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GATE Mathematics Mock Test - 2 - Mathematics MCQ


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30 Questions MCQ Test GATE Mathematics Mock Tests - GATE Mathematics Mock Test - 2

GATE Mathematics Mock Test - 2 for Mathematics 2024 is part of GATE Mathematics Mock Tests preparation. The GATE Mathematics Mock Test - 2 questions and answers have been prepared according to the Mathematics exam syllabus.The GATE Mathematics Mock Test - 2 MCQs are made for Mathematics 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for GATE Mathematics Mock Test - 2 below.
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GATE Mathematics Mock Test - 2 - Question 1

The question given below consists of a pair of related words followed by four pain of words. Select the pair that best expresses the relation in the original pair.

LEGEND : MAP ::

Detailed Solution for GATE Mathematics Mock Test - 2 - Question 1
(THINGE AND PURPOSE) A legend (meaning an explanatory list of symbols used in a map) explains the symbols used in a map; a glossary explains the technical terms used in a text.
GATE Mathematics Mock Test - 2 - Question 2

Evaluate 

Detailed Solution for GATE Mathematics Mock Test - 2 - Question 2

Adding (i) and (ii), we get

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GATE Mathematics Mock Test - 2 - Question 3

Find the solution of the differential equation .

Detailed Solution for GATE Mathematics Mock Test - 2 - Question 3

Given,

By integrating both sides we get,

GATE Mathematics Mock Test - 2 - Question 4

Consider the differential equestion and y = 0 and  x → ∞ then  is

Detailed Solution for GATE Mathematics Mock Test - 2 - Question 4

Given differential equestion is 

multiply bothe of side of equation by ey we get

Take ey = t

Integrating factor = 

solutions is

where c is an arbitrary constant

where u = ex

so 

now 

when x = log2

= not defind

GATE Mathematics Mock Test - 2 - Question 5

Solution of the following differential equation .

Detailed Solution for GATE Mathematics Mock Test - 2 - Question 5

Leibnitz’s linear equation:

Solution is,

GATE Mathematics Mock Test - 2 - Question 6

If  and if  is

Detailed Solution for GATE Mathematics Mock Test - 2 - Question 6

GATE Mathematics Mock Test - 2 - Question 7

Convert Cartesian coordinates (2, 6, 9) to Cylindrical and Spherical Coordinates.

Detailed Solution for GATE Mathematics Mock Test - 2 - Question 7

The Cylindrical coordinates is of the form ( ρ, φ, z) where ρ =  and   and z = z where (x, y, z) is the Cartesian coordinates. The Spherical coordinates is of the form (r, θ, φ) where   and  .Now, substituting the values for x as 2, y as 6 and z as 9, we get the answer as (6.32, 71.565., 9) and (11, 35.097., 71.565.).

GATE Mathematics Mock Test - 2 - Question 8

Curl of  is

Detailed Solution for GATE Mathematics Mock Test - 2 - Question 8

Hence F is irrotational field as Curl 

*Answer can only contain numeric values
GATE Mathematics Mock Test - 2 - Question 9

Evaluate .


Detailed Solution for GATE Mathematics Mock Test - 2 - Question 9

Let 

Put cos 2θ = t and -2 sin 2θ dθ = dt

Where θ = 0, t = 1

θ = 

= 0.75

*Answer can only contain numeric values
GATE Mathematics Mock Test - 2 - Question 10

Let  for all real x and y. If f’(0) exists and equals – 1 and f(0) = 1, then find f(2). 


Detailed Solution for GATE Mathematics Mock Test - 2 - Question 10

Analytic Method : since 

Replacing x by 2x and y by 0, then 

= f’(0) = –1 x ∈ R   (given) 

Integrating, we get f(x) = –x + c 

Putting x = 0, then f(0) = 0 + c = 1     (given)

∴ c = 1 

then f(x) = 1 – x 

∴ f(2) = 1 – 2 = –1 

*Answer can only contain numeric values
GATE Mathematics Mock Test - 2 - Question 11

What is the number of subgroups of Sof order 12 ? 


Detailed Solution for GATE Mathematics Mock Test - 2 - Question 11

Let H be any subgroup of order 12 in S4

Let H ≠ A4

and let if possible that H contains and odd permutation thus H has 6 odd and 6 even permutation

 ⇒ H ∩ A4 is a subgroup of A4 of order 6 

⇒ A4 has subgroup of order 6 contradiction by a well know theorem 

Hence Ais the only subgroup of S4 of order 12. 

*Answer can only contain numeric values
GATE Mathematics Mock Test - 2 - Question 12

If  what is the value of this integral   the circular path x2 + y2 = 1 ?


Detailed Solution for GATE Mathematics Mock Test - 2 - Question 12

We have

⇒ for every closed path .

GATE Mathematics Mock Test - 2 - Question 13

Choose the curl of  at the point (2,1,-2).

Detailed Solution for GATE Mathematics Mock Test - 2 - Question 13

GATE Mathematics Mock Test - 2 - Question 14

Let there be a vector X = yz2 ax + zx2 ay + xy2 az. Find X at P(3,6,9) in cylindrical coordinates.

Detailed Solution for GATE Mathematics Mock Test - 2 - Question 14

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 ax – 398 ay + 108 az.

GATE Mathematics Mock Test - 2 - Question 15

In conics, the _____ is revolving to form two anti-parallel cones joined at the apex.

Detailed Solution for GATE Mathematics Mock Test - 2 - Question 15

In conics, the generator is revolving to form two anti-parallel cones joined at the apex. The plane is then made to cut these cones and we get different conic sections. If we cut at right angles with respect to the axis of the cone we get a circle.

GATE Mathematics Mock Test - 2 - Question 16

Given f (x) = ex cosy, what is the value of the fifth term in Taylor's series near (1, π/4) where it is expanded in increasing order of degree & by following algebraic identity rule?

Detailed Solution for GATE Mathematics Mock Test - 2 - Question 16

GATE Mathematics Mock Test - 2 - Question 17

In a Binomial Distribution, if p, q and n are probability of success, failure and number of trials respectively then variance is given by

Detailed Solution for GATE Mathematics Mock Test - 2 - Question 17

For a discrete probability function, the variance is given by

Variance(V) = 

Where µ is the mean, substitute P(x)=nCx px q(n-x) in the above equation and put µ = np to obtain

V = npq.

GATE Mathematics Mock Test - 2 - Question 18

Find the distance between two points A(5,60.,0) and B(10,90.,0) where the points are given in Cylindrical coordinates.

Detailed Solution for GATE Mathematics Mock Test - 2 - Question 18

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.

GATE Mathematics Mock Test - 2 - Question 19

Find the value of x: 3 x2 alogax = 348?

Detailed Solution for GATE Mathematics Mock Test - 2 - Question 19

Since, alogax = x . The given equation may be written as: 3x2 x = 348 ⇒ x = (116)1/3 = 4.8.

GATE Mathematics Mock Test - 2 - Question 20

For a third degree monic polynomial, it is seen that the sum of roots are zero. What is the relation between the minimum angle to be rotated to have a Rolles point (α in Radians) and the cyclic sum of the roots taken two at a time c

Detailed Solution for GATE Mathematics Mock Test - 2 - Question 20

From Vietas formulas we can deduce that the x2 coefficient of the monic polynomial is zero (Sum of roots = zero). Hence, we can rewrite our third degree polynomial as c

Now the question asks us to relate α and c

Where c is indeed the cyclic sum of two roots taken at a time by Vietas formulae

As usual, Rolles point in the rotated domain equals the Lagrange point in the existing domain. Hence, we must have

y ‘ = tan(α)

3x2 + c = tan(α)

To find the minimum angle, we have to find the minimum value of α

such that the equation formed above has real roots when solved for x So, we can write

tan(α) – c > 0

tan(α) > c

α > tan-1(c)

Thus, the minimum required angle is

α = tan-1(c).

GATE Mathematics Mock Test - 2 - Question 21

Which of the following series converges?

Detailed Solution for GATE Mathematics Mock Test - 2 - Question 21

Hence, series  converges and sum is 5e.

GATE Mathematics Mock Test - 2 - Question 22

If x3y2 is an integrating factor of (6y2 + axy)dx + (6xy + bx2)dy =0, a, b ∈ ℝ then

Detailed Solution for GATE Mathematics Mock Test - 2 - Question 22

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

*Multiple options can be correct
GATE Mathematics Mock Test - 2 - Question 23

If f(x) = x3 + 3x2 + Sin(x) and g(x) = ex – 1 than find value of 

*Multiple options can be correct
GATE Mathematics Mock Test - 2 - Question 24

What is the formula used to find the area surrounded by the curves in the following diagram?

Detailed Solution for GATE Mathematics Mock Test - 2 - Question 24

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.

*Multiple options can be correct
GATE Mathematics Mock Test - 2 - Question 25

A Function f(x) has the property f(a) = f(b) for ∀a,b…∈….I and a + b = 20 then which of the following even degree polynomials could be f(x)

Detailed Solution for GATE Mathematics Mock Test - 2 - Question 25

Given that f(a) = f(b) for all integral values of a, b, one additional constraint is that a +  b / 2 = 10

This means that the graph is symmetric about the line x = 10

This means that the even degree polynomial has a Rolle point(the only Rolle point) at x = 10

We need to differentiate the functions given in the options and observe which of these has a single(OR repeated root), which is equal to 10.

The first Option when differentiated yields

f'(x) = x3 – 30x2 + 300x – 1000

Equating to zero, we see that x = 10

satisfies the equation

(10)3 – 30(10)2 + 300(10) – 1000 = 0.

*Multiple options can be correct
GATE Mathematics Mock Test - 2 - Question 26

If f(x) = x2 – 3x + 2 and g(x) = x3 – x2 + x – 1 than find value of 

Detailed Solution for GATE Mathematics Mock Test - 2 - Question 26

*Multiple options can be correct
GATE Mathematics Mock Test - 2 - Question 27

The graph of f(x) passes through the point (-1; 4). The slope of the line tangent to the graph at the point (x; f(x)) is -2x -3. Find f(0).

Detailed Solution for GATE Mathematics Mock Test - 2 - Question 27

The slope of the tangent line is -2x -3 means f0(x) = -2x -3. Thus, f(x) =

-x2 -3x + C for some C. Plugging in the poing (-1; 4) gives C = 2 and thus f(x) =

-x2  -3x + 2 and f(0) = 2.

*Multiple options can be correct
GATE Mathematics Mock Test - 2 - Question 28

Find the limit of Riemann sums that is equal to the defnite integral 

Detailed Solution for GATE Mathematics Mock Test - 2 - Question 28

We must have . Iff we let  we have 

*Answer can only contain numeric values
GATE Mathematics Mock Test - 2 - Question 29

Find 


Detailed Solution for GATE Mathematics Mock Test - 2 - Question 29

*Answer can only contain numeric values
GATE Mathematics Mock Test - 2 - Question 30

The extremium value of the function z = xy over the plane x + y = 1 is (upto two decimal places) _________.


Detailed Solution for GATE Mathematics Mock Test - 2 - Question 30

The lagrangian function is given by

L = xy+ λ (x + y - 1)

Then Lx = y + λ = 0 i.e. λ = -y             ........(i)

Ly = x + λ  = 0 i.e. λ = -x                   ........(ii)

and x+ y - 1= 0                                  .......(iii)

From (i) and (ii), we have x = y

The from (iii), we have 2x - 1= 0 i.e. x=1/2

The critical point is x = y = 1/2, hence

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