Mathematics Model Question Paper 1 PDF

GOVERNMENT OF KARNATAKA DEPARTMENT OF SCHOOL EDUCATION ( PRE UNIVERSITY ) Model Question Paper -1 I P.U.C: MATHEMATICS (35) : 2024-25 Time : 3 hours Max....

GOVERNMENT OF KARNATAKA DEPARTMENT OF SCHOOL EDUCATION ( PRE UNIVERSITY ) Model Question Paper -1 I P.U.C: MATHEMATICS (35) : 2024-25 Time : 3 hours Max. Marks : 80 Instructions : 1) The question paper has five parts namely A, B, C, D and E. Answer all the parts. 2) PART A has 15 MCQ’s ,5Fill in the blanks of 1 mark each. 3) For questions having figure/graph, alternate questions are given at the end of question paper in separate section for visually challenged students. PART A I. Answer ALL the Multiple Choice Questions 15 ×1 = 15 1. If U is universal set and A ⊂ 𝑈, then U′∩A = A) A B) U C) A′ D) ∅. 2. If (x - 1, y + 2) = (1 , 5), then (x, y) is A) (1, 2) B) (2, 3) C) (3 , 2 ) D) (2 , 1) 3. Match List I with List II List I List II a) Domain of 𝑠𝑖𝑛𝑥 i) (−∞ , ∞) − {𝑛𝜋: 𝑛 ∈ 𝑍} b) Domain of 𝑐𝑜𝑡𝑥 ii) [−1 , 1] c)Range of 𝑐𝑜𝑠𝑥 iii) (−∞ , ∞) Choose the correct answer from the options given below: A) a-i , b-ii, c-iii B) a-iii, b-ii, c-i C) a-ii, b-i, c-iii D) a-iii, b-i, c-ii 4. The conjugate of a complex number −5 + 3𝑖 (A) -5-3i (B) 5-3i (C) ) - 5 + 3i (D) 5+3i. 5. The solution of -8 ≤ 5𝑥 − 3 < 7 is (A) -1 ≤ 𝒙 < 𝟐. B) -1 < 𝒙 ≤ 𝟐 C) -1 ≤ 𝒙 ≤ 𝟐 D) 2 < 𝒙 ≤ -1. 1  1 1 x 6. If + 7! = 8! , then x = 6! (A) 56 (B)49 (C)64 (D) 81. 7. The value of 10𝐶1 + 10𝐶2 + 10𝐶3 + ⋯ … … ….. +10𝐶10 is (A) 1024 (B) 1023 (C) 1022 (D) 512 8. The 4th term of the sequence defined by 𝑎1 = 1 = 𝑎2 𝑎𝑛 = 𝑎𝑛−1 + 𝑎𝑛−2 , 𝑛 > 2 is (A) 9 (B) 5 (C) 2 (D) 3. 9. Equation of the line parallel to x- axis and passing through (-2,3) is (A) 𝑥 = 3 (B) 𝑥 = −2 (C) 𝑦 = −2 (D) 𝑦 = 3. 10. The length of the latus rectum of the parabola 𝑦 2 = -9x 9 9 ( A) 9 (B) −9 C) D). 4 2 𝑥2 𝑦2 11. The eccentricity of the hyperbolas: − 16 = 1 9 5 5 18 32 ( A) (B) (C) (D). 4 3 4 3 12. Statement 1: The perpendicular distance from the point P( 6, 7, 8) to zx- plane is 7 Statement 2 : The shortest distance of the point (a, b, c) from the x-axis is √𝑏 2 + 𝑐 2 A) Statement 1 is true and Statement 2 is false. B) Statement 1 is false and Statement 2 is false. C) Statement 1 is true and Statement 2 is true, Statement 2 is not a correct explanation for Statement 1 D) Statement 1 is true and Statement 2 is true, Statement 2 is a correct explanation for Statement 1 13. For the figure given below, consider the following statements 1 and 2 Statement 1: The given function limit exists at x =1,x = 2 and x = 4 Statement 2: The given function limit exists at at x=3 2  A)Statement 1 is true and Statement 2 is false B) Statement 1 is false and Statement 2 is true C)Both Statement 1and 2 are true D)Both Statement 1 and 2 are false 14. Mean deviation about median for first 5 natural numbers is 6 5 (A) 5 (B) (C) 6 (D) 6. 5 15. The number of simple events corresponding to the sample space “two coins are tossed once” is A) 1 B) 2 C) 3 D) 4. II. Fill in the blanks by choosing the appropriate answer from those given in the bracket. (0,1, 2, 3, 4, 5,) 5 ×1 = 5 16. If A = { 2, 4, 6, 8} and B = { 6, 8, 10} ,then n( A - B).…… 17. If 5x – 3 < 3x +1, when x is a natural number. The number of values of x is …… 18. If 15C3r = 15C3+r then 𝑟 --------- 19. The values of k for which the line (k–5)x–(4–𝑘 2 )y+𝑘 2 –7k+ 6 = 0 is parallel to the y-axis is --------- 20. The derivative of f(x) = 4x at x = 0 is --------- PART-B ANSWER ANY SIX QUESTIONS 6 × 2 = 12 21. If A= { 3, 5, 7, 9, 11}, B= { 7, 9, 11, 13}, C= {15, 17} find A∩(B∪C). 22 In a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length of minor arc of the chord. 23. Prove that (sin3x+ sin x) sin x + (cos 3x – cos x) cos x = 0. 𝑎+𝑖𝑏 24. If x + iy = , prove that x2 + y2 = 1. 𝑎−𝑖𝑏 25. In how many ways can a team of 3 boys and 3 girls be selected from 5 boys and 4 girls? 26. Using Binomial Theorem evaluate (99)5. 27. In a G.P, the third term is 24 and the 6th term is 192. Find the 10th term. 28. Find the equation of the circle with centre (2,2) and passes through the point (4,5). 29. A fair coin with 1 marked on one face and 6 on the other and a fair die are both tossed find the probability that the sum of numbers that turn up is 12. 3  PART –C ANSWER ANY SIX QUESTIONS : 6 × 3 = 18. 30. If U={ 1,2,3,4,5,6 }, A={ 2,3 } and B={ 3,4,5 } verify (𝐴 ∪ 𝐵)/ = 𝐴/ ∩ 𝐵 /. 31. Determine the domain and range of the relation R defined by R = {(x, x + 5): x ∈ {0, 1,2, 3, 4, 5}}. 32. Show that tan 3 x tan 2 x tan x = tan 3x – tan 2 x – tan x. 𝑢 𝑣 33. If (x + iy) 3 = u + iv, then show that + 𝑦 = 4(𝑥 2 − 𝑦 2 ). 𝑥 34. Find all pairs of consecutive odd positive integers both of which are smaller than 10 such that their sum is more than 11. 𝟔 𝟔 35. Evaluate(√𝟑 + √𝟐) − (√𝟑 − √𝟐). 36. If the angle between two lines is π/4 and slope of one of the lines is 1/2 find the slope of the other line. 37. Find the equation of the set of points P such that its distances from the points A (3, 4,-5) and B (-2, 1, 4) are equal. 38. Find the derivative of tan 𝑥 w. r. t x from first principle. PART – D Answer any FOUR questions 4 × 5 = 20 39. Define Signum function. Draw the graph of it. Also write its domain and range. sin 3𝑥 + sin 5𝑥 + sin 7𝑥 + sin 9𝑥 40. Prove that = 𝑡𝑎𝑛6𝑥. cos 3𝑥+ cos 5𝑥+cos 7𝑥+cos 9𝑥 41. In how many ways can the letters of the word PERMUTATIONS be arranged if the i) words start with P and end with S, (ii) vowels are all together, iii) there are always 4 letters between P and S? 42. Derive an expression for the perpendicular distance between a point(𝑥1 , 𝑦1 ) and a line 𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0. 𝑥 𝑛 −𝑎𝑛 43. Prove that ,for any positive integer n, lim [ ] = 𝑛𝑎𝑛−1 , and hence 𝑥→0 𝑥−𝑎 𝑥 15 −1 evaluate lim [𝑥 10 −1] 𝑥→1 4 44. Find the variance and standard deviation for the following data 𝑥𝑖 4 8 11 17 30 24 32 𝑓𝑖 3 5 9 5 4 3 1 45. Three coins are tossed once. Find the probability of getting (i) 3 tails (ii) exactly two tails (iii) no tail (iv) atmost two tails PART-E Answer the following question. 1 𝑥 𝑥 𝑥 46. If 𝑠𝑖𝑛𝑥 = 4 , 𝑥 in quadrant II , then find sin 2 , cos 2 , tan 2. 6 OR 𝑥2 𝑦2 Define ellipse and derive the equation of the ellipse in standard form as + 𝑏2 = 1. 𝑎2 𝑥 5 −𝑐𝑜𝑠 𝑥 47. Find the derivative of 𝑓(𝑥) = with respect to x. 4 𝑠𝑖𝑛 𝑥 OR Find the sum of the series up to n terms 5 + 55 +555 + … … PART F (For Visually Challenged Students only) |𝑥| 13. 𝐿𝑒𝑡 𝑓𝑥) = , 𝑥 ≠ 0 and 𝑓𝑥) = 0, 𝑥 = 0. 𝑥 Statement 1: The given function limit exists at x =1 and x = -1 Statement 2: The given function limit exists at at x= 0 A)Statement 1 is true and Statement 2 is false B) Statement 1 is false and Statement 2 is true C)Both Statement 1and 2 are true D)Both Statement 1 and 2 are false *************************** 5  GOVERNMENT OF KARNATAKA DEPARTMENT OF SCHOOL EDUCATION ( PRE UNIVERSITY ) Model Question Paper -2 I P.U.C.MATHEMATICS (35) :2024-25 Time: 3 hours Max. Marks: 80 Instructions: 1) The question paper has five parts namely A, B, C, D and E. Answer all the parts. 2) PART A has 15MCQ’s ,5 Fill in the blanks of 1 mark each. 3) For questions having figure/graph, alternate questions are given at the end of question paper in separate section for visually challenged students. PART – A I Answer all the multiple-choice questions: 15 × 1 = 15 1. The interval form of {𝑥: 𝑥  𝑅, − 5 < 𝑥  7} is A) (-5, 7) B) [-5, 7] C) {-5, 7} D) (-5, 7] 2. A function f is defined by 𝑓(𝑥) = 2𝑥 − 5, then the value of f(0) is A) − 3 B) −4 C) −5 D) 7 3. The range of f (x) = sin x is A) [0, ) B] (-, ) C) [-1, 1] D) (-1, 1) 4. The additive identity of 5 + 3i is 5 − 3i A) 0 + 1i B) 0 + 0i C) 5 - 3i D) 14 1  5. The standard form of (-5i)  i  is 8  −5 5 5 A) + i0 B) 0 + i C) 5 + 8i D) + i0 8 8 8 6 6. The solution of 3x + 8 > 2, when x is a real number is A) (-2, ) B) (-, -2) C) (-, -2] D) (-, -1) 7. The equation of the line, which has slope 2 and y-intercept -5 is. A) 2𝑥 − 𝑦 − 5 = 0 B) 2𝑥 + 𝑦 − 5 = 0 C) 2𝑥 − 𝑦 + 5 = 0 D) 2𝑥 + 𝑦 + 5 = 0 8. Match List I with List II List I List II a) 5𝐶0 i) 20 b) 5𝑃2 ii) 10 c) 5𝐶2 iii) 1 Choose the correct answer from the options given below: A) a-i , b-ii, c-iii B) a-iii, b-ii, c-i C) a-ii, b-i, c-iii D) a-iii, b-i, c-ii 9. The equation of line in the figure is A) 5𝑥 + 3𝑦 = 15 B) 3𝑥 + 5𝑦 = 15 C) 3𝑥 + 5𝑦 + 15 = 0 D) 5𝑥 + 3𝑦 + 15 = 0 10. The fifth term whose nth term is an = n(n + 2) is A) 30 B) 35 C) 40 D) 45 5 11. Statement 1:The eccentricity of hyperbola 9𝑥 2 − 16𝑦 2 = 144 is 4 𝑥2 𝑦2 √𝑎 2 +𝑏2 Statement 2:The eccentricity of hyperbola − = 1 is. 𝑎2 𝑏2 𝑎 A) Statement 1 is true and Statement 2 is false. B) Statement 1 is false and Statement 2 is false. C) Statement 1 is true and Statement 2 is true, Statement 2 is a correct explanation for Statement 1 D) Statement 1 is true and Statement 2 is true, Statement 2 is not a correct explanation for Statement 1 7 12. The axis in which the point (0, 5, 0) lies is A) 𝑥 − 𝑎𝑥𝑖𝑠 B) 𝑦 − 𝑎𝑥𝑖𝑠 C) 𝑧 − 𝑎𝑥𝑖𝑠 D) 𝑥 + 𝑦 = 0 13. lim (x3 – x2 + 1) is x → −1 A) -1 B) 0 C) 1 D) 2 14. The median of 3, 10, 6, 7, 11, 13, 15 is A) 7 B) 9 C) 10 D) 11 15. The probability of getting exactly two heads on tossing a coin thrice is 2 2 3 1 A) 3 B) 5 C) 8 D) 2 II. Fill in the blanks by choosing the appropriate answer from those given in the bracket (1, -1, 64, 2, 4, 20) 5 ×1 = 5 16. Let A = {x, y, z} and B {1, 2}, then the number of relations from A to B is ______. 17. The slope of the line passing through the points (4, 0) and (6, 4) is _______.  5  18. The value of sin   is ________.  2  5 19. The second term in the expansion of (√2 + 1) is _______. 20. The number of solutions of 24x < 100 when x is a natural number is _______. PART – B Answer any SIX questions: 6 × 2 = 12 21. If A  B = (a,1) (a,2) (a,3) (b,1) (b,2) (b,3) , find the sets A and B and hence find 𝐵 × 𝐴. 25 22. 18 1 Express i +   in a + i b form. i 23. Find the multiplicative inverse of 2 – 3i. 24. Using binomial theorem evaluate (102)5. 13 25. The sum of first three terms of a G.P. is 12 and their product is – 1. Find the common ratio and the terms. 26. Find the angle between the lines √3 x + y = 1 and x + √3 y = 1. 8   x15 − 1  27. Evaluate: lim  10  x →1  x − 1  28. Find the derivative of f(x) w. r. t x from first principal given that 𝑓(𝑥) = 𝑠𝑖𝑛𝑥. 2 1 29. If P(A) = and P(B) = , find P(A or B) and P(A and B) if A and B are mutually exclusive. 3 2 PART – C Answer any SIX questions: 6 ×3 = 18 30. If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}. Verity that (A  B)| = A| B|. 31. Draw the Venn diagram for (i) 𝐴 ∪ 𝐵 (ii) 𝐴 − 𝐵 (iii) (𝐴 ∩ 𝐵)| −5 32. Find the values of other five trigonometric functions if cot x = , x lies in second 12 quadrant. 33. Prove that sin 3x = 3 sin x – 4 sin3x. 34. Solve the inequality and show the graph of the solution on the number line 3x − 4 x + 1  - 1. 2 4 35. Find the number of Permutations of the letters of the word PERMUTATIONS. Among them how many have vowels are all together? 𝑥 2 4 36. Expand using binomial theorem (1 + 2 − 𝑥) , 𝑥 ≠ 0. 37. Reduce the equation of the circle 𝑥 2 + 𝑦 2 − 4𝑥 − 8𝑦 − 45 = 0 into Centre-radius form and hence find its centre and radius. 38. If the origin is the centroid of the triangle PQR with vertices P (2a, 4, 6), Q(−4,3b, −10) and R (8,14, 2c) then find the values of a, b, c. PART – D Answer any FOUR questions: 4 × 5 = 20 39. Define modulus function, draw the graph. Write the domain and the range. cos 4 x + cos 3 x + cos 2 x 40. Prove that = cot 3x. sin 4 x + sin 3 x + sin 2 x 9 41. A group consists of 7 boys and 5 girls. Find the number of ways in which a team of 5 members can be selected so as to have at least one boy and one girl. x y 42. Derive the equation of a line with x-intercept ‘a’ and y-intercept ‘b’ in the form of + a b = 1. Hence find the equation of a line that cuts off equal intercepts on the coordinate axes and passes through the point (2, 3). sin x 43. Prove geometrically that lim = 1, x being measured in radians. x→0 x 44. Find the mean deviation about median for the following data Marks 0-10 10-20 20-30 30-40 40-50 50-60 No.of girls 6 8 14 16 4 2 45. One card is drawn from a well shuffled deck of 52 cards. If each outcome is equally likely, calculate the probability that the card will be (i) a diamond (ii) not an ace (iii) a black card (i.e., a club or, a spade) (v) not a black card. PART - E Answer the following questions: 46. Prove geometrically that cos (A + B) = cosAcosB – sin A sin B. Hence prove that cos 2A = cos2A – sin2 A. OR (6) 𝑥2 𝑦2 Define Hyperbola. Derive its equation in the form − =1 𝑎2 𝑏2 47. If A.M. and G.M. of two positive numbers a and b are 10 and 8, respectively, find the numbers. (4) OR sin x + cos x Differentiate with respect to ‘ x’. sin x − cos x PART F (For Visually Challenged Students only) 9. The Equations for x and y axes are (A) 𝑥 = 1 , 𝑦 = 1 (B) 𝑦 = 1 (C) 𝑥 = 0 𝑎𝑛𝑑 𝑦 = 0 (D) 𝑥 = 1 𝑎𝑛𝑛 𝑦 = 0. ********************************** 10  GOVERNMENT OF KARNATAKA DEPARTMENT OF SCHOOL EDUCATION ( PRE UNIVERSITY ) Model Question Paper -3 I P.U.C: MATHEMATICS (35) :2024-25 Time : 3 hours Max. Marks : 80 Instructions : 1) The question paper has five parts namely A, B, C, D and E. Answer all the parts. 2) PART A has 15 MCQ’s ,5Fill in the blanks of 1 mark each. 3) For questions having figure/graph, alternate questions are given at the end of question paper in separate section for visually challenged students. PART A I. Answer ALL the Multiple Choice Questions 15 ×1 = 15 1. Which of the following is not correct (A) The set Q of rational numbers is a subset of the set R of real numbers (B) {x :x is an even natural number}  {x : x is an integer} (C) If A is the set of all prime divisors of 56 and B the set of all divisors of 56 then B A (D) 1,5,9  1,3,5,7,9 2. Let A = { 1, 2, { 3, 4 }, 5 }. Which of the following statement is correct? A){3, 4} ⊂A B){3, 4} ∈ A C) 3 ∈ A D) 4 ∈ A. 3. The terminal side of an angle 𝜃 in standard position passes through the point (3, −4)as in the following figure, then sinθ = 3 3 4 3 A) − 4 B] C) − 5 D) − 5 5 4. If sin2x= 2sinxcosx then sin6x is equal to A) 2sinx cosx B) 2 sin6x cos6x C)2 sin3x cos3x D) sin6x cos6x 4𝑛+2 5. The value of (− √−1 ) , 𝑛 ∈ 𝑁 is (A) 1 (B) – 𝑖 (C) 𝑖 (D) – 1. 6. ∑𝑛𝑘=0 3𝑟 𝑛𝐶𝑟 = ( A) 2𝑛 (B) 3𝑛 (C) 4𝑛 (D) n. 11 7. The diagram represents two simultaneous linear inequalities on a number line. Which of the following inequalities is the solution? (A) −1 ≤ 𝑥 < 2 (B) 2 ≤ 𝑥 < −1 (C ) −1 > 𝑥 ≥ 2 (D ) −1 < 𝑥 ≤ 2 8. Statement I: The point (0,2) is at 2 units distance from X-axis above the origin Statement II: The point (2,0) is at 2 units distance from the Y- axis left of origin (A) Both statements are true (B) Statement I is true and statement II is false (C ) Statement I is false and statement II is true (D) Both statements are false. 9. The two lines 𝑎1 𝑥 + 𝑏1 𝑦 + 𝑐1 = 0 and 𝑎2 𝑥 + 𝑏2 𝑦 + 𝑐2 = 0 where 𝑏1 , 𝑏2 ≠ 0 are 𝑎1 𝑏 I. Parallel if = 𝑏1 𝑎2 2 II. Perpendicular if 𝑎1. 𝑎2 − 𝑏1 𝑏2=0 (A) Both I and II are true (B) only I is true (C) only II is true (D) Both I and II are false 10. The radius of the circle x2 + y2 + 8x + 10y – 8 = 0 is A) 8 B) 5 C) 6 D) 7 11. The derivative of f(x)=1+ x + x2 + x3 + x4 +………x50 at x =1 is (A) 1275 (B) 1200 (C) 1326 (D) 1542 12. The point (-2, -3, -4) lies in the A) First Octant B)Seventh octant C) Second octant D) Eighth octant 13. Consider the function f(x)={ 12 𝑥≤0 𝑥>0 , then left hand limit of f(x) at x=0 is A) 1 B) 2 C) 0 D) 3 14. The median of the observations 3, 9, 5, 3, 12,10, 18, 4, 7, 19, 21 is A) 7 B) 9 C) 10 D) 11 15. If A is any event associated with a sample space 𝑆 then A) 0 ≤ 𝑃(𝐴) ≤ 1 B) 1 < 𝑃(𝐴) < 2 C) 𝑃(𝐴) ≥ 1 D) 𝑃(𝐴) ≤ 0. 12  II. Fill in the blanks by choosing the appropriate answer from those given in the bracket (0,1, 2,3, 4,5 ) 51=5 𝑥 2 5 1 16. If ( + 2, 𝑦 − ) = ( , ), then the value of y is ______. 3 3 3 3 17. The value of sin 4π is _______. 18. The total number of possible subsets of set A={ a, b} is ________. 19. If 15C3r = 15C3+r then 𝑟 is _______. 20. If the sequence 𝑎𝑛 is defined as 𝑎1 = 1 and 𝑎𝑛 = 𝑎𝑛−1 + 2 for n≥2 , then 𝑎3 is _____ PART-B ANSWER ANY SIX QUESTIONS 6 × 2 = 12 21. If U= {1, 2, 3, 4, 5, 6, 7, 8, 9}, A= {3, 4, 5, 6} B= {2, 4, 6, 8},then find (A-B)∁ 22 If A = { 3, 5, 7, 9, 11 }, B = {7, 9, 11, 13}, C = {11, 13, 15}, then find A ∪ (B ∩ C). 3𝜋 3𝜋 23. Prove that 𝑐𝑜𝑠 ( 2 + 𝑥) 𝑐𝑜𝑠(2𝜋 + 𝑥) [𝑐𝑜𝑡 ( 2 − 𝑥) + 𝑐𝑜𝑡(2𝜋 + 𝑥)] = 1 24. Express the complex number (1 − 𝑖)4 in a + ib form 25. Solve the inequalities 5x – 3 ≥ 3x – 5 and show the graph of the solution on number line. 26. Ravi obtained 70 and 75 marks in first two unit test. Find the minimum marks he should get in the third test to have an average of at least 60 marks. 27. Find an approximation of (0.99)5 using the first three terms of its expansion.. 28. A.M and G.M. of roots of a quadratic equation are 8 and 5, respectively, then find the quadratic equation. 29. Consider the experiment of rolling a die.Let A be the event ‘getting a prime number’, B be the event ‘getting an odd number’. Write the sets representing the events (a) A or B (b) A but not B 13  PART –C ANSWER ANY SIX QUESTIONS : 6 × 3 = 18. 30. Let A = {1, 2, 3, 4, 6}. Let R be the relation on A defined by R={(a, b): a, b ∈ A, b is exactly divisible by a}.(i) Write R in roster form, (ii) Find the domain of R (iii) Find the range of R. 31. Prove that 𝑠𝑖𝑛2 6𝑥 − 𝑠𝑖𝑛2 4𝑥 = 𝑠𝑖𝑛2𝑥 sin 10𝑥. (3−2𝑖)(2+3𝑖) 32. Find the conjugate of (1+2𝑖)(2−𝑖). 33. In how many of the distinct permutations of the letters in MISSISSIPPI do the four I’s not come together? 34. Using binomial theorem, show that 9n+1 – 8n – 9 is divisible by 64, whenever n is a positive integer. 35. If p is the length of perpendicular from the origin to the line whose intercepts on 1 1 1 the axes are a and b, then show that + 𝑏 2 = 𝑝2. 𝑎2 36. Find the coordinates of the focus,the equation of the directrix and latus rectum of the parabola 𝑥 2 − 8𝑦 = 0 37. Show that the points P (-2, 3, 5), Q (1, 2, 3) and R (7, 0, -1) are collinear. 1 38. Find the derivative of 𝑓(𝑥) = 𝑥 2 , from first principal. PART – D Answer any FOUR questions 4 × 5 = 20 39. Define polynomial function.. If the function f: 𝑅 → 𝑅 is defined by f(x) = 𝑥 3 , draw the graph of it. Also write its domain and range. 𝜋 𝜋 3 40. Prove that 𝑐𝑜𝑠 2 𝑥 + 𝑐𝑜𝑠 2 (𝑥 + 3 ) + 𝑐𝑜𝑠 2 (𝑥 − 3 ) = 2 41. A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of: (i) exactly 3 girls ? (ii) atleast 3 girls ? (iii) atmost 3 girls ? 42. Derive an expression for the acute angle between two lines having slopes m1 and m2 and hence find the slopes of the lines, if the slope of a line is double of the slope 1 of another line. If tangent of the angle between them is 3. 14  sin  43. Prove geometrically that lim = 1, 𝜃 being measured in radians. x→0  44. Find the mean deviation about the mean for the following data Marks 50- 60- 70- 10-20 20-30 30-40 40-50 Obtained 60 70 80 Number of 2 3 8 14 8 3 2 Students 45. On her vacations Veena visits four cities (A, B, C and D) in a random order. What is the probability that she visits (i) A before B ? (ii) A before B and B before C ? PART-E Answer the following question. 46. Prove geometrically that 𝑐𝑜𝑠(𝑥 + 𝑦) = 𝑐𝑜𝑠𝑥𝑐𝑜𝑠𝑦 − 𝑠𝑖𝑛𝑥𝑠𝑖𝑛𝑦 and hence prove that √3−1 𝑐𝑜𝑠750 =. 2√2 OR Define and derive the equation of parabola in the standard form 𝑦 2 = 4ax and find the latus rectum of the parabola 𝑦 2 = -9x 6 𝑥+𝑐𝑜𝑠 𝑥 47. Find the derivative of 𝑓(𝑥) = with respect to x. 𝑡𝑎𝑛 𝑥 OR The sum of first three terms of a G.P. is 16 and the sum of next three terms is 128. Determine the first term and the common ratio. 4 PART F (For Visually Challenged Students only) 3. The terminal side of an angle 𝜃 in standard position passes through the point (3, −4)as in the 2rd quadrant , then sinθ = 4 3 4 3 A) B] C) − 5 D) − 5 5 5 7. The solution of the inequality -1 ≥ x and x ≥ 1 for real x. A) [ -1 , 1]. B) (−∞, −1] ∪ [ 1, ∞) C) (−1,1 ) D) (−∞, −1) ∪ (1, ∞). ************************** 15