MODEL TEST PAPER I Time : 3 hours Maimum Marks : 00 General Instructions : (i) (ii) (iii) (iv) (v) All questions are compulsory. Q. to Q. 0 of Section A are of mark each. Q. to Q. of Section B are of 4 marks each. Q. 3 to Q. 9 of Section C are of 6 marks each. There is no overall choice. However an internal choice has been provided in some questions. SECTION A. A = {,, 3, 4, 5, 6}, B = {, 3, 5, 7, 9} U = {,, 3, 4,...0}, Write (A B). Epress ( i) in the standard form a + ib. 3. Find 0 th term from end of the A.P. 3, 7,,... 407. 4. Evaluate 5 + 6 + 7 +... + 0 5. Evaluate lim 0 e e 6. Evaluate lim 0 7. A bag contains 9 red, 7 white and 4 black balls. If two balls are drawn at random, find the probability that both balls are red. 8. What is the probability that an ordinary year has 53 Sundays? 5 [XI Mathematics]
9. Write the contrapositive of the following statement : it two lines are parallel, then they do not intersect in the same plane. 0. Check the validity of the compound statement 80 is a multiple of 5 and 4. SECTION B. Find the derivative of sin with respect to from first principle. Find the derivative of sin cos sin cos with respect to.. Two students Ajay and Aman appeared in an interview. The probability that Ajay will qualify the interview is 0.6 and that Aman will qualify the interview is 0.. The probability that both will qualify is 0.04. Find the probability that (a) (b) Both Ajay and Aman will not qualify. Only Aman qualifies. 3 3. Find domain and range of the real function f 4. Let R be a relation in set A = {,, 3, 4, 5, 6, 7} defined as R = {(a, b): a divides b, a b}. Write R in Roster form and hence write its domain and range. Draw graph of f() = +. 5. Solve : sin cos. 4 6. Prove that 9 5 cos. cos cos 3 cos sin 5 sin. 6 [XI Mathematics]
7. If and y are any two distinct integers, then prove by mathematical induction that n y n is divisible by ( y) n N. 8. If + iy = (a + ib) /3, then show that 4 y Find the square roots of the comple number 7 4i 9. Find the equation of the circle passing through points (, ) and (4, 3) and has its centre on the line 3 + 4y = 7. The foci of a hyperbola coincide with of the foci of the ellipse y. Find the equation of the hyperbola, if its eccentricity is. 5 9 0. Find the coordinates of the point, at which yz plane divides the line segment joining points (4, 8, 0) and (6, 0, 8).. How many words can be made from the letters of the word Mathematics, in which all vowels are never together.. From a class of 0 students, 8 are to be chosen for an ecusion party. There are two students who decide that either both of them will join or none of the two will join. In how many ways can they be choosen? a b y SECTION C 3. In a survey of 5 students, it was found that 5 had taken mathematics, had taken physics and had taken chemistry, 5 had taken mathematics and chemistry, 9 had taken mathematics and physics, 4 had taken physics and chemistry and 3 had taken all the three subjects. Find the number of students who had taken (i) (ii) atleast one of the three subjects, only one of the three subjects. 7 [XI Mathematics]
4. Prove that 3 3 3 4 3 cos A cos A cos A cos 3 A. 3 3 4 5. Solve the following system of inequations graphically + y 40, 3 + y 30, 4 + 3y 60, 0, y 0 A manufacturer has 600 litres of a % solution of acid. How many litres of a 30% acid solution must be added to it so that acid content in the resulting miture will be more than 5% but less than 8%? 6. Find n, it the ratio of the fifth term from the beginning to the fifth term from the end in the epansion of 4 4 n is 6 :. 3 7. The sum of two numbers is 6 times their geometric mean. Show that the 3 : 3. numbers are in the ratio 8. Find the image of the point (3, 8) with respect to the line + 3y = 7 assuming the line to be a plane mirror. 9. Calculate mean and standard deviation for the following data Age Number of persons 0 30 3 30 40 5 40 50 50 60 4 60 70 30 70 80 5 80 90 The mean and standard deviation of 0 observations are found to be 0 and respectively. On rechecking it was found that an observation was misread as 8. Calculate correct mean and correct standard deviation. 8 [XI Mathematics]
ANSWERS SECTION A. (A B) = {, 3, 5, 7, 8, 9, 0}. 3 4 i 5 5 3. 33 4. 840 5. 6. 7. 8 95 8. 7 9. If two lines intersect in zone plane then they are not parallel. 0. Statement in true. SECTION B. cos sin or sin cos. (a) 0.76 (b) 0.08 3. Domain = R {, } Range = (, 0), [3, ) 4. R = {(, ), (, 3), (, 4), (, 5), (, 6), (, 7), (, 4), (, 6), (3, 6)} Domain = {,, 3} Range = {, 3, 4, 5, 6, 7} y = 3 y y = + O y 9 [XI Mathematics]
5., n z 3 8. 4 3i and 4 + 3i 9. 5 + 5y 94 + 8y + 55 = 0 y 4 0. (0, 4, 46). 4868640. 63 3. (i) 3; (ii) 5. y 40 30 0 0 O 003040 y 6. n = 0 8. (, 4) 9. Mean = 55. S.D. =.874 Correct Mech = 0. Correct S.D. =.99 0 [XI Mathematics]
MODEL TEST PAPER II Time : 3 hours Maimum Marks : 00 General Instructions : (i) (ii) (iii) (iv) All questions are compulsory. The question paper consists of 9 questions divided into three Sections A, B and C. Section A comprises of 0 questions of one mark each. Section B comprises of questions of four marks each and Section C comprises of 7 questions of si marks each. There is no overall choice. However, an internal choice has been provided in 4 questions of four marks each and questions of si marks each. You have to attempt only one of the alternatives in all such questions. SECTION A. Determine the range of the relation R defined by R = {(, + 5) : {0,,, 3, 4, 5}}. What is the probability that a letter chosen at random from a word 'EQUALITY' is a vowel? 3. Write the value of sin 75. 4. Find the derivative of a b with respect to. 5. Find : lim 6. A coin is tossed twice, then find the probability of getting at least one head. [XI Mathematics]
7. Find the value of k for which the line (k 3) (4 k ) y + K 7k + 6 = 0 is parallel to the -ais. 8. Find the value of k for which 7, k, are in G.P. 7 9. Epress i 9 + i 0 + i + i in the form of a + ib. 0. Write the general solution of cos =. SECTION B. Find the derivative of f () = cosec with respect to from the first principle. Evaluate : lim 3 9. 6 5. At what point the origin be shifted, if the co-ordinates of a point (4, 5) becomes ( 3, 9)? 3. Find the euqation of the circle passing through (0, 0) and making intercepts a and b on the co-ordinate eis. Find the co-ordinates of the foci, the vertices, the eccentricity and the y length of the latus-rectum of the ellipse. 49 36 4. Fine the co-ordinates of the points which trisect the line segment joining the point P (4,, 6) and Q (0, 6, 6). 5. A youngman visits a hospital for medical check-up. The probability that he has lungs problem is 0.45, heart problem is 0.9 and either lungs or heart problem is 0.47. What is the probability that he has both types of problems : lungs as well as heart? Out of 000 persons, how many are epected to have both types of problem? What should be done to keep good health and the hospital away? Describe briefly. [XI Mathematics]
6. Find the confficient of 5 in the product ( + ) 6 ( ) 7 using binomial theorem. Show that the coefficient of the middle term in the epanision of ( + ) n is equal to the sum of the coefficients of two middle terms in the epansion of ( + ) n. 7. Find the sum of sequence 7, 77, 777, 7777,... to n terms. 8. Determine the number of 5 card combinations out of a deck of 5 cards if each selection of 5 cards has eactly one king. 9. Convert 7i ( i ) in the polar form. 0. Prove that : sin 5 sin 3 sin cos 5 cos tan If sin 3, cos y, were any y both lie in second quadrant, find 5 3 the value of sin ( + y).. Write the contrapositive of (i) convere of (ii) negation of (iii) and identify the quantifier in (iv) (i) If a number is divisible by 9, then it is divisibily by 3. (ii) (iii) if is a prime number, then is odd. is not a comple number. (iv) For every prime number P, P is an irrational number.. If U = {,, 3,..., 5}, A = {3, 6, 9,, 5}, B = {,, 3, 4, 5}, C = {, 4, 6, 8, 0,, 4}, the find. (i) A (ii) A B (iii) A B (iv) B C 3 [XI Mathematics]
SECTION C 3. Calculate mean and variance for the following distribution : Classes 0-30 30-60 60-90 90-0 0-50 50-80 80-0 Frequency 3 5 0 3 5 The mean of 5 observations is 4.4 and their variance is 8.4. If three of the observation are, and 6, find the other two obervations. 4. The ratio of the A.M. and G.M. of two positie number a and b is m : n. show that : a : b m m n : m m n 5. Prove the following by using the principle of mathematical induction for all n N : n (n + ) (n + 5) is a multiple of 3. 6. In any triangle ABC, prove that : (b c ) cot A + (c a ) cot B + (a b ) cot C = 0 Prove that : cos + cos 3 cos. 3 3 7. A solution of 8% boric acid is to be diluted by adding a % boric acid solution to it. The resulting miture is to be more than 4% but less than 6% boric acid. If we have 640 litres of the 8% solution, how many litres of the % solution will have to be added? 8. In a survey, it is found that 05 people take X brand pan-masala, 30 take Y brand pan-masala and 45 take Z brand pan-masala. If 70 people take X brand as well as Y brand, 75 take Y brand as well as Z brand as well as Z brand 60 take X brand as well as Z brand and 40 take all the three, find how many people are surveyed who take the pan-masala of any kind? How many take Z brand pan-masala only. As a student what measures you take to spread awareness against pan-masala in society? 9. Prove that : cos 0 cos 30 cos 50 cos 70 3 6 4 [XI Mathematics]
MODEL TEST PAPER I Time : 3 hours Maimum Marks : 00 SOLUTIONS AND MARKING SCHEME SECTION A Marks. {5, 6, 7, 8, 9,0}. 3. 3 4. a ( a b) 5. 0 6. 3 4 7. 3 8. ± 9. 0 + 0i 0. n, n z 3. d f ( h) f ( ) f ( ) lim d h 0 h SECTION B 5 [XI Mathematics]
d (cosec ) d cosec ( h) cosec lim h h0 lim h0 sin( h) sin h sin sin( h) h lim 0 h sin sin( h ) h h sin cos lim h sin sin( h) h0 h h sin cos h h ( ) lim. lim h0 h0 sin sin( ) cos ( ) cot cosec sin sin lim 3 9 6 5 lim 3 9 6 5 6 5 6 5 lim 3 9 6 5 ( 6) 5 3 lim 6 5 9 6 5 6 [XI Mathematics]
= 5 + 5 = 0. Let the origin be shifted at a point (h, k). The Original co-ordinates of a point are (4, 5) =(, y) The new co-ordinates of a point are ( 3, 9) = (X, Y) X + h = 3 +h and Y + k = 9 + k for = 4 and y = 5 3 + h = 4 and 9 + k = 5 h = 7, k = 4 Hence, the origin be shifted at (7, 4). 3. Y (0, b) B C C(0, 0) A( a, 0) X Obiviously the circle passing through O(0,0), A (a, 0) and B (0, b) So AOB / So (A (a, 0), B(0.b) are the co-ordinates of end points of diameter of circle. So equation of circle is ( a) ( 0) + (y 0) (y b) = 0 + y a by = 0 y (7) (6) 7 [XI Mathematics]
Co-ordinates of foci = 3, 0 Co-ordinates of vertices = (+ 7, 0) e 3 7 Length of latus-rectum 7 7 : : 4. P A B P (4,, 6) (0, 6, 6) Let A and B be the points of triesection of PQ. A divides PQ in the ratio :. B dividaas PQ in the ratio :. A 0 4 ( 6) 6 ( 6),, i.e. A is (6, 4, ). and B 0 4 ( 6) 6 ( 6),, i.e., B is (8, 0, ). 5. let E be the event for lungs problem and E be the event for heart problem. P(E ) = 0.45, P (E ) = 0.9 P( E E ) 0.47 P( E E ) P( E ) P( E ) P( E E ) 0.47 0.45 0.9 P( E E ) P( E E ) 0.74 0.47 0.7 8 [XI Mathematics]
The epectation = 0.7 000 = 70 persons. One should do (i) regular physical eercise, (ii) walking, (iii) playing some games, (iv) avoid junk food and take healthy food, (v) avoid tension and worry. 6. (+) 6 ( ) 7 = { 6 C 0 + 6 C () + 6 C () + 6 C 3 () 3 + 6 C 4 () 4 + 6 C 5 () 5 + 6 C 6 () 6 } { 7 C 0 + 7 C ( ) + 7 C 3 ( ) 3 + 7 C 4 ( ) 4 + 7 C 4 ( ) 5 + 7 C 6 ( ) 6 + 7 C 7 ( ) 7 } = ( + + 60 + 60 3 + 40 4 + 9 5 + 9 5 + 64 6 ) ( 7 + + 35 3 + 35 4 5 + 7 6 7 ) Coefficient of 5 as n is even = ( ) + (35) + 60 ( 35) + 60 + () + 40 ( 7) + 9 = 397 380 = 7 So middle term (of ( + ) n = (n + )th term. = n C n X n Coefficient of n = n C n similarly, middle term of ( + ) n = nth and (n+ ) th term The coefficient of these terms are n C n and n C n respectively. For showing n C n + n C n = n C n 7. S n = 7 + 77+ 777 + 7777 +...to n terms 7 [9 99 999 9999...to n terms] 9 7 3 4 0 0 0 0... to n terms 9 9 [XI Mathematics]
0 0 0 3... n terms... n terms 7 9 n n 7 0(0 ) 7 0(0 ) n n 9 0 9 9 8. Required number of ways = 4 C 48 C 4 4 48 47 46 45 3 4 = 4 47 46 45 = 77830 9. Comple number 7i i ( i ) z r z amplitude = 3 4 Required polar form 3 3 cos i sin 4 4 0. L.H.S. sin 5 sin 3 sin cos 5 cos sin 5 sin sin 3 cos 5 cos sin 3 cos sin 3 sin 3 sin sin 3 (cos ) sin 3 sin cos sin 30 [XI Mathematics]
sin sin cos = tan = R.H.S. sin ( + y) = sin X cos y + cos sin y...(i) cos sin 9 5 5 cos 4 5 Sicne lies in second quadrat. cos 4 5 sin cos y 44 5 69 69 sin y 5 3 sin y 5 3 From (i), 3 4 5 sin( y ) 5 3 5 3 56 65. (i) If a number is not divisible by 3, it is not divisible by 9. (ii) If a number is odd, then it is a prime number. (iii) is a comple number. (iv) For every. 3 [XI Mathematics]
. (i) A = {,, 4, 5, 7, 8, 0,, 3, 4} (ii) A B = { 6, 9,, 5} (iii) A B = {,, 3, 4, 5, 6, 9,, 5} (iv) B C = {, 4} SECTION C 3. Classes Mid-point i f 05 u i fu u 30 fu 0-30 5 3 6 9 8 30-60 45 3 6 4 60-90 75 5 5 5 90-0 05 0 0 0 0 0 0-50 35 3 3 3 50-80 65 5 0 4 0 80-0 95 3 6 9 8 N = 30 fu = fu = 76 ( marks for above calculation) Mean, X A h fu N 05 30 30 = 07 Variance ( ) fu fu N N 3 [XI Mathematics]
= 900 (76) 30 30 = 76 Let the other two observation be any 6. The series is,, 6, y. 6 y Mean, X 4.4 5 + y = 3...(i) Variance 5 8.4 i n i (3.4) (.4) (.6) y 8.4 5 4.4( y ) (4.4) y 97 Solving (i) and (ii), we get 9, y 4 or 4, y 9 Hence, two observation are 4 and 9. a b 4. A.M. G.M. ab a b m ab n 33 [XI Mathematics]
Applying componendo and dividendo property, we get a b ab m n a b ab m n a b m n a b m n a b m n a b m n Applying comonendo and dividendo property again, a b a b a b a b m n m n m n m n a m n m n b m n m n a m n m n b m n m n Squaring, a m n m n m n b b n m n m n m m n m m n m m n m m n a : b m m n : m m n Proved. 5. Let P (n) (n + ) (n + 5) is a multiple of 3. 34 [XI Mathematics]
P () is ( + ) ( + 5) is a multiple of 3. i.e. is multiple of 3 which is true. So, P() is true. Let P(m) be true, m N. m(m + ) (m + 5) is a multiple of 3. m(m + ) (m + 5) = 3 (let), where is an integer. We shall prove that P (m + ) is true. i.e., (m + ) (m + +)(m + + 5) is a multiple of 3. Now, (m + ) (m + ) (m + 6) = (m + ) (m + 8m + } = (m + ) {(m + 5m) + (3m + )} = (m + ) (m + 5m) + (m + ) (3m + ) = (m + ) m(m + 5) + 3(m + ) (m + 4) = 3 + 3 (3m + ) (m + 4) = 3{ + (m + ) (m + 4)} = a multiple of 3 P (m + ) is true. So by induction P(n) is true for all a b c 6. By sine formula, k( let) sin A sin B sin C n N a k sin A, b k sin B, c k sin C L.H.S ( b c ) cot A ( c a ) cot B ( a b ) cot C cos A cos B cos C ( b c ) ( c a ) ( a b ) sin A sin B sin C 35 [XI Mathematics]
b c a k c a b k ( b c ). ( c a ). bc a ca b a b c k ( a b ). ab c k ( b c )( b c a ) ( c a ) c a b abc ( a b )( a b c ) k 0 0 R. H. S ab L.H.S cos cos cos 3 3 3 cos cos cos 3 3 3 cos cos cos 3 3 cos cos cos 3 3 cos cos cos 3 3 cos cos 3 R. H. S 36 [XI Mathematics]
7. Suppose litre of % solution is added for dilution Total miture = (640 + ) litre 4 640 8 6 (640 ) (640 ) 00 00 00 00 4 50 6 00 00(640 ) 00 50 4 6 640 4(640 ) 50 6(640 ) 560 4 50 and 50 3840 6 4 50 560 and 50 3840 6 560 and 80 4 80 and 30 30 80 Hence the volume of % solution to be added lies between 30 litres and 80 litres. 8. X-brand Y-brand e b f c a d g Z-brand 37 [XI Mathematics]
a b 70 since a 40 b 30 a d 75 d 35 a c 60 c 0 e c a b 05 e 5 a b d f 30 f 5 g c a d 45 g 50 Total people surveyed a b c d e f g = 5 No. of people taking Z-brand = g = 50 Taking pan-masala is very injurious to health. It causes cancer, mental disorder, high blood pressure and various other diseases. There is also a wastage of money in taking it. Compaign against pan-masala is alo required. 9. L.H.S. = cos 0 cos 30 cos 50 cos 70 = cos 30 (cos 70 cos 50 ) cos 0 3 ( cos 70 cos 50 ) cos 0 3 (cos 0 cos 0 ) cos 0 4 3 ( cos 0 cos 0 4 3 ( cos 0 cos 0 cos 0 4 38 [XI Mathematics]
3 ( cos 0 cos 0 cos 0 8 3 ( cos 0 cos 30 cos 0 8 3 cos 30 8 3 3 8 3 6 R. H. S 39 [XI Mathematics]