Board Revision Maths Paper I

8/14/2019 Board Revision Maths Paper I

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Mathematics Paper I


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Out line of Paper I Logic of mathematics(6)

3 marks each 2 out of 3 = 6/9 6/9 Matrices(7)

4 marks each 1 out of 2 = 4/8

3 marks each 1 out of 2 = 3/67/14

Vectors and three dimensional geometry(12) 4 marks each 1 out of 2 = 4/8



12/21 Linear Programming(7)



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Out line of Paper I

Two dimensional coordinate geometry(12) Pair of straight line

Circle Parabola Ellipse and hyperbola =

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Probability(6) 3 marks each 2 out of 3 =6/9


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Out line of Paper I

In all 5 questions of 10marks each

10 = 6 + 4

Knowledge based

questions 2+2+3 on Vector &geometry

2+2 Lines, circle andconic sections

Understanding base 3+3 logic

3+3 Probability

3 Vectors & geometry

3+3+3 (2D)

Application base 3 logic

3+3+4+4 matrices

3 probability

3+4+4 vectors &geometry

4 (2D)

3+3 Linear

programming Skill based

4+4 Linearprograming


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Mathematical Logic

Symbolical representation and (), or (), implication (),

equivalence (), negation ( or )

Definitions:Tautology: A statement which is always

true for any truth values of component

statements is called as tautology Contradiction: A statement which is

always false for any truth values ofcomponent statements is called as

contradiction


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Mathematical Logic

Converse: converse of implicationp q is q p

Inverse: Inverse of implicationp q is p q

Contra positive: Contra positive ofp q is q p

Possible questions are1. Convert given statement into symbols

2. Convert given statements into indicatedsymbolic form

3. Verify using truth table

4. Verify or obtain truth values, if truth values ofcomponent statements are known


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1. Represent given statement usingVein diagrams ( just use three facts)

U

Y

All x are y


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U

Y

No x are y


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U

Y

Some x are y some x are not y


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Mathematical Logic

1. Write negation of the followingstatements ( to do so use (pq) (pq)

(pq) (pq) (p q) (p q) (pq) [(p q) (q p)]

[(p q) (q p)]6. Note that negation of all is at least one

and of some is no

2. Write the dual: means replace by


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MatricesTypes of matrices:

Row matrix If a matrix has only one row(horizontal arrangement) then is called asrow matrix its order is 1xn

Column matrix-If a matrix has only onecolumn (vertical arrangement) then iscalled as column matrix its order is nx1

Square matrix- A matrix is square matrix ifnumber of rows equals number of column.

Diagonal matrix- A square matrix is calledas diagonal matrics if aij = 0 if i j


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Symmetric matrix- A square matrix iscalled as symmetric matrics if aij = aji for

all i & j Skew symmetric matrix- A square matrix

is called as skew symmetric matrics if aij =

-aji

for all i &j

(diagonal entries must be zero)

Null matrix- A matrix of any order is calledas null matrix if aij= o i,j

Transpose of a matrix- A matrix AT or A iscalled as transpose of A if it is obtained byinterchanging row by column and columnby row.

Singular matrix- A square matrix is said to


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Possible types of questions

=

61

23

42

A

BAABthatshowBAcomputing

ABwithoutFind

=503

634B

Here A is of order 3x2 B of 2x3 then AB isof order 3x3 and BA of 2x2 hence cant

be equal


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12493

A

Prove that (A+B)2 = A2 +AB+B2

For solving this problem do not showRHS = LHS instead you may try as

(A+B)2 = A2 +AB+BA+B2 butrequired result implies that BA = 0hence show BA = 0

=68

34B


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=20

21A

Prove that (A+B)2 A2 +2AB+B2 For solving this problem do not show

RHS LHS instead you may do

as,

The inequality is because of AB BAthus just show that AB BA

01

12B


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2021

A

Prove that (A+B)(A-B) A2 - B2 For solving this problem do not show

RHS LHS instead you may do

as,

The inequality is because of AB BAthus just show that AB BA

01

13B


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= 1143

AIf

To prove above result you need to use

principle of finite induction as you did inprevious problem.

=

nn

nn

21

421nAthen

= ba

0

0

AIf

=

n

nn

b0

0aAthen


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To obtain inverse of matrix1) by elementary row operations

R i R j interchanging two rows

R i R j multiply every element by non zeroelement

R i R i+ R j

To obtain inverse of A consider A.A-1 =I

Go on performing elementary

operations till A changes to I and at-


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To obtain inverse of matrix

1) by using matrix polynomial

For given A either a relation will begiven or we will need to prove thesame as A2+2A-3I=0 using thisrelation we can operate A-1 to get

A + 2I-3A-1

hence A-1 = 1/3(A + 2I)


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To solve system of equations

1) By using reduction method

The given system of equation is tobe converted in matrix equation asAX=B for example: x-y + 4z = 4;2x+ 6y z = 3 & x + y 2x = -1 will

have the form

=

1

3

4

z

y

x

211

162

411


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Now reduce the coefficient matrixto triangular matrix ( if fractionsare not appearing then reduce it

to unit matrix) get the values ofx, y and z.

=

1

34

z

yx

211

162411


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Points to remember for matrices

Copy the matrix by proper care.

Be careful about calculations whichyou are performing.

After finding inverse or values of x,yand z substitute and verify the same.

Take care of order of matrix

You are good at calculationsbelieve me you are!


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Linear

programmin

g2 questions on find maxima/ minima of3 marks each

1 question on form LPP of 4 marks


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Possible type of questions

Define Convex set: A set of points is said to be convex

set if the line joining any two points of the setentirely lies in the set

Convex polygon: A bounded polygonal convexset is called convex polygon

Corner points/ extreme points: The point ofintersection of any two boundaries of the halfplanes determined by a system of linear

inequalities is called an extreme point or cornerpoint.

Convex polygon theorem: z = f(x, y) be givenlinear function defined over a convex polygon X,maximum or minimum values will be atextreme points.


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Draw the graph for solution set ofinequalities 3x + 5y 15, 5x + 2y 10,

x 0, y 0 maximize Z = x + y

(0, 3)

(0, 5)

(5, 0)

(2, 0)

(20/19, 45/19)

Draw the figure,find cornerpoints,

substitute thevalue andcompare the

result foradjoining figuremaxima is at

(20/19,45/19)


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Possible type of questions

Using word problem form the LPP andsolve graphically. You may start first towrite LPP, check whether you want to

maximize profit or production or tominimize expenses, choose propervariables and form the problem.

Note that objects, days can never benegative

Use tabular form for formation of LPP


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Pair of straight line,

circle, conic section


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Important formulae

The equation ax2 + 2hxy + by2 = 0represents

1. Two real and different lines if h2 > a b2. Two coincident lines if h2 = a b Consider by2 +2hxy + ax2 = 0 then dividing

by ax2 and slope of lines passing throughorigin is y/x we get bm2 + 2hm + 1 = 0 then

using relation between roots we have m1 + m2 = -2h/b and m1m2 = b/a then angle

between two lines will be

ba

abh2tan

.mm1

mm

21

21

21 +=+

1 = tan


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Important formulae

The equation ax2 + 2hxy + by2 = 0represents Pair of two perpendicularlines ifa+b = 0

The difference of the slopes of linesgiven by ax2 + 2hxy + by2 = 0 is givenby

b

abh2mm

2

12

=


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Important formulae

The equation ax2 + 2hxy + by2 = 0represents Pair of two lines and theequation of pair of lines which are

bisectors is given by Pair of bisector is

h

x.

ba

yx 22 =


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Important formulae

The equationax2 + 2hxy + by2+2gx

+2fy+c = 0 represents Pair of two

lines not passing through origin if

0

cfg

fbh

gha


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Important formulae



lines not passing through originangle between these lines will besame as angle between linesrepresented by ax2 + 2hxy +by2=0


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Important formulae



parallel lines if

f

g

b

h

h

a

=


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Standard types of questions

A term will beunknown inequation ax2 + 2hxy+ by2+2gx +2fy+c

= 0 what must bethe term so that theequation representspair of straight lines

use determinant

Find the equation ofthe pair of linesthrough origininclined at 60o to

line x + 2y = 1 Let slope of required

line be m, usingrelation between

angle and slopes weget 3=(2m+1)/(2-m) replace m by y/xto get solution


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d d i


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Standard equation x2 + y2 = a2

(x-h)2 + (y k)2 = a2

General equation x2 + y 2 +2 g x +2 f y + c= 0

Center (-g, -f) radius g2 + f2 + c Diameter form (x-x1)(x-x2) + (y-y1)(y-y2)=0

Parametric form x = a cos(t), y = a.sin(t) Parametric form x-h = a cos(t), y-k = a.sin(t)

Equation of tangent to x2 + y2 +2g x +2 fy +c = 0 at (x1 ,y1) is x.x1+yy1+ g(x+x1)+

f(y+y1)+c=0

Condition of tangency: y = m x + c istangent to std. Circle if c = am2+1

Equation of director circle x2

+ y2

= 2a2

Th bl b


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The problems may be 1. Find equation of circle with center at (2,3)

and passing through (1,-1): Find radius using distanceformula and get the answer

and 2x + 3y+1 =0 is tangent : Find distance fromtangent which equals radius and get - -

and a line x + 3y =7 cuts a chord AB of length 8: Finddistance from chord, OM is bisector, usePythagoras theorem to find radius and get-

2. End points of diameter use formula

3. Three points given: Use general equation orconsider (h,k) as center and equate radius toget (h, k) and using distance formula obtain

radius.1. These three points are (0,0), (a,0) and (0,b) then AB

will be diameter and we get result using diameter form

1 T h k h th i li i t t t


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1. To check whether given line is tangent to

General circle then condition of tangencyNOTapplicable find point of intersection or distance fromline of center is equal to radius

Standard circle: use condition of tangencyor you canfind point of intersection, if is one, then tangent.

To find equation of tangents from external pointof circle: use slope point form and the fact that distance from line of center is equal to radiususing this relation you will get a quadraticequation whose roots are slopes of requiredtangents.

To find angle between tangents from external

point of circle: use slope point form and the factthat distance from line of center is equal toradius using this relation you will get aquadratic equation whose roots are slopes ofrequired tangents. Dont find equations of

tangent use relation between roots of QE and


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To check whether given circles areorthogonal to each other : find and writevalues of g1, f1, c1 and g2, f2, c2 use the

relation 2 g1 g2 + 2 f1f2 = c1 + c2

To find equation of director circle togeneral circle

1. Find circle with same center but radiusdouble

2. Find locus of points from which tangents areperpendicular

3. Shift the origin at center of circle and usedirector circles std form, go back to originalvariables by substitution.


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Conic section


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(a2+b2)/a >1Hyperbola

(a2-b2)/a


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X2/a2 - y2/b2=1Hyperbola

X2/a2 + y2/b2=1Ellipse

Y2 = 4axParabola

Std . EquationConic


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X = a sec()y = b tan()Hyperbola

X = a cos()y = b sin()

Ellipse

X=at2 ,Y= 2atParabola

ParametricEquation

Conic


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2b2/aHyperbola

2b2/aEllipse

4aParabola

Length of latusrectum

Conic


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(ae,0);(-ae,0)Hyperbola

(ae,0);(-ae,0)Ellipse

(a,0)Parabola

focus/fociConic


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X = a/eHyperbola

X = a/eEllipse

X = -aParabola

Equation ofdirectrix

Conic


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xx1

/a2 -

yy1/b2=1Hyperbola

xx1/a2 +

yy1/b2=1

Ellipse

yy1= 2a(x+x1)Parabola

Eq. of tangentConic


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C = a2m2 - b2Hyperbola

C = a2m2+b2Ellipse

C = a/mParabola

Condition Oftangency

Conic


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(-a2m2 /c,-b2/c)Hyperbola

(-a2m2 /c,b2/c)Ellipse

(a/m2, 2a/m)Parabola

Point ofcontact

Conic


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X2 + y2 = a2- b2Hyperbola

X2 + y2 = a2+b2Ellipse

X = -aParabola

Locus of pointsfrom which

tangents are Conic


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(x-x1)(x1+x2)/a2 + (y-

y1)(y1+y2)/b2=1

Hyperbola

(x-x1)(x1+x2)/a2 + (y-

y1)(y1+y2)/b2=1

Ellipse

Y = mx 2aParabola

Equation of chord joining points(x

1

,y1

)(x2,

y2

)Conic


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a x cos - b y cot = a2 - b2Hyperbola

a x sec - b y cosec = a2 - b2Ellipse

Y = m.x 2am am3Parabola

Equation of normalConic


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Important definitions

Set of all outcomes in a random experimentis called as sample space.

Any subset of sample space is called as anevent

If A is an event then S\A = A is called ascomplementary event

A and B are said to be mutually exclusiveevents if A B =

A and B are mutually exclusive andexhaustive events ifA B = and A B= S

Probability of event A = n(A) / n(S)


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Important definitions

P(A) = 0 means event is impossibleevent

P(A) = 1 means event is certain event

0 P(A) 1 P(A B ) = P(A) + P(B) P(A B)

P(A B C ) = P(A) + P(B) +P(C)+ P(A B C )

[P(AB )+P(BC )+P(C A)]

P(A) = 1 P(A)


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Tips and tricks

To determine number of elements inevent space or sample space usefollowing

Throwing of one die n(S) = 6 Throwing of k dies n(S) = 6k

Choosing a card n(S) = 52

Event A or B n(Event) = n(A) + n(B) Event A & B n(Event) = n(A) x n(B)


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Tips and tricks

importaisorderr)!(n

n!Prn

importanotisorderr)!(nr!

n!Cr

n

Can be used when to form

numbers, arrange different books,

Can be used when to form groupwithout order, types of books


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Important results Vector: directed segmentTwo vectors are equal if have same direction

and magnitude

Triangle law of vector addition

Parallelogram law of vector addition

Dot product and their properties

Cross product and their properties

Scalar multiplication, generates parallelvector

Scalar triple product

Two vectors are perpendicular if their dot

product is zero AO

l


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Important results

A vector is unit vector if its magnitudeis one

To determine unit vector along AB if

point A & B are given use Find position vector of A and B, find b a

thus you have vector AB now divide it byits magnitude to get the answer.

To determine vector perpendicular togiven two vectors

To find this take cross product of the two

l


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Important results

To check whether given three points A,B and C are collinear you can use

Cross product of AB and AC must be zero

Using section formula we can provecolinearity by showing that c = a + b

Prove that AB = .AC

To check whether A,B,C and D arecoplanar you can use

AB.(AC x AD) = 0

AB = .AC + .AD

I l


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Important results

Section formula: If P divides AB in ratio m:nthen position vector of P is (m.b+n.a)/(m+n): to prove the result use AP.n=PB.m (note the

direction). Use OA + AP = OP substitute AP =m.PB/n and now use position vectors for relationOA + m.PB/n =OP

If P divides externally then AP.n=BP.m (note thedirection).

Ifris any vector coplanar to a and b (a andb are non zero) then ris uniquely expressesas linear combination ofa and b

I l


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Important results

Applications in geometry Area ofABC is I AB x AC I = I AB x BC I Area of parallelogram ABCD is = IAB X AC I

Volume of parallelepiped =AB . (AC X AD) Application in trigonometry:

Rules of T- ratios of sum and difference of

angles can be proved by vectors as

Consider cross product ofOA and OB

by definition of cross product and by

analytical method take modulus and

you have the result, use the same for +

I l


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Important results Sine Rule: in triangle ABC

a/sin A = b/sin B =c/ sin C use AB + BC + CA = 0

and consider cross product with AB to get one

equality and then with AC to get other equate

one side of these equality and get the result bydividing by proper fraction

Cosine rule : Use AB + BC + CA = 0 hence

AB + BC = AC and equate magnitude ofboth sides considering IABI = c, IBCI = a

and ICAI = b we get,

cos(A) = (b2 + c2 a2)/2bc lly other results

I t t lt


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Important results Application to Physics:

Work done : F.s

Angular momentum: moment of momentum

means ifM is momentum and P is any point on

line of action then moment about O is crossproduct ofOP and M

Torque: moment of force means torque

Projection of a vectora along a vectorb meansprojba = a.b / b = a.eb

Resolution ofa along b means b)b

b.a(

2

I t t lt


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Important results Direction angles: If L is any line then its angle

with X+ ,Y+,Z + are called as direction angles andare denoted by , and

Direction cosine: If, and are directionangles then cos, cos and cos are called asdirection cosine denoted by l, m and n

l2 + m2 + n2 = 1 to prove this result letP(x,y,z) be any point, let IOPI = r then usingdefinition of dot product OP.i = r cos = (xi+yj+zk).i = x

lly y = rcos and z = r cos square these results and add to get the

required result.

I t t lt


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Important results Direction ratio: Ifl, m and n are

direction cosine then a,b and c are calledas direction ratio if a/l=b/m = c/n

Relation between dr s and dcs: If a,b

and c are direction ratio and l, m and nare direction cosine then a/l=b/m = c/n =k say then using l2 + m2 + n2 = 1 weget required result as

l = a /a2+ b2 + c2 m = b /a2+ b2 + c2

n c / a2+ b2 + c2

Board Revision Maths Paper I

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