B.Sc. (Data Science) SEM I 2022 2023 Dec 2023 DS PRECALCULUS Question Paper - Mumbai University | munotes

Questions asked in this paper

• (ii) Figures to the right indicate marks,

1. Q1 Attempt any three of the following: 15 marks
   • a. Check whether the function f:R R defined as f(x) = 3x — 2 is bijective or not. If yes, find the inverse of f
   • b. Divide polynomial P(x) = 3x3 + 9x? — 5x — 1 by D(x) = x + 4 and write the quotient and remainder. Also, use the remainder theorem to find P(—4) c Find the equation of the line that passes through (1, —6) and is parallel to the line
   • d. Find the equation of the circle that has the points P(1,8) and Q(5, —6) as the endpoints of diameter
   • e. Let f(x) = — 12x + i3. Express f in standard form and find the vertex. Sketch the graph and find minimum value of f
   • f. Solve the equation 2x
2. Q2 Attempt any three of the following: 15 marks
   • a. Evaluate
   • (i) log,
   • c. Solve =0
   • d. Draw the graph of sinx,cosx ,tan x
   • e. (I) State law of sines and law cf cosines (II) Use law of sines to find the angle B
   • f. Ifcot9 = 3/4 and @ is in the third quadrant. Find the value of 5 other trigonometric
3. Q3 Attempt any three of the following: 15 marks
   • a. Prove: 2tanxsecx =
   • b. Ifcosx = and x is in the second quadrant, find cos 2x and sin 2x
   • c. Find the exact value of the sin
   • d. Verify the identity:
   • e. Solve the equation 1 + sin 6 = 2 6
   • f. Write x) as an algebraic expression in x only, where <x < 1
4. Q4 Attempt any three of the followiag: 15 marks
   • a. Use Cramer’s rule to solve the System of equations:
   • b. Solve the following System of equations using Gaussian Elimination
   • c. Express the complex no. in the polar form,
   • d. Find the three cube roots of z = = 4 5; + 2k be two vectors. Find a unit vector which js Perpendicular to both y and be two vectors. Find the angle between
5. Q5 Attempt any three of the following: 15 marks
   • a. Find the vertices, foci, length of transverse axis and asymptotes of the hyperbola and sketch its graph x2 — 9y7+9=9
   • b. Find the of the ellipse whose are (+4,0), vertices are (+5,0). Also sketch it’s graph Using principal of mathematical induction, Prove that
   • d. The third term of the sequence jis — and the sixth term is 9. Tind the first and second term Evaluate the limit if it exists Find the derivative of the function f aty =

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