1.
The differential of 9x⁻⁶ is:
Correct Answer
C. -54x⁻⁷
Explanation
The correct answer is -54x⁻⁷ because when we take the differential of 9x⁻⁶, we bring down the exponent as a coefficient and decrease the exponent by 1. So the differential of x⁻⁶ is -6x⁻⁷. Since there is a coefficient of 9, the final answer is -54x⁻⁷.
2.
D/dx cosecx
Correct Answer
A. -cosecx cotx
Explanation
The given expression is the derivative of cosec(x) with respect to x. The derivative of cosec(x) is -cosec(x) cot(x). This can be obtained using the quotient rule of differentiation. The negative sign indicates that the derivative is negative. The cosec(x) term represents the reciprocal of the sine function, and the cot(x) term represents the reciprocal of the tangent function. Therefore, the correct answer is -cosec(x) cot(x).
3.
The differential of tanx+1/tanx is
Correct Answer
C. Sec²x-cosec²x
Explanation
The differential of a function represents the rate of change of the function with respect to its independent variable. In this case, the given function is tanx+1/tanx. To find its differential, we need to differentiate each term separately. The derivative of tanx is sec²x, and the derivative of 1/tanx is -cosec²x. Therefore, the differential of tanx+1/tanx is sec²x - cosec²x.
4.
D/dx log_{e}x
Correct Answer
B. 1/x
Explanation
The derivative of logex with respect to x can be found using the chain rule. The derivative of logex is 1/x, as the derivative of ex is ex and the derivative of loge(u) is 1/u. Therefore, the correct answer is 1/x.
5.
D/dx(x^{2}log x)
Correct Answer
A. X(1+2log x)
Explanation
The given expression is the derivative of x^2 * log x with respect to x. To find the derivative, we can use the product rule, which states that the derivative of the product of two functions is equal to the derivative of the first function times the second function, plus the first function times the derivative of the second function. Applying this rule, we get x(1+2log x) as the derivative of x^2 * log x.
6.
D/dx(tan^{-1}x)
Correct Answer
A. 1/1+x²
Explanation
The derivative of arctan(x) is 1/(1+x²). This can be derived using the chain rule, where the derivative of the inner function x is 1, and the derivative of arctan(u) is 1/(1+u²). Therefore, the derivative of arctan(x) is 1/(1+x²).
7.
∫secx tanx dx
Correct Answer
C. Secx
Explanation
The integral of secx tanx is equal to secx. This can be derived using the trigonometric identity that states secx = 1/cosx. By rewriting the integral in terms of sinx and cosx, we can simplify it to 1/cosx * sinx, which is equal to secx. Therefore, the correct answer is secx.
8.
∫(1/x√x²-1)
Correct Answer
B. Sec⁻¹x
Explanation
The integral of (1/x√(x^2-1)) can be solved using trigonometric substitution. Let x = secθ, then dx = secθtanθ dθ. Substituting these values, the integral becomes ∫(secθtanθ)(secθ)(secθtanθ) dθ. Simplifying, we get ∫(sec^2θtan^2θ) dθ. Using the identity sec^2θ = 1 + tan^2θ, we can rewrite the integral as ∫(tan^2θ + tan^4θ) dθ. Integrating term by term, we get (1/3)tan^3θ + (1/5)tan^5θ + C. Substituting back x = secθ, the final answer is (1/3)tan^3(sec⁻¹x) + (1/5)tan^5(sec⁻¹x) + C, which is equivalent to sec⁻¹x.
9.
∫6 sinx dx
Correct Answer
A. -6 cosx+c
Explanation
The given integral is ∫6 sinx dx. By using the integral of sinx, which is -cosx, we can find the antiderivative of the function. Multiplying the sinx term by the constant 6 gives us -6 cosx. Adding the constant of integration, c, gives us the final answer of -6 cosx + c.
10.
∫ cosec^{2}x dx
Correct Answer
D. -cotx
Explanation
The integral of cosec^2x can be found using the trigonometric identity 1 + cot^2x = cosec^2x. By rearranging the terms, we get cot^2x = cosec^2x - 1. Taking the integral of both sides, we have the integral of cot^2x dx = integral of (cosec^2x - 1) dx. The integral of cot^2x is -cotx, and the integral of 1 dx is x. Therefore, the integral of cosec^2x dx is -cotx + C, where C is the constant of integration.
11.
I^{3}
Correct Answer
B. -i
Explanation
The given sequence represents the powers of the imaginary unit i. The first term is i, which represents i to the power of 1. The second term is -i, which represents i to the power of 2. The third term is i³, which represents i to the power of 3. Finally, the fourth term is i², which represents i to the power of 4. Since i to the power of 4 is equal to 1, i² is also equal to 1. Therefore, the correct answer is -i, which represents i to the power of 2.
12.
If (x+3i)=(4+yi) then value of x and y is:
Correct Answer
B. X=4 ,y=3
Explanation
The given equation states that x + 3i is equal to 4 + yi. By comparing the real and imaginary parts of both sides of the equation, we can determine that x = 4 and y = 3.
13.
Find out the sum of a complex numbers (2+3i) with its conjugate complex number.
Correct Answer
A. 4
Explanation
The sum of a complex number with its conjugate complex number is equal to twice the real part of the complex number. In this case, the real part of the complex number (2+3i) is 2. Therefore, the sum of (2+3i) with its conjugate complex number is 2 + 2 = 4.
14.
Find out the multiplication of complex numbers (2+3i) with its conjugate complex number.
Correct Answer
C. 13
Explanation
When multiplying a complex number with its conjugate, we first need to find the conjugate of the given complex number. The conjugate of (2+3i) is (2-3i).
To multiply the complex number with its conjugate, we use the formula (a+bi)(a-bi) = a^2 + b^2, where a and b are the real and imaginary parts of the complex number respectively.
In this case, (2+3i)(2-3i) = 2^2 + 3^2 = 4 + 9 = 13.
Therefore, the multiplication of the complex number (2+3i) with its conjugate complex number is 13.
15.
I^{61}
Correct Answer
C. I
Explanation
The given expression is a representation of the imaginary unit, which is denoted by the symbol "i". The imaginary unit is defined as the square root of -1. In mathematics, "i" is used to represent numbers that cannot be expressed as real numbers. Therefore, the correct answer is "i".
16.
Find out the standard equation of Parabola.
Correct Answer
B. Y²=4ax
Explanation
The given equation y²=4ax represents the standard equation of a parabola. In this equation, "a" represents the distance between the vertex and the focus of the parabola. The vertex of the parabola is at the origin (0,0), and the axis of symmetry is the y-axis. The equation shows that for every x-value, there is a corresponding y-value that satisfies the equation, resulting in a parabolic shape. The coefficient of "a" determines the steepness of the parabola.
17.
Find out the value of x if the point(x,3) has a distance of 5 from the origin point.
Correct Answer
C. ±4
Explanation
The point (x,3) has a distance of 5 from the origin point. This means that the distance between (x,3) and (0,0) is 5 units. Using the distance formula, we can calculate the distance as √((x-0)^2 + (3-0)^2) = √(x^2 + 9). Equating this distance to 5, we get √(x^2 + 9) = 5. Squaring both sides, we get x^2 + 9 = 25. Solving this equation, we find that x = ±4. Therefore, ±4 is the correct answer.
18.
Find out the distance between point (2,3) and (4,1)
Correct Answer
C. 2√2
Explanation
The distance between two points in a coordinate plane can be found using the distance formula, which states that the distance between two points (x1, y1) and (x2, y2) is equal to the square root of the sum of the squares of the differences between their corresponding coordinates. In this case, the coordinates of the two points are (2, 3) and (4, 1). Using the distance formula, we can calculate the distance as follows: √((4-2)^2 + (1-3)^2) = √(2^2 + (-2)^2) = √(4 + 4) = √8 = 2√2. Therefore, the correct answer is 2√2.
19.
Which one is the Ellipse equation?
Correct Answer
A. X²⁄a²+y²/b²=1
Explanation
The equation x²⁄a²+y²/b²=1 represents the equation of an ellipse. In this equation, the variables x and y represent the coordinates of points on the ellipse, while the constants a and b determine the size and shape of the ellipse. The equation states that the sum of the squared x-coordinate divided by the square of a plus the squared y-coordinate divided by the square of b is equal to 1. This equation is a standard form of the ellipse equation and is commonly used to represent ellipses in mathematics.
20.
Find out the distance of the point (5,-2) from the y-axis.
Correct Answer
A. 5
Explanation
The distance of a point from the y-axis is the perpendicular distance between the point and the y-axis. In this case, the point (5,-2) lies on the x-axis, which is perpendicular to the y-axis. Therefore, the distance of the point (5,-2) from the y-axis is 5 units.
21.
Laplace transform of cosh at?
Correct Answer
A. S/s²-a²
Explanation
The Laplace transform of cosh(t) is given by s/(s²-a²). This can be derived using the definition of the Laplace transform and the properties of the hyperbolic cosine function. The Laplace transform of cosh(t) is an important result in the field of mathematics and engineering, and it is used in various applications such as solving differential equations and analyzing systems in the frequency domain.
22.
Laplace transform of a impulse function is:
Correct Answer
A. 1/s
Explanation
The Laplace transform of an impulse function is 1/s. This is because the impulse function represents an infinitesimally narrow and infinitely tall pulse, which has an area of 1. The Laplace transform of a function represents the area under the curve, so the Laplace transform of the impulse function is 1 divided by the Laplace variable s.
23.
Laplace transform of e^{2t }sin 4t
Correct Answer
A. 4/(s-2)²-a²
Explanation
The given expression is the Laplace transform of e2t sin 4t. The Laplace transform of e2t is 1/(s-2) and the Laplace transform of sin 4t is 4/(s²+16). Using the properties of Laplace transform, the overall Laplace transform of e2t sin 4t can be obtained by multiplying the individual transforms. Multiplying 1/(s-2) and 4/(s²+16), we get 4/(s-2)(s²+16). Simplifying further, we can rewrite it as 4/(s-2)²-4/(s²+16). Hence, the correct answer is 4/(s-2)²-a².
24.
Laplace of e^{bt}cos at:
Correct Answer
C. S-b/(s-b)²+a²
Explanation
The Laplace transform of e^btcos(at) is given by (s-b)/(s-b)²+a².
25.
Laplace of e^{-5t}
Correct Answer
C. 1/s+5
Explanation
The Laplace transform of e-5t is given by 1/(s+5). This is because the Laplace transform of e-at is 1/(s+a), where a is a constant. In this case, a = -5, so the Laplace transform of e-5t is 1/(s+(-5)), which simplifies to 1/(s+5).