GATE Mathematics Mock Test - 2 - Mathematics MCQ

The above transparent figure, when folded along the dotted line, becomes

Adding (i) and (ii), we get

Given differential equestion is 

multiply bothe of side of equation by e^y we get

Take e^y = t

Integrating factor = 

solutions is

where c is an arbitrary constant

where u = e^x

so 

now 

when x = log2

= not defind

Leibnitz’s linear equation:

Solution is,

The formula to convert any vector from Spherical coordinates to Cartesian coordinates is given by

after substituting the values of the vector. Now, solving the matrix we get the answer

Hence F is irrotational field as Curl 

The variance (V) for a Binomial Distribution is given by V = npq

Standard Deviation = 

Circle is a conic section. When the plane cuts the right circular cone at right angles with the axis of the cone, the shape obtained is called as a circle. If the angle is oblique we get the other parts of the conic sections.

Let 

Put cos 2θ = t and -2 sin 2θ dθ = dt

Where θ = 0, t = 1

θ = 

= 0.75

Hence, the correct answer is 14.

Since let 

By L hospital rule 

The formula for converting a vector from Cartesian coordinates to Cylindrical coordinates is

Substituting the column matrix in right hand side by the given the vector, and solving the matrix we get a vector in cylindrical coordinates. Now change the point P which is in Cartesian coordinates to Cylindrical coordinates. Now, we should substitute the point P in X thus finding the value of X at P. Hence we get the value 100 a_x – 398 a_y + 108 a_z.

Differentiating f_xx = -y².sin(xy)

f_yy = -x².sin(xy)

f_xy = -yx.sin(xy)

Observe that f_xx. f_yy – (fxy)²

Hence, it is a saddle point.

If the plane cuts at an angle with respect to the axis and does not cut all the generators then the conics formed is a parabola. If the plane cuts all the generators then the conic section formed is called as ellipse.

For a discrete probability function, the mean value or the expected value is given by

Mean(μ) =

For Binomial Distribution P(x)= ⁿC_x p^x q^(n-x), substitute in above equation and solve to get

µ = np.

First convert each point which is in cylindrical coordinates to Cartesian coordinates. Now using the formula, distance =   and substituting the values of x, y, and z in it, we get the required answer as 6.19 units. This sum can also be solved using a direct formula to find distance using two points in Cylindrical coordinates.

Hence, series  converges and sum is 5e.

Given (6y² + axy)dx + (6xy + bx²)dy = 0

I. F. = x³y²

multiplying by I.F. in (1). We get

(6y⁴x³ + ax⁴y³)dx + (6x⁴y³ + bx⁵y²)dy = 0 be exact. Then 

⇒ 24x³y³ + 3ax⁴y² = 24a³y³ + 5bx⁴y²

⇒ 3a = 5b

⇒ 3a - 5b = 0

Take the derivative implicity

The area is present above the x-axis. Area above the x-axis is positive. The area is bounded by x-axis, curve y = f(x), straight lines x=a and x=b. Hence, area is found by integrating the curve with the lines as limits.

∇² in spherical coordinate is given by

Since  is function of r only