HPAS 2014 Maths Optional Paper-1 Question 1(a)
Find rank of the matrix : \[ A=[\begin{matrix}1&1&1&-1\\ 1&2&3&4\\ 3&4&5&2\end{matrix}] \]For solution: Click here
HPAS 2014 Maths Optional Paper-1 Question 1(b)
Show that the set : \(W=\{(a,b,c):a-3b+4c=0;a,b,c\in R\}\), is a subspace of the vector space \(V_{3}(R)\).For solution: Click here
HPAS 2014 Maths Optional Paper-1 Question 1(c)
Examine for continuity of the following function at \(x=0\) : \[ f(x)=\begin{cases}\frac{x-|x|}{x}&,&x\ne0\\ 1&,&x=0\end{cases} \] Is it differentiable ?For solution: Click here
HPAS 2014 Maths Optional Paper-1 Question 1(d)
Find the surface represented by the equation : \(x^{2}+4y^{2}+z^{2}-4yz+2zx-4xy\) \(-2x+4y-2z-3=0\) .For solution: Click here
HPAS 2014 Maths Optional Paper-1 Question 1(e)
Solve : \((D^{4}+2D^{3}-3D^{2})y=x^{3}+2~sin~x\), where \(D\equiv\frac{d}{dx}\)For solution: Click here
HPAS 2014 Maths Optional Paper-1 Question 1(f)
Two rods, each of length 2a, have their ends united at an angle \(\alpha\), and are placed in a vertical plane on a sphere of radius r. Prove that the equilibrium is stable or unstable according as \(sin~\alpha>or<\frac{2r}{a}\)For solution: Click here
HPAS 2014 Maths Optional Paper-1 Question 2(a)
Find the characteristic roots and characteristic vectors of the following matrix : \[ [\begin{matrix}1&2&3\\ 0&-4&2\\ 0&0&7\end{matrix}]. \]For solution: Click here
HPAS 2014 Maths Optional Paper-1 Question 2(b)
Solve the following equations using matrix method: \[ \begin{cases} 2x_{1}+3x_{2}+x_{3}=9 \\ x_{1}+2x_{2}+3x_{3}=6 \\ 3x_{1}+x_{2}+2x_{3}=8. \end{cases} \]For solution: Click here
HPAS 2014 Maths Optional Paper-1 Question 2(c)
Prove that the mapping : \(t:V_{2}(R)\rightarrow V_{3}(R)\), which is defined by \(t(x,y)=(x,y,0)\) is a linear transformation from vector space \(V_{2}(R)\) to the vector space \(V_{3}(R)\).For solution: Click here
HPAS 2014 Maths Optional Paper-1 Question 3(a)
Given the sum of the perimeters of a square and a circle, show that the sum of their areas is least when the side of the square is equal to the diameter of the circle.For solution: Click here
HPAS 2014 Maths Optional Paper-1 Question 3(b)
Fully examine the nature of the origin on the curve : \(x^{2}(x^{2}-4a^{2})=y^{2}(x^{2}-a^{2})\) and trace the curve.For solution: Click here
HPAS 2014 Maths Optional Paper-1 Question 3(c)
Find the area of surface of a cone whose semi- vertical angle is \(\alpha\) and the base is a circle of radius \(r\).For solution: Click here
HPAS 2014 Maths Optional Paper-1 Question 4(a)
Evaluate : \(\iint_{A}(xy)(x+y)dxdy\) where region A is the area between the parabola \(y=x^{2}\) and the line \(y=x\) .For solution: Click here
HPAS 2014 Maths Optional Paper-1 Question 4(b)
If \(u=tan^{-1}(\frac{x^{3}+y^{3}}{x-y})\) then prove that: \(x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}=sin~2u\) .For solution: Click here
HPAS 2014 Maths Optional Paper-1 Question 4(c)
Prove that the limit of used in Lagrange’s mean value theorem tends to \(\frac{1}{2}\) when \(h\rightarrow0\), provided \(f^{\prime\prime}(x)\) is continuous and \(f^{\prime\prime}(x)\ne0\).For solution: Click here
HPAS 2014 Maths Optional Paper-1 Question 5(a)
Find the equation of the sphere which passes through the points : (1, 0, 0); (0, 1, 0) and (0, 0, 1), and has its radius as small as possible.For solution: Click here
HPAS 2014 Maths Optional Paper-1 Question 5(b)
What conic does equation : \(13x^{2}-18xy+37y^{2}+2x+14y-2=0\) represent? Find its centre and the equation to the conic referred to the centre as origin.For solution: Click here
HPAS 2014 Maths Optional Paper-1 Question 5(c)
Find length and the equations to the shortest distance between the following lines : \(\frac{x-3}{3}=\frac{y-8}{-1}=\frac{z-3}{1}\) and \(\frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-6}{4}\)For solution: Click here
HPAS 2014 Maths Optional Paper-1 Question 6(a)
Discuss the solutions of the following equation : \(x^{3}p^{2}+x^{2}yp+a^{3}=0\) , where \(p\equiv\frac{d}{dx}\)For solution: Click here
HPAS 2014 Maths Optional Paper-1 Question 6(b)
Solve by the method of variation of parameters : \(x^{2}\frac{d^{2}y}{dx^{2}}+x\frac{dy}{dx}-y=x^{2}e^{x}\)For solution: Click here
HPAS 2014 Maths Optional Paper-1 Question 6(c)
For Bessel function, prove that : \(x~J_{n}^{\prime}(x)=n~J_{n}(x)-x~J_{n+1}(x)\).For solution: Click here
HPAS 2014 Maths Optional Paper-1 Question 7(a)
For Beta function, prove that : \(B(m,n)=B(m+1,n)+B(m,n+1)\).For solution: Click here
HPAS 2014 Maths Optional Paper-1 Question 7(b)
If \(\alpha=sin~\theta i+cos~\theta j+\theta k\), \(b=cos~\theta i-sin~\theta j-3k\) and \(c=2i+3j-k\), then evaluate \(\frac{d}{d\theta}[a\times(b\times c)]\) for \(\theta=0\).For solution: Click here
HPAS 2014 Maths Optional Paper-1 Question 7(c)
Apply Gauss theorem to evaluate : \(\int_{s}\{(x^{3}-yz)dydz-2x^{2}y~dzdx+zdxdy\}\) over the surface of a cube bounded by the coordinate planes and the planes \(x=y=z=a\).For solution: Click here
HPAS 2014 Maths Optional Paper-1 Question 8(a)
The moments of a given system of coplanar forces about three points: (2,0), (0, 2) and (2, 2) in their plane are 3, 4 and 10 units respectively. Find the magnitude of the resultant force and the equation of its line of action.For solution: Click here
HPAS 2014 Maths Optional Paper-1 Question 8(b)
A particle moves with simple harmonic motion in a straight line. If in the first second after starting from rest it travels a distance a and in the next second it travels a distance b in the same direction, then find the amplitude and period of the motion.For solution: Click here
HPAS 2014 Maths Optional Paper-1 Question 8(c)
A particle of mass m is falling under gravity through a medium whose resistance is \(\mu\) times the velocity. If the particle is released from rest, show that the distance fallen through in time t is : \(\frac{gm^{2}}{\mu^{2}}(e^{\frac{\mu t}{m}}-1+\frac{\mu t}{m})\)For solution: Click here