Board Revision Maths Paper Ii.

Board Pattern Mathematics Paper II By Nitin Oke For Safe Hands

By Nitin Oke for SAFE HANDS

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) By Nitin Oke for SAFE HANDS

Out line of paper • • • • • • • • •

Limit and continuity (3+3+3) Differentiation (3+3+3) Application of differentiation (2+2) Indefinite & definite integrals (3+3+2) Application of integration (3) Differential equation (2) Application of differential equation(3) Numerical methods (3+3) Boolean Algebra (2+2)

By Nitin Oke for SAFE HANDS

06/09 09/18 04/08 08/16 03/06 02/04 03/06 03/06 02/04

Limit (3+3) and Continuity (3) • Theory questions are – Prove that

• • • • •

C

sin x Lim =1 x →0 x

Use the fact A(∆OAB) ≤ Area of sector OAC ≤ A(∆OAC) 1 > (sinx)/x > cosx taking limit of both sides We get

sin x Lim =1 x →0 x

By Nitin Oke for SAFE HANDS

B

O

A

Some slandered formulae sin x Lim =1 x →0 x

tan x Lim =1 x →0 x

ax − 1 Lim = log a x →0 x

1 1 log(1 + x) Lim =0 ( ) Lim =1 x Lim (1 + x) x = e x →∞ x x →0 x →0 x −1 xn − an e Lim = nxn −1 Lim =1 x →0 x − a x →0 x 1 − cos x 1 Lim = 2 x →0 x 2 By Nitin Oke for SAFE HANDS

You need to remember L’ Hospital’s 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 proper term 1 a ( ) ( ) sin ax a Lim = Lim (1 + ax) bx = e b b x →0 bx x →0 • • • •

By Nitin Oke for SAFE HANDS

Differentiation (3+3+3+2) 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’ + u’v If y = u/v then prove that y’ = (vu’ – u’v)/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. By Nitin Oke for SAFE HANDS

Some important results of inverse trigonometric functions • T-1(T(x)) = x

 x±y  − 1 − 1 − 1  tan (x) ± tan ( y) = tan   1  xy  

• T (T-1(x)) = x • (CoT-1(x)) = T-1(1/x) • Sin-1(-x) = -sin-1(x) • Tan (-x) = -tan (x) -1

-1

• Cos-1(-x) = π - cos-1(x)

a sin x ± b cos x − 1 sin ( ) = sin −1( sin( α ± x ) a 2 + b2 b where α = tan −1( ) a

• Sin-1(x) + cos-1(x) = π/2 • Tan-1(x) + cot-1(x) = π/2 • Sec-1(x) + cosec-1(x) = π/2 By Nitin Oke for SAFE HANDS

You need to know! Derivatives by first principle Derivative of parametric function

Derivative of composite function by chain rule

Derivatives

Derivative of inverse trigonometric functions using proper substitution

Derivative of u.v.w OR u.v/w.z OR uv by taking log of both sides

Higher order Derivatives and relation between derivatives

By Nitin Oke for SAFE HANDS

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 By Nitin Oke for SAFE HANDS

Integration (3+3+3+2) Indefinite(3+3), Definite (3 + 2) & application • One proof of indefinite integral or one property of definite integral (out of two) & two problems of 3 marks ( one out of two each on I and D) one problem of 2 marks (out of 2 on I ) • Proof will be of – Integration by parts – ∫ e x (f( x) + f'(x))dx = e x f(x) + c 2 x a 2 2 a2 + x2 + log[x + a 2 + x 2 + c ∫ a + x dx = 2 2 x a2 2 2 2 2 x −a − log[x + x 2 − a 2 + c ∫ x − a dx = 2 2 x 2 a2 x 2 2 2 − 1 a −x − sin   + c ∫ a − x dx = 2 2 a

By Nitin Oke for SAFE HANDS

Integration (3+3+3+2) Indefinite(3+3), Definite (3 + 2) & application • One proof of indefinite integral or one property of definite integral (out of two) & two problems of 3 marks ( one out of two each on I and D) one problem of 2 marks (out of 2 on I ) • Proof will be of b

a

∫ f(x) = − ∫ f(x) a

2a

a

0

0

∫ f(x) = 2∫ f(x) if f(2a - x) = f(x)

b

b

c

b

a

a

c

= 0 if f(2a - x) = - f(x)

∫ f(x) = ∫ f(x) + ∫ f(x) a

a

∫ f(x) = 2∫ f(x) if f(x) is

−a

even

0

= 0 if odd. By Nitin Oke for SAFE HANDS

Integration (3+3+3+2) Indefinite(3+3), Definite (3 + 2) & application • One proof of indefinite integral or one property of definite integral (out of two) & two problems of 3 marks ( one out of two 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.

By Nitin Oke for SAFE HANDS

Numerical Methods (3+3) • There will be no problem on solving integrals by numerical method. The problems will be based on relations of E, E-1,∆, and ∇ You know that – E(f(x) = f(x+h) ∆(f(x)) = f(x+h) – f(x) ∆=E–I ∆∇=∇∆ ∇E = ∆ ∆E =E∆ ∆-∇=∆∇ – E-1 = I -∇ ∆/∇ + ∆/∇ = ∆ + ∇ – (I+∆) (I-∇)= I – E(I-∇) = I By Nitin Oke for SAFE HANDS

Boolean Algebra (2+2) • Questions will be based on only properties of Boolean algebra or on duals. One question will be on logic gates or switching circuits • If x, y, z are elements of Boolean algebra then with usual 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’)

By Nitin Oke for SAFE HANDS

Boolean Algebra (2+2) • Logic gates

x y

AND

x y

OR

y

x . y (1,1 is 1 all other zero) x + y (0,0 is 0 all other one

NOT

y’

By Nitin Oke for SAFE HANDS

By Nitin Oke for SAFE HANDS