DESIGN OF THE QUESTION PAPER MATHEMATICS - CLASS XI Time : 3 Hours Max. Marks : 00 The weightage of marks over different dimensions of the question paper shall be as follows:. Weigtage of Type of Questions Marks (i) Objective Type Questions : (0) 0 = 0 (ii) Short Answer Type questions : () 4 = 48 (viii) Long Answer Type Questions : (7) 7 6 =4 Total Questions : (9) 00. Weightage to Different Topics S.No. Topic Objective Type S.A. Type L.A. Type Total Questions Questions Questions. Sets - (4) - 4(). Relations and Functions - - (6) 6() 3. Trigonometric Functions () (4) (6) (4) 4. Principle of Mathematical - (4) - 4() Induction 5. Complex Numbers and () (4) - 6(3) Quadratic Equations - 6. Linear Inequalities () (4) - 5() 7. Permutations and Combinations - (4) - 4() 8. Binomial Theorem - - (6) 6() 9. Sequences and Series - (4) - 4() 0. Straight Lines () (4) (6) (4). Conic Section - - (6) 6(). Introduction to three - (4) - 4() dimensional geometry 3. Limits and Derivatives () (4) - 5() 4. Mathematical Reasoning () (4) - 5() 5. Statistics - (4) (6) 0() 6. Probability () - (6) 7() Total 0(0) 48() 4(7) 00(9)
3 EXEMPLAR PROBLEMS MATHEMATICS SAMPLE QUESTION PAPER Mathematics Class XI General Instructions (i) The question paper consists of three parts A, B and C. Each question of each part is compulsory. (ii) Part A (Objective Type) consists of 0 questions of mark each. (iii) Part B (Short Answer Type) consists of questions of 4 marks each. (iv) Part C (Long Answer Type) consists of 7 questions of 6 marks each. PART - A. If tan θ = and tan φ =, then what is the value of (θ + φ)? 3. For a complex number z, what is the value of arg. z + arg. z, z 0? 3. Three identical dice are rolled. What is the probability that the same number will appear an each of them? Fill in the blanks in questions number 4 and 5. 4. The intercept of the line x + 3y 6 = 0 on the x-axis is.... cosx 5. lim x0 is equal to.... x In Questions 6 and 7, state whether the given statements are True or False: 6. x, x0 x 7. The lines 3x + 4y + 7 = 0 and 4x + 3y + 5 = 0 are perpendicular to each other. In Question 8 to 9, choose the correct option from the given 4 options, out of which only one is correct. 8. The solution of the equation cos θ + sinθ + = 0, lies in the interval (A), 4 4 (B) 3, 4 4 (C) 3 5, 4 4 (D) 5 7, 4 4
DESIGN OF THE QUESTION PAPER 33 9. If z = + 3i, the value of z z is (A) 7 (B) 8 (C) 3i (D) 0. What is the contrapositive of the statement? If a number is divisible by 6, then it is divisible by 3. PART - B. If A B = U, show by using laws of algebra of sets that A B, where A denotes the complement of A and U is the universal set.. If cos x = 7 3 and cos y =, x, y being acute angles, prove that x y = 60. 4 3. Using the principle of mathematical induction, show that 3n is divisible by 7 for all n N. 4. Write z = 4 + i 4 3 in the polar form. 5. Solve the system of linear inequations and represent the solution on the number line: 3x 7 > (x 6) and 6 x > x b c c a a b 6. If a + b + c 0 and,, a b c also in A.P. are in A.P., prove that,, a b c are 7. A mathematics question paper consists of 0 questions divided into two parts I and II, each containing 5 questions. A student is required to attempt 6 questions in all, taking at least questions from each part. In how many ways can the student select the questions? 8. Find the equation of the line which passes through the point ( 3, ) and cuts off intercepts on x and y axes which are in the ratio 4 : 3. 9. Find the coordinates of the point R which divides the join of the points P(0, 0, 0) and Q(4,, ) in the ratio : externally and verify that P is the mid point of RQ. 3 x 0. Differentiate f(x) = 3 4x with respect to x, by first principle.
34 EXEMPLAR PROBLEMS MATHEMATICS. Verify by method of contradiction that p = 3 is irrational.. Find the mean deviation about the mean for the following data: x i 0 30 50 70 90 f i 4 4 8 6 8 PART C 3. Let f(x) = x and g(x) = x be two functions defined over the set of nonnegative real numbers. Find: f (i) (f + g) (4) (ii) (f g) (9) (iii) (fg) (4) (iv) (9) g (sin 7x sin 5 x) (sin 9x sin3 x) 4. Prove that: tan 6x (cos7x cos5 x) (cos9x cos3 x) 5. Find the fourth term from the beginning and the 5th term from the end in the 0 3 x 3 expansion of. 3 x 6. A line is such that its segment between the lines 5x y + 4 = 0 and 3x + 4y 4 = 0 is bisected at the point (, 5). Find the equation of this line. 7. Find the lengths of the major and minor axes, the coordinates of foci, the vertices, the ecentricity and the length of the latus rectum of the ellipse. 69 44 x y 8. Find the mean, variance and standard deviation for the following data: Class interval: 30-40 40-50 50-60 60-70 70-80 80-90 90-00 Frequency: 3 7 5 8 3 9. What is the probability that (i) a non-leap year have 53 Sundays. (ii) a leap year have 53 Fridays (iii) a leap year have 53 Sundays and 53 Mondays.
DESIGN OF THE QUESTION PAPER 35 MARKING SCHEME MATHEMATICS CLASS XI PART - A Q. No. Answer Marks. 4. Zero 3. 36 4. 3 5. 6. True 7. False 8. D 9. A 0. If a number is not divisible by 3, then it is not divisible by 6. PART - B. B = B φ = B (A A ) = (B A) (B A ) = (B A) (A B) = (B A) U (Given) = B A A B.
36 EXEMPLAR PROBLEMS MATHEMATICS. cos x = 7 sin x = 4 3 cos x 49 7 cos y = 3 4 sin y = 69 3 3 96 4 cos(x y) =cosx cosy + sinx siny = 3 4 3 3 3 7 4 7 4 x y = 3 3. Let P(n) : 3n is divisble by 7 P() = 3 = 8 = 7 is divisible by 7 P() is true. Let P(k) be true, i.e, 3k is divisible by 7, 3k = 7a, a Z We have : 3(k + ) = 3k. 3 = ( 3k ) 8 + 7 = 7a. 8 + 7 = 7(8a + ) P(k + ) is true, hence P(n) is true n N 4. Let 4 + i 4 3 = r (cosθ + i sinθ) r cosθ = 4, r sinθ = 4 3 r = 6 + 48 = 64 r = 8.
DESIGN OF THE QUESTION PAPER 37 tanθ = 3 θ = π z = 4 + i 4 3 = 8 cos i sin 3 3 5. The given in equations are : 3 3 3x 7 > (x 6)... (i) and 6 x > x... (ii) (i) 3x x > + 7 or x > 5... (A) (ii) x + x > 6 or x > 5... (B) From A and B, the solutions of the given system are x > 5 Graphical representation is as under: b c c a a b 6. Given,, a b c are in A.P. b c c a a b,, a b c will also be in A.P. a b c a b c a b c,, will be in A.P. a b c Since,a + b + c 0,, a b c will also be in A.P. 7. Following are possible choices: Choice Part I Part II } (i) 4 (ii) 3 3 (iii) 4
38 EXEMPLAR PROBLEMS MATHEMATICS Total number of ways of selecting the questions are: = 5 C 5 5 5 5 5 C4 C3 C3 C4 C =0 5 + 0 0 + 5 0 = 00 8. Let the intercepts on x-axis and y-axis be 4a, 3a respectively x y Equation of line is : 4a 3a or 3x + 4y = a ( 3, ) lies on it a = 7 Hence, the equation of the line is 3x + 4y + 7 = 0 9. Let the coordinates of R be (x, y, z) x = (4) (0) 4 y = ( ) (0) z = ( ) (0) R is ( 4,, ) 4 4 Mid point of QR is,, i.e., (0, 0, 0) Hence verified. 0. f (x) = 3 x f (x + Δx) = 3 ( x x ) 3 4x 3 4( x x) f (x) = lim x 0 3 x x 3 x lim f( xx) f( x) x 0 3 4x 4x 3 4x x x
DESIGN OF THE QUESTION PAPER 39 = lim x 0 (3 x x)(3 4 x) (3 4x 4 x)(3 x) ( x)(3 4x 4 x) (3 4 x) = x 0 9 x 3x 4x 3x 4xx 9 3x x 4x x 4xx li m ( x)(3 4x 4 x)(3 4 x) 5x 5 lim ( x ) (3 4 x 4 x )(3 4 x ) (3 4 x ) = x 0. Assume that p is false, i.e., ~p is true i.e., 3 is rational There exist two positive integers a and b such that 3 a, a and b are coprime b a = 3b 3 divides a 3 divides a a = 3c, c is a positive integer, 9c = 3b b = 3c 3 divides b also 3 is a common factor of a and b which is a contradiction as a, b are coprimes. Hence p : 3 is irrational is true.. x i : 0 30 50 70 90 f i : 4 4 8 6 8 f i 80 f i x i : 40 70 400 0 70 fx i i 4000 d x x : 40 0 0 0 40 Mean = 50 i i f i d i : 60 480 0 30 30 fi di 80 Mean deviation = 80 6 80
330 EXEMPLAR PROBLEMS MATHEMATICS PART C 3. (f + g) (4) = f(4) + g(4) = (4) + 4 = 6 + = 8 (f g) (9) = f(9) g(9) = (9) 9 = 8 3 = 78 (f. g) (4) = f(4). g(4) = (4). (4) = (6) () = 3 f g (9) = f (9) (9) 8 7 g(9) 9 3 4. sin 7x + sin 5x = sin 6x cosx sin 9x + sin 3x = sin 6x cos 3x cos 7x + cos 5x = cos 6x cosx cos 9x + cos 3x = cos 6x cos 3x sin 6x cos x sin 6xcos3x L.H.S = cos 6x cos x cos 6x cos 3x sin 6 x (cos3x cos x) sin 6x = cos 6 x (cos3x cos x) cos6x = tan 6x 5. Using T r = C n n r r r x y we have T 4 = 0C 7 3 3 x 3 3 3 x 5 5 = 0.9.8 x 40 x 4 3.. 3 7 5 th term from end = ( 5 + ) = 7 th term from beginning
DESIGN OF THE QUESTION PAPER 33 T 7 = 0C 4 3 6 x 3 6 3 x = 0.9.8.7 3 890 4.3.. 6. Let the required line intersects the line 5x y + 4 = 0 at (x, y ) and the line 3x + 4y 4 = 0 at (x, y ). 5x y + 4 = 0 y = 5x + 4 3x + 4y 4 = 0 y = 4 3x 4 Points of inter section are (x, 5x + 4), 4 3x x, 4 x x and 43x 4 5x 4 5 x + x = and 0x 3x = 0 6 0 Solving to get x, x 3 3 y = 3, y = 8 3 Equation of line is y 5 = or 07x 3y 9 = 0 5 3 ( x ) 6 3
33 EXEMPLAR PROBLEMS MATHEMATICS 7. Here a = 69 and b = 44 a = 3, b = Length of major axis = 6 Length of minor axis = 4 Since e = b 44 5 5 e a 69 69 3 5 foci are (± ae, 0) = 3,0 3 = (± 5, 0) vertices are (± a, 0) = (± 3, 0) latus rectum = b (44) 88 a 3 3 8. Classes: 30-40 40-50 50-60 60-70 70-80 80-90 90-00 f: 3 7 5 8 3 f 50 x i : 35 45 55 65 75 85 95 d i : = xi 65 0 3 0 3 f i d i : 9 4 0 8 6 6 fd i i 5 fd i i : +7 8 0 8 8, fd i i 05 Mean x = 5 65 0 65 3 6 50 05 5 Variance σ = 0 0 50 50 S.D. σ = 0 4.7 9. (i) Total number of days in a non leap year = 365 = 5 weeks + day
DESIGN OF THE QUESTION PAPER 333 P(53 sun days) = 7 (ii) Total number of days in a leap year = 366 = 5 weeks + days These two days can be Monday and Tuesday, Tuesday and Wednesday, Wednesday and Thursday, Thursday and Friday, Friday and Saturday, Saturday and Sunday, Sunday and Monday P(53 Fridays) = 7 (iii) P(53 Sunday and 53 Mondays) = 7 (from ii)
334 EXEMPLAR PROBLEMS MATHEMATICS Notes