STUPIDSID
Solve any one question from Q1 & Q1 (e)
1 (a)
Define curvature of a curve at a point and find the radius of curvature at any point (s, ψ) of the curve s=4 a sin ψ.
2 M
1 (b)
If \(u=f \left( \dfrac {y}{x} \right)\), then show that \(x \dfrac{\partial u}{\partial x} + y \dfrac {\partial u}{\partial y}=0\)
2 M
1 (c)
Discuss the maxima and minima of the function x3 + y3 -3axy.
3 M
1 (d)
Compute the approximate value of √11 to four decimal place by taking the first five terms of an approximate Taylor's expansion.
7 M
1 (e)
If \(x^x y^y z^z=c\) then show that \(\dfrac {\partial^2 z}{\partial x \partial y}= - [x log (ex)]^{-1}\)
14 M
Solve any one question from Q2 & Q2 (e)
2 (a)
Using Gamma function, evaluate \(\displaystyle \int^{\infty}_0 \sqrt{x}e^{-3\sqrt{x}}dx\)
2 M
2 (b)
Evaluate: \(\displaystyle \int^2_0 \int^1_0 (x^2+y^2)dxdy\)
2 M
2 (c)
Evaluate:\(\displaystyle \int^{1}_{-1} \int^{z}_0 \int^{x+z}_{x-z} (x+y+z)dx dy dz\)
3 M
2 (d)
Evaluate: \(\lim_{n\rightarrow \infty} \left[ \left(1+ \dfrac{1}{n^2}
\right) \left(1+ \dfrac {2^2}{n^2} \right) \left(1+ \dfrac {3^2}{n^2} \right ) \cdots \cdots \left( 1+ \dfrac {n^2}{n^2}
\right)
\right]\)
7 M
2 (e)
Prove the Legendre's duplication formula \(\Gamma (m) \Gamma \left (m+ \dfrac {1}{2}
\right )= \dfrac {\sqrt{\pi}}{2^{2m-1}} \Gamma (2m)\)
14 M
Solve any one question from Q3 & Q3 (e)
3 (a)
State whether the differential equation (ey+1) cos x dx+ey sin x dy=0 is exact differential
equation or not.
2 M
3 (b)
Solve the differential equation p = sin ( y - x )
2 M
3 (c)
Solve the differential equation \(\dfrac {dy}{dx}- \dfrac {dx}{dy}= \dfrac {x}{y}- \dfrac {y}{x}\)
3 M
3 (d)
Solve \(x^2 \dfrac {dy}{dx}- 3x \dfrac {dy}{dx}+4y=(1+x)^2\)
7 M
3 (e)
Solve the simultaneous equations: \(\dfrac {dx}{dt} + 5x+y=e^r \dfrac {dy}{dt}-x+3y=e^{2t}\)
14 M
Solve any one question from Q4 & Q4 (e)
4 (a)
Find one non zero minor of highest order of the matrix \(A= \begin{bmatrix}-1 &- 2 &3 \\-2 &4 &-1 \\-1 &2 &7 \end{bmatrix}\) and hence find the rank of the matrix A.
2 M
4 (b)
Find the sum and product of eigen values of the matrix \(A= \begin{bmatrix}6 &- 2&2 \\-2 &3 &1 \\2 &-1 &3 \end{bmatrix}\) without actually computing them.
2 M
4 (c)
Find the characteristic equation of the matrix \(A= \begin{bmatrix}2 &2 &1 \\1 &3 &1 \\1 &2 &2
\end{bmatrix}\)
3 M
4 (d)
Find the normal form of the matrix \(A=\begin{bmatrix}2 &3 &-1 &-1 \\1 &-1 &-2 &-4 \\3
&1 &3 &-2 \\6 &3 &0 &- 7 \end{bmatrix}\) and hence find its rank.
7 M
4 (e)
For what values of λ, the equations x+y+z=1, x+2y+4z=λ, x+4y+10z=λ2
14 M
Solve any one question from Q5 & Q5 (e)
5 (a)
Let p ≡ Raju is tall, q ≡ Raju is handsome and r ≡ People like Raju then write the following statements in language.
i) (p⇒q)∨(p⇒r)
ii) p⇒(q∨r)
iii) ∼ p∨∼q
iv) ∼ (∼p∨∼q)
i) (p⇒q)∨(p⇒r)
ii) p⇒(q∨r)
iii) ∼ p∨∼q
iv) ∼ (∼p∨∼q)
2 M
5 (b)
In a Boolean algebra B, prove that a + b = b⇒a, b=a, ∀a, b∈B.
2 M
5 (c)
Draw the switching circuit for the following functions and replace it by simpler one:
F(x,y,z)=x,y,z+(x+y),(x+z)
F(x,y,z)=x,y,z+(x+y),(x+z)
3 M
5 (d)
Prove that a tree with n vertical has (n-1) edges.
7 M
5 (e)
If p, q, r are three statement then show that (p⇔q)∧(q⇔r) ⇒(p⇔r) is a tautology.
14 M
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