Board Revision Maths Paper II.

8/14/2019 Board Revision Maths Paper II.

1/17

By Nitin Oke for SAFE HANDS

Board Pattern Mathematics

Paper II

By

Nitin OkeFor Safe Hands


2/17


Out line of paper

Qu.1

(A) Two out of three (3+3+3) (B) One out of two (2 + 2)

Qu.2

(A) Two out of three (3+3+3) (B) One out of two (2 + 2)

Qu.3, Qu.4, Qu.5 (A) (a) One out of two (3+3) (b) One out of two (3 + 3) (B) One out of two (2 + 2)


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Out line of paper

Limit and continuity(3+3+3) 06/09

Differentiation (3+3+3) 09/18 Application of differentiation (2+2) 04/08

Indefinite & definite integrals (3+3+2) 08/16

Application of integration (3) 03/06 Differential equation (2) 02/04

Application of differential equation(3) 03/06

Numerical methods (3+3) 03/06 Boolean Algebra (2+2) 02/04


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Theory questions are

Prove that

Use the fact

A(OAB) Area of sector OAC A(OAC) 1 > (sinx)/x > cosx taking limit of both sides

We get

Limit (3+3) and Continuity (3)

1

0

= x

xsinxLim

A

B

C

O

1

0=

xxsin

xLim


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You need to remember

L Hospitals rule is not allowed in board examination

Write standard result before using at end Trigonometric functions must have angle in radian

Be careful about problem of continuity whether it

is at point or on interval. Please note that following results are not standard

result you need to divide and multiply by properterm

b

a

bx

axsin

xLim =0

)ba(

e)

bx()ax(

xLim =+

1

1

0

i i i


7/17By Nitin Oke for SAFE HANDS

i erentiation + + +Derivative & application of derivatives

One proof (out of two) & two problems of 3 marks

( out of three) one problem of 2 marks (out of two) Proof will be of

Chain rule y= f(u) & u = g(x) then dy/dx = (dy/du).(du/dx)

If y = u+v then prove that y = u + v If y = u.v then prove that y = uv + uv

If y = u/v then prove that y = (vu uv)/v2

If y = f(x) then y = 1/(dx/dy)

If y = f(u) and x = g(u) then dy/dx = (dy/du)/(dx/du) If f(x) is derivable at x = a then f(x) is continuous at x=a

Derivatives of inverse circular functions.

Derivative by first principle.

i i



ome important resu ts o inversetrigonometric functions

T-1(T(x)) = x

T (T-1(x)) = x

(CoT -1(x)) = T-1(1/x)

Sin-1(-x) = -sin-1(x)

Tan-1(-x) = -tan-1(x)

Cos-1(-x) = - cos-1(x)

Sin-1

(x) + cos-1

(x) =

/2 Tan-1(x) + cot-1(x) = /2

Sec-1(x) + cosec-1(x) = /2

=

xy

yxtan)y(tan)x(tan

1

111

( )

)ab(tanwhere

xsin(sin)

ba

xcosbxsina(sin

1

1

22

1

=

=+


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Application of derivatives

Geometrical applications

Geometrical meaning of derivative Tangent at a point of y = f(x), As y y1 = f(x).(x-x1)

Normal at a point of y = f(x) As y y1 = (x x1)/f(x)

Rate of change measure

Meaning of growth and decay rate Physical meaning

Approximation F (a + h) = h. f (a) + f (a) You need to identify function, value of a & h

Maxima minima Identification of critical points Single derivative test

Double derivative test

I t ti (3 3 3 2)


11/17


One proof of indefinite integral or one property of definite

integral (out of two) & two problems of 3 marks ( one out oftwo each on I and D) one problem of 2 marks (out of 2 on I )

Proof will be of Integration by parts

Integration (3+3+3+2)Indefinite(3+3), Definite (3 + 2) & application

c)x(fxedx))x('f)x(f(xe + =+

cxaxlog[a

xax

dxxa + ++++=+ 222

222

2

22

caxxlog[a

axx

dxax + += 222

222

2

22

ca

xsin

axa

xdxxa +

= 1

2

222

2

22

I t ti (3 3 3 2)


12/17



integral (out of two) & two problems of 3 marks ( one out oftwo each on I and D) one problem of 2 marks (out of 2 on I )

Proof will be of


=

a

b

b

a)x(f)x(f

+=b

c

c

a

b

a

)x(f)x(f)x(f

odd.if

evenis)x(fif)x(f)x(f

aa

a

0

2

0

=

=

f(x)-x)-f(2aif0

f(x)x)-f(2aif)x(f)x(f

aa

==

== 0

2

0

2

I t ti (3 3 3 2)


13/17



integral (out of two) & two problems of 3 marks ( one out oftwo each on I and D) one problem of 2 marks (out of 2 on I ) Problem to find area or volume of solid of revolution.



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Boolean Algebra (2+2)

Questions will be based on

only properties of Booleanalgebra or on duals. Onequestion will be on logicgates or switching circuits

If x, y, z are elements ofBoolean algebra then withusual notations

x + x = x

x.x = x

x . x = 0

x. 1 = x

x + 1 = x x + x = 1

(x + y) = x . Y

(x . Y) = x + y

x + (x . Y ) = x x . ( x + y ) = x

x + x . y = x + y

(x) = x

(x + y) . (x + z) = x + y. z

x.y + x.y = (x + y) . (x + y)


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Boolean Algebra (2+2)

Logic gates

ANDx

yx . y (1,1 is 1 all other zero)

x

yx + y (0,0 is 0 all other one

OR

NOTy

y


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Board Revision Maths Paper II.

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