Optional Mathematics
Question paper
SEE 2081
Computer Engineering (Technical)
SEE 2081 (2025)
Optional Mathematics
Time: 3 hours
Full Marks: 75
Answer all the questions:
Group 'A' [10×1=10]
1. What type of the function is represented by f(x) = x² + 5x + 6? Write it.
2. Write the statement of remainder theorem.
3. In which condition a function f(x) is continuous at the point x = a? Write it.
4. What is the determinant value of singular matrix? Write it.
5. Write the condition of parallelism of two lines having slopes m₁ and m₂.
6. In which condition a circle is formed when a plane surface intersects a cone? Write it.
7. Express Cos2θ in terms of tanθ.
8. If cosA = 1, write the acute angle value of A.
9. If β be the angle between two vectors \(\vec{p}\) and \(\vec{q}\) then write the formula to find the value of Cosβ.
10. The inversion point of a point P is P' in a circle having centre O and radius r. If the distance of the point P and P' from the centre O are OP and OP', write the relationship among OP, OP' and r.
Group 'B' [8×2=16]
11. If f(x) = (x-3)/5 and g(x) = 5x+3, find the value of fog(2).
12. Find the vertex of parabola formed from an equation y = x² – 5x + 6.
13. Find the inverse matrix of the matrix A = [[3,4],[2,3]].
14. The line passing through the points (-2, -5) and (1, a) is perpendicular to the line having equation 2x - y + 5 = 0, find the value of a.
15. Prove that √(1-sinA) = sin(A/2) – cos(A/2).
16. Solve: 2Cos²Î¸ - 2Sin²Î¸ = 1, [0 ≤ θ ≤ Ï€].
17. Prove that vector \(\vec{a}\) = 3i + 2j and vector \(\vec{b}\) = 4i – 6j are perpendicular to each other.
18. The interquartile range of a continuous data is 20 and the first quartile (Q₁) = 10. Find the coefficient of quartile deviation.
Group 'C' [11×3=33]
19. If f(x) = 3x – 5, g(x) = (2x+7)/3 and f₀g⁻¹(x) = f(x) then find the value of x.
20. If the third term and sixth term of a geometric series are 12 and 96 respectively, what is the arithmetic mean between first term and the sum of first four terms of the series? Find it.
21. Given f(x)=3x-5 is a real valued function.
(a) Find f(3.9), f(3.99), f(4.01) and f(4.001).
(b) Find lim(x→4⁻)f(x) and lim(x→4⁺)f(x).
(c) Is the function f(x) continuous at the point x=4? Give reason.
22. Solve using Cramer's rule: 4x + 3y = –18, 2x – 5y = 4.
23. Find the angle between a pair of lines represented by the equation 2x² + 7xy + 3y² = 0.
24. Prove that: Cos20° Cos40° Cos60° Cos80° = 1/16.
25. If A + B + C = π, prove that Sin2A + Sin2B + Sin2C = 4SinA.SinB.SinC.
26. One end of a diameter of the circle having equation x² + y² – 4x – 6y – 12 = 0 is (5,4). Find the coordinates of the other end of the diameter.
27. A triangle PQR having the vertices P(1,2), Q(4,1) and R(2,5) is transformed by a 2×2 matrix so that the co-ordinates of the vertices of its image are P'(5,2), Q'(6,1) and R'(12,5) respectively. Find the 2×2 matrix.
28. Find the mean deviation from mean of the given data:
Marks: 5-15, 15-25, 25-35, 35-45, 45-55
Frequency: 3, 5, 4, 5, 3
29. Find the standard deviation from the given data:
Class Interval: 2-4, 4-6, 6-8, 8-10
Frequency: 3, 4, 2, 1
Group 'D' [4×4=16]
30. Find the maximum value of the objective function P = 4x + 6y under constraints x+2y ≤ 8, 3x+2y ≤ 12, x ≥ 0, y ≥ 0.
31. One end of a diameter of the circle having equation x² + y² – 4x – 6y – 12 = 0 is (5,4). Find the coordinates of the other end of the diameter.
32. In a triangle XYZ, the midpoints of sides XY and YZ are A and B respectively. Prove by vector method: AB || XZ.
33. A triangle ABC with vertices A(1,2), B(4,-1) and C(2,5) is reflected successively on the lines x = 5 and y = -2. Find vertices of the images so obtained. Plot the triangle ABC and images on graph paper.
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