Diploma 1st Sem Mathematics 1 Past Paper January 2024 PDF

# JANUARY 2024 ## MATHEMATICS-I - **Time Allowed:** 2.5 Hours - **Full Marks:** 60 - **104(N)** ### Instructions: - Answer to Question No. 1 of Group A must be written in the main answer script. - In Question No. 1, out of 2 marks for each MCQ, 1 mark is allotted for right answer and 1 mark is all...

# JANUARY 2024 ## MATHEMATICS-I - **Time Allowed:** 2.5 Hours - **Full Marks:** 60 - **104(N)** ### Instructions: - Answer to Question No. 1 of Group A must be written in the main answer script. - In Question No. 1, out of 2 marks for each MCQ, 1 mark is allotted for right answer and 1 mark is allotted for correct explanation of the answer. - Answer any Five (05) Questions from Group-B. ## Group-A 1. Choose the correct answer from the given alternatives and explain your answer (any ten) - **2×10=20** i) If for the vectors *a* and *b*, |*a*| = 1, |*b*| = 2 and *a* · *b* = √3, then angle between the vectors *a* and *b* is - (a) 90° - (b) 60° - (c) 45° - (d) 30°. ii) If one root of the equation x² - 6x + m = 0 be double the other, then the value of m is - (a) 4 - (b) 6 - (c) 8 - (d) -8. iii) The value of 2log₂5 + 9log₃√3 is - (a) 9 - (b) 7 - (c) 8 - (d) none of these. iv) The value of the expression ω²(1+i)(iω-1) is - (a) 1 - (b) -2 - (c) -1 - (d) 0. v) The value of *k* ⋅ (*i* × *j*) is - (a) 1 - (b) 0 - (c) -1 - (d) none of these. vi) If *z* = 2 + *i*√3, then *z* · *z*¯ is - (a) 7 - (b) 1 - (c) -7 - (d) 0. vii) The coefficient of x³ in the expansion of (1 + 3x + 3x² + x³)¹⁰ is - (a) ¹⁰C₃ - (b) ¹⁰C₂ - (c) ³⁰C₃ - (d) ³⁰C₂. viii) If the vectors 2*i* - 3*j* + *k* and *m* *i* - *j* + *m* *k* are perpendicular to each other, then the value of *m* is - (a) 1 - (b) -1 - (c) 2 - (d) -2. ix) If cos(sin⁻¹(1/5) + cos⁻¹x) = 0, then the value of x is - (a) 0 - (b) 1 - (c) 4/5 - (d) 1/5 x) If cos 3x = sin 2x, then x = - (a) 15° - (b) 18° - (c) 30° - (d) 22.5° xi) If f(x - 2) = 2x² + 3x - 5, then f(-1) = - (a) 0 - (b) 1 - (c) -1 - (d) 2. xii) The domain of the function 1/√(x-2)(3-x) is - (a) 2 ≤ x ≤ 3 - (b) 2 < x ≤ 3 - (c) 2 ≤ x < 3 - (d) 2 < x < 3. xiii) lim_(x→π/2) (cot x)/(π/2 - x) = - (a) -1 - (b) 0 - (c) 1 - (d) none of these. xiv) If f(x) = logₑ*x* + e^lox, then f'(x) is - (a) e^x + 1 - (b) e^x + x - (c) 2 - (d) none of these. xv) The function (3-x)(x-1) is maximum for x = - (a) 1 - (b) 2 - (c) 3 - (d) 4. ## Group-B **Answer any Five (05) Questions** 2. i) If *α* and *β* be the roots of the equation x² - 3x + 2 = 0, find the equation whose roots are 1/*α* and 1/*β*. ii) The fifth term in the expansion of (x² - 1/x)^n is independent of x. Find n. iii) Prove that √1 + √-1 = √2, where i = √-1. **(3 + 3 + 2)** 3. i) If *a* = 2*i* + *j* - *k*, *b* = *i* - 2*j* - 2*k* and *c* = 3*i* - 4*j* + 2*k*, find the projection of *a* + *c* in the direction of *b*. ii) Prove that 2log(*a* + *b*) = 2loga + log(1 + *b*²/*a*²). iii) If ω³ = 1 and 1 + ω + ω² = 0, find the value of ω²⁰²² + ω²⁰²³ + ω²⁰²⁴. iv) if tan⁻¹(1/2) + 3 + ω + ω² = 0, find the value of sin θ. **(2 + 3 + 1 + 2)** 4. i) If log₃x = 1/9, find the value of x. ii) Find the number of terms in the expansion of (x + y)⁷(x - y)⁷. iii) Find the modulus of (*a* - *i*b)², where *i* = √-1. iv) Prove that sec²(tan√5) + cosec²(cot⁻¹5) = 32. **(2 + 2 + 2 + 2)** 5. i) Find a unit vector perpendicular to both the vectors *i* - 2*j* + 3*k* and 2*i* + *j* + *k*. ii) If one root of the equation x² + *a*x + 8 = 0 is 4 and the roots of the equation x² + *a*x + *b* = 0 are equal, find the value of *b*. iii) If tan x tan 5x = 1, prove that tan 3x = 1. **(3 + 3 + 2)** 6. i) The position vectors of A, B, C, D are given by the vectors *i* + *j* + *k*, 2*i* + 3*j*, 3*i* + 5*j* - 2*k* and *k* - *j*. Prove that AB and CD are parallel vectors. ii) If tan(*A* + *B*) = 1/2 and tan(*A* - *B*) = 1, find the value of tan 2*A*. iii) Show that sin(*x* + *y*) / sin(*x* - *y*) = (tan *x* + tan *y*) / (tan *x* - tan *y*). **(3 + 2 + 3)** 7. i) If f(*x*) = log₂ *x* and *q*(*x*) = x², find f(*q*(2)). ii) If y = x² and x² d²y/dx² = *a*y, find the value of *a*. iii) Find the derivative of x² with respect to x. iv) If y = log cot x tan x, prove that dy/dx = 0. **(2 + 2 + 2 + 2)** 8. i) Evaluate: lim_(x→0) (3*x* - 1) / (√9 + *x* - 3) ii) Prove that sin 3x cos ecx - cos 3x sec x = 2. iii) Prove that the function log(x + √x² + 1) is an odd function. iv) Find the value of (1/2) sin⁻¹(1/2) - (1/2) cos⁻¹(1/2) **(3 + 2 + 2 + 1)** 9. i) A parachutist falls through a distance x = logₑ(6 - 5e⁻ᵗ) in the tth second of its motion. Find dx/dt at t = 0. ii) If sin x + sin²x = 1, prove that cot x + cot²x = 1. iii) If y = e^sin⁻¹t and x = e^cos⁻¹t, prove that dy/dx is constant. **(3 + 3 + 2)**