BSc Data Science SEM I 2023 2024 Dec 2024 PRECALCULUS Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q1 All questions are compulsory
2. Q2 All questions carry equal marks
3. Q1 Attempt the following (Any three) (15 M)
   • a. Find fog, gof, fof and gog of the functions f(x) =
   • b. Find the inverse of the function f (x) =
   • c. A quadratic function f(x) = 2x2 + 12x + 10 is given. (i) Express f in standard form. (ii) Sketch a graph of f. (iii) Find the maximum or minimum value of f
   • d. Express the polynomial f(x) = x* — 3x? + factor form and also find all its
   • e. Find an equation of the circle that has the points P(1, 8) and Q(5, -6) as the endpoints of a
   • f. Solve : (x +2)? <0,,
4. Q2 Attempt the following (Any three) (15 M)
   • b. A certain culture of the bacterium Rhodo-bactersphaeroides initially has 25 bacteria and is observed to double every 5 hours.(a) Find an exponential model for the number of bacteria in the culture after t hours. (b) Estimate the number of bacteria after 18 hours. (c) After how many hours will the bacteria count reach | million?
   • c. Find the exact value of the following:
   • i) ii) tan
   • d. Find the values of all the trigonometric functions of t from tant = , terminal point of t is in Quadrant III ae e. Find the value of the following: i) ii) ‘ f. Find the area of the triangle having sides of length 7 and 9 and included
5. Q3 Attempt the following (Any three) (15 M)
   • a. Verify the identity:
   • b. Prove the identity: = (secx — tanx)
   • c. Find tan26 if cos@ where @ is in quadrant
   • d. Find the value of the following: i) ii) tan =
   • e. Solve the equation: 2sin?@ — 7sin@ + 3 = 0
   • f. Find all the solution of 2sin3@ — 1 = 0
6. Q4 Attempt the following (Any three) (15 M)
   • a. Express complex number 2V3 — 2i into polar form
   • b. Find tHe fifth root of SEM I PRE-CALCULUS MARKS
   • c. Solve the following system of equation by using Gaussian Elimination method
   • d. Solve the following system of equation by using Cramer’s rule
   • e. Ifu = 2i + j,v = 3i — 2j find u. v and the angle between the vectors u and v
   • f. If u= —j + 3k,v = 2i — k, find a unit vector that is orthogonal to the plane containing the vectors u and v
7. Q5 Attempt the following (Any three) (15 M)
   • a. Find the equation of the parabola that has its vertex at the origin and directrix x = —5 And sketch its graph.
   • b. Find the vertices., foci, asymptotes , length of transverse axis of the hyperbola 25y? — 9x* = 225 and sketch its graph The 3rd term of a geometric sequence is 63/4 and the 6th term is 1701/32 . Find the fifth
   • d. Use Mathematical induction prove that 1? + 2? = for all natural number ‘n’
   • e. Find the limit exist: i) ii)
   • f. Find the derivative of f(x) = Vx atx =a

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