Maths 1a Guess Paper 5q6l3o
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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) (ii)
1. If
Answer all Questions Each question carries 2 marks.
= 0,
,
,
,
10×2 = 20
and f: A → B is a surjection defined by f ( ) = cos x then find B.
2. Find the domain of the function f ( ) = √4 − −2
3. If A = 5 −1
1 −2 0 and B = 4 4
3 1 then find 2A + B T and 3B T – A 0 2
1 4. Define trace of a matrix and find the trace of A If A = 0 −
5. If ⃗ = 2i + 5j + ⃗ and ⃗ = 4i + mj +
2 − −1 2 2 1
⃗ are collinear vectors then find the values of m and n.
6. Find the vector equation of the line ing through point 2i + 3j + ⃗ and parallel to the vector 4i – 2j + ⃗
7. Find the area of the parallelogram having 2i – 3j and 3i - ⃗ as adjacent sides. 1 8. Prove that cos340°cos40° + sin 200°sin 140° = 2 9. If cos + sin = √2 cos prove that cos − sin = √2 sin 5
10. If cosh x= 2 find 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
5×4 = 20
11. If I = 1 0 and E = 0 1 then show that (aI+bE)3 = a3 I+ 3a2 bE
0 1 0 0 12. If ⃗, ⃗, ⃗, are non-coplanar vectors. Prove that the four points −a⃗+ 4b⃗ - 3c⃗ , 3 a⃗+ 2b⃗ - 5c⃗ , -3a⃗+ 8b⃗ - 5c⃗ and 3 a⃗+ 2b⃗+ c⃗ are coplanar.
13. Prove that the smaller angle
between any two diagonals of a circle is given by sin
=
manabadi.com is not responsible for any inadvertent error that may have crept in the guess paper being 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 all )(1 +available 14.have If A +been B = 45° then + tanpaper tan = 2onand hence deduceas theauthentic value of tanas 22 possible. ° efforts made toprove make that the 1 guess this website Manabadi or any staff persons will not be responsible for any loss to persons caused by any shortcoming, defect or inaccuracy in the Guess Papers provided by Manabadi.com website.
This Document is provided by www.manabadi.com for FREE for the benefit of Intermediate students. Copying and redistribution by any company/ website is illegal and Manabadi.com has all rights to claim on such type of website or company.
15. Solve sin + √3 cos = √2 16. Prove that tan-1 12 + tan-1 15 + tan-1 18 = 17. Prove that cot A + cot B + cot C =
π 4
a2 + b2 + c2 4∆
Section C II
Long answer type questions
(i) Answer any five questions (ii) Each question carries 7 marks
18. If f: A→ B , g : B→ C
are two bijections, then (gof)-1 = f -1 og -1
19. By mathematical Induction
20. Show that
− − 2 2
5× 7 = 35
∀ ∈
, prove that 1 + (1 + 2 ) + (1 + 2 + 3 ) + …………..n brackets = ( + 1) ( + 2) 12
2 − − 2
2 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 r⃗ = 6i+2J+2k + t i-2j+2k and r⃗ = -4i -k + 5 3i-2j+2k where s, t are scalars
23. If A, B, C are angles in a triangle, then prove that 24. If a = B, b = 14, C = 15 show that R =
A
B
C
A
2
2
2
2
sin2 + sin2 - sin2 =1-2 cos
, r = 4, r1 =
cos
B 2
sin
C 2
, r2 = 12 and r3 = 14.
Scheme of valuation Maths – IA (1) Given f: A → B is surjection i.e (codomain off) B = range of f = f (A) = cos 0° , cos
= 1,
π 6
1M π
π
π
4
3
2
cos , cos , cos
√3 1 1 , , ,D 2 √2 2
(2) f ( ) is defined ⟺ 4x - x 2 ≥ 0
1M
1M
⟺ x ( x – 4) ≤ 0 manabadi.com is not responsible for any inadvertent error that may have crept in the guess paper being published on NET. The published on net is for the information to the examinees. ] a paper ⟺ x ∈ [to 0, guess 4be This does not constitute Main Question paper 1M and should NOT follow the same. While all efforts 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 any shortcoming, defect or inaccuracy in the Guess Papers provided by Manabadi.com website.
This Document is provided by www.manabadi.com for FREE for the benefit of Intermediate students. Copying and redistribution by any company/ website is illegal and Manabadi.com has all rights to claim on such type of website or company.
(3) First find 2A and BT ⇒ next find 2A + BT
1M
⇒ next find 3BT – A
1M
Find
(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 + k⃗ = t 4i + mj + nk⃗ after simplification m = 10, n = 2
1M
(6) Vector eqn of the line ing through the point A( ⃗) and parallel to the vector ⃗ is r⃗ = ⃗ +
∴ required equation is
1M
r⃗ = 2i+3J+k + t 4i-2j+3E ; t ∈ R
∴ area of parallelogram is √94 Sq. units.
1M
1M
⃗ is a⃗ × b⃗
(7) area of the parallelogram having the sides ⃗
⃗ ;t∈R
1M
1M
(8). cos(360° − 20°) cos 40° + sin(180° + 20°) sin(180° − 40°) cos 20° cos 40° + - sin 20° sin 40° 1M cos(40° + 20°) cos cos B − sin A sin B = cos(A + B) = cos 60° 1
=2
1M
(9) cos + sin
= √2 cos
sin
= √2 − 1cos Multiplying √2 + 1 on both sides √2 + 1 sin
1M
= (2 – 1 ) cos
√2 sin + sin = cos cos − sin = √2 sin
1M
10. cosh x = cosh2 xis = 2not coshresponsible −1 manabadi.com for any inadvertent error that may have crept in the guess paper being published on NET. The guess paper published on net is for the information to the examinees. = 2 1 This does not constitute to be a Main Question paper and should NOT follow the same. While all efforts 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 any shortcoming, defect or inaccuracy in the Guess Papers provided by Manabadi.com website.
This Document is provided by www.manabadi.com for FREE for the benefit of Intermediate students. Copying and redistribution by any company/ website is illegal and Manabadi.com has all rights to claim on such type of website or company.
=
1M
cosh 2 - sinh (2 ) = 1 sinh (2 ) = cosh (2 ) - 1 =
-1
sinh 2
5√21
=
1M
2
Section - B (11) aI + bE =
1M
0
(aI + bE) =
3
2M
0
a3 I+3a2 bE =
3
1M
0
(12) AB = 4 a⃗ - 2b⃗ - 2c⃗ AC = -2 a⃗ - 4b⃗ - 2c⃗
2M
AD = -2 a⃗ - 2b⃗ + 4c⃗ [AB
4 −2 −2 AD ] = −2 4 −2 a b c −2 −2 4
AC
=0 1M ∴ four points A, B, C, D are coplanar.
1M B R
(13) Take the unit cube Let OP and AS are two diagonals of a cube OP = I + j + E’ 1M Ab = -I + j + E’ cos =
|
. ||
|
=
S A Q e
1M
(14) A + B = 45° tan(A+B) = tan 45° tan A+ tan B 1- tan A tan B
=1
tan A + tan B =1- tan A tan B
1M
(1)
LHS (1 + tan )(1 + tan ) manabadi.com is not responsible for any inadvertent error that may have crept in the guess paper 1 + tan A +on tanNET. B + tan A tan B paper published on net is for the information to the examinees. being published The guess This does 1 + 1not constitute to be a Main Question paper and should NOT follow the same. While all
efforts 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 any shortcoming, defect or inaccuracy in the Guess Papers provided by Manabadi.com website.
This Document is provided by www.manabadi.com for FREE for the benefit of Intermediate students. Copying and redistribution by any company/ website is illegal and Manabadi.com has all rights to claim on such type of website or company.
=2 2M Take A = B = 22 (1 + tan )(1 + tan ) = 2 tan
= √2 - 1
tan 22 ° = √2 – 1 (15) sin + √3 cos Divide with 2 sin + cos
√
−
G.S is −
1M
= √2
cos
=
1M
√
= cos
1M
= 2n ±
1M
X = 2n ±
+
= 2n +
, 2n -
16. tan
+ tan
1M
+ tan
1M
. .
tan
.
.
1M
.
tan (1)
1M
=
(17)
= tan
1M
ω+A cos A b2+ c 2- a2 2bc a 2R
1M
b2 + c 2 - a2
∆
1M
(a + b + c )
∆
1M 1M
(18) since f : A → B, g : B →c are bijections Then gof: A→ c is also bijection manabadi.com responsible ( )is :not C →A is a bijection for any inadvertent error that may have crept in the guess paper being published on NET. The guess paper published on net is for the information to the examinees. : B constitute → A, :C→ are abijections This does not toB be Main Question paper and should NOT follow the same. While all efforts 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 any shortcoming, defect or inaccuracy in the Guess Papers provided by Manabadi.com website.
This Document is provided by www.manabadi.com for FREE for the benefit of Intermediate students. Copying and redistribution by any company/ website is illegal and Manabadi.com has all rights to claim on such type of website or company.
Then o : C → A is also bijection ( ) & o have same domain, namely C. ) ( )=( Next we P.T ( o )( ) ∀ c C ∴( ) =& o 1M
3M 3M
(19) nth term 1 +2 + …………
=
(
)(
)
1M
Let 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 true adding (k + 1)th term on both the sides in eqn 2
∴ S(k + 1) is true
3M Conclusion: by the principle of finite m.I S(n) is rue for all n N
(20)
→ + + + 2 2
+ + + + + − − 2 2 − − 1 1 1 ( + + ) 2 − − 2 2 2 − − → − → − 1 0 0 −( + + ) 0 ( + + ) 2 2 0 −( + + ) =( +
+ )
(21) The augmented matrix is 2 −1 3 9 [ ]~ 1 1 1 6 1 −1 1 2 ↔ 1 1 1 6 [ ] ~ 2 −1 3 9 1 −1 1 2
1M
2M 2M
1M
2M
1M
1M
↔ not- 2responsible↔for any - inadvertent error that may have crept in the guess paper manabadi.com is being 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 all efforts 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 any shortcoming, defect or inaccuracy in the Guess Papers provided by Manabadi.com website.
This Document is provided by www.manabadi.com for FREE for the benefit of Intermediate students. Copying and redistribution by any company/ website is illegal and Manabadi.com has all rights to claim on such type of website or company.
1 ~ 0 0
1 −3 −2
1 6 1 −3 0 −4
1M
1 1 1 6 ~ 0 −3 1 −3 0 1 0 2 ↔ 1 1 1 6 ~ 0 1 0 2 0 −3 1 −3 → − → 1 0 1 4 ~ 0 1 0 2 0 0 1 3 → − 1 0 0 1 0 1 0 2 0 0 1 3 X=1 y=2 z=3
1M
1M − 3 1M
1M
(22) Compare with r̅ = a + r̅ = ̅ + ̅ a − ̅ = 10i + 2J + = i - 2J +2 ̅ = 3 i - 2J - 2 Find × ̅ = 12
1M
1M ̅
and a − ̅ Shortest distance =
[
=108 ̅ | × |
Shortest distance 9 units
2M ]
2M
1M
(23) A + B + C = 180° sin
+ sin
sin
+ sin
sin
+ sin 90° −
1 - cos
+ sin sin
1M sin
+ cos sin
1 - cos + cos
1M
− sin
1 - cos + sin 1 - cos + 2 cos
1M
1M
− sin sin
1M 1M
manabadi.com is not responsible for any inadvertent error that may have crept in the guess paper being published on NET. The guess paper published on net is for the information to the examinees. 1 – 2not cos constitute cos sin to be a Main1M This does Question paper and should NOT follow the same. While all efforts 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 any shortcoming, defect or inaccuracy in the Guess Papers provided by Manabadi.com website.
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(24) S =
= 21
− ( − )( − )( − ) = 84
∆= R= r=
1M
∆ ∆
= = =
=
1M
=4 ∆ ∆ ∆
1M
1M =
1M
= 12
1M
= 14
1M
The End
manabadi.com is not responsible for any inadvertent error that may have crept in the guess paper being 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 all efforts 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 any shortcoming, defect or inaccuracy in the Guess Papers provided by Manabadi.com website.
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) (ii)
1. If
Answer all Questions Each question carries 2 marks.
= 0,
,
,
,
10×2 = 20
and f: A → B is a surjection defined by f ( ) = cos x then find B.
2. Find the domain of the function f ( ) = √4 − −2
3. If A = 5 −1
1 −2 0 and B = 4 4
3 1 then find 2A + B T and 3B T – A 0 2
1 4. Define trace of a matrix and find the trace of A If A = 0 −
5. If ⃗ = 2i + 5j + ⃗ and ⃗ = 4i + mj +
2 − −1 2 2 1
⃗ are collinear vectors then find the values of m and n.
6. Find the vector equation of the line ing through point 2i + 3j + ⃗ and parallel to the vector 4i – 2j + ⃗
7. Find the area of the parallelogram having 2i – 3j and 3i - ⃗ as adjacent sides. 1 8. Prove that cos340°cos40° + sin 200°sin 140° = 2 9. If cos + sin = √2 cos prove that cos − sin = √2 sin 5
10. If cosh x= 2 find 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
5×4 = 20
11. If I = 1 0 and E = 0 1 then show that (aI+bE)3 = a3 I+ 3a2 bE
0 1 0 0 12. If ⃗, ⃗, ⃗, are non-coplanar vectors. Prove that the four points −a⃗+ 4b⃗ - 3c⃗ , 3 a⃗+ 2b⃗ - 5c⃗ , -3a⃗+ 8b⃗ - 5c⃗ and 3 a⃗+ 2b⃗+ c⃗ are coplanar.
13. Prove that the smaller angle
between any two diagonals of a circle is given by sin
=
manabadi.com is not responsible for any inadvertent error that may have crept in the guess paper being 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 all )(1 +available 14.have If A +been B = 45° then + tanpaper tan = 2onand hence deduceas theauthentic value of tanas 22 possible. ° efforts made toprove make that the 1 guess this website Manabadi or any staff persons will not be responsible for any loss to persons caused by any shortcoming, defect or inaccuracy in the Guess Papers provided by Manabadi.com website.
This Document is provided by www.manabadi.com for FREE for the benefit of Intermediate students. Copying and redistribution by any company/ website is illegal and Manabadi.com has all rights to claim on such type of website or company.
15. Solve sin + √3 cos = √2 16. Prove that tan-1 12 + tan-1 15 + tan-1 18 = 17. Prove that cot A + cot B + cot C =
π 4
a2 + b2 + c2 4∆
Section C II
Long answer type questions
(i) Answer any five questions (ii) Each question carries 7 marks
18. If f: A→ B , g : B→ C
are two bijections, then (gof)-1 = f -1 og -1
19. By mathematical Induction
20. Show that
− − 2 2
5× 7 = 35
∀ ∈
, prove that 1 + (1 + 2 ) + (1 + 2 + 3 ) + …………..n brackets = ( + 1) ( + 2) 12
2 − − 2
2 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 r⃗ = 6i+2J+2k + t i-2j+2k and r⃗ = -4i -k + 5 3i-2j+2k where s, t are scalars
23. If A, B, C are angles in a triangle, then prove that 24. If a = B, b = 14, C = 15 show that R =
A
B
C
A
2
2
2
2
sin2 + sin2 - sin2 =1-2 cos
, r = 4, r1 =
cos
B 2
sin
C 2
, r2 = 12 and r3 = 14.
Scheme of valuation Maths – IA (1) Given f: A → B is surjection i.e (codomain off) B = range of f = f (A) = cos 0° , cos
= 1,
π 6
1M π
π
π
4
3
2
cos , cos , cos
√3 1 1 , , ,D 2 √2 2
(2) f ( ) is defined ⟺ 4x - x 2 ≥ 0
1M
1M
⟺ x ( x – 4) ≤ 0 manabadi.com is not responsible for any inadvertent error that may have crept in the guess paper being published on NET. The published on net is for the information to the examinees. ] a paper ⟺ x ∈ [to 0, guess 4be This does not constitute Main Question paper 1M and should NOT follow the same. While all efforts 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 any shortcoming, defect or inaccuracy in the Guess Papers provided by Manabadi.com website.
This Document is provided by www.manabadi.com for FREE for the benefit of Intermediate students. Copying and redistribution by any company/ website is illegal and Manabadi.com has all rights to claim on such type of website or company.
(3) First find 2A and BT ⇒ next find 2A + BT
1M
⇒ next find 3BT – A
1M
Find
(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 + k⃗ = t 4i + mj + nk⃗ after simplification m = 10, n = 2
1M
(6) Vector eqn of the line ing through the point A( ⃗) and parallel to the vector ⃗ is r⃗ = ⃗ +
∴ required equation is
1M
r⃗ = 2i+3J+k + t 4i-2j+3E ; t ∈ R
∴ area of parallelogram is √94 Sq. units.
1M
1M
⃗ is a⃗ × b⃗
(7) area of the parallelogram having the sides ⃗
⃗ ;t∈R
1M
1M
(8). cos(360° − 20°) cos 40° + sin(180° + 20°) sin(180° − 40°) cos 20° cos 40° + - sin 20° sin 40° 1M cos(40° + 20°) cos cos B − sin A sin B = cos(A + B) = cos 60° 1
=2
1M
(9) cos + sin
= √2 cos
sin
= √2 − 1cos Multiplying √2 + 1 on both sides √2 + 1 sin
1M
= (2 – 1 ) cos
√2 sin + sin = cos cos − sin = √2 sin
1M
10. cosh x = cosh2 xis = 2not coshresponsible −1 manabadi.com for any inadvertent error that may have crept in the guess paper being published on NET. The guess paper published on net is for the information to the examinees. = 2 1 This does not constitute to be a Main Question paper and should NOT follow the same. While all efforts 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 any shortcoming, defect or inaccuracy in the Guess Papers provided by Manabadi.com website.
This Document is provided by www.manabadi.com for FREE for the benefit of Intermediate students. Copying and redistribution by any company/ website is illegal and Manabadi.com has all rights to claim on such type of website or company.
=
1M
cosh 2 - sinh (2 ) = 1 sinh (2 ) = cosh (2 ) - 1 =
-1
sinh 2
5√21
=
1M
2
Section - B (11) aI + bE =
1M
0
(aI + bE) =
3
2M
0
a3 I+3a2 bE =
3
1M
0
(12) AB = 4 a⃗ - 2b⃗ - 2c⃗ AC = -2 a⃗ - 4b⃗ - 2c⃗
2M
AD = -2 a⃗ - 2b⃗ + 4c⃗ [AB
4 −2 −2 AD ] = −2 4 −2 a b c −2 −2 4
AC
=0 1M ∴ four points A, B, C, D are coplanar.
1M B R
(13) Take the unit cube Let OP and AS are two diagonals of a cube OP = I + j + E’ 1M Ab = -I + j + E’ cos =
|
. ||
|
=
S A Q e
1M
(14) A + B = 45° tan(A+B) = tan 45° tan A+ tan B 1- tan A tan B
=1
tan A + tan B =1- tan A tan B
1M
(1)
LHS (1 + tan )(1 + tan ) manabadi.com is not responsible for any inadvertent error that may have crept in the guess paper 1 + tan A +on tanNET. B + tan A tan B paper published on net is for the information to the examinees. being published The guess This does 1 + 1not constitute to be a Main Question paper and should NOT follow the same. While all
efforts 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 any shortcoming, defect or inaccuracy in the Guess Papers provided by Manabadi.com website.
This Document is provided by www.manabadi.com for FREE for the benefit of Intermediate students. Copying and redistribution by any company/ website is illegal and Manabadi.com has all rights to claim on such type of website or company.
=2 2M Take A = B = 22 (1 + tan )(1 + tan ) = 2 tan
= √2 - 1
tan 22 ° = √2 – 1 (15) sin + √3 cos Divide with 2 sin + cos
√
−
G.S is −
1M
= √2
cos
=
1M
√
= cos
1M
= 2n ±
1M
X = 2n ±
+
= 2n +
, 2n -
16. tan
+ tan
1M
+ tan
1M
. .
tan
.
.
1M
.
tan (1)
1M
=
(17)
= tan
1M
ω+A cos A b2+ c 2- a2 2bc a 2R
1M
b2 + c 2 - a2
∆
1M
(a + b + c )
∆
1M 1M
(18) since f : A → B, g : B →c are bijections Then gof: A→ c is also bijection manabadi.com responsible ( )is :not C →A is a bijection for any inadvertent error that may have crept in the guess paper being published on NET. The guess paper published on net is for the information to the examinees. : B constitute → A, :C→ are abijections This does not toB be Main Question paper and should NOT follow the same. While all efforts 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 any shortcoming, defect or inaccuracy in the Guess Papers provided by Manabadi.com website.
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Then o : C → A is also bijection ( ) & o have same domain, namely C. ) ( )=( Next we P.T ( o )( ) ∀ c C ∴( ) =& o 1M
3M 3M
(19) nth term 1 +2 + …………
=
(
)(
)
1M
Let 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 true adding (k + 1)th term on both the sides in eqn 2
∴ S(k + 1) is true
3M Conclusion: by the principle of finite m.I S(n) is rue for all n N
(20)
→ + + + 2 2
+ + + + + − − 2 2 − − 1 1 1 ( + + ) 2 − − 2 2 2 − − → − → − 1 0 0 −( + + ) 0 ( + + ) 2 2 0 −( + + ) =( +
+ )
(21) The augmented matrix is 2 −1 3 9 [ ]~ 1 1 1 6 1 −1 1 2 ↔ 1 1 1 6 [ ] ~ 2 −1 3 9 1 −1 1 2
1M
2M 2M
1M
2M
1M
1M
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This Document is provided by www.manabadi.com for FREE for the benefit of Intermediate students. Copying and redistribution by any company/ website is illegal and Manabadi.com has all rights to claim on such type of website or company.
1 ~ 0 0
1 −3 −2
1 6 1 −3 0 −4
1M
1 1 1 6 ~ 0 −3 1 −3 0 1 0 2 ↔ 1 1 1 6 ~ 0 1 0 2 0 −3 1 −3 → − → 1 0 1 4 ~ 0 1 0 2 0 0 1 3 → − 1 0 0 1 0 1 0 2 0 0 1 3 X=1 y=2 z=3
1M
1M − 3 1M
1M
(22) Compare with r̅ = a + r̅ = ̅ + ̅ a − ̅ = 10i + 2J + = i - 2J +2 ̅ = 3 i - 2J - 2 Find × ̅ = 12
1M
1M ̅
and a − ̅ Shortest distance =
[
=108 ̅ | × |
Shortest distance 9 units
2M ]
2M
1M
(23) A + B + C = 180° sin
+ sin
sin
+ sin
sin
+ sin 90° −
1 - cos
+ sin sin
1M sin
+ cos sin
1 - cos + cos
1M
− sin
1 - cos + sin 1 - cos + 2 cos
1M
1M
− sin sin
1M 1M
manabadi.com is not responsible for any inadvertent error that may have crept in the guess paper being published on NET. The guess paper published on net is for the information to the examinees. 1 – 2not cos constitute cos sin to be a Main1M This does Question paper and should NOT follow the same. While all efforts 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 any shortcoming, defect or inaccuracy in the Guess Papers provided by Manabadi.com website.
This Document is provided by www.manabadi.com for FREE for the benefit of Intermediate students. Copying and redistribution by any company/ website is illegal and Manabadi.com has all rights to claim on such type of website or company.
(24) S =
= 21
− ( − )( − )( − ) = 84
∆= R= r=
1M
∆ ∆
= = =
=
1M
=4 ∆ ∆ ∆
1M
1M =
1M
= 12
1M
= 14
1M
The End
manabadi.com is not responsible for any inadvertent error that may have crept in the guess paper being 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 all efforts 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 any shortcoming, defect or inaccuracy in the Guess Papers provided by Manabadi.com website.
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