooms in each storey is 35, 32, 31, 34, 38, 33. Find the

standard deviation of the data. (2007)

• The marks obtained by 100 students are given below. Find their arithmetic mean. (2006,03,01)

Marks 30-34 35-39 40-44 43-49 50-54 55-59

No. of Students 14 16 18 23 18 11

• The weight measurements of 12 medicines (in grams) are 43, 54, 45, 44, 58, 47, 50, 52, 51, 45,

48, 46. Calculate their standard deviation. (2006)

• Ten students took a test in Mathematics (out of 100) they got 66, 46, 50, 52, 60, 63, 64, 51, 61

and 55 marks. Find the variance of their marks. (2005)

• A set of data contains the values as 148, 145, 160, 157, 156 and 160. Prove that Mode > Median

> Mean. (2005)

• The following are the marks obtained by 10 students in English. Find standard deviation of

marks. (2004)

46, 50, 52, 60, 63, 64, 51, 55, 66

• Following are the heights of 40 students in inches: Find the mode of the heights of students.

(2004)

Heights (inches) 48-50 50-52 52-54 54-56 56-58 58-60

No. of Students 5 7 10 9 6 3

• The heights of 11 players of a football team are as under. Find variance of the heights. (2003)

57, 61, 60, 64, 59, 55, 58, 63, 65, 61, 56

• Find S.D from the following information: (2002)

219.5, 195, 5555x x x= ∑ = ∑ == ∑ = ∑ == ∑ = ∑ == ∑ = ∑ =

• The following are the marks obtained by 10 students in English. Find the median. (2002)

23,15,48,41,5,8,9,11,51, 3X ====

• Find the variance 5,13,15, 25,12,18,17,19, 20,16, 3X ==== (2001)

• On the prize distribution day 84 students of a school brought pocket money with them as

under. Find A.M. (2000)

Rupees 15-19 20-24 25-29 30-34 35-39


B L A N K S

• The median of 0,2,4,6,8,9 is _______. (4,6,8,5)

• Of –2, –1, 0, 1, 2, the mean is 0.

• In series 0, 1, 4, 6, 7, 9, 12 the median is 6.

• The variance is the square of the standard deviation.

• The sum of 5 observations is 125, the mean is 25.

• /x n∑∑∑∑ is the formula of arithmetic mean.

• The median of 2, 4, 6, 8, 10, 12 is 7.

• 3 median – 2 mean = mode.

• The value which appears most in the data is called MODE.

• If the arithmetic mean of 20 numbers is 100 then their sum is 2000.

• 4 6 5 6 9 0

2 8 7 4 9 4

−−−− + =+ =+ =+ = −−−−

• The median 2, 4, 6, 8, 10, 12 is 7.

T R I G O N O M E T R Y

• Prove that: �9(: + (�+: = �9(:;*�: (2012)

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• Prove that: sin2< +cos

2 <= 1 (2011)

• Find the value of the trigonometric ratios of 30o. (2010)

• Find all the trigonometric ratios of 45o. (2009, 2004)

• Find the trigonometric ratios of 30o. (2005)

• Find the trigonometric ratios of 60o. (2006)

• Prove that:

1 cos

1 cos sin

Sinθ θθ θθ θθ θθ θθ θθ θθ θ

++++====

−−−− (2009)

• sin 260 cos 260 1o o+ =+ =+ =+ = . (2008)

• (((( ))))2cos sin 2sin .cos 1θ θ θ θθ θ θ θθ θ θ θθ θ θ θ− + =− + =− + =− + = . (2007)

• 1 cos cos

sincos 1

θ θθ θθ θθ θθθθθ

θθθθ−−−−

= += += += + (2006)

• (((( ))))tan cot sin cos 1θ θ θ θθ θ θ θθ θ θ θθ θ θ θ+ =+ =+ =+ = . (2005, 2002)

• (((( ))))2 2 2cosec 1 sin cosθ θ θθ θ θθ θ θθ θ θ− =− =− =− = . (2004)

• 2 2

1 cot cosecθ θθ θθ θθ θ+ =+ =+ =+ = . (2003)

• Derive 2 2sin cos 1θ θθ θθ θθ θ+ =+ =+ =+ = . (2001)

• The foot of a tower is at a distance of 210 dm from a point on the earth. The angle of elevation of the

tower from the point is 60o. Find the height of the tower. (2008)

• A tree 90 dm high on the bank of a river makes an angle of 30o from a point directly on the opposite

bank of a river. Find the width of a river. (2002, 2000)

• A ladder makes an angle of 60o with the floor and reaches a height of 6 meters on the wall. Find the

length of the ladder. (2001)

• Solve the triangle ABC in which m<C = 90o, m<B = 60

o and b = 4 cm. (2007)

B L A N K S

• The reciprocal of sin cosecθ θθ θθ θθ θ==== . • If in a rt. Triangle ABC m<B = 90o and measures of sides a, b, c are 6, 10 and 8 respectively then tan

m<A = ¾.

• 2 1 cot 2Cosec θ θθ θθ θθ θ− =− =− =− =

(1) 2

603

oCosec ==== (2) .Sin Secθ θθ θθ θθ θ ==== (3) tan 60 3

o ====

(4) tan .cot 1θ θθ θθ θθ θ ==== (5) cosec .tan secθ θ θθ θ θθ θ θθ θ θ==== (6) 1

cot 603

o ====

(7) 2

1 cos sinθ θθ θθ θθ θ− =− =− =− = (8) 30 2o

Cosec ==== (9) 1

tan 303

o ====

BLANKS FROM GEOMETRY (10 Years)

• The sum of all angles of a cyclic triangle is 360o.

• A line cannot be PARALLEL to two intersecting lines.

• Each of the supplementary angles can be a right angle.

• The angle inscribed in major arc is an acute angle.

• The line which meets circle in only one point is called TANGENT.

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• An angle inscribed in a semi circle is right angle.

• The angle inscribed in minor arc is obtuse.

• If a st. line intersects a circle in two points the st. line is called SECANT.

• A line segments whose end points lie on circle is called CHORD.

• The measure of each angle of an equilateral triangle is 60o.

• The vertical angles formed by two intersecting lines are Congruent.

• The circle passing through the vertices of a triangle is called Circum circle.

• A chord passing through the centre of a circle is called the DIAMETER of a circle.

• If the sum of measures of two angles is 180o then the angles are called SUPPLEMENTARY.

I M P O R T A N T Q U E S T I O N S

• CHAPTER # 1: Examples 1.22, 1.6 Ex. 1.1 (6, 9) Ex. 1.2 (12, 14, 17, 18, 20) Ex. 1.3 (1, 3, 5)

MISCELL. (4, 11, 12)

• CHAPTER # 2: Examples (pg. 20, 3 (pg. 39), 2, 3 (pg. 43), 1, 2, 3 (pg 44-45) 2 (pg. 47)

Ex.2.1 (3) Ex. 2.3 (10, 11, 12, 13) Ex. 2.4 (7-12) Ex. 2.6 (10-14) 2.7 (5-14)

Ex.2.8 (2-10) Miscellaneous Ex. (6, 8, 10)

• CHAPTER # 3: Examples 3, 4 (pg. 55) 5, 6 (very important) (pg. 56) 1 (pg. 62) 2, 3 (pg. 64)

2 (pg. 66) 3, 1 (pg. 67) 2.3 (pg. 68).

Ex. 3.1 (Objectives 1 – 12) Ex.3.2 (11-25 especially 20, 21, 22) Ex.3.4, 3.5,

3.6 Miscellaneous Ex. (5, 6, 7, 8, 10)

• CHAPTER # 4: Example 1 (pg. 84) 2, 3 (pg. 85) 4 (pg. 86) 3, 4 (pg. 87) 3, 4 (pg. 89)

2 (pg.92) Ex. 4.1 (4) Ex. 4.4 (9-12) Ex. 4.5 (3) Ex. 4.7

Ex. 4.8 (2) Ex. 4.9 (2-5) Ex. 4.10 (7, 8, 9) Miscell. (4, 6, 8, 9, 11, 12)

• CHAPTER # 5: Examples 2, 3, 4, 5, 6 (pg. 96-97) 1, 2 (pg. 98) 1, 2 (pg. 100) 1, 2, 3 (pg.101-

102) 1, 2 (pg. 103-104) 1, 2 (pg. 106) 1, 2 (pg. 118) 2 (pg. 120) 1(pg.122), 3,

4 (pg.127-128) Ex.5.1 to Ex.5.7, Ex.5.10 (11-15) 5.11(4, 5, 6, 8), 10 (2, 3, 4,

5) 11(1, 2, 3) 12 (1, 3).

Ex. 5.12 (1, 4, 5) Ex. 5.14 (9-14) Miscell. (1, 2, 3, 5, 6, 7, 10, 11)

• CHAPTER # 6: Example 1 (pg.148) example (pg. 154) example (pg.156) example 1

(pg. 159) example 1 (pg.163) Ex.6.4 (4, 5, 6) Ex. 6.5 (By Cramer’s Rules)

Miscell. (1, 8).

• CHAPTER # 1: Examples 1 (pg.1) 2 (pg.5) 3 (pg.6) 1 (pg.7) 2 (pg.8) example (pg.9) 3

(pg.10) 4 (pg.11) 1, 2 (pg.13) 1, 2 (pg.14-15) 1 (pg.17) 2 (pg.18) Ex.1.1 (1)

1.2, 1.3, 1.4, 1.5 (1-10) 1.6 (9-16) 1.8 Miscell. (1, 2, 3, 4, 10).

• CHAPTER # 2: Example 1 (pg.21) 4, 1 (pg.4) 1, 2 (pg. 24-25) Ex.2.1 {1 (i, iv, v) 2, 3, 4 (i, ii,

iii, iv), 5, 7(i, ii), 8}

• CHAPTER # 3: Example (pg.30) 1, 2, 3 (pg.3) 1, 2, 4, 5 (pg.40-41) 1, 2, 3 (pg.43-44) Ex.3.1

(2, 3, 9, 10, 11, 12, 16, 17) Ex. 3.2, Ex. 3.3 (1-7) Miscell. (8, 15, 17).

• CHAPTER # 4: Examples 1 (pg.72), 2 (pg.73) 3, 4 (pg.74) 10 (pg.78) 1, 2 (pg.80-81) 3,

4(pg.82) 1, 2 (pg.84) 3, 4 (pg.85) 1 (93) Ex.4.3 (2, 4, 6, 7, 8, 10) Ex. 4.4 (1, 2,

4, 6, 9) Miscell. (4, 12 (i, ii, iii), 19, 21).

• CHAPTER # 8: Ratios of 60o, Examples 1, 2, 3, 4 (pg.174-175) 1, 2, 3, 4 (pg.176-179) 1, 2,

3, 4, 7 (pg.181-185) Ex.8.3 (2(ii, ii, vi, vii, viii, ix, xi), 3, 4) Ex. 8.4 (2, 3)

Ex.8.5 (1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 14, 17, 18) Miscell. (2(iv,vi), 3, 4, 5, 7, 8)

F O R M U L A E

• (((( )))) ' ' 'A B A B====U IU IU IU I • (((( )))) ' ' 'A B A B====I UI UI UI U

• NO. OF SUBSETS = 2n

• NO. OF BINARY RELATIONS: m ×××× n

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• A ∆∆∆∆ B = (((( )))) (((( ))))A B A B A B∆ = −∆ = −∆ = −∆ = −U IU IU IU I • 1oa ====

• 1 1

aa

−−−− ==== • 1

1a

a−−−−====

• log log loga a a

m n mn+ =+ =+ =+ = • log log loga a a

mm n

n− =− =− =− =

• log logn

a am n m==== •

loglog

log

n

b

n

aa

b====

• (((( )))) (((( )))) (((( ))))2 2 2 22a b a b a b+ + − = ++ + − = ++ + − = ++ + − = + • (((( )))) (((( ))))2 24a b a b ab+ − − =+ − − =+ − − =+ − − =

• (((( )))) (((( ))))2 24a b a b ab+ = − ++ = − ++ = − ++ = − + • (((( )))) (((( ))))2 2

4a b a b ab− = + −− = + −− = + −− = + −

• (((( ))))2 2 2 22 2 2a b c a b c ab bc ca+ + = + + + + ++ + = + + + + ++ + = + + + + ++ + = + + + + + • (((( )))) (((( ))))3 3 2 2

a b a b a ab b+ = + − ++ = + − ++ = + − ++ = + − +

• (((( )))) (((( ))))3 3 2 2a b a b a ab b− = − − +− = − − +− = − − +− = − − + • (((( )))) (((( ))))3 3 33a b a b ab a b+ = + + ++ = + + ++ = + + ++ = + + +

• (((( )))) (((( ))))3 3 33a b a b ab a b− = − − −− = − − −− = − − −− = − − −

• (((( )))) (((( ))))3 3 3 2 2 23a b c abc a b c a b c ab bc ca+ + − = + + + + − − −+ + − = + + + + − − −+ + − = + + + + − − −+ + − = + + + + − − −

PRODUCT OF POLYNOMIALS = H.C.F ×××× L.C.M 1

/A Adj A A−−−− ====

DETERMINANT OF MATRIX = ad – bc

• Quadratic Formula: 2 4

2

b b acx

a

− ± −− ± −− ± −− ± −====

• Arithmetic mean for grouped data /fx f∑ ∑∑ ∑∑ ∑∑ ∑

• Median for grouped data /2

nl h f c

+ −+ −+ −+ −

• Mode for grouped data (((( ))))

(((( )))) (((( ))))1

1 2

l fm f h

fm f fm f

+ − ×+ − ×+ − ×+ − ×

− + −− + −− + −− + −

• Mean for ungrouped data = /x n∑∑∑∑

• Median for ungrouped data = If n is odd then Median 1

2

n ++++====

• If n is even then Median = 1 2

2 2 2

n nth th

++++ ++++

item.

• 1

cosecSinθθθθ

θθθθ====

• 1

secCosθθθθ

θθθθ====

• sin

cosTan

θθθθθθθθ

θθθθ====

• 2 2cos 1Sin θ θθ θθ θθ θ+ =+ =+ =+ =

• 2 21 tan secθ θθ θθ θθ θ+ =+ =+ =+ =

• 2 21 cot cosecθ θθ θθ θθ θ+ =+ =+ =+ =

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