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Assertion (A): The acute angle between the line `barr = hati + hatj + 2hatk + λ(hati - hatj)` and the x-axis is `π/4`
Reason(R): The acute angle 𝜃 between the lines `barr = x_1hati + y_1hatj + z_1hatk + λ(a_1hati + b_1hatj + c_1hatk)` and `barr = x_2hati + y_2hatj + z_2hatk + μ(a_2hati + b_2hatj + c_2hatk)` is given by cosθ = `(|a_1a_2 + b_1b_2 + c_1c_2|)/sqrt(a_1^2 + b_1^2 + c_1^2 sqrt(a_2^2 + b_2^2 + c_2^2)`
Concept: undefined >> undefined
Find `(dy)/(dx)` if x + sin(x + y) = y – cos(x – y)
Concept: undefined >> undefined
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If `tan ((x + y)/(x - y))` = k, then `dy/dx` is equal to ______.
Concept: undefined >> undefined
Assertion (A): Maximum value of (cos–1 x)2 is π2.
Reason (R): Range of the principal value branch of cos–1 x is `[(-π)/2, π/2]`.
Concept: undefined >> undefined
Evaluate `sin^-1 (sin (3π)/4) + cos^-1 (cos π) + tan^-1 (1)`.
Concept: undefined >> undefined
Find the general solution of the differential equation:
(xy – x2) dy = y2 dx
Concept: undefined >> undefined
The angle between the lines 2x = 3y = – z and 6x = – y = – 4z is ______.
Concept: undefined >> undefined
Read the following passage:
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An equation involving derivatives of the dependent variable with respect to the independent variables is called a differential equation. A differential equation of the form `dy/dx` = F(x, y) is said to be homogeneous if F(x, y) is a homogeneous function of degree zero, whereas a function F(x, y) is a homogeneous function of degree n if F(λx, λy) = λn F(x, y). To solve a homogeneous differential equation of the type `dy/dx` = F(x, y) = `g(y/x)`, we make the substitution y = vx and then separate the variables. |
Based on the above, answer the following questions:
- Show that (x2 – y2) dx + 2xy dy = 0 is a differential equation of the type `dy/dx = g(y/x)`. (2)
- Solve the above equation to find its general solution. (2)
Concept: undefined >> undefined
Find the angle between the following two lines:
`vecr = 2hati - 5hatj + hatk + λ(3hati + 2hatj + 6hatk)`
`vecr = 7hati - 6hatk + μ(hati + 2hatj + 2hatk)`
Concept: undefined >> undefined
Using the matrix method, solve the following system of linear equations:
`2/x + 3/y + 10/z` = 4, `4/x - 6/y + 5/z` = 1, `6/x + 9/y - 20/z` = 2.
Concept: undefined >> undefined
Differentiate `tan^(-1)(sqrt(1-x^2)/x)` with respect to `cos^(-1)(2xsqrt(1-x^2))` ,when `x!=0`
Concept: undefined >> undefined
If y = xx, prove that `(d^2y)/(dx^2)−1/y(dy/dx)^2−y/x=0.`
Concept: undefined >> undefined
Solve the following differential equation: `(x^2-1)dy/dx+2xy=2/(x^2-1)`
Concept: undefined >> undefined
Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is `cos^(-1)(1/sqrt3)`
Concept: undefined >> undefined
Evaluate: `int(5x-2)/(1+2x+3x^2)dx`
Concept: undefined >> undefined
Evaluate : ` int x^2/((x^2+4)(x^2+9))dx`
Concept: undefined >> undefined
Use elementary column operation C2 → C2 + 2C1 in the following matrix equation :
`[[2,1],[2,0]] = [[3,1],[2,0]] [[1,0],[-1,1]]`
Concept: undefined >> undefined
find : `int(3x+1)sqrt(4-3x-2x^2)dx`
Concept: undefined >> undefined
For what values of k, the system of linear equations
x + y + z = 2
2x + y – z = 3
3x + 2y + kz = 4
has a unique solution?
Concept: undefined >> undefined
Find : ` d/dx cos^−1 ((x−x^(−1))/(x+x^(−1)))`
Concept: undefined >> undefined
