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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
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
The value of 2log2 5 + 9log3 √3 is
The value of 2log2 5 + 9log3 √3 is
The value of the expression ω²(1+i)(iω-1) is
The value of the expression ω²(1+i)(iω-1) is
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The value of k(ij) is
The value of k(ij) is
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If z = 2 + i√3, then zz is
If z = 2 + i√3, then zz is
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The coefficient of x³ in the expansion of (1 + 3x + 3x² + x³)¹º is
The coefficient of x³ in the expansion of (1 + 3x + 3x² + x³)¹º is
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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
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If cos (sin+cos x) = 0, then the value of x is
If cos (sin+cos x) = 0, then the value of x is
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If cos 3x = sin 2x, then x is
If cos 3x = sin 2x, then x is
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If f(x - 2) = 2x² + 3x - 5, then f(-1) is
If f(x - 2) = 2x² + 3x - 5, then f(-1) is
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The domain of the function 1/√((x-2)(3-x)) is
The domain of the function 1/√((x-2)(3-x)) is
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Lim┬(x→(π/2)) (cot x)/(π/2 - x) is
Lim┬(x→(π/2)) (cot x)/(π/2 - x) is
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If f(x)= loge* + elox, then f'(x) is
If f(x)= loge* + elox, then f'(x) is
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The function (3-x)(x-1) is maximum for x =
The function (3-x)(x-1) is maximum for x =
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If a and β be the roots of the equation x² - 3x + 2 = 0, find the equation whose roots are 1/a and 1/β.
If a and β be the roots of the equation x² - 3x + 2 = 0, find the equation whose roots are 1/a and 1/β.
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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.
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Prove that √i + √(-i) = √2, where i = √-1.
Prove that √i + √(-i) = √2, where i = √-1.
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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.
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Prove that 2log(a+b) = 2loga+log (1+a²/b²).
Prove that 2log(a+b) = 2loga+log (1+a²/b²).
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If w³ = 1 and 1+w+w² = 0, find the value of w^2022 + w^2023 + w^2024.
If w³ = 1 and 1+w+w² = 0, find the value of w^2022 + w^2023 + w^2024.
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If tan(θ/2) = 1/3, find the value of sin(θ).
If tan(θ/2) = 1/3, find the value of sin(θ).
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If log3 x = 1/9, find the value of x.
If log3 x = 1/9, find the value of x.
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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.
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Find the modulus of (a - ib)², where i = √-1.
Find the modulus of (a - ib)², where i = √-1.
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Prove that sec²(tan√5) + cosec²(cot⁻¹5) = 32.
Prove that sec²(tan√5) + cosec²(cot⁻¹5) = 32.
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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.
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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.
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If tan x tan 5x = 1, prove that tan 3x = 1.
If tan x tan 5x = 1, prove that tan 3x = 1.
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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.
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If tan(A+B) = 1/2 and tan(A - B) = 1/3, find the value of tan(2A).
If tan(A+B) = 1/2 and tan(A - B) = 1/3, find the value of tan(2A).
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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).
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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)).
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If y = x² and x² d²y/dx² = ay, find the value of a.
If y = x² and x² d²y/dx² = ay, find the value of a.
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Find the derivative of x with respect to x².
Find the derivative of x with respect to x².
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Study Notes
Mathematics-I, January 2024
- Time Allowed: 2.5 hours
- Full Marks: 60
Group A - Multiple Choice Questions (MCQs)
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Instructions: Answer question 1. Each MCQ carries 2 marks (1 mark for correct answer, 1 mark for correct explanation).
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Question 1: (various parts - multiple choice)
- Part i: Angle between two vectors given their magnitudes and dot product.
- Part ii: Finding the value of 'm' in a quadratic equation where one root is double the other.
- Part iii: Evaluating a logarithmic expression involving √3.
- Part iv: Simplifying a complex number expression.
- Part v: Finding the value of a vector expression (ij).
- Part vi: Finding the product of a complex number with its conjugate.
- Part vii: Finding the coefficient of x³ in an expansion
- Part viii: Finding the value of 'm' if given vectors are perpendicular.
Group B - Extended Answer Questions
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Instructions: Answer any five questions from this group.
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Question 2: (various parts)
- Part i: Finding the equation with roots related to roots of a given quadratic equation.
- Part ii: Finding 'n' (in an expression) where a given term is independent of x.
- Part iii: Proving an identity involving square roots of complex numbers.
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Question 3: (various parts)
- Part i: Finding the projection of a vector sum in the direction of another.
- Part ii: Proving an equation involving logarithms and a complex number term.
- Part iii: Proving a trigonometric identity.
- Part iv: Evaluating an expression.
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Question 4: (various parts)
- Part i: Evaluating the value of 'x' in an equation with logarithms.
- Part ii: Finding the number of terms in a binomial expansion
- Part iii: Finding the modulus of a complex number.
- Part iv: Proving an equation involving trigonometric functions.
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Question 5: (various parts)
- Part i: Finding a unit vector perpendicular to two given vectors.
- Part ii: Solving a quadratic equation where one root is known and another is specific.
- Part iii: Proving an equation involving trigonometric functions.
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Question 6: (various parts)
- Part i: Proving that two given vectors are parallel.
- Part ii: Evaluating the value of a trigonometric function based on given values of trigonometric tangents.
- Part iii: Proving/showing a trigonometric identity.
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Question 7: (various parts)
- Part i: Evaluating a composite function involving logarithm.
- Part ii: Finding the value of 'a' in a differential equation.
- Part iii: Finding the derivative of a function.
- Part iv.: Solving a differential equation
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Question 8: (various parts) - Part i: Evaluating a limit of a function
- Part ii: Proving a trigonometric identity/property
- Part iii: Showing an algebraic function is odd
- Part iv: Evaluating an expression involving trigonometric functions
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Question 9: (various parts)
- Part i: Finding the derivative of a function.
- Part ii: Proving an equation involving trigonometric functions.
- Part iii: Proving that a derivative of a function is constant.
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Description
Test your knowledge in Mathematics-I with this comprehensive quiz. It includes multiple choice questions on vectors, quadratic equations, logarithms, complex numbers, and more. Perfect for students preparing for their upcoming exams.
