If A = (1 2 3), then the rank of AA^T is ________.

(a)

0

(b)

2

(c)

3

(d)

1

If  \(T=\begin{array}{l} A \\ B \end{array}\left(\begin{array}{ll} 0.7 & 0.3 \\ 0.6 & x \end{array}\right)\) is a transition probability matrix, then the value of x is ________.

(a)

0.2

(b)

0.3

(c)

0.4

(d)

0.7

For what value of k, the matrix \(A=\left( \begin{matrix} 2 & k \\ 3 & 5 \end{matrix} \right) \) has no inverse?

(a)

\(\frac { 3 }{ 10 } \)

(b)

\(\frac { 10 }{ 3 } \)

(c)

3

(d)

10

If A, B are two n x n non-singular matrices, then ___________

(a)

AB is non-singular

(b)

AB is singular

(c)

(AB)^-1 = A^-1 B^-1

(d)

(AB)^-1 does not exit

\(\int _{ -1 }^{ 1 }{ { x }^{ 3 }{ e }^{ { x }^{ 4 } } } \) dx is _______.

(a)

1

(b)

2\(\int _{ -1 }^{ 1 }{ { x }^{ 3 }{ e }^{ { x }^{ 4 } } } \)dx

(c)

0

(d)

\({ e }^{ { x }^{ 4 } }\)

If ∫ x sin x dx = - x cos x + α then α = __________ +c

(a)

sin x

(b)

cos x

(c)

C

(d)

none of these

∫ sec² (7-4x) dx = _____________ +c

(a)

\(-\frac { 1 }{ 4 } tan(7-4x)\)

(b)

\(\frac { 1 }{ 7 } tan(7-4x)\)

(c)

\(\frac { 1 }{ 4 } tan(7-4x)\)

(d)

\(\frac { 1 }{ 7 } tan(7-4x)\)

\(\int { \frac { { e }^{ x }+1 }{ { e }^{ x }+x } } \) dx = ______________ +c

(a)

log |e^x +1|

(b)

log |e^x +x|

(c)

log |e^x -1|

(d)

log |e^x -x|

The demand function for the marginal function MR = 100 − 9x² is ________.

(a)

100 − 3x²

(b)

100x − 3x²

(c)

100x − 9x²

(d)

100 + 9x²

If MR and MC denote the marginal revenue and marginal cost and MR − MC = 36x − 3x² − 81 , then the maximum profit at x is equal to ________.

(a)

3

(b)

6

(c)

9

(d)

5

The area of the region bounded by the curve y² = 2y - x and the y-axis _____ sq. units

(a)

\(\frac{4}{3}\)

(b)

\(\frac{2}{3}\)

(c)

4

(d)

\(\frac{16}{3}\)

The P.I of (3D² + D − 14)y = 13e^2x is ______.

(a)

\(\frac {x}{2}\)e^2x

(b)

xe^2x

(c)

\(\frac {x^2}{2}\)e^2x

(d)

13xe^2x

A homogeneous differential equation of the form \(\frac { dy }{ dx } \) = f\(\left( \frac { y }{ x } \right) \) can be solved by making substitution, ______.

(a)

y = v x

(b)

v = y x

(c)

x = v y

(d)

x = v

The variable separable form of \(\frac { dy }{ dx } =\frac { y(x-y) }{ x(x+y) } \) by taking y = vx and \(\frac { dy }{ dx } =v+x\frac { dv }{ dx } \) is ______.

(a)

\(\frac { 2{ v }^{ 2 } }{ 1+v } dv=\frac { dx }{ x } \)

(b)

\(\frac { 2{ v }^{ 2 } }{ 1+v } dv=-\frac { dx }{ x } \)

(c)

\(\frac { 2{ v }^{ 2 } }{ 1-v } dv=\frac { dx }{ x } \)

(d)

\(\frac { 1+v }{ 2{ v }^{ 2 } } dv=-\frac { dx }{ x } \)

The differential equation obtained by eliminating a and b from y = a e^3x + b e^-3x is _____________

(a)

\(\frac { { d }^{ 2 }y }{ dx^{ 2 } } \)+ay = 0

(b)

\(\frac { { d }^{ 2 }y }{ dx^{ 2 } } \)-9y = 0

(c)

\(\frac { { d }^{ 2 }y }{ dx^{ 2 } } -9\frac { dy }{ dx } \)

(d)

\(\frac { { d }^{ 2 }y }{ dx^{ 2 } } \)+9x = 0

The particular integral of the differential equation \(\frac { d^{ 2 }y }{ { dx }^{ 2 } } -5\frac { dy }{ dx } \)+6y=e^5x is _______

(a)

\(\frac { e^{ 5x } }{ 6 } \)

(b)

\(\frac { xe^{ 5x } }{ 21 } \)

(c)

6e^5x

(d)

\(\frac { { e }^{ 5x } }{ 25 } \)

For the given points (x₀, y₀) and (x₁, y₁) the Lagrange’s formula is _______.

(a)

\(y(x)=\frac{x-x_{1}}{x_{0}-x_{1}} y_{0}+\frac{x-x_{0}}{x_{1}-x_{0}} y_{1}\)

(b)

\(y(x)=\frac{x_{1}-x}{x_{0}-x_{1}} y_{0}+\frac{x-x_{0}}{x_{1}-x_{0}} y_{1}\)

(c)

\(y(x)=\frac{x-x_{1}}{x_{0}-x_{1}} y_{1}+\frac{x-x_{0}}{x_{1}-x_{0}} y_{0}\)

(d)

\(y(x)=\frac{x_{1}-x}{x_{0}-x_{1}} y_{1}+\frac{x-x_{0}}{x_{1}-x_{0}} y_{0}\)

If f (x)=x²  + 2x + 2 and the interval of differencing is unity then Δf (x) _______.

(a)

2x −3

(b)

2x +3

(c)

x + 3

(d)

x − 3

E [f(x₀)] is ______________

(a)

f(x_o + h)

(b)

f(x_o - h)

(c)

f(x_o) + h

(d)

f(x_o) - h

If y is to be estimated for the value of x between two extreme points in a set of values, it is called ___________

(a)

Interpolation

(b)

extrapolation

(c)

Forward interpolation

(d)

backward interpolation

For what values of the parameter λ, will the following equations fail to have unique solution: 3x − y+λz = 1, 2x + y + z = 2, x + 2y − λz = −1 by rank method.

Integrate the following with respect to x.
sin³ x

Evaluate the following integrals:
\(\int _{ -1 }^{ 1 }{ { x }^{ 2 }{ e }^{ -2x } } dx\)

If \(\int _{ 0 }^{ a }{ { 3x }^{ 2 } } dx=8\) find the value of a

Solve the following:
\(\frac { dy }{ dx } +ycosx=sinx\ cosx\).

Write down the order and degree of the following differential equations.
\(\sqrt { 1+\left( \frac { dy }{ dx } \right) ^{ 2 } } \)= 4x

When h = 1, find Δ (x³).

Find the rank of the matrix \(\left( \begin{matrix} 5 & 3 & 0 \\ 1 & 2 & -4 \\ -2 & -4 & 8 \end{matrix} \right) \)

Show that the equations x- 3y + 4z = 3, 2x - 5y + 7z = 6, 3x - 8y + 11z = 1 are inconsistent

Evaluate ഽ\(\sqrt { { x }^{ 2 }-16 } \)dx

Solve 9y'' − 12y' + 4y = 0

Solve: (x+y)²\(\frac { dy }{ dx } \) = 1

From the following table find the missing value

x	2	3	4	5	6
f(x)	45.0	49.2	54.1	-	67.4

Using graphic method, find the value of y when x=27.

x	10	15	20	25	30
y	35	32	29	26	23

The total cost of 11 pencils and 3 erasers is Rs. 64 and the total cost of 8 pencils and 3 erasers is Rs. 49. Find the cost of each pencil and each eraser by Cramer’s rule.

If \(\int _{ a }^{ b }{ dx } =1\) and \(\int _{ a }^{ b }{ xdx } =1\), then find a and b

Evaluate ഽ x³ sin (x⁴) dx

The price of a machine is 6,40,000 if the rate of cost saving is represented by the function f(t) = 20,000 t. Find out the number of years required to recoup the cost of the function.

The sum of Rs. 2,000 is compounded continuously, the nominal rate of interest being 5% per annum. In how many years will the amount be double the original principal? (log_e2 = 0.6931)

Equipment maintenance and operating costs (are related to the overhaul interval x by the equation \({ x }^{ 2 }\frac { dc }{ dx } -10xc=-10\) with c = c₀ and x = x₀. Find c as a function of x.

Using Lagrange's formula find the value of y when x = 4 from the following table.

x	0	3	5	6	8
y	276	460	414	343	110