MATHEMATICS - CLASS XI Time : 3 Hours Max. Marks : 100 The weightage of marks over different dimensions of the question paper shall be as follows: 1. Weigtage of Type of Questions Marks (i) Objective Type Questions : (10) 10 × 1 = 10 (ii) Short Answer Type questions : (12) 12 × 4 = 48 (viii) Long Answer Type Questions : (7) 7 × 6 =42 Total Questions : (29) 100 2. Weightage to Different Topics S.No. Topic Objective Type Questions S.A. Type Questions L.A. Type Questions Total 1. Sets - 1(4) - 4(1) 2. Relations and Functions - - 1(6) 6(1) 3. Trigonometric Functions 2(2) 1(4) 1(6) 12(4) 4. Principle of Mathematical Induction - 1(4) - 4(1) 5. Complex Numbers and Quadratic Equations 2(2) 1(4) -- 6(3) 6. Linear Inequalities 1(1) 1(4) - 5(2) 7. Permutations and Combinations - 1(4) - 4(1) 8. Binomial Theorem - - 1(6) 6(1) 9. Sequences and Series - 1(4) - 4(1) 10. Straight Lines 2(2) 1(4) 1(6) 12(4) 11. Conic Section - - 1(6) 6(1) 12. Introduction to three dimensional geometry - 1(4) - 4(1) 13. Limits and Derivatives 1(1) 1(4) - 5(2) 14. Mathematical Reasoning 1(1) 1(4) - 5(2) 15. Statistics - 1(4) 1(6) 10(2) 16. Probability 1(1) - 1(6) 7(2) Total 10(10) 48(12) 42(7) 100(29) 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 10 questions of 1 mark each. (iii) Part B (Short Answer Type) consists of 12 questions of 4 marks each. (iv) Part C (Long Answer Type) consists of 7 questions of 6 marks each. PART -A 111. If tan θ = and tan φ = , then what is the value of (θ + φ)?2. 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 2x + 3y – 6 = 0 on the x-axis is ................. . 23 1cos x−5. lim 2 is equal to ................. .x→0 x In Questions 6 and 7, state whether the given statements are True or False: 1 x 06. x +≥2, ∀> 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 cos2θ + sinθ + 1 = 0, lies in the interval ⎛ ππ⎞ π π ⎞ 35 ⎞ 57 ⎞⎛ 3 ⎛ππ⎛ππ(A) ⎜− ,⎟ (B) ⎜ ,⎟ (C) ⎜ , ⎟ (D) ⎜ , ⎟⎝ 44 ⎠⎝44 ⎠⎝44 ⎠⎝44 ⎠ 9. If z= 2 + 3i, the value of z⋅ zis (A)7 (B) 8 (C) 2 − 3i (D) 1 10. What is the contrapositive of the statement? “If a number is divisible by 6, then it is divisible by 3. PART - B 11. 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. 1 13 12. If cos x= 7 and cos y= 14 , x, ybeing acute angles, prove that x– y= 60°. 13. Using the principle of mathematical induction, show that 23n– 1 is divisible by 7 for all n∈ N. 14. Write z= – 4 +i43 in the polar form. 15. Solve the system of linear inequations and represent the solution on the number line: 3x– 7 > 2 (x– 6) and 6 – x > 11 – 2x bccaab +++11116. If a+ b+ c≠ 0 and a , b , c are in A.P., prove that ,,abcare also in A.P. 17. A mathematics question paper consists of 10 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 2 questions from each part. In how many ways can the student select the questions? 18. Find the equation of the line which passes through the point (–3, –2) and cuts off intercepts on xand yaxes which are in the ratio 4 : 3. 19. Find the coordinates of the point R which divides the join of the points P(0, 0, 0) and Q(4, –1, –2)in the ratio 1 : 2 externally and verify that P is the mid point of RQ. 3 − x 20. Differentiate f(x) = + with respect to x, by first principle.34x 21. Verify by method of contradiction that p = 3 is irrational. 22. Find the mean deviation about the mean for the following data: xi 10 30 50 70 90 fi 4 24 28 16 8 PART C 23. Let f(x) = x2 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 7 x +sin5 ) +(sin 9 x +sin3 ) xx24. Prove that: =tan 6 x(cos7 x +cos5 ) +(cos9 x +cos3 ) xx 25. Find the fourth term from the beginning and the 5th term from the end in the 3⎛x 3 ⎞10 expansion of ⎜−2 ⎟.⎝3 x ⎠26. A line is such that its segment between the lines 5x –y + 4 = 0 and 3x + 4y – 4 = 0 is bisected at the point (1, 5). Find the equation of this line. 27. Find the lengths of the major and minor axes, the coordinates of foci, the vertix2 y2 ces, the ecentricity and the length of the latus rectum of the ellipse +=1.169 144 28. Find the mean, variance and standard deviation for the following data: 29. What is the probability that (i) a non-leap year have 53 Sundays.(ii) a leap year have 53 FridaysClass interval: 30 - 40 40 - 50 50 - 60 60 - 70 70 - 80 80 - 90 90 -100 Frequency: 3 7 12 15 8 3 2 (iii) a leap year have 53 Sundays and 53 Mondays. MARKING SCHEME MATHEMATICS CLASS XI Q. No. Answer 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. PART -A Marks π 4 Zero 1 36 3 1 2 True False D A If a number is not divisible by 3, then it is not divisible by 6. PART - B 11. B = B ∪φ = B ∪ (A ∩ A′) = (B ∪ A) ∩ (B ∪ A′)1 = (B ∪ A) ∩ (A′∪ B) = (B ∪ A) ∩ U (Given) =B ∪ A ⇒ A ⊂ B. 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 1 1 4312. cos x = ⇒ sin x = 1 −cos 2 x =1 −= 7 49 7 1 13 169 33 cos y = ⇒ sin y = 1 −= 114 196 14 1 cos(x – y) =cosx cosy + sinx siny 2 1 ⎛⎞13 333 1⎛⎞4 ⋅= 1= ⎜⎟⎜⎟+⎝⎠⎝⎠714 7142 π 1⇒ x – y = 32 113. Let P(n) : “23n – 1 is divisble by 7” 2 1P(1) = 23 – 1 = 8 – 1 = 7 is divisible by 7 ⇒ P(1) is true. 2 Let P(k) be true, i.e, “23k – 1 is divisible by 7”, ∴ 23k – 1 = 7a, a ∈ Z 1 We have : 23(k + 1) – 1 = 23k . 23 – 1 1 1 =(23k – 1) 8 + 7 = 7a . 8 + 7 = 7(8a + 1) 2 1 ⇒ P(k + 1) is true, hence P(n) is true ∀∈ Nn2 1 14. Let – 4 + i 4 = r (cosθ + i sinθ) 2 1 ⇒r cosθ =– 4, r sinθ = 43 ⇒ r2 = 16 + 48 = 64 ⇒ r = 8. 1 2 DESIGN OF THE QUESTION PAPER 327 tanθ =– 3 ⇒ θ = π – 2 3 3 π π = 1 1 2 ∴ z= – 4 + i43 = 8 2 2 cos sin 3 3iπ π⎛ ⎞+⎜ ⎟⎝ ⎠ 1 2 15. The given in equations are : 3x– 7 > 2(x– 6) ... (i) and 6 – x > 11 – 2x... (ii) (i) ⇒ 3x– 2x> – 12 + 7 or x> –5 ... (A) 1 (ii) ⇒ –x+ 2x> 11 – 6 or x> 5 ... (B) 1 From A and B, the solutions of the given system are x> 5 1 Graphical representation is as under: 1 16. Given , , bccaab a b c + + + are in A.P. ∴1 ,1 ,1 bc ca ab a b c + + + + + + will also be in A.P. 1 1 2 ⇒ , , abcabcabc a b c ++ ++ ++ will be in A.P. Since,a+ b+ c≠ 0 ⇒ 11 1 ,,abc will also be in A.P. 17. Following are possible choices: Choice Part I Part II (i) 2 4 (ii) 3 3 (iii) 4 2 } 1 1 1 2 1 ∴Total number of ways of selecting the questions are: 55 555 5= (C2 × C4 + C3 × C3 + C4 × C2 ) =10 × 5 + 10 × 10 + 5 × 10 = 200 18. Let the intercepts on x-axis and y-axis be 4a, 3arespectively xy∴Equation of line is : + =1 4a 3a or 3x+ 4y= 12a (–3, –2) lies on it ⇒ 12a= –17 Hence, the equation of the line is 3x+ 4y+ 17 = 0 19. Let the coordinates of R be (x, y, z) 1(4) −2(0) =−4∴x= 12− 1( 1) 2(0)−− =1y= 12− 1( 2) 2(0)−− z= =2 ∴ R is (– 4, 1, 2)12− ⎛− + 441 −12 −2⎞Mid point of QR is ⎜ ,, ⎠⎟i.e., (0, 0, 0)⎝ 2 22 Hence verified. 3 −x −x3( +Δ x)20. f(x) = ∴ f(x+ Δx) =34x +Δ x)+ 34( + x 3 xx 3−−Δ −xlim −fx( +Δx) − () Δ→0 + +Δ x 34f′(x) = Δ→ x fx x 34x 4 + xlim = x 0 Δ Δx 11 2 11 2 1 2 11 2 11 2 1 2 1 1 1 1 1 2 1 (3 xx)(34)(34 −+ x 4 x) (3 −x)1−−Δ + x +Δ = lim x Δ +Δ xΔ→0 ()(3 x +4x 4 x) (3 +4) 2 9 +12 x− x−4x23 x 4xx 93 −12 x+4x2 −12 x 43 −Δ− Δ−+ x Δ+ xΔx = lim = x Δ +Δ xΔ→0 ()(3 x +4x 4 x) (3 +4) 1 15 x 15−Δ −lim == = 21x0 ()(3 Δx +4x 4 x) (3 +4) (3 +4) xΔ→ +Δ x 21. Assume thatpis false, i.e., ~pis true 1 i.e., 3 is rational 2 ∴ There exist two positive integersaand bsuch that a 13 = , aand bare coprimeb 2 ⇒ a2 = 3b2 ⇒ 3 divides a2 ⇒ 3 divides a 1 ∴ a= 3c, c is a positive integer, ∴ 9c2 = 3b2 ⇒ b2 = 3c2 ⇒ 3 divides balso 1 ∴ 3 is a common factor of aand bwhich is a contradiction as a, bare coprimes. 1 Hence p: 3 is irrational is true. 22. xi: 10 30 50 70 90 fi: 4 242816 8 ∴ ∑fi=80 21 fx: 40 720 1400 1120 720 ∴ fx=4000 1ii∑ii1| d| =| xx|:40 20 0 20 40 ∴ Mean = 50ii− 2 fi| d|160 480 0 320 320 ∴ ∑i| | 1280 1i: fdi= 1280 ∴ Mean deviation = =16 180 PART C 23. (f + g) (4) = f(4) + g(4) = (4)2 + 4 = 16 + 2 = 18 (f – g) (9) = f(9) – g(9) = (9)2 – 9 = 81 – 3 = 78 (f . g) (4) = f(4) . g(4) = (4)2. (4) = (16) (2) = 32 ⎛⎞ff (9) (9)2 81 ⎜⎟(9) = = ==27⎝⎠gg(9) 93 24. sin 7x + sin 5x = 2 sin 6x cosx sin 9x + sin 3x = 2 sin 6x cos 3x cos 7x + cos 5x = 2 cos 6x cosx cos 9x + cos 3x = 2 cos 6x cos 3x 2 sin 6 x cos x +2sin 6 x cos3 x ∴ L.H.S = 2cos 6 x cos x +2cos6 x cos3 x sin 6 x (cos3 x +cos x) sin 6 x = = cos 6 x (cos3 x +cos x) cos6 x = tan 6x r25. Using Tr +1 = n Cr xn −⋅yr we have 33⎛⎞ x 7 ⎛ ⎞ −3T43 ⎜⎟⎜ ⎟=10C ⋅2⎝⎠⎝ ⎠ 3 x 10.9.8 1 4015 15= − ⋅⋅ x =− x3.2.1 34 27 5th term from end = (11 – 5 + 1) = 7th term from beginning 11 2 11 2 11 2 11 2 1 1 1 1 1 2 1 1 2 1 1 1 1 3⎛⎞x 4 ⎛⎞ 36 ∴ T7 =10C6 ⋅21⎜⎟⎜⎟⎝⎠⎝⎠ 3 x 10.9.8.7 32 = ⋅=1890 14.3.2.1 1 26. Let the required line intersects the line 5x – y + 4 = 0 at (x1, y1) and the line 3x + 4y – 4 = 0 at (x2, y2). ∴ 5x1 – y1 + 4 = 0 ⇒ y1 = 5x1 + 4 43x2−3x2 + 4y2 – 4 = 0 ⇒y2 = 4 ⎛ 43−x2 ⎞ 1∴ Points of inter section are (x1, 5x1 + 4), ⎜x2, ⎟⎝ 4 ⎠ 2 43x2− ++ x +x 5x 412 1∴ =1 and 4 1=52 2 1 ⇒ x1 + x2 = 2 and20x1 – 3x2 = 20 2 26 20Solving to get x1 = , x2 = 123 23 2228 1 ∴ y1 = y2 = 23,23 2 222 −523∴ Equation of line is y – 5 = (x −1) 126 −123 1 or 107x – 3y – 92 = 0 2 332 EXEMPLAR PROBLEMS – MATHEMATICS 27. Here a2 = 169 and b2 = 144 ⇒ a= 13, b= 12 1 ∴ Length of major axis = 26 Length of minor axis = 24 Since e2 = 2 2 144 25 51 1 169 169 13 b e a − = − = ∴ = 1 foci are (± ae, 0) = 513 ,0 13 ⎛ ⎞± ⋅⎜ ⎟⎝ ⎠ = (± 5, 0) 1 vertices are (± a, 0) = (± 13, 0) 1 latus rectum = 2 2 2(144) 288 13 13 b a = = 1 28. Classes: 30-40 40-50 50-60 60-70 70-80 80-90 90-100 f: 3 7 12 15 8 3 2∴ 50f =∑ 1 2 xi: 35 45 55 65 75 85 95 di: = 65 10 ix − –3 –2 –1 0 1 2 3 fidi: –9 –14 –12 0 8 6 6 15iifd =−∑ 1 2 iifd : +27 28 12 0 8 12 18, 2 105iifd =∑ 1 Mean x = 1565 50− 10 65 3 62× = − = 1 Variance σ2 = 2 2105 15 10 20150 50 ⎡ ⎤⎛− ⎞− ⋅ =⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦ 1 1 2 S.D. σ = 201 14.17= 1 29. (i) Total number of days in a non leap year = 365 = 52 weeks + 1 day 1 1 ∴ P(53 sun days) = 17 (ii) Total number of days in a leap year = 366 = 52 weeks + 2 days 1 ∴ 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 2 1 ∴ P(53 Fridays) = 7 2 1 1 (iii) P(53 Sunday and 53 Mondays) = 7 (from ii) 1 2