Annual Examination (2014-15) Mathematics (Set A) ANSWER KEY Date: 23 /02/15 Class: XI Time: 3 hours M. M: 100 1 2 Section A Find the component statements of the following and check whether they are true or not. All prime numbers are either even or odd Ans. p : All prime number are even q:all prime number are odd. False Find the multiplicative inverse of the complex number 3 2i. Ans. 3 Reduce the equation Ans. normal form 4 5 Three vertices of a parallelogram ABCD are A3, 1,2, B1,2, 4 and C 1,1,2 coordinates of the fourth vertex. Ans. (1,-2,8) Write the contrapositive of the following statement : If you are taking part in violent processions then you are not true Indian Ans If you are true Indian then you are not taking part in the violent processions. Find the n n x 2 6 If lim 80 and n N, find n. x2 x 2 Ans. n = 5 Section B 7 8 9 A survey shows that 63% of the American like cheese where 76% like apples. If x% of the Americans like both cheese and apples, find the value of x. In a survey of 600 students in a school, 150 students were found to be drinking Tea and 225 drinking coffee,100 were drinking both Tea and coffee. Find how many students were drinking neither Tea nor Coffee. Ans. Apply formula We get 39 (2+2 mark) Neither Tea nor Coffee = 600-275 = 325 (2+2) Two dice are thrown once. Find the probability of getting an even number on the first die or a total of 8. Ans. Let A denotes getting an even number on the first die and B denotes a total of 8 P(A) = 18/36 P(B) = 5/36 P(A B) = 3/36..(1) Using formula we get P(A B) 5/9. Find n,if the ratio of fifth term from the beginning to the fifth term from the end in the expansion of ( is ) Ans. 5 th term from end =(n-3)th term from the beginning 5th term from beginning = C(n,4)
5th term from end =.(2) fifth term from the beginning to the fifth term from the end in the expansion of ( so using the above two equations we have n =10..(2) x ysec x y x sec x 10 Evaluate: lim y0 y Evaluate: i Ans. First use secx = 1/cosx and then formula of cosa-cosb and using the concept of limits we get..(2+2) Using the identity of in the denominator we get the limit as ½.(2+2) A 1, 2,3, 4,5,6.Let R be a relation on A defined by 11 Let a, b : a, b A, bis exactly divisible by a (i) Write R in Roster form (ii) Find the domain of R (iii) Find the range of R. Ans. For roster form (2 marks) Domain = {1,2,3,4,5,6} Range = {1,2,3,4,5,6}..(1+1) 12 How many four letter words can be formed using the letter of the word INEFFECTIVE. Ans. There are 11 letters in the given word The four letter words may consist of (i) 3 alike and 1 distinct (ii) 2 alike letters of one kind and two of other kind (iii) 2 alike letters and 2 distinct (iv) All different letters..(3) Required answer = C (6,1) 4+18+540+840 =1422 (1) 13 tan A tan A 4 4 Prove that : cosec 2. tan A tan A 4 4 In any triangle ABC, If acos A = bcos B, Show that the triangle is either isosceles or right angled. Ans. = = cosec2a.. (1.5+1.5+1) Using the sine formula a = ksina,b = ksinb we get equation as acos A = bcos B ksinacos A =ksinbcos B..(1) multiply both side by ½ we get sin2a Sin2B=0.(1) solving we get A=B or angle C =90 14 Find the equation of the circle which passes through the points (1,3) and (2,-1), and has its centre on the line 2x+ y -4 =0. Find the length of axes; the coordinates of the vertices and the foci; the eccentricity and the length of latus rectum of the hyperbola 9. Ans.Let (-g,-f) be the centre of the circle -2g-f-4 =0.. (.5) ) is
Let equation of circle be 2g+6f+c = -10 4g-2f+c = -5 solving center (3/2,1).. (2.5) Equation of the circle be (1) Compare with the equation of Hyperbola 15 16 9 Vertices (0,±4) Foci(0,± ) Eccentricity = / Length of conjugate axis = 14 Length of transverse axis = 8 Length f atu re tu 9/.(.5 ea h) Prove that the lengths of the perpendiculars drawn from the points (, 0) and (, 0) to the line is. Ans. Let D,d denotes the length of perpendicular from the points (, 0) and (, 0) to the line D d = ( =..(3+1) Prove the following by the principle of mathematical induction for all n : Ans. Prove it for n = 1.5 5... Suppose the statement is true for n = k and prove it for n =k+1..(1.5) Hence by PMI the statement is true for all natural numbers..(2.5) 17 If then show that. Ans. From the given equation we have..(3) Adding them we get the result (1) 18 Find the ratio in which the join of A (1,2,3) and B(-3,4,-5) is divided by the plane Also,find the coordinates of the point of division. Ans.Let the required ratio is k:1 Let the point be (a,b,0) Using section formula k = 3/5 and the point is (-1/2,11/4,0) (1+3) 19 Find the equation of line passing through the point of intersection of two lines and having x intercept equal to 3. Ans. ( 2x +3y -2)+α(x -3y +1) =0.(1) -(2+α)/(2+α) = 3 α - equation is x+5y -.. Section C 20 Solve the following system of linear inequality graphically: Ans. Draw the three lines and do individual shading..(4.5) 5
Shade the feasible region..(1.5) 21 Find the sum of n term of the series: to n terms Show that Ans. First we find the nth term =..(2) Sn = =..(4) The nth term of the numerator = The nth term of the denominator = L.H.S. = (1) Using the sum of n terms,sum of the squares of n term and the sum of the cube of n terms identities we get the R.H.S (2.5+2.5) 22 Find the mean deviation about median of the following data: Height 95-105 105-115 115-125 125-135 135-145 145-155 (in cm) Number of 9 13 25 30 13 10 boys Given below the diameters of the circles (in mm) drawn in a design. Diameter 33-36 37-40 41-44 45-48 49-52 Number of circles 15 11517 21 22 25 Calculate the mean diameter of the circles and variance. Ans. CLASS fi C xi fi 95-105 9 9 100 26 234 105-115 13 22 110 16 208 115-125 25 47 120 6 10 125-135 30 77 130 4 120 135-145 13 90 140 14 182 145-155 10 100 150 24 240 Median = 126.(3) Mean deviation about median = 11.34..(3) Diameter fi xi yi fiyi fi 32.5-36.5 15 34.5-2 4-30 60 36.5-40.5 17 38.5-1 1-17 17 40.5-44.5 21 42.5 0 0 0 0 44.5-48.5 22 46.5 1 1 22 22 48.5-52.5 25 50.5 2 4 50 100 100 25 199 Mean = 43.5. (3 including table) Variance = 30.84 (3)
23 24 (i) Differentiate the following w.r.t. x : (ii) Differentiate the following w.r.t. x : (iii)differentiate the following w.r.t. x : Ans. (i) Using quotient rule we have derivatives as (ii) Using product rule we get result (iii)use quotient rule and get the result (2 each part) (a) If 4 digit numbers greater than 5000 are randomly formed from the digits 0,1,3,5,and 7 what is the probability of forming a number divisible by 5 when (i) the digits are repeated? (ii) the repetition of digits is not allowed? (b) The probability that a person will get an electric contract is and the probability that he will not get a plumbing contract is. If the probability of getting atleast one contract is the probability that he will get both? Ans.(a)(i) 99/249 (ii) 18/48. (1.5) each (b) 17/105. (3) what is 25 Prove that : Ans.L.H.S = = * ( )+ (1) = * ( )+..(2) = in (3) 26 (i) Differentiate ( 5) using first principle. (ii) For the function given by prove that (5) Ans.(i) Using first principle derivative is -4 sin(4x+5) (4) (ii) f (x) = 2x- 3 (5) use this and get the result (2)