MATHS

S. No	Topic	Division	Marks
1.	Functions	7 + 2 + 2	11
2.	Mathematical Induction	7	07
3.	Addition Of Vectors		08
4.	Multiplication Of Vectors	7 + 4 + 2	14
5.	Trigonometric Equations (upto transformations)	7 + 4 + 2x2	15
6.	Trigonometric Equations (remaining)	4	04
7.	Inverse Trigonometric Functions	4	04
8.	Hyperbolic Functions	2	02
9.	Properties Of Triangles	7 + 4 + 2	13
10.	Heights and distances	7	07
11.	Complex Numbers	2	02
12.	Demoviours Theorem	7	07
13.	Trigonometric Expansion	4	04
	Total		97

MATHS INTER 2nd YEAR II B BLUE PRINT

S.No	TOPIC	DIVISION	MARKS
1.	Circles	2x7 + 2	16
2.	System Of Circles	1X7	07
3.	Parabola	7 + 2	09
4.	Ellipse	4 + 2	06
5.	Hyperbola	4 + 2	06
6.	Polar Co-ordinates	4 + 2	06
7.	Successive Differentiations	4 + 2	06
8.	Integrations	7 + 4 + 2x2	15
9.	Definite integration	7 + 2	09
10	Numerical Integration	7 + 2	09
11.	Differential Equations	4 + 4	08
	Total	97	97

II B BLUE PRINT

S.No	TOPIC	DIVISION	MARKS
1.	Circles	2x7 + 2	16
2.	System Of Circles	1X7	07
3.	Parabola	7 + 2	09
4.	Ellipse	4 + 2	06
5.	Hyperbola	4 + 2	06
6.	Polar Co-ordinates	4 + 2	06
7.	Successive Differentiations	4 + 2	06
8.	Integrations	7 + 4 + 2x2	15
9.	Definite integration	7 + 2	09
10	Numerical Integration	7 + 2	09
11.	Differential Equations	4 + 4	08
	Total	97	97

MATHS INTER 2nd YEAR IIA BLUE PRINT

S.No	TOPIC	DIVISION	MARKS
1.	Quadratic Expressions	4+2	06
2.	Theory of Equations	7+2	09
3.	Matrices	2x7 + 4 + 2x2	22
4.	Permutations & Combinations	4 + 2 + 7	15
5.	Binomial Theorems	7 + 4 + 2	13
6.	Partial Fractions	4	04
7.	Exponential & Logarithmic series	4 + 2	06
8.	Probability	4 + 7 + 2	13
	Total

INTERMEDIATE PUBLIC EXAMINATION

MARCH 2006

MATHEMATICS PAPER 1(A)

ALGEBRA, VECTOR ALGEBRA AND TRIGONOMETRY

TIME: 3 Hrs. Max. Marks: 75.

SECTION – A

VERY SHORT ANSWER TYPE QUESTIONS
Answer all questions. Each question carries 2 marks.

1. Find the inverse of the function f(x) = 5^x.

2. Find the value of cos²45 – sin ²15.

3. Prove that sin (π/5). Sin (2π/5). Sin (3π/5) Sin (4π/5) = 5/16.

4. Prove that (cosh x – sinh x )ⁿ = cosh nx – sinh nx.

5. In a triangle ABC, So that ( b – a cos C ) sin A = a cos A sin C.

6. Simplify

( cos θ – i sin θ)⁷
(Sin 2θ – i cos 2θ)⁴

7. Show that sin 6θ = 6 cos⁵ θ sin θ – 20 cos³ sin³ θ + 6 cos θ sin ⁵ θ.

8. The position vector of the two points A and B are i + j + k
and i + j – 4 k respectively. Find the position vector of the point
which divides AB in the ratio 2: 3 internally.

9. Find the vector equation of the plane passing through the points
(1, -2, 5) (0, -5, -1) (-3. 5. 0)

10. Find the angle made by the straight line through the points
(1, -3, 2) and ( 3, -5, 1) with the x – axis.

SECTION – B

SHORT ANSWER TYPE QUESTIONS.

Attempt any 5 questions, Each question carries 4 marks.

11. A functioned is defined as follows

3x – 2 if x > 3

f ( x) = x² – 2 if -2 ≤ x ≤ 2.

2x + 1 if x < - 3

then find the values of f(4) , f( 2.5) f(-2) , f( 0).

12.Show that

__1____ _ ______1___ _ ____2_____ = 0
√(12- √140) √(8 – √60) √ (10+ √ 84)

13. If (3.7)^x = (0.037)y = 10,000, then find the value of 1/x – 1/y.

14. solve the equation tan θ + tan 2θ + √3 tan θ tan 2θ = √3

15. If sin ^{– 1} x + sin ^–1 y + sin ^–1 z = π, prove that
x √ (1 – x²) + y√ (1 – y²) + z√ (1-z²) = 2 xyz.

16. if A = (2, 4, -1) B = ( 4, 5, 1) and C = ( 3, 6, -3 ) are the vertices
of a triangle ABC, find the lengths of the sides and
show that it is a right angled triangle.

17.Find the area of triangle with the points A ( 1,2,3) B ( 2, 3, 1)
and C ( 3,1,2)

SECTION – C

LONG ANSWER TYPE QUESTIONS
Attempt any 5 questions, each question carries 7 marks.

18. If f: A → B , and g: B → C , are two bijective functions, then
prove that ( g o f ) ^-1 = f ^-1 o g ^-1 .

19, By using mathematical induction show that
1/1.4 + 1/ 4.7 + 1/ 7. 10 + ………… upto n terms. = n / ( 3n + 1 )
for all n є N .

20. If A + B + C = 180 ^o then, prove that
sin ² A/2 + sin ² B/2 – sin ² C/2 = 1 – cos A/2 cos B/2 cos C/2.

21. in ∆ ABC, show that r₁ + r₂ + r₃ = 4 R.

22. from the top of a tree on the bank of a lake, an aeroplane
in the sky makes an angle of elevation ‘α’ and its image in the
river makes an angle of depression ‘ β’ . If the height of the tree
from the water surface is ‘a’ and that of the height of the aeroplane h,
show that h = a sin (α + β) /sin(β-α).

23.a) if (x+iy)=1/1+cosθ+I sin θ,then show that 4x²=1.

b) if the point ‘P’ denotes the complex number z=x+iy
in the argand plane and if z-i/z-1 is a purely imaginary number,
find the locus of ‘P’.

24.By vector method , prove that ∆=√{s(s-a)(s-b)(s-c)}.

INTERMEDIATE PUBLIC EXAMINATION, MAY 2006

MATHEMATICS PAPER 1(A)

ALGEBRA, VECTOR ALGEBRA AND TRIGONOMETRY

TIME: 3 Hrs. Max. Marks: 75.

SECTION – A

VERY SHORT ANSWER TYPE QUESTIONS

Answer all questions. Each question carries 2 marks.

1. Find the range of the function f: A à R where A = {1, 2, 3, 4} and f(x) = x²+x-2.

2. Find a unit vector parallel to the resultant of the vectors, r₁= 2i + 4j – 5k and
r₂ = i + 2j + 3k.

3. If the position vectors of the vertices A, B, C of ∆ ABC, are 7j + 10k, -i + 6j + 6k
and -4i +9j +6k respectively. Prove that the triangle is right angled and isosceles.

4. If a = I + j + k and b = 2i + 3j + k , find the length of the projection of b on a and
the length of the projection of a on b.

5. Prove that tan (A + 135) tan (A-135) = -1.

6. If tan A = 8/25, find the values of sin 2A and cos 2A.

7. If cosh x = 5/2, find the value of cosh 2x.

8. In ∆ ABC, express ∑ r1 cot ( A/2) in terms of ‘s’.

9. Find the values of (√3/2 – i/2)¹².

10. Expand cos 4A in powers of cos A.

SECTION – B

SHORT ANSWER TYPE QUESTIONS.

Attempt any 5 questions, Each question carries 4 marks.

11. f: R à R are defined by f(x) = 3x – 2 and g(x) = x2 + 1, then find the following

i) (g o f-1) (2) ii) (g o f ) ( x-1)

12. If x = (√3 – √2)/ (√3 +√2), y =(√3 + √2)/ (√3 +√2) then
show that x² + xy + y² = 99.

13. If x = log _2a a, y = log _3a 2a and z = log _4a 3a, then
show that xyz + 1 = 2yz.

14. If, a, b, c are non coplanar vectors, show that a + a2b + c,
-a + 3b – 4c, a – b + 2c are non coplanar.

15. If a = 2i + 3j + 4k , b = I + j – k , compute a X (b X c)
and verify that is perpendicular to a.

16. If tan ( π sin A ) = cot (π cot A), then show that
2 2

4sin ( A + π ) = ±1/√2 .
4

17. Show that Tan ^‑1 1/8 + Tan ^‑1 1/2 + Tan ^‑1 1/5 = π/4.

SECTION – C

LONG ANSWER TYPE QUESTIONS

Attempt any 5 questions, each question carries 7 marks.

18. Let f : A à B and g: B à C be bisections, Prove that
g o f : A à C is also bijection.

19. using the principles of Mathematical Induction, prove that 2.3+ 3.4 + 4. 5 + ……….
upto π Terms = n (n2 + 6n + 11)/3, for all n € N.

20. For any vectors,a, b , c prove that a X ( b X c) = ( a. c ) b – ( a . b ) c.

21. If A + B + C = 180⁰, prove that
cos A + cos B + - cos C = - 1 + 4 cos A/2 cos B/2 cos C/2 .

22. In ∆ ABC, prove that r + r₁+ r₂ - r₃ = 4 R cos C.

23. From the top of a tree on the bank of a lake, an aeroplane in the
sky makes an angle of elevation A and the of the height of the aeroplane
is ‘h’. Show that h = a Sin (A + B)
Sin (A - B)

24. If the amplitude of ( z – 2 )/ ( 2 – 6i ) is π/2 , find the
equation of locus of z.

FIRST YEAR MATHEMATICS – I B
(Blue print)

TYPE OF QUESTION	VSA	SA	LA	TOTAL
No. of questions to be given	10	7	7	24
No. of questions to be answered	10	5	5	20
Marks Allotted	20/20	20/28	35/49	75/97
Estimated time (In minutes)	10 x 4 = 40	5 x 9 = 45	5 x 18 = 90	175

S. No	Weightage to content: units / Sub-units:	marks
1	Locus	04
2	Change of Axes	04
3	Straight Lines	15
4	Pair of Straight Lines	14
5	3D- Coordinates	02
6	Direction cosines & Direction ratios – 3D Geometry	07
7	The Plane – 3D Geometry	02
8	Functions, Limits, Continuity	06
9	Differentiations	17
10	Errors – Approximations	02
11	Rate measure	04
12	Tangent – Normal	09
13	Maxima – Minima	07
14	Partial Differentiation	04
	Total	97

Scheme of Options:

Section – A: No Option

Section – B: 5 out of 7

Section – C: 5 out of 7

INTER 1stYEAR MATHS (1 A) PREVIOUS PAPERS

2007	2007		MARCH 2006	MAY 2006
2005	2005 june		2004	2004

INTER FIRST YEAR 1 A BLUE PRINT

1st year intermediate model papers by board of intermediate of education


INTER 1stYEAR MATHS (1 B) PREVIOUS PAPERS

2007	2007		2006	2006
2005	2005		2004	2004

INTER FIRST YERAR I B BLUE PRINT