Podcast
Questions and Answers
If for the vectors a and b, |a|=1, |b| = 2 and a.b=√3, then angle between the vectors a and b is
If for the vectors a and b, |a|=1, |b| = 2 and a.b=√3, then angle between the vectors a and b is
- 60° (correct)
- 90°
- 45°
- 30°
If one root of the equation x² - 6x + m = 0 be double the other, then the value of m is
If one root of the equation x² - 6x + m = 0 be double the other, then the value of m is
- -8
- 8 (correct)
- 4
- 6
The value of 2log₂5 + 9log₃√3 is
The value of 2log₂5 + 9log₃√3 is
- 8 (correct)
- 9
- 7
- none of these
The value of the expression ω²(1 + i)(iω -1) is
The value of the expression ω²(1 + i)(iω -1) is
If z=2+i√3, then z.z* is
If z=2+i√3, then z.z* is
The coefficient of x³ in the expansion of (1 + 3x + 3x² + x³)¹⁰ is
The coefficient of x³ in the expansion of (1 + 3x + 3x² + x³)¹⁰ is
If the vectors 2i-3j+k and mi-j+mk are perpendicular to each other, then the value of m is
If the vectors 2i-3j+k and mi-j+mk are perpendicular to each other, then the value of m is
If cos(sin⁻¹(1/5) + cos⁻¹(x)) = 0, then the value of x is
If cos(sin⁻¹(1/5) + cos⁻¹(x)) = 0, then the value of x is
If cos 3x = sin 2x, then x is
If cos 3x = sin 2x, then x is
If f(x-2)=2x²+3x-5, then f(-1) is
If f(x-2)=2x²+3x-5, then f(-1) is
The domain of the function 1/√(x-2)(3-x) is
The domain of the function 1/√(x-2)(3-x) is
Lim_(x→(π/2)) (cot x) / (π/2 -x) is
Lim_(x→(π/2)) (cot x) / (π/2 -x) is
If f(x) = logeˣ + e^lox , then f'(x) is
If f(x) = logeˣ + e^lox , then f'(x) is
The function (3-x)(x-1) is maximum for x =
The function (3-x)(x-1) is maximum for x =
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.
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.
The fifth term in the expansion of (x² - 1/x)ⁿ is independent of x. Find n.
The fifth term in the expansion of (x² - 1/x)ⁿ is independent of x. Find n.
Prove that √i + √-i = √2 where i = √-1.
Prove that √i + √-i = √2 where i = √-1.
If a = 2i+j-k, b=i-2j-2k and c=3i-4j+2k, find the projection of a + c in the direction of b.
If a = 2i+j-k, b=i-2j-2k and c=3i-4j+2k, find the projection of a + c in the direction of b.
Prove that 2log(a+b) = 2loga+log(1 + b²/a²).
Prove that 2log(a+b) = 2loga+log(1 + b²/a²).
If ω³ = 1 and 1 + ω + ω² = 0, find the value of ω²⁰²² + ω²⁰²³ +ω²⁰²⁴
If ω³ = 1 and 1 + ω + ω² = 0, find the value of ω²⁰²² + ω²⁰²³ +ω²⁰²⁴
If tan⁻¹(1/2)+ 4 tan⁻¹(1/3) = θ/2, find the value of sin θ.
If tan⁻¹(1/2)+ 4 tan⁻¹(1/3) = θ/2, find the value of sin θ.
If log₃(x) = 1/9, find the value of x.
If log₃(x) = 1/9, find the value of x.
Find the number of terms in the expansion of (x + y)⁷ (x - y)⁷.
Find the number of terms in the expansion of (x + y)⁷ (x - y)⁷.
Find the modulus of (a-ib)², where i = √-1.
Find the modulus of (a-ib)², where i = √-1.
Prove that sec²(tan⁻¹√5) + cosec²(cot⁻¹5) = 32.
Prove that sec²(tan⁻¹√5) + cosec²(cot⁻¹5) = 32.
Find a unit vector perpendicular to both the vectors i -2j+3k and 2i+j+k.
Find a unit vector perpendicular to both the vectors i -2j+3k and 2i+j+k.
If one root of the equation x² + ax + 8 = 0 is 4 and the roots of the equation x² + ax + b = 0 are equal, find the value of b.
If one root of the equation x² + ax + 8 = 0 is 4 and the roots of the equation x² + ax + b = 0 are equal, find the value of b.
If tan x tan 5x = 1, prove that tan 3x = 1.
If tan x tan 5x = 1, prove that tan 3x = 1.
The position vectors of A, B, C, D are given by the vectors i+j+k, 2i+3j, 3i+5j-2k and k-j. Prove that AB and CD are parallel vectors.
The position vectors of A, B, C, D are given by the vectors i+j+k, 2i+3j, 3i+5j-2k and k-j. Prove that AB and CD are parallel vectors.
If tan(A+B) = 1/2 and tan(A-B) = 1, find the value of tan 2A.
If tan(A+B) = 1/2 and tan(A-B) = 1, find the value of tan 2A.
Show that sin(x + y) / sin(x - y) = (tan x + tan y) / (tan x - tan y).
Show that sin(x + y) / sin(x - y) = (tan x + tan y) / (tan x - tan y).
If f(x) = log₂ x and q(x) = x², find f(q(2)).
If f(x) = log₂ x and q(x) = x², find f(q(2)).
If y = x² e^x and x² d²y / dx² = ay, find the value of a.
If y = x² e^x and x² d²y / dx² = ay, find the value of a.
Find the derivative of x² with respect to x.
Find the derivative of x² with respect to x.
If y = logₓ(cot x tan x), prove that dy / dx = 0.
If y = logₓ(cot x tan x), prove that dy / dx = 0.
Evaluate: lim_(x→0) (3x -1) / (√9 + x - 3)
Evaluate: lim_(x→0) (3x -1) / (√9 + x - 3)
Prove that sin 3x cosec x - cos 3x sec x = 2.
Prove that sin 3x cosec x - cos 3x sec x = 2.
Prove that the function log(x + √(x² + 1)) is an odd function.
Prove that the function log(x + √(x² + 1)) is an odd function.
Find the value of (1/2)sin⁻¹(1/2) - (1/2)cos⁻¹(1/2)
Find the value of (1/2)sin⁻¹(1/2) - (1/2)cos⁻¹(1/2)
A parachutist falls through a distance x = log(6 - 5e⁻ᵗ) in the tᵗʰ second of its motion. Find dx / dt at t = 0.
A parachutist falls through a distance x = log(6 - 5e⁻ᵗ) in the tᵗʰ second of its motion. Find dx / dt at t = 0.
If sin⁴ x + sin² x = 1, prove that cot⁴ x + cot² x = 1.
If sin⁴ x + sin² x = 1, prove that cot⁴ x + cot² x = 1.
If y = e^sin⁻¹x and x = e^cos⁻¹t, prove that dy / dx is constant.
If y = e^sin⁻¹x and x = e^cos⁻¹t, prove that dy / dx is constant.
Flashcards
Dot Product
Dot Product
The dot product of two vectors is the product of their magnitudes and the cosine of the angle between them.
Angle Between Vectors
Angle Between Vectors
The angle between two vectors can be found using the dot product formula: cos(theta) = (a · b) / (|a| |b|).
Double Root in Quadratic Equation
Double Root in Quadratic Equation
A quadratic equation with one root double the other can be solved using the relationship between roots and coefficients.
Logarithm Simplification
Logarithm Simplification
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Cube Root of Unity (ω)
Cube Root of Unity (ω)
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Cross Product of Vectors
Cross Product of Vectors
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Conjugate of a Complex Number
Conjugate of a Complex Number
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Binomial Theorem
Binomial Theorem
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Perpendicular Vectors
Perpendicular Vectors
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Inverse Trigonometric Functions
Inverse Trigonometric Functions
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Trigonometric Identities
Trigonometric Identities
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Domain of a Function
Domain of a Function
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Limit of a Function
Limit of a Function
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Derivative of a Function
Derivative of a Function
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Function
Function
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Odd Function
Odd Function
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Function Transformation
Function Transformation
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Parabola
Parabola
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Vertex of a Parabola
Vertex of a Parabola
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Axis of Symmetry
Axis of Symmetry
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Quadratic Equation
Quadratic Equation
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Roots of a Quadratic Equation
Roots of a Quadratic Equation
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Product of Sum and Difference
Product of Sum and Difference
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Quadratic Formula
Quadratic Formula
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Complex Number
Complex Number
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Modulus of a Complex Number
Modulus of a Complex Number
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Argument of a Complex Number
Argument of a Complex Number
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Polar Form of a Complex Number
Polar Form of a Complex Number
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Vector
Vector
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Magnitude of a Vector
Magnitude of a Vector
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Direction of a Vector
Direction of a Vector
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Equality of Vectors
Equality of Vectors
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Study Notes
Mathematics-I - January 2024 Exam
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Time Allowed: 2.5 hours
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Full Marks: 60
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Group A: Answer Question 1. Answer any 10 parts. Each part is worth 2 marks; one mark for the correct answer, and one mark for the explanation.
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Group B: Answer any five questions
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Group A - Question 1: Include relevant questions and answer choices (multiple-choice questions) from the provided text.
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Vector Angles: Find the angle between two vectors given their magnitudes and dot product.
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Quadratic Equation Roots: Find the value of m if one quadratic root is double the other; find the relationship between the roots and the equation's coefficients.
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Logarithms and Square Roots: Evaluate expressions involving logarithms and square roots, including the relationship between radicals/logarithms.
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Complex Numbers (Magnitude): Given a complex number, find its product of the numbers and its magnitude in multiple parts.
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Coefficients: Find the coefficient of x3 in a binomial expansion of (1 + 3x + 3x² + x³)² .
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Vectors and Perpendicularity: Find the value of m where two vectors are perpendicular to each other.
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Functions (Domain): Determine the domain of a function dependent on a square root expression.
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Trigonometric Equations: Find an angle (x) when cos(3x) = sin(2x)
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Function Evaluations: Evaluate a function at a given point (e.g., f(-1)) given the general expression for f(x - 2))
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Detailed Questions for Group B: Complete the questions in Group B from the provided text. These questions are more complex and require a deep understanding of different concepts.
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Additional Topics: The document presents diverse mathematical problems, and concepts include vector geometry, complex numbers, binomial expansions, trigonometric functions, calculus, and possibly others that could be in the exam.
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