MATHS 1A Guess Paper

5/24/2018 MATHS 1A Guess Paper

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Intermediate Public Examination (March 2014)

Maths IA

Time: 3 Hrs Max. Marks: 75

Note: This question paper consists of three sections A, B, and C.

Section A

I very short Answer type questions(i) Answer all Questions(ii) Each question carries 2 marks. 102 = 20

1.If = 0, , , , and f: A B is a surjection defined by f ()= cos x then find B.2.Find the domain of the function f ()= 4 3.If A =

2 15 01 4and B = 2 3 14 0 2then find 2A + BTand 3BT A

4.Define trace of a matrix and find the trace of A If A = 1 2 01 2 2 1 5.If = 2i + 5j + and = 4i + mj + are collinear vectors then find the values of m and n.6.Find the vector equation of the line passing through point 2i + 3j + and parallel to the vector4i 2j + 7.Find the area of the parallelogram having 2i 3j and 3i - as adjacent sides.8.Prove that cos340cos40+ sin 200sin 140=

1

2

9.If cos + sin = 2cosprove that cos sin = 2sin10. If coshx=

5

2find the values of (i) cosh(2x)

(ii) sinh(2x)

Section B

II Short answer type questions

(i) Answer any five questions

(ii) Each question carries 4 marks 54 = 20

11.If I =1 00 1

and E = 0 10 0

then show that (aI+bE)3= a3I+ 3a2bE12.If

,

, ,

are non-coplanar vectors. Prove that the four points

a

+ 4b - 3c

, 3 a

+ 2b - 5c

,

-3a+ 8b - 5c and 3 a+ 2b+ c are coplanar.13.Prove that the smaller angle between any two diagonals of a circle is given by sin = 14.If A + B = 45then prove that (1+ tan)(1+ tan) = 2and hence deduce the value of tan22

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15.Solve sin+ 3cos = 216.Prove that tan-1

1

2+ tan-1

1

5+ tan-1

1

8=4

17.Prove that cotA+ cotB+ cotC=a2+ b2+ c2

4 Section C

II Long answer type questions

(i) Answer any five questions

(ii) Each question carries 7 marks 5 7 = 35

18.If f: A B, g : B C are two bijections, then (gof)-1= f-1og-119.By mathematical Induction

, prove that 1 +(1 +2) + (1 +2 + 3)+ ..n brackets =

(

+1)

(

+2)

12

20.Show that 2 22 22 2 = (a + b + c)3

21.Solve the following equations by using Gauss-Jordan method. 2x y + 3z = 9 , x + y + z + = 6 , x y + z = 2

22.Find the shortest distance between the skew lines

r= 6i+2J+2k+ t i-2j+2kandr= -4i -k+ 5 3i-2j+2k where s, t are scalars23.If A, B, C are angles in a triangle, then prove that sin2

A

2+ sin2

B

2- sin2

C

2=1-2cos

A

2cos

B

2sin

C

2

24.If a = B, b = 14, C = 15 show that R = , r = 4, r1= , r2= 12 and r3= 14.

Scheme of valuation

Maths IA

(1) Given f: AB is surjectioni.e (codomain off) B = range of f= f (A) 1M

=cos0, cos6 cos

4, cos

3, cos

2

=1, 32

,12 , 12 , D 1M

(2) f (

)is defined

4x -

x

2

0 1M

x ( x 4) 0x [0, 4] 1M


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This does not constitute to be a Main Question paper and should NOT follow the same. While allefforts have been made to make the guess paper available on this website as authentic as possible.

Manabadi or any staff persons will not be responsible for any loss to persons caused by anyshortcoming, defect or inaccuracy in the Guess Papers provided by Manabadi.com website.


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(3)First find 2A and BTnext find 2A + BT 1MFind next find 3BT A 1M

(4)Sum of the principal diagonal elements of the matrix A is called trace of the matrix A 1M

Trace of A = a11+ a22+ a33

= 1 + -1+ 1= 1 1M

(5) Given

,

are collinear

x = t

for some t

R

2i + 5j + k

= t

4i + mj + nk

1M

after simplification m = 10, n = 2 1M

(6)Vector eqn of the line passing through the point A() and parallel to the vector is r= + ; t R 1Mrequired equation is r= 2i+3J+k+ t 4i-2j+3E; t R 1M(7)area of the parallelogram having the sides is a b 1M

area of parallelogram is

94Sq. units. 1M

(8).cos(360 20) cos40+ sin(180+ 20) sin(180 40)cos20cos40+- sin20sin40 1Mcos(40+ 20)

= cos60

=1

2 1M

(9)cos + sin = 2cossin = 21cos 1MMultiplying 2+ 1 on both sides2+ 1 sin = (2 1 ) cos2sin+ sin = coscos sin = 2sin 1M

10.coshx = cosh2x= 2cosh 1

= 2 - 1

cos cosB sinAsinB = cos(A+B)





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B

R

A

S

Q

= 1M

cosh 2 - sinh(2)= 1sinh(2) = cosh(2)- 1

= - 1sinh2

=

5

21

2

1M

Section - B

(11)aI + bE = 0 1M

(aI + bE) = 30

2M

a3

I+3a2

bE = 3

0 1M(12) AB= 4 a - 2b - 2c

AC = -2 a - 4b - 2c 2MAD = -2 a - 2b + 4c[AB

AC

AD

]=

422

2 4

2

22 4

a

b

c

= 0 1M four points A, B, C, D are coplanar. 1M(13)Take the unit cube

Let OP and AS are two diagonals of a cube

OP= I + j + E 1MAb

= -I + j + E

cos= . ||||= 1M(14)A + B = 45

tan(A+B)= tan45tanA+ tanB

1- tanAtanB = 1

1M

(1)

LHS

(1+ tan)(1+ tan)1 + tanA+ tanB+ tanAtanB

1 + 1

tanA+ tanB=1- tanAtanB

e





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= 2 2M

Take A = B = 22(1+ tan)(1+ tan)= 2tan = 2- 1tan22= 2 1 1M

(15)sin+ 3cos = 2Divide with 2 sin + cos = 1Mcos = cos 1M

G.S is = 2n 1MX = 2n

+

= 2n+ , 2n - 1M16. tan + tan + tan = tan 1M

tan ..... 1Mtan(1) 1M= 1M

(17)+ AcosA

b2+ c2- a2

2bca

2R

1M

b2+ c2- a2 1M(a + b + c) 1M 1M

(18) since f : AB, g : B c are bijectionsThen gof: Ac is also bijection(): CA is a bijection: BA, : CB are bijections





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Theno: CA is also bijection() &ohave same domain, namely C. 3MNext we P.T () ()= (o)()cC 3M()= &o 1M

(19)nth

term

1 +2 + = ()() 1MLet S(n) be the statement

i.e. 1+ (1 +2)+ ----------+ ()() = ()() case (i) if n = 1

LHS = RHS = 1 1M

Case (ii) Assume that S(k) is true.

i.e. 1+ (1 +2)+ ----------+ ()()

=()()

1M

case (iii) now we P.T S(k + 1) is trueadding (k + 1)

thterm on both the sides in eqn 2S(k + 1) is true 3M

Conclusion: by the principle of finite m.I S(n) is rue for all nN 1M(20) + +

++ ++ + +

2

2

2 2 2M

(++ ) 1 1 12 22 2 2M

(++) 1 0 02 (++ ) 02 0 (++ ) 1M

=(+ +) 2M(21)The augmented matrix is

[]~21 31 1 111 1

962 1M

[

]~

1 1 12

1 3

11 1

69

2 1M

- 2 -





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~1 1 103 102 0

634 1M~

1 1 10

3 1

0 1 0

6

3

2

1M

~1 1 10 1 0

03 1623 1M 3

~1 0 10 1 00 0 1

423 1M

1 0 0

0 1 00 0 1

1

23X = 1 y = 2 z = 3 1M

(22)Compare with r= a +r= +

a = 10i + 2J + 1M

= i - 2J +2

= 3 i - 2J - 2 Find = 12 1Mand a =108 2MShortest distance =

[ ]|| 2M

Shortest distance 9 units 1M

(23)A + B + C = 180

sin + sin + sinsin + sinsin 1Msin + sin90 sin 1M1 - cos + cos sin 1M1 - cos

+ cos sin 1M1 - cos

+

sin

sin

1M

1 - cos + 2cos sin 1M1 2 cos

cos sin 1M


manabadi.com is not responsible for any inadvertent error that may have crept in the guess paperbeing published on NET. The guess paper published on net is for the information to the examinees.

This does not constitute to be a Main Question paper and should NOT follow the same. While allefforts have been made to make the guess paper available on this website as authentic as possible.

Manabadi or any staff persons will not be responsible for any loss to persons caused by anyshortcoming, defect or inaccuracy in the Guess Papers provided by Manabadi.com website.


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(24) S = = 21 1M= ( )( )( )= 84 1M

R = = 1M

r == 4 1M

= = 1M= = 12 1M= = 14 1MThe End




MATHS 1A Guess Paper

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