Generate Quiz 6973A8C198975

We can rewrite sin(x)cos(x) as 1/2 sin(2x)

Maximum value of sin(2x) is at x = 45° because sin 90° = 1 (max).

Therefore, maximum value of sin(x)cos(x) = (1/2)×1 = 1/2 or 0.5.

The right response is b) Cauchy. It was the mathematician Cauchy who introduced the derivative notation D'(x).

The complex number that remains unaltered when added to any other complex number is known as the additive identity in the context of complex numbers. (0 + 0i) is this additive identity.

Therefore, (c) (0 + 0i) is the right response.

Positive integers beginning with 1 are known as natural numbers (1, 2, 3,…). Since you can always add one to any natural number to get a larger natural number, there is no such thing as a largest natural number.

C is the right response.

A circle must have a center on the y-axis and a radius equal to the distance between the center and the y-axis in order for it to touch it.

The given equation can be rewritten as follows:

(x² – 6x) + (y² – 4y) = 7

When the square is finished for both the x and y terms, we obtain:

(x² – 6x + 9) + (y² – 4y + 4) = 7 + 9 + 4

(x – 3)² + (y – 2)² = 20

A circle with a radius of -20 and a center of (3, 2) is represented by this equation.

Statement I is untrue because the center (3, 2) is on the x-axis.

Statement II: the circle x² + y²+6x+4y-7=0 touches the x-axis

We can rewrite the provided equation as follows, using the same procedure as for statement I:

y2 + 4y + (x2 + 6x) = 7.

When the square is finished for both the x and y terms, we obtain:

(x² + 6x + 9) + (y² + 4y + 4) = 7 + 9 + 4

(x + 3)² + (y + 2)² = 20

A circle with radius √20 and center (-3, -2) is represented by this equation.

Statement II is accurate since the center (-3, -2) is on the x-axis.

As a result, (c) is the right response: I am false, II is true.

Answer:

C is the right response.

The midpoint between the focus and the focus projection onto the directrix is known as the vertex of a parabola.

The directrix in this instance is the line x = 2, and the focus is at (0, 0). The point (2, 0) is where the focus is projected onto the directrix.

Consequently, the following represents the halfway point between the focus and its projection:

((0 + 2)/2, (0 + 0)/2) = (1, 0)

As a result, the parabola’s vertex is (1, 0).

A vector that is perpendicular to both a and b is produced when two vectors, a and b, are cross-producted. We must divide this cross product by its magnitude to get the unit vector, which is a×b/|a×b|.

z + z̄ = x + ίy + x – ίy

= 2x

= 2 real parts of z

= 2 Re (z)

Both polar form (r, θ) and rectangular form (a + bi) can be used to represent complex numbers, where:

The actual part is a.

I represents the imaginary unit, and b represents the imaginary part.

R stands for the modulus, or absolute value (distance from the origin).

Measured counter-clockwise from the positive x-axis, θ is the angle (argument).

The complex number in this instance is 1 + √3i, where:

a = 1 (real part)

b = √3 (partially imaginary)

The following formulas can be used to convert to polar form:

r = √(a² + b²)

θ = tan⁻¹(b / a)

Calculating the absolute value of r:

r = √((1)² + (√3)²

r = √(1 + 3)

r = √4

r = 2.

Calculating θ (angle):

θ = tan⁻¹((√3) / (1))

θ = 60° (arctan(√3) ≈ 60°)

Note: The angle of a complex number in polar form is usually limited to the range of 0° to 360°, even though tan⁻¹(√3) can also produce -30° + 180° = 150°. The same point on the complex plane is represented in this instance by 60° and 150°, but with different angle references. In order to adhere to standard procedure, we select the smaller angle (60°).

Along a closed path, the sum of the vectors is always zero.

Answer:

The right response is d.

Any point on a parabola is always the same distance from the directrix as the focus is from the vertex. The y-axis is the directrix in this instance, and the vertex is located at (2, 0).

The focus must be on the same horizontal line as the vertex and at the same distance to the right of the vertex as the directrix is to the left of the vertex because the parabola is symmetric around its vertex. As a result, the focus and vertex will both have a y-coordinate of 0.

Since negative numbers are to the left on a number line, the y-axis, which is two units to the left of the vertex, is the directrix. Consequently, the focus needs to be two units to the vertex’s right. Thus, (2 + 2, 0) = (4, 0) is the focus.

Therefore, (d) (4, 0) is the right response.

φy=f(x+φc)-f(x)

The reciprocal of the number is the multiplicative inverse.

7 = 17 × 77 = 772

= 7 7

The right response is a

The distance between a focus (F) and the hyperbola’s center (C) divided by the distance between a focus (F) and a vertex (V) is known as the hyperbola’s eccentricity (e).

The equation can be rewritten as follows: [\frac{x^{2}}{4^{2}}-\frac{y^{2}}{5^{2}}=1].

This demonstrates that a = 2 since the transverse axis is horizontal and has length 2a = 4. Likewise, b = 5 since the conjugate axis has length 2b = 5.

As we know, e2=1+a2b2.Since e2=1+2252=441 in this instance, e=441.It’s

Response: (a) 241It’s

D is the right response.We call this kind of locus a “hyperbola.” Because it depicts the location of places where the distance difference between two fixed points, known as the foci, is constant, a hyperbola is the right response. As a result, (d) a hyperbola is the right choice.

Answer:

The right response is d.

The point at which a parabola shifts from opening upward to opening downward, or the other way around, is known as the vertex. The point (h, k) is typically the vertex of a parabola in vertex form, which is expressed as y = A(x – h)^2 + k. [How did you get this vertex form?]

We are told that the vertex is on the x-axis and that the parabola’s equation is y = x^2 – 16x + k. This indicates that the vertex’s y-coordinate is 0. In vertex form, k stands for the vertex’s y-coordinate. Therefore, k must equal 0 for the vertex to lie on the x-axis.

Response:

The ratio of the y-component to the x-component is the tangent of the angle formed by the vector and the x-axis.

tan(θ) = y/x = 1/√3 = 1/√(3/3) = 1/√3

We obtain θ = 30° by taking the inverse tangent.

Two vectors u and v are either parallel or antiparallel (pointing in opposite directions) if their cross product is zero.

The angle between the vectors is either 0° or 180° when the cross product is zero.

The right choice is (C).

Answer:

Definition of chord

When vectors are added, corresponding components are added one at a time.

Vector addition for points A = (a, b) and B = (c, d) is:

(a, b) + (c, d) = (a + c, b + d)

Applying this to the provided vectors in this case:

Vector 1 + Vector 2 = (7, 9) + (3, -5)

Resulting vector = (7 + 3, 9 – 5)

Resulting vector = (10, 4)

The ability to eliminate common factors on opposing sides of an equation under specific circumstances is known as the cancellation property. We can observe that “a” shows up on both sides of the equation in this instance.

To understand how cancellation works, let’s simplify the equation:

Ra + c = b + cab

Taking “c” away from each side:

Ra = b + cab – c

Taking “a” out of the right side:

Ra = b + a(c – b)

Only when a ≠ 0 can we divide both sides by “a” because we are working with real numbers (R). The primary requirement for cancellation is this. It is not defined to divide by zero.

Dividing both sides by a, assuming a ≠ 0, yields:

R = b/a + (c – b)

Consequently, cancellation permits R to be isolated provided that a ≠ 0.

Answer:

<VECTORS’ SCALAR TRIPLE PRODUCT> The absolute value of the scalar triple product provides the volume of a parallelepiped made up of three vectors, u, v, and w:

Volume = |(u × v) · w|

Where:

u = i + 2j – k

v = i – 2j + 3k

w = i – 7j – 4k

Scalar triple product calculation:

(u × v) = (1) (3) – (2)(-2) – (1)(2) + (2)(1) – (-1)(1) + (-k)(1) = i – 2j + 3k

(u × v) · w = (1)(1) + (-2)(-7) + (3)(-4) = 1 + 14 – 12 = 3

Consequently, the parallelepiped’s volume is:

Volume = |(u × v) · w| = 3

(d) 48 is the appropriate choice.