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SAMPLE QP FOR XI HALF-YEARLY EXAMINATION SUB: MATHEMATICS CLASS - XI TIME: 3 HOURS M.M: 100 General Instructions: 1. All questions are compulsory. 2. The question paper consists of 26 questions divided into three sections viz. A, B and C. Section A comprises of 6 questions of one mark each, Section B comprises of 13 questions of four marks each and Section C comprises of 7 questions of six marks each. 3. There is no overall choice .However, internal choice has been provided in 4 questions of four marks each and 2 questions of six marks each. You have to attempt only one of the alternatives in all such questions. 4. Use of calculator is not permitted. SECTION-A 1. What is the number of proper subsets of a set containing n elements? (1) 2. If R is a relation from a finite set A having 4 elements to a finite set B having 3 elements, then what is the number of relations from A to B? (1) 3. What is the value of (1+i)(1+ )(1+ )(1+ ) ? (1) 4. Write the solution of 4x+3 6x+7 in interval form. (1) 5. Find the coefficient of (1) in . 6. Find the sum to infinity of the G.P. . (1) SECTION B Page 1 of 8 7. If U={1,2,3,4,5,6,7,8,9,10} ,A={2,4,6,8,10}, B={1,3,5,7,9} and C={3,4,7,8,!0}, (i) ( B C ) Find : (ii) ( A C ) 8. If A,B and C are any three sets, then prove that : A - (B (4) ) = (A-B) (A-C) (4) 9. Let R be the relation on the set N of natural numbers defined by R (a, b) : a 3b 12, a, b N Find: (a)R (b) Domain of R (c) Range of R (4) OR Find the domain and range of the function f(x) given by f(x) = . 10. Let f,g : R→R be defined , respectively by f(x) = x+1, g(x) = 2x-3 . Find f+g, f-g, fg and . (4) 11. Prove that: = (4) 12. Solve for x : 4 sin 2 x 4 cos x 1 (4) OR Solve: =√ 13. In a triangle ABC , if a=18, b=24 and c=30, find cosA, cosB and cosC (4) 14. Find the square root of the complex number 5 12i (4) 15. If = a ib, x, y, a, b R , then prove that : 4( ) (4) OR Convert the complex number √ into polar form. Page 2 of 8 16. a). How many different words can be formed using the letters of the word HARYANA? (4) b).How many of these begin with H and end with N ? c).In how many of these H and N are together? 17. Find the term independent of x in the expansion of . (4) 18. Find the image of the point (3,8) with respect to the line x+3y=7 assuming the line to be a plane mirror. (4) 19. Find the equation of the line passing through the intersection of the lines (4) 3x-4y+1=0 and 5x+y-1 =0 and cutting equal intercepts on the coordinate axes. OR Verify that the area of the triangle with vertices (4,6),(7,10) and (1,-2) remains invariant under the translation of axes when origin is shifted to the point (-2,1). SECTION - C 20. In a survey of 25 students, it was found that 15 had taken Mathematics,12 had taken Physics and 11 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 that had taken (a) at least one of the three subjects, (b) only one of the subjects. (6) Page 3 of 8 21. . Prove that : = (6) OR Prove that: A+ = 22. A committee of 8 persons is to be constituted from a group of 5 women and 7 men. In how many ways can this be done? In how many ways can the committee be formed if it consists of atleast 3 women and 3 men? What value is depicted by taking men and women in forming the committee? (6) 23. Solve the following system of inequalities graphically: (6) x+2y 10, x+y 1, x-y 0, x 0 , y 0 24. Prove the following by principle of mathematical induction for all n N : 1.3+3.5+5.7+……………………….+(2n-1)(2n+1) = (6) 25. Find the equations of the medians of a triangle formed by the lines x+y-6=0, x-3y-2=0 and 5x-3y+2=0. 26. If a and b are the roots of (6) -3x+p =0 and c,d are the roots of -12x+q =0,where a,b,c,d form a G.P.. Prove that (q+p):(q-p) = 17:15. (6) OR Find the sum of the first n terms of the series: 3+7+13+21+31+……………… Page 4 of 8 Marking Scheme Answers: 1. SECTION-A -1 4. (-2, ) 2. 3.0 5.1512 6. 1 For each correct answer, 1mark SECTION-B 7. (B ) ={1,3,4,5,7,8,9,10} (A-C) = {2,6} )’ = (A (1) =A (1) = (A-B) 9. R= {(9,1),(6,2),(3,3)} For domain 3-x (1) (1) (A-C) (1) 0 x= (1) Range of R={1,2,3} (1) –x – 3 (1) domain of R = R- {3] (1) 10. f+g = 3x-2 Fg = 2 (1) (2) Domain of R={9,6,3} OR )’ ={2,6} (B (A-C)’ = {1,3,4,5,7,8,9,10} (1) ’ 8. LHS=A (1) (1) Range of R= R-{-1} (1) f-g = -x+4 (1) 11. = (1) (1) ,x (1) ( ) = 2sin4x .cos2x +2sin4x (1) = 2sin4x(cos2x+1) (1) =4 ( x sin4x ) 12. 4 sin 2 x 4 cos x 1 4 cos 2 x 4 cos x 3 0 (1) 3 cos x ( Not Acceptable) 2 1 x 2n , n N 1 3 2 cos x OR 1 1 , ( ) 2 2 cosθ -√ sinθ = -1 (1) 1 √ = (1) Page 5 of 8 Cos( +θ) = cos (1) 13. cosA = = cosB = = cosC = =0 14. Let √ ( +1) ( = x+iy =5+12i (1) OR ( ) 2 √ 16. a. and y= ±2 17. Let it be 12-3r =0 y= - (1) (1) (1 ) r= | | = 8 (1) (1) (1) polar form= 8(cos b. term r= 4 (1) x+iy= = -4+4i√ = 840 (1) ( ) = arg(z) = 18. (2) Solving x= = 3+2i or -3-2i 15. (x+iy) = (a+ib)3 = ( ) =5 and 2xy=12 = 13 x= (1) ( +1) +2xyi =5+12i √ θ = 2nπ + (1) (1) ) (1) c. (1) X2 (2) term = C(n,r) (1) term (independent of x)=C(6,4) . (2) x+3y=7 Equation of vertical line 3x-y =1 (3,8) Foot of the perpendicular =(1,2) (1) 19. Finding point of intersection (3/23,8/23) Considering the equation in intercept form image = (-1,-4) (1) (1) (1) Page 6 of 8 To find the intercept 11/23 (1) (1) Finding the equation of line 23x+23y=11 OR Let P(4,6), Q(7,10) and R(-1,2) be the given points. Area of triangle PQR= 6 sq. units When origin is (-2,1) (1 ) New coordinates are (6,5), (9,9) and (1,1) (1) Now the area of the triangle = 6 sq. units (1 ) SECTION-C 20. n(P n(U) = 25 n(M) =15 n(P)=12 n(C)=11 )=4 n(M ) =3 (2) (a). n(M n(M )=15+12+11-5-9-4+3=23 ) =5 n(M (2) (b). Taken only one of the subjects= 23-5-9-4+6=11 21. = (cos20 √ = cos20 sin80 - √ √ OR sin80 + LHS= √ √ (1 ) cos60 .sin80 )- - () √ ).sin80 = (sin100 = (2) ) sin 80 sin 60 (2sin40 √ (1) sin80 sin80 + ( (1) ( ) + ( = (3+cos2A+2cos2A( )) (2) ) (1) (3+cos2A +2cos2A.cos240 ) 22. C(12,8) =495 (3) ) = = (3+cos2A+cos(2A+240 )+cos(2A-240 )) = ) =9 (1 ) (1) = (1) .C(5,3).C(7,5)+ C(5,4)C(7,4) +C(5,5)C(7,3) =420 For any one value one mark. 23. For each correct line 1 mark (1X3) For correct region 3 marks Page 7 of 8 24. To prove for n=1 1mark To assume that the statement is true for n=k 1mark To prove that the statement is true for n=k+1 4marks 25. Finding the vertices A(2,4),B(5,1) and C(-1,-1) ( ) Finding the mid points of AB,BC and AC i.e. F(7/2,5/2),D(2,0) and E(1/2,3/2) ( ) Equation of the median AD x=2 (1) Equation of the median BE x+9y -14 =0 (1) Equation of the median CF 7x-9y-2=0 (1) 26. a+b= 3 ,ab= p b =ar , c = a r= 2 OR (1) ,d =a c+d =12 , cd = q (1) ( ) a(1+r) = 3 , a (1+r) = 12 = = = (1) (1) ( ) S =3+7+13+21+…………………………………….+. S= 3+7+13+21+………………………………………….+ 0 = 3+4+6+8+……………………………………………………….(1) = 3+4+6+8+……………………………………….(n-1)terms = =∑ +n+1 ∑ = = ∑ + ( (1) (1) +n ) ( ) ( ) Page 8 of 8