HPAS 2015 Maths Optional Paper-1 Question 1(a)
Find characteristic equation and roots of the following matrix: \[ A=[\begin{matrix}8&-6&2\\ -6&7&-4\\ 2&-4&3\end{matrix}] \]For solution: Click here
HPAS 2015 Maths Optional Paper-1 Question 1(b)
Prove that the set S = {(1, 2, 1), (2, 1, 0), (1, -1, 2)} forms a basis of vector space \(V_{3}(R)\).For solution: Click here
HPAS 2015 Maths Optional Paper-1 Question 1(c)
If \(f(x)=\sin x\) and \(g(x)=\cos x\), \(\forall x\in[0,\frac{\pi}{2}]\), then find the value of c with the help of Cauchy’s mean value theorem.For solution: Click here
HPAS 2015 Maths Optional Paper-1 Question 1(d)
Solve : \[ (D^{3}-7D-6)y = e^{2x}(1+x) \] where \(D=\frac{d}{dx}\)For solution: Click here
HPAS 2015 Maths Optional Paper-1 Question 1(e)
If : u = x + y + z, v = x² + y² + z² and \(w=xy+yz+zx\), then find : \[ (grad~u).\{(grad~v)\times(grad~w)\} \text{, where grad } \equiv\nabla \]For solution: Click here
HPAS 2015 Maths Optional Paper-1 Question 1(f)
Forces 13, 10 and 5 kg weight act along the sides BC, CA and AB of an equilateral triangle ABC. Find the direction and the magnitude of their resultant.For solution: Click here
HPAS 2015 Maths Optional Paper-1 Question 2(a)
Apply matrix theory to solve the following system of equations : \[ x+y+z=6 \] \[ x-y+z=2 \] \[ 2x+2y-z=1 \]For solution: Click here
HPAS 2015 Maths Optional Paper-1 Question 2(b)
Prove that the mapping \(f:V_{3}(R)\rightarrow V_{2}(R)\) defined by : \[ f(u_{1},u_{2},u_{3})=(u_{1}-u_{2},u_{1}-u_{3}) \] is a linear transformation. (Note: Source file has a typo, \(V_1(R)\), which is inconsistent with the function definition. Corrected to \(V_3(R)\).)For solution: Click here
HPAS 2015 Maths Optional Paper-1 Question 2(c)
State and prove Cauchy-Schwarz inequality.For solution: Click here
HPAS 2015 Maths Optional Paper-1 Question 3(a)
Evaluate : \[ \lim_{x\rightarrow0}(\frac{1}{x^{2}}-\cot^{2}x) \]For solution: Click here
HPAS 2015 Maths Optional Paper-1 Question 3(b)
Prove that the radius of curvature at any point (x, y) on the Astroid: \[ x^{2/3}+y^{2/3}=a^{2/3} \] is three times the length of the perpendicular from the origin on the tangent at that point.For solution: Click here
HPAS 2015 Maths Optional Paper-1 Question 3(c)
If : \[ u=x~\phi(y/x)+\psi(y/x) \], then prove that: \[ x^{2}\frac{\partial^{2}u}{\partial x^{2}}+2xy\frac{\partial^{2}u}{\partial x\partial y}+y^{2}\frac{\partial^{2}u}{{\partial y^{2}}}=0 \]For solution: Click here
HPAS 2015 Maths Optional Paper-1 Question 4(a)
If : \[ u^{3}+v^{3}=x+y \text{ and } u^{2}+v^{2}=x^{3}+y^{3} \], then find the value of : \[ \frac{\partial(u,v)}{\partial(x,y)} \]For solution: Click here
HPAS 2015 Maths Optional Paper-1 Question 4(b)
Evaluate : \[ \iint_{V}2z~dxdydz \] where region of integration V is a cone enclosed by the following surface : x² + y² = z², \(z=1\)For solution: Click here
HPAS 2015 Maths Optional Paper-1 Question 4(c)
Prove that a rectangular solid of maximum volume within a sphere is a cube.For solution: Click here
HPAS 2015 Maths Optional Paper-1 Question 5(a)
Reduce the equation : \[ 11y^{2}+14yz+8zx+14xy-6x-16y+2z-2=0 \] to canonical form and state the nature of the surface.For solution: Click here
HPAS 2015 Maths Optional Paper-1 Question 5(b)
Find the equation of the sphere having the circle: \[ x^{2}+y^{2}+z^{2}+10y-4z=8, x+y+z=3 \] as a great circle.For solution: Click here
HPAS 2015 Maths Optional Paper-1 Question 5(c)
Find the equation of the cylinder whose generators are parallel to the line : \[ \frac{x}{1}=\frac{y}{-2}=\frac{z}{3} \] and whose guiding curve is the ellipse : x² + 2y² = 1, \(z=0\)For solution: Click here
HPAS 2015 Maths Optional Paper-1 Question 6(a)
Discuss the solutions of the following equation: \[ p^{2}(2-3y)^{2}=4(1-y) \]For solution: Click here
HPAS 2015 Maths Optional Paper-1 Question 6(b)
Solve : \[ x^{2}\frac{d^{2}y}{dx^{2}}-(x^{2}+2x)\frac{dy}{dx}+(x+2)y=x^{3}e^{x} \]For solution: Click here
HPAS 2015 Maths Optional Paper-1 Question 6(c)
For Bessel function, prove that: \[ 2n~J_{n}(x)=x[J_{n-1}(x)+J_{n+1}(x)] \]For solution: Click here
HPAS 2015 Maths Optional Paper-1 Question 7(a)
If f and g are two scalar point functions, prove that: \[ div(f\nabla g)=f\nabla^{2}g+\nabla f.\nabla g \]For solution: Click here
HPAS 2015 Maths Optional Paper-1 Question 7(b)
Prove that : \[ \nabla\times(\nabla\times a)=\nabla(\nabla.a)-\nabla^{2}a \]For solution: Click here
HPAS 2015 Maths Optional Paper-1 Question 7(c)
Evaluate by Green’s theorem: \[ \int_{C}[(cos~x~sin~y-xy)dx+sin~x~cos~y~dy] \] where C is the circle \(x^{2}+y^{2}=1\)For solution: Click here
HPAS 2015 Maths Optional Paper-1 Question 8(a)
Six equal heavy rods, freely hinged at their ends form a regular hexagon ABCDEF which when hung-up by the point A is kept from altering its shape by two light rods BF and CE. Find the thrusts of these rods.For solution: Click here
HPAS 2015 Maths Optional Paper-1 Question 8(b)
A particle is moving vertically downwards from rest through a medium whose resistance varying as velocity, discuss its motion.For solution: Click here
HPAS 2015 Maths Optional Paper-1 Question 8(c)
The greatest and least velocities of a certain planet in its orbit round the sun are 30 and 29.2 km/sec. Find the eccentricity of the orbit.For solution: Click here