22MATS11 Set-2 Solved Model Question Paper (CSE) 22 Scheme

22MATS11 Set – 2 Solved Model Question Paper 1st semester P cycle for Computer Science and Engineering (CSE) Stream 22 Scheme

Module – 1

1.A] With usual notation prove that $\frac{1}{P^{2}}=\frac{1}{r^{2}}+\frac{1}{r^{4}}\left( \frac{dr}{d\theta} \right)^{2}$

1 b] Find the angle between the curves $r=\frac{a}{1+\cos\theta}$ and $r=\frac{b}{1-\cos\theta}$

1 c] Find the radius of curvature of the curve $y=x^{3}(x-a)$ at the point (a,0)

or

2 a] Show that the $r=a(1+\cos\theta)$ $r=a(1-\cos\theta)$ curves and cuts each other orthogonally.

2 b] Find the pedal equation of the curve $r(1-\cos\theta)=2a$ .

2 c] Find the radius of curvature for the curve $y^{2}=\frac{a^{2}(a-x)}{x}$ , where the curve meets the x-axis.

Module -2

3 a] Expand $\log(1+\sin x)$ Up to the term containing $x^{4}$ using maclaurin’s Series.

3 b] If $u=\log(\tan x+\tan y+\tan z)$ Show that $\sin2x\frac{\partial u}{\partial x}+\sin2y\frac{\partial u}{\partial y}+\sin2z \frac{\partial u}{\partial z}=2$

3 c] Find the extreme values of the function $f(x,y)=x^{2}+y^{2}+6x-12$ .

or

4 a] Evaluate $\lim_{x \to 0} \left( \frac{a^{x}+b^{x}+c^{x}}{3} \right)^{1/x}$

4 b] If $u=f\left( \frac{x}{y},\frac{y}{z},\frac{z}{x} \right)$ Show that $x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}+z\frac{\partial u}{\partial z}=0$

4 c] If $x=r\sin\theta\cos\varphi$ , $y=r\sin\theta\sin\varphi$ and $z=r\cos\theta$ ,then find value of $\frac{\partial (x,y,z)}{\partial (r,\theta,\varphi)}$

Module 3

5 a] Solve $\frac{d y}{dx}+\frac{y}{x}=y^{2}x$ .

5 b] Find the orthogonal trajectories of $r=a(1+\cos\theta)$ , where a is parameter.

5 c] Solve $p^{2}+2py\cot x -y^{2}=0$ .

or

6 a] Solve $y(2xy+1)dx -xdy=0$ .

6 b] Find the orthogonal trajectories of the family $r^{n}\sin n\theta=a^{n}$ .

6 c] Find the general solution of the equation $(py+x)=2p$ by reducing into Clairaut’s form by taking the substitution $X=x^{2},Y=y^{2}$

Module – 4

7 a] (i) Find the remainder when 223 is divided by 47. (ii) Find the last digit in 7118 .

7 b] Find the solutions of the linear congruence $11x=4(mod25)$ .

7 c] Encrypt the message STOP using RSA with key (2537 . 13) using the prime number 43 and 59.

or

8 a] Using Fermat’s Little Theorem, show that 830 -1 is divisible by 31.

8 b] Solve the system of linear congruence ,x=3(mod 5) , y= 2(mod 6), z= 4(mod 7) Using Remainder Theorem.

8 c] (i) Find the remainder when 175 × 113 × 53 is divided by 11. (ii) Solve 𝑥3 + 5𝑥 + 1 = 0(mod 27).

Module – 5

9 a] Find the rank of the matrix

9 b] By Using Gauss-Jordan method.
x+y+z = 9
2x+y-z = 0
2x+5y+7z= 52

9 c] Using power method, find the largest eigenvalue and corresponding eigenvector of the matrix

or

10 a] Solve the following system of equation by Gauss-Seidel method:
20x+y-2z = 17
3x+20y-z = -18
2x-3y+20z = 25

10 b] Test for consistency
x-2y+3z = 2
3x-y+4z= 4
2x+y-2z = 5 and hence solve

10 c] Solve the system of equations by Gauss elimination method
2x+y+4z = 12
4x+11y-z= 33
8x-3y+2z = 20

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