22MATS11 Set-1 Solved Model Question Paper (CSE) 22 Scheme

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


1.A] With usual notation prove that \tan\phi=r\frac{d \theta}{dx}

2.B] Find the angle between the curves π‘Ÿ = π‘Žπ‘™π‘œπ‘”πœƒ and r=a/logπœƒ

1.C] x=a(\theta+\sin\theta), y=a(1-cos\theta) is 4acos(\frac{\theta}{2}) Show that the radius of curvature at any point πœƒ on the cycloid


2.A] Show that the curves π‘Ÿ = π‘Ž(1 + π‘ π‘–π‘›πœƒ) and π‘Ÿ = π‘Ž(1 βˆ’ π‘ π‘–π‘›πœƒ) cuts each other orthogonally

2.B] Find the pedal equation of the curve \frac{2a}{r}=(1+cos\theta)

2.C] Find the radius of curvature for the y^{2}=\frac{4a^{2}\left( 2a-x \right)}{x} curve, where the curve meets the x-axis.


3.A] Expand log(𝑠𝑒𝑐π‘₯) up to the term containing π‘₯ 4 using Maclaurin’s series.

3.B] If 𝑒 = 𝑒 π‘Žπ‘₯+𝑏𝑦𝑓(π‘Žπ‘₯ βˆ’ 𝑏𝑦) prove that 𝑏 (πœ•π‘’/πœ•π‘₯) + π‘Ž (πœ•π‘’/πœ•π‘¦) = 2π‘Žπ‘π‘’ by using the concept of composite functions.

3.C] Find the extreme values of the function 𝑓(π‘₯, 𝑦) = π‘₯ 3 + 3π‘₯𝑦 2 βˆ’ 3𝑦 2 βˆ’ 3π‘₯ 2 + 4


4.A] Evaluate i) \lim_{x \to 0} \left( \frac{a^{x}}{b^{x}} \right)^{\frac{1}{x}}. ii) \lim_{x \to 0} \left( \frac{tanx}{x} \right)^{\frac{1}{x}}

4.B] If 𝑒 = 𝑓(π‘₯ βˆ’ 𝑦, 𝑦 βˆ’ 𝑧, 𝑧 βˆ’ π‘₯) show that πœ•π‘’/πœ•π‘₯ + πœ•π‘’/πœ•π‘¦ +πœ•π‘’/πœ•π‘§ = 0

4.C] If π‘₯ + 𝑦 + 𝑧 = 𝑒, 𝑦 + 𝑧 = 𝑣 π‘Žπ‘›π‘‘ 𝑧 = 𝑒𝑣𝑀, find the values of πœ•(π‘₯,𝑦,𝑧)/πœ•(𝑒,𝑣,𝑀).


5.A] Solve 𝑑𝑦/𝑑π‘₯ + 𝑦/π‘₯= π‘₯2𝑦6

5.B] Find the orthogonal trajectories of π‘₯2 /π‘Ž2 +𝑦2/𝑏2+πœ†= 1, where Ξ» is a parameter.

5.C] Solve π‘₯𝑦𝑝2βˆ’(π‘₯2 + 𝑦2)𝑝+π‘₯𝑦 = 0


6.A] Solve (π‘₯2 + 𝑦2 + π‘₯)𝑑π‘₯ + π‘₯𝑦𝑑𝑦 = 0

6.B] When a switch is closed in a circuit containing a battery E, a resistance R and an inductance L, the current i build up at a rate given by L di dt Ri = E. Find i as a function of t. How long will it be, before the current has reached one-half its final value, if E = 6 volts, R = 100 ohms and L = 0.1 henry?

6.C] Find the general solution of the equation (𝑝π‘₯ βˆ’ 𝑦)(𝑝𝑦 + π‘₯) = π‘Ž2𝑝 by reducing into Clairaut’s form by taking the substitution 𝑋 = π‘₯2, π‘Œ = 𝑦2


7.a] Find the least positive values of x such that
i) 71 ≑ π‘₯(π‘šπ‘œπ‘‘8)
ii) 78 + π‘₯ ≑ 3(π‘šπ‘œπ‘‘5)
iii) 89 ≑ (π‘₯ + 3)(π‘šπ‘œπ‘‘4)

7.B] Find the remainder when (349 Γ— 74 Γ— 36) is divided by 3

7.C] Solve 2π‘₯ + 6𝑦 ≑ 1(π‘šπ‘œπ‘‘7)
4π‘₯ + 3𝑦 ≑ 2(π‘šπ‘œπ‘‘7)


8.A] i) Find the last digit of 72013
ii) Find the last digit of 1337

8.B] Find the remainder when the number 21000 is divided by 13

8.C] Find the remainder when 14! is divided by 17


9.A] Find the rank of the matrix

9.B] Solve the system of equations by Gauss-Jordan method x+y+z+=10, 2x-y+3z=19, x+2y+3z=22.

9.C] Using power method find the largest eigenvalue and the corresponding eigenvector of the matrix A =


10.A] Solve the following system of equations by Gauss – Seidel method 10x+y+z=12, x+10y+z=12, x+y+10z=12.

10.B] For what values of a and b the system of equation x + y + z = 6: x + 2y + 3z = 10: x + 2y + az = b has i) no solution
ii) a unique solution and
iii) infinite number of solution

10.C] Solve the system of equations by Gauss elimination method
x + y + z = 9, x βˆ’ 2y + 3z = 8, 2x + y βˆ’ z = 3


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