Download OU Intermediate 2018 Core Subjects 0166 Mathematics Question Paper

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Total No. of Questions—24

Total No. of Printed Pages—4 Regd. No.

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Part I

MATHEMATICS Paper I(A)

(English Version)

Time : 3 Hours] [Max. Marks : 75

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

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SECTION A

I. Very Short Answer Type Questions : 10x2=20

(i) Answer ALL questions.

(ii) Each question carries TWO marks.

1. If a ∈ R, f: R → R defined by f(x) = ax + b (a ≠ 0), find f^-1.

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2. Find the domain of real valued function : f(x) = √(x - x²).
3. If A =
   

   2	4
   -1	k

   and A² = 0, then find the value of P.
4. Find the rank of the following matrix :
   

   1	1	1
   1	1	1
   1	1	1

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5. Find the vector equation of the line joining the points 2i + j + 3k and -4i + 8j - k.
6. If a = 2i + 5j + k and b = 4i + mj + nk are collinear Vectors then find m and n.
7. If a = 2i - 3j + k and b = i + 4j - 2k, then find (a+b)X(a-b).
8. Prove that : (cos9° + sin9°) / (cos9° - sin9°) = cot36°.
9. Prove that : sin50° sin70° + sin10° = 0.

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10. Show that : (coshx + sinhx)ⁿ = cosh (nx) + sinh (nx), for any n ∈ R.

SECTION B

II. Short Answer Type Questions : 5x4=20

(i) Answer ANY FIVE questions.

(ii) Each question carries FOUR marks.

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11. If a, b, c are non-coplanar vectors and if
    A =

    a	b	c
    a1	b1	c1
    a2	b2	c2

    is a non-singular matrix, then prove that A^-1 = ?:z::
12. If a, b, c are non-coplanar vectors, then prove that : - a + 4b - 3c, 3a + 2b - 5c, - 3a + 8b - 5c, -3a+2b+c are coplanar
13. For any two Vectors a and b, show that : (1+|a|²)(1+|b|²) = |1-a.b|² +|a+b+axb|²
14. Prove that : sin²18° = (5 - √5) / 8

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15. Solve the equation : sinx + √3 cosx = √5
16. Prove that : tan^-1(1/2) + tan^-1(1/5) + tan^-1(1/8) = π/4
17. Prove that : cotA + cotB + cotC = (a²+b²+c²) / (4Δ)

SECTION C

III. Long Answer Type Questions : 5x7=35

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(i) Answer ANY FIVE questions.

(ii) Each question carries SEVEN marks.

18. Let f : A → B, I_A and I_B be identity functions on A and B respectively, then prove that f o I_A = f = I_B o f.
19. Show that 49ⁿ + 16n - 1 is divisible by 64 for all positive integers ‘n’.
20. Show that :
    

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    a	b	c
    b	c	a
    c	a	b

    = a³ + b³ + c³ - 3abc
21. Solve the following system of equations by Cramer’s rule :
    2x — y + 3z = 8
    -x + 2y + z = 4
    3x + y - 4z = 0.

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22. If a = i - 2j + 3k, b = 2i + j + k, c = i + j + 2k, then find |(axb)xc|.
23. If A + B + C = 2S then prove that cos(s - A) + cos(s - B) + cos(s - C) + coss = 4cos(A/2) cos(B/2) cos(C/2)
24. If P₁, P₂, P₃ are altitudes drawn from vertices A, B, C to the opposite sides of a triangle respectively, then show that :
    (1/P₁² + 1/P₂² + 1/P₃²) = (a²+b²+c²) / (8Δ³)

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