Mathematics-I Quiz, January 2024

The angle between two vectors is 60°. This can be found using the dot product formula: a · b = |a| |b| cos θ, where θ is the angle between the vectors.

If one root of a quadratic equation is twice the other, then the roots can be expressed as 'r' and '2r'. Use the sum and product of roots formulas to solve for the value of 'm'.

Simplify the logarithmic expression by using the properties of logarithms: logₐb + logₐc = logₐ(b*c) and logₐbⁿ = nlogₐb.

The given expression is 1 + iω - ω² - iω³. Substitute ω² = -1 - ω and simplify. The value of the expression is -1.

The cross product of two orthogonal unit vectors (i and j) results in another unit vector (k). The dot product of any vector with a unit vector in a different direction yields 0.

The product of a complex number and its conjugate is always equal to the square of its magnitude. Find the magnitude of z and square it to get the answer.

The coefficient of x³ in the expansion of (1 + 3x + 3x² + x³)¹⁰ can be found using the multinomial theorem. The coefficient is given by ¹⁰C₃, where ¹⁰C₃ represents the number of ways to choose 3 x³ terms out of 10.

Two vectors are perpendicular if their dot product is zero. Find the dot product of the given vectors and set it equal to zero. Solve the resulting equation for 'm'.

Use the trigonometric identity cos(A + B) = cosA cosB - sinA sinB. Since cos(sin⁻¹(1/5) + cos⁻¹x) = 0, the angle inside the cosine function must be 90°. Solve for x.

Convert the given equation cos3x = sin2x to a form that can be solved. Use the trigonometric identities sin(90 - θ) = cosθ and cos(90 - θ) = sinθ. Solve for x.

Substitute x - 2 = -1 in the given expression f(x - 2) = 2x² + 3x - 5. Solve for f(-1).

The function is undefined when the denominator, √(x-2)(3-x), is zero or negative. Find the values of x that make the denominator zero or negative. The domain is all values of x except those that make the denominator zero or negative.

Find the limit by using L'Hopital's rule: if lim_(x→a) f(x) / g(x) = 0/0 or ±∞/±∞, then lim_(x→a) f(x) / g(x) = lim_(x→a) f'(x) / g'(x).

Differentiate the function f(x) = logₑx + e^lox with respect to x. Use the properties of logarithms and derivatives: d/dx(logₐu) = 1/(ulnₐ) and d/dx(e^u) = e^udu/dx.

The function (3-x)(x-1) is a quadratic function with its maximum point at the vertex. Find the x-coordinate of the vertex by using the formula x = -b / 2a, where a = -1 and b = 4.

Find the equation with roots 1/α and 1/β by using the relationships between roots and coefficients of a quadratic equation. The sum of roots of the new equation will be α + β, and the product of roots will be αβ.

The fifth term in the expansion of (x² - 1/x)^n is given by (n choose 4) * (x²)^(n-4) * (-1/x)⁴. For the term to be independent of x, the powers of x in the numerator and denominator must be equal. Set n - 4 = 4 and solve.

Use the property of complex numbers that √(a + bi) = √((√(a² + b²)+a)/2) + (b/√(2√(a² + b²)+2a))i. Substitute a = 1 and b = -1 and simplify.

Calculate the projection of vector a + c onto b using the formula: proj_b(a + c) = ((a + c) · b) / |b|². Find the dot product of (a + c) and b, calculate the magnitude of b, and plug the values into the formula.

Use the logarithmic property logₐ(m/n) = logₐm - logₐn to simplify the expression. Then, use the logarithmic property logₐmⁿ = nlogₐm to further simplify the expression.

Use the property ω³ = 1 to simplify the expression. Substitute ω³ = 1 and simplify the resulting expression. The value of the expression is 3ω².

Find the value of log₃x by substituting the given value of log₃x = 1/9 into the equation. Solve for x by using the definition of logarithms: logₐb = c ⇔ a^c = b.

The number of terms in the expansion of (x + y)⁷(x - y)⁷ is given by (n + 1), where n is the highest power of x or y in the expansion. The highest power of x or y in the expansion is 14, so the number of terms is 15.

Find the modulus of (a - ib)² by squaring the complex number and then finding its magnitude. Remember that the magnitude of a complex number z = a + bi is |z| = √(a² + b²).

The unit vector perpendicular to both the vectors can be found by calculating their cross product and then normalizing the resulting vector. Find the cross product of the two vectors and then divide by its magnitude to get the unit vector.

Use the fact that if one root of a quadratic equation is 4, then the other root can be found by using the sum and product of roots formulas. Use the fact that the roots of the second equation are equal to find the value of b.

The position vectors of A, B, C, and D can be used to find the vectors AB and CD. Two vectors are parallel if one is a scalar multiple of the other. Check if AB and CD are scalar multiples of each other.

Use the tangent addition formula: tan(A + B) = (tanA + tanB) / (1 - tanA tanB) and tan(A - B) = (tanA - tanB) / (1 + tanA tanB). Solve for tanA and tanB and then calculate tan2A using the double angle formula.

Use the trigonometric identities sin(x + y) = sinx cosy + cosx siny and sin(x - y) = sinx cosy - cosx siny. Substitute these identities into the given expression and simplify.

The composite function f(q(2)) is found by first calculating q(2) and then finding the value of f at the result. Calculate q(2) and then evaluate f at the calculated value.

Find the value of a by substituting y = x² and its second derivative into the given equation. Simplify the equation and solve for a.

Differentiate the function x² with respect to x using the power rule of differentiation: d/dx (x^n) = n*x^(n-1).

Find the derivative of y = log cot x tan x using the properties of logarithms and derivatives. Use the chain rule and the product rule to differentiate the function. Simplify the resulting derivative to show that it equals 0.

Evaluate the limit by direct substitution. Substitute x = 0 into the expression and simplify. If the result is an indeterminate form, use L'Hopital's rule to evaluate the limit.

Use trigonometric identities to simplify the expression. Remember that sin(90 - θ) = cosθ and cos(90 - θ) = sinθ. Simplify the expression to prove that it is equal to 2.

A function is odd if f(-x) = -f(x) for all x in its domain. Substitute -x into the function and simplify. If the result is equal to -f(x), then the function is odd.

Evaluate the expression using the property sin⁻¹(1/2) = π/6 and cos⁻¹(1/2) = π/3. Substitute the values and simplify.

Find the derivative of x = logₑ(6 - 5e⁻ᵗ) with respect to t using the chain rule and properties of derivatives. Evaluate the derivative at t = 0 to find the value of the function at that time.

Use the trigonometric identity sin²x + cos²x = 1 to find the value of cos²x. Substitute the value of cos²x into the expression and simplify. Use the definition of cotangent to prove the equality.

Use the chain rule and properties of differentiation to find the derivative of y with respect to x. Express the derivative in terms of t. Simplify the derivative to show that it is constant.