Choose the correct answer for the following :-

1 (a) (i)
The Leibnitz theorem is the formula to find the n

(A) trigonometric function

(B) exponential function

(C) product of two algebraic functions

(D) product of two functions

^{th}derivative of(A) trigonometric function

(B) exponential function

(C) product of two algebraic functions

(D) product of two functions

1 M

1 (a) (ii)
The n

(A) log 5.5

(B) (log 5)

(C) e

(D) (log 5)

^{th}derivative of 5^{x}is :(A) log 5.5

^{x}(B) (log 5)

^{n}5^{x}(C) e

^{(log 5)x }(D) (log 5)

^{2}e^{(log 5)x}
1 M

1 (a) (iii)
The value of 'c' of the Cauchy mean value theorem for f(x)=e

(A) 5

(B) 3

(C) 0

(D) 4

^{x}, g(x)=e^{-x}in (3, 7) is:(A) 5

(B) 3

(C) 0

(D) 4

1 M

1 (a) (iv)
The generalized series of Maclaurin's series expansion is

(A) Taylor series

(B) Exponential series

(C) Logarithmic series

(D) Trigonometric series

(A) Taylor series

(B) Exponential series

(C) Logarithmic series

(D) Trigonometric series

1 M

1 (b)
Verify Rolle's theorem for the function f(x)=x

^{2}(1-x)^{2}in 0 ? x ? 1 and also find the values of c.
4 M

1 (c)
If sin

^{-1}y=2log (x+1), prove that (x+1)^{2}y_{n+2}+ (2n+1)(x+1) y_{n+1}+(n^{2}+4)y_{n}=0
6 M

1 (d)
Expand by using Maclaurin's series, the function log (1+sin x) upto fifth degree terms.

6 M

Choose the correct answer for the following :-

2 (a) (i)
The curve \[r=\dfrac {a}{1+\cos \theta}\] intersect orthogonally with the following curve: \[ (A) \ r=\dfrac {b}{1-\cos\theta}\\(B) \ r=\dfrac {c}{1+\sin\theta}\\(C) \ r=\dfrac {b}{1+\sin\theta}\\(D) \ r=\dfrac {d}{\cos \theta} \]

1 M

2 (a) (ii)
If ? be the angle between the tangent and radius vector at any point on the curve r=f(?), then sin ? equals to

\[ (A) \ \dfrac {dr}{ds}\\(B) \ r\dfrac {d\theta}{ds}\\(C) \ r\dfrac {d\theta}{dr}\\(D) \ r\dfrac {dr}{d\theta} \]

\[ (A) \ \dfrac {dr}{ds}\\(B) \ r\dfrac {d\theta}{ds}\\(C) \ r\dfrac {d\theta}{dr}\\(D) \ r\dfrac {dr}{d\theta} \]

1 M

2 (a) (iii)
L Hospital's Rule can be applied to the limits of the form :

(A) 0/0

(B) 0×?

(C) ?-?

(D) ?

(A) 0/0

(B) 0×?

(C) ?-?

(D) ?

^{0}
1 M

2 (a) (iv)
\[ \lim_{x\rightarrow x}(a^{1/x}-1)x \] is of the following form

(A) 0×?

(B) ?-?

(C) ?

(D) 0

(A) 0×?

(B) ?-?

(C) ?

^{0}(D) 0

^{2}
1 M

2 (b)
\[ Evaluate \ \lim_{x \rightarrow \rho/2 }(\tan x) \]

4 M

2 (c)
Find the radius curvature for the curve x

^{2}y=a(x^{2}+y^{2}) at the point (-2a.2a).
6 M

2 (d)
Find the Pedal equation for the curve r(1-cos ?)=2a

6 M

Choose the correct answer for the following :-

3 (a) (i)
\[ If \ f(x,y)=\dfrac {1}{x^3}+\dfrac {1}{y^3}+\dfrac {1}{x^3+y^3}, \ then \ x\dfrac {\partial f}{\partial x}+y\dfrac {\partial f}{\partial y}\ is : \]

(A) 0

(B) 9

(C) 1

(D) -3f

(A) 0

(B) 9

(C) 1

(D) -3f

1 M

3 (a) (ii)
If\ x=\rho \cos \theta, \ y=\rho \sin \theta, \ z=z \ then \ \dfrac {\partial(x,y,z)}{\partial(\rho, \theta,z)}:

(A) ?

(B) 1

(C) 0

(D) ?

(A) ?

(B) 1

(C) 0

(D) ?

1 M

3 (a) (iii)
If an error of 1% is made in measuring its base and height, the percentage error in the area of a triabgle is

(A) 0.2%

(B) 0.02%

(C) 1%

(D) 2%

(A) 0.2%

(B) 0.02%

(C) 1%

(D) 2%

1 M

3 (a) (iv)
One of the necessary and sufficient condition for a function to have a maximum value is

(A) AC-B

(B) AC-B

(C) AC-B

(D) AC-B

(A) AC-B

^{2}>0.A<0(B) AC-B

^{2}=0.A=0(C) AC-B

^{2}<0.A<0(D) AC-B

^{2}>0.A>0
1 M

3 (b)
\[ If \ V=e^{a\theta} \cos (a \log r), \ prove \ that \ \dfrac {\partial^2v}{\partial r^2}+\dfrac {1}{r}\dfrac {\partial v}{\partial r}+ \dfrac {1}{r^2}\dfrac{\partial^2v}{\partial \theta^2}=0 \]

6 M

3 (c)
Examine the function f(x,y)=1 + sin(x

^{2}+y^{2}) for extreme values.
5 M

3 (d)
In calculating the volume of right circular cone, error of 2% and 1% are made in height and radius of the base respectively find the percentage error in the volume.

5 M

Choose the correct answer for the following :-

4 (a) (i)
If F=??, then the curl F :

(A) solenoidal

(B) irrotational

(C) rotational

(D) none of these

(A) solenoidal

(B) irrotational

(C) rotational

(D) none of these

1 M

4 (a) (ii)
if V=x

(A) 2xi+2yi

(B) 2x+2y

(C) 2xi+2yj+k

(D) xi+yj

^{2}+y^{2}+3 then grad V is:(A) 2xi+2yi

(B) 2x+2y

(C) 2xi+2yj+k

(D) xi+yj

1 M

4 (a) (iii)
The value of 'a' of the vector F=(x+3y)i+(x-2z)j+(x+az)k, which is solenoidal :

(A) -2

(B) -1

(C) 0

(D) 3

(A) -2

(B) -1

(C) 0

(D) 3

1 M

4 (a) (iv)
If R=x

(A) x+y+z

(B) x-y-z

(C) (x+y+z)

(D) 2(x-y+z)

^{2}y ? y^{2}z ? z^{2}, then Laplacian of R is:(A) x+y+z

(B) x-y-z

(C) (x+y+z)

(D) 2(x-y+z)

1 M

4 (b)
Find div F and curl F, where F=? (x

^{3}+y^{3}+z^{3}-3xyz)
6 M

4 (c)
Prove that curl (?u)=? curl u + grad ? × u

6 M

4 (d)
Show that the cylindrical system is orthogonal.

4 M

5 (a) (i)
\[ The \ value \ of \ \int^{\pi/2}_0 \cos x \sin^{99}x \ dx \ is \]

(A) 1/99

(B) 1/100

(C) ?/100

(D) 99/100

(A) 1/99

(B) 1/100

(C) ?/100

(D) 99/100

1 M

5 (a) (ii)
The curve y

(A) symmetric about the x-axis

(B) symmetric about the x and y axis

(C) symmetric about the y-axis

(D) none of these

^{2}(a^{2}+x^{2})=x^{2}(a^{2}-x^{2}) is(A) symmetric about the x-axis

(B) symmetric about the x and y axis

(C) symmetric about the y-axis

(D) none of these

1 M

5 (a) (iii)
The length of the arc y=f(x) from x=a to x=b is

\[ (A) \ \int^b_a \sqrt{1+\left ( \dfrac {dy}{dx} \right )^2}dx \\(B) \ \int^b_a \sqrt{1+ \left ( \dfrac {dx}{dy} \right )^2}dx \\(C) \ \int^b_a \sqrt{\left (\dfrac {dx}{dy} \right )^2 + \left ( \dfrac {dy}{dx} \right )^2}dx \\(D) \ none \ of \ these \]

\[ (A) \ \int^b_a \sqrt{1+\left ( \dfrac {dy}{dx} \right )^2}dx \\(B) \ \int^b_a \sqrt{1+ \left ( \dfrac {dx}{dy} \right )^2}dx \\(C) \ \int^b_a \sqrt{\left (\dfrac {dx}{dy} \right )^2 + \left ( \dfrac {dy}{dx} \right )^2}dx \\(D) \ none \ of \ these \]

1 M

5 (a) (iv)
The value of \[ \int^\pi_0 \sin^4 x \ dx \] is equal to:

(A) 3?/8

(B) 3/8

(C) ?/16

(D) ?/4

(A) 3?/8

(B) 3/8

(C) ?/16

(D) ?/4

1 M

5 (b)
Obtain the reduction formula for ? sin

^{n}x dx
4 M

5 (c)
\[ Evaluate \ \int^a_0 x\sqrt{ax-x^2 dx} \]

6 M

5 (d)
Find the area of an arch of the cycloid x=a(? - sin ?), y=a(1-cos ?).

6 M

Choose the correct answer for the following :-

6 (a) (i)
The order and degree of the differential equation \[ \left [ 1+ \left ( \dfrac {dy}{dx} \right )^2 \right ]^2=c\dfrac {d^2y}{dx^2} \] respectively is

(A) one, two

(B) one, one

(C) two, one

(D) three, two

(A) one, two

(B) one, one

(C) two, one

(D) three, two

1 M

6 (a) (ii)
The differential equation \[ [ 1+e^{x/y}]dx+e^{x/y}\left [1-\dfrac {x}{y} \right ]dy=0 \ is \]

(A) homogeneous and linear

(B) homogeneous and exact

(C) non-homogenenous and exact

(D) none of these

(A) homogeneous and linear

(B) homogeneous and exact

(C) non-homogenenous and exact

(D) none of these

1 M

6 (a) (iii)
The solution of the differential equation \[ \dfrac {dy}{dx}=e^{x+y}: \]

(A) e

(B) e

(C) e

(D) e

(A) e

^{x}+e^{y}=c(B) e

^{x}+e^{-y}=c(C) e

^{x}-e^{-y}=c(D) e

^{x+y}=c
1 M

6 (a) (iv)
Replacing dy/dx by -dx/dy in the differential equation of (x,y, dy/dx)=0, we get the differential equation of

(A) polar trajectory

(B) orthogonal trajectory

(C) trajectory

(D) none of these

(A) polar trajectory

(B) orthogonal trajectory

(C) trajectory

(D) none of these

1 M

6 (b)
\[ Solve \ \dfrac {dy}{dx}=\dfrac {2x-y+1}{x+2y-3}\]

6 M

6 (c)
Solve dr+(2r cot ?+ sin 2?)d?=0

6 M

6 (d)
Find the orthogonal trajectory of the family of coaxial circles \[ \dfrac {x^2}{a^2}+\dfrac {y^2}{b^2+\lambda}=1 \]

4 M

Choose the correct answer for the following :-

7 (a) (i)
The normal form of the matrix are

\[ (A) \ [1_3, 0]\\(B) \ \begin{bmatrix}1\\2\end{bmatrix} \\(C) \ \begin{bmatrix}1_3 &0 \\ 0&0 \end{bmatrix}\\(D) \ all \ of \ these \]

\[ (A) \ [1_3, 0]\\(B) \ \begin{bmatrix}1\\2\end{bmatrix} \\(C) \ \begin{bmatrix}1_3 &0 \\ 0&0 \end{bmatrix}\\(D) \ all \ of \ these \]

1 M

7 (a) (ii)
The solution of the simultaneous equation x+y=3, x-y=3 is

(A) only trivial

(B) only unique

(C) unique and trivial

(D) none of these

(A) only trivial

(B) only unique

(C) unique and trivial

(D) none of these

1 M

7 (a) (iii)
In Gauss Jordan method, the coefficient matrix reduces to matrix

(A) diagonal

(B) unit matrix

(C) triangular matrix

(D) none of these

(A) diagonal

(B) unit matrix

(C) triangular matrix

(D) none of these

1 M

7 (a) (iv)
If r is the rank of the matrix [A] of order m × n then r is:

(A) r?m

(B) r?n

(C) r?n

(D) r?m

(A) r?m

(B) r?n

(C) r?n

(D) r?m

1 M

7 (b)
Find the rank of following matrix by elementary transform: \[ A=\begin{bmatrix}0 &2 &3 &4 \\ 2&3 &5 &4 \\ 4&8 &13 &12 \\ \end{bmatrix}\]

4 M

7 (c)
Find for what value of k the system of equations x+y+z=1, x+2y+4z=k, x+4y+6z=k

^{2}. Posses a solution solve completely in each case.
6 M

7 (d)
Solve the following system of equations by Gauss elimination method: x+y+z=9; x-2y+3z=8; 2x+y-z=3

6 M

Choose the correct answer for the following :-

8 (a) (i)
If the determine of the coeffcient matrix is zero, then there exist

(A) trivial solution

(B) non-trivial solution

(C) unique solution

(D) no solution

(A) trivial solution

(B) non-trivial solution

(C) unique solution

(D) no solution

1 M

8 (a) (ii)
If P is the modal matrix of an orthogonal matrix, then its inverse matrix is equal to

(A) P

(B) P

(C) diagonal matrix

(D) none of these

(A) P

^{-1}(B) P

(C) diagonal matrix

(D) none of these

1 M

8 (a) (iii)
The quadratic form for the matrix \[ A=\begin{bmatrix} a&h \\ h&b \\ \end{bmatrix}\ is \]

(A) ax

(B) ax

(C) ax

(D) none of these

(A) ax

^{2}+2hxy+by^{2}(B) ax

^{2}+by^{2}(C) ax

^{2}+2bxy+2by^{2}(D) none of these

1 M

8 (a) (iv)
The nature of the quadratic function of the matrix having the eigen values [0, 2, 4] is

(A) positive definite

(B) positive semi-definite

(C) negative definite

(D) negative semi-definite

(A) positive definite

(B) positive semi-definite

(C) negative definite

(D) negative semi-definite

1 M

8 (b)
Reduce the matrix \[ A=\begin{bmatrix}-1 &3 \\ -2&4 \end{bmatrix}\] to the digonal form and hence find A

^{4}.
6 M

8 (c)
Find all the eigen values of the matrix \[ A=\begin{bmatrix}8 &-6 &2 \\ -6&7 &-4 \\ 2&-4 &3 \end{bmatrix} \]

4 M

8 (d)
Reduce the quadratic form 3x

^{3}+3y^{2}+3z^{2}+2xy+2zx into canonical form.
6 M

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