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Model Test Paper, Math, Feb. - 2017 - Question 1

is differentiable, then f is strictly increasing on the interval

Model Test Paper, Math, Feb. - 2017 - Question 2

Let f : [a, b] → R be a differentiable function s.t. f(x) ≠ 0 for x ∈ [a, b] then is equal to

Model Test Paper, Math, Feb. - 2017 - Question 4

The area of the region that is inside the circle r = 2 cos q and outside the cardiod r = 2(1 – cos q).

Model Test Paper, Math, Feb. - 2017 - Question 5

Fg and Fe represents gravitational and electrostatic forces respectively, between the two electrons situated at a distance of 10 m. The ratio Fg/Fe is of the order of

Model Test Paper, Math, Feb. - 2017 - Question 6

Consider the funnel formed by revolving the curve y = 1/x about the x- axis, between x = 1 and x = a, where a > 1 If Sa denote the surface area of the funnel then

Model Test Paper, Math, Feb. - 2017 - Question 7

Let x ∈ R, n ∈ N then there exist a unique y ∈ R s.t. y^{n} = x if

Model Test Paper, Math, Feb. - 2017 - Question 8

If both f, g : R → R are discontinuous at 0, then the product function fg : R → R is

Model Test Paper, Math, Feb. - 2017 - Question 9

Let A be a n × n matrix and y be a n × 1 matrix (vector) such that the equation Ax = y for a n × 1 matrix (vector) y admits no solution then the rank of A is

Model Test Paper, Math, Feb. - 2017 - Question 10

Let b be an n × 1 (column) vector and A be an n × n positive definite matrix. Define P : R^{n} → R by

Here t stands for transpose. If x_{0} is a vector such that Ax_{0} = B then which of the following is true? (where t is used for transpose)

Model Test Paper, Math, Feb. - 2017 - Question 11

U = {(x, y, z) | x = y = z}, V = {(x, y, z) | x = 0} what is U + V?

Model Test Paper, Math, Feb. - 2017 - Question 12

If A ⊂ R^{2} is a countable set, then R^{2} \ A is

Model Test Paper, Math, Feb. - 2017 - Question 13

Let A, B ⊂ R^{n} and define, A + B = {a + b; a ∈ A, b ∈ B}. If A and B are open then which of the following statement is TRUE ?

Model Test Paper, Math, Feb. - 2017 - Question 14

The value of the integral where R is the following solid region

Model Test Paper, Math, Feb. - 2017 - Question 15

Let S be the surface in R^{3} defined by the parametric equation

r = 2 + cos t + cos u

θ = t 0 ≤ t ≤ 2π

z = sin u 0 ≤ u ≤ 2π

then the using the divergence theorem what is the volume of the region inside S.

Model Test Paper, Math, Feb. - 2017 - Question 16

The following picture shows the parametric curve (x, y) = (t – t^{3}, t^{2})

By using Green’s theorem what will be the area of the shaded region.

Model Test Paper, Math, Feb. - 2017 - Question 17

Let G be a finite group of order 2n for some integer n. Consider the map Ø : G → G given by Ø(a) = a^{2}. Then which of the following statement holds?

Model Test Paper, Math, Feb. - 2017 - Question 18

If F is a conservative vector field where then what should be the value of λ

Model Test Paper, Math, Feb. - 2017 - Question 19

A rigid body is rotating with constant angular velocity ω about fixed axis. If V be the velocity at any point of the body then the value of curl V will be

Model Test Paper, Math, Feb. - 2017 - Question 20

An ideal gas has molecules with 5 degrees of freedom. The ratio of specific heats at constant pressure (Cp ) and at constant volume (Cv ) is :

Model Test Paper, Math, Feb. - 2017 - Question 21

For what values of λ and μ the system of equations

2x + 3y + 5z = 9

7x + 3y – 2z = 8

2x + 3y + λz = μ

has unique solution

Model Test Paper, Math, Feb. - 2017 - Question 22

Which of the following mapping is not linear ?

Model Test Paper, Math, Feb. - 2017 - Question 25

The value of y as t → ∞ for an initial value of y(1) = 0, for the differential equation

Model Test Paper, Math, Feb. - 2017 - Question 26

If y_{1} = sinx and y_{2} = sinx – cosx are linearly independent solutions of y” + y = 0. Then determine the constants c_{1 }and c_{2} so that the solution sin x + 3cosx = c_{1}y_{1} + c_{2}y_{2}.

Model Test Paper, Math, Feb. - 2017 - Question 27

General solution of the differential equation

Model Test Paper, Math, Feb. - 2017 - Question 28

Let u_{1}, ... u_{n }be a linearly dependent set of functions on a ≤ x ≤ b, and let each function be(n – 1) times differentiable in (a, b). Then

*Multiple options can be correct

Model Test Paper, Math, Feb. - 2017 - Question 29

Consider the series and let S_{n} denote the nth partial sum of the given series then the sequence <Sn> is

*Multiple options can be correct

Model Test Paper, Math, Feb. - 2017 - Question 30

At which of the following points f(x) = x^{5 }– 5x^{4} + 5x^{3 }+ 12 for all x ∉ R has neither a local maximum nor a local minimum

*Multiple options can be correct

Model Test Paper, Math, Feb. - 2017 - Question 31

If f(x) = x^{3} + x^{2} – 5x + 3 for all x ∈ R then which of the following statements are TRUE?

*Multiple options can be correct

Model Test Paper, Math, Feb. - 2017 - Question 32

If f(x) = 1 – x^{2/3} for all x ∈R then which of the following statements are FALSE?

*Multiple options can be correct

Model Test Paper, Math, Feb. - 2017 - Question 33

Let 0 < a < 1 and let <xn> satisfy the condition |x_{n+1} – x_{n}| ≤ a_{n} for n ∈ N then which of the following statements are TRUE?

*Multiple options can be correct

Model Test Paper, Math, Feb. - 2017 - Question 34

Let T be the linear transform of R^{3} given by T(1, 0, 0) = (1, 0, 0); T(1, 1, 0) = (1, 1, 0); T(1, 1, 1) = (1, 1, 0) then which of the following is \ are true ?

*Multiple options can be correct

Model Test Paper, Math, Feb. - 2017 - Question 35

The function f : R^{n} → R defined as f(x_{1}, ..x_{n} ) = Max{|xi|}, i = 1, ... n is

*Multiple options can be correct

Model Test Paper, Math, Feb. - 2017 - Question 36

Let A be a subset of R^{2} with the property that every continuous function f : A → R has a maximum in A. Then which of the following is TRUE ?

*Multiple options can be correct

Model Test Paper, Math, Feb. - 2017 - Question 37

then which of the following statements are TRUE ?

*Multiple options can be correct

Model Test Paper, Math, Feb. - 2017 - Question 38

Determine which of the following functions defined on the interval (-∞,∞) are solution of the differential equation y' = y - t.

*Answer can only contain numeric values

Model Test Paper, Math, Feb. - 2017 - Question 41

If the region bounded by the parabola y = x^{2} + 1 and the line y = x + 3 is revolved about the x- axis to generate a solid then the volume of the solid correct upto two decimal values is ____

*Answer can only contain numeric values

*Answer can only contain numeric values

Model Test Paper, Math, Feb. - 2017 - Question 43

The number of (distinct) real roots of the equation x^{4} + 2x^{2} – 6x + 2 = 0 is\are _____

*Answer can only contain numeric values

Model Test Paper, Math, Feb. - 2017 - Question 44

Is (–3, 5] the internal of convergence of the power series (select 1 if yes

and 0 if no)

*Answer can only contain numeric values

Model Test Paper, Math, Feb. - 2017 - Question 45

Let A = (a_{ij}) be a 100 × 100 matrix defined by a_{ij} = i^{2} + j^{2} then the rank of A is _____

*Answer can only contain numeric values

Model Test Paper, Math, Feb. - 2017 - Question 46

Let T : R^{3} → R^{3} be a linear transformation such that T ≠ 0 but T^{2} ≠ 0 then dim (Range (T)) is _____

*Answer can only contain numeric values

Model Test Paper, Math, Feb. - 2017 - Question 47

Let G be a finite group, p the smallest prime divisor of |G| and x ∈ G and element of order p.

Suppose H ∈G is s.t. h x h^{–1} = x^{10} then the value of p is _______

*Answer can only contain numeric values

Model Test Paper, Math, Feb. - 2017 - Question 48

Let R be the region in R^{2} determined by in equalities x^{2 }+ y^{2 } ≤ 4 and y^{2} ≤ x^{2}. Then value of the following integral corrected upto three decimal places is _______

*Answer can only contain numeric values

Model Test Paper, Math, Feb. - 2017 - Question 49

Let R be the region in R^{3} determined by the in equalities r ≤ 1, and 0 ≤ z ≤ y - r^{2} where r =

then the value of the integral correct upto three decimal values is ____

*Answer can only contain numeric values

Model Test Paper, Math, Feb. - 2017 - Question 50

Let C be the rectangle in R^{3 }with vertices (0, 0, 0), (1, 0, 0), (1, 1, 1) and (0, 1, 1), oriented in the given order by using stoke’s theorem the value of the following integral

correct upto three decimal is ____

*Answer can only contain numeric values

Model Test Paper, Math, Feb. - 2017 - Question 51

If a function f is defined as

then the value of the integral where R is rectangle R = [0, 1, 0, 1] correct upto three decimal places is ______

*Answer can only contain numeric values

Model Test Paper, Math, Feb. - 2017 - Question 52

If a particle is moving once round a square c formed by the lines y = ± 1, x = ± 1 in the xy-plane with a force F = (x^{2} + xy + z)i + (x^{2} + y^{2} – z)j + xy k then the work done is ______

*Answer can only contain numeric values

Model Test Paper, Math, Feb. - 2017 - Question 53

If function f is defined as then the value of

*Answer can only contain numeric values

*Answer can only contain numeric values

Model Test Paper, Math, Feb. - 2017 - Question 55

If the equations ax + hy + g = 0, hx + by + f = 0, gx + fy + c = λ are consistent then the value of

select 1 for true and 0 for false

*Answer can only contain numeric values

Model Test Paper, Math, Feb. - 2017 - Question 56

is nilpotent mater then the order of the matrix is _____

*Answer can only contain numeric values

Model Test Paper, Math, Feb. - 2017 - Question 57

If (D^{5} – D) y = 12e^{x} + 8 sin x – 2x then its PI = axe^{x} + bx sin x + x^{2} then a equals to ____

*Answer can only contain numeric values

Model Test Paper, Math, Feb. - 2017 - Question 58

If the differential equation has a solution y(x) = a cos (βx) + b sin (βx) for some non- zero real numbers a, b, β then the value of α must be _____

*Answer can only contain numeric values

Model Test Paper, Math, Feb. - 2017 - Question 59

If the function x, x^{2,}, x^{3} are independent then the equation of differential equation formed with these independent solution x3y”’ – 3x2y” + 6xy’ – 6y equals to_______

*Answer can only contain numeric values

Model Test Paper, Math, Feb. - 2017 - Question 60

The sets x^{2} – x + 1, x^{2} – 1, 3x^{2} – x – 1 is linearly independent_______ (select 1 for yes and 0 for no)

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