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HPBSE Model Test Paper Class-XII Maths 1

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Parush Thakur
Cambridge International School, Kulu

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Downloaded From: http://www.cbseportal.com MATHEMATICS Max. Marks 85 I. II. Unit-I (8) 1. Relations and Functions, 3 2. Inverse Trignometric Function, 1+3 (C) = 3 = 4 Unit-II (11 ) 1. Matrices, 1+2 (C) 2. Determinants, 3 + 5 =3 = 8 III. Unit-III (37 ) 1. Continuity and Differentiability, 1+2+3 (C) 2. Applications of Derivatives, 1+2+5 (C) 3. Integrals, 1 + 3 + 3 + 3 4. Application of integrals, 5 (C) 5. Differential Equations, 1+2+3 = = = = = IV. Unit-IV (15) 1. Vectors, 1+1+3 2. Three-Dimensional Geometry, 1 + 3 + 5(C) = 5 = 9 V. Unit-V (5) 1. Linear Programming , 5 =5 VI. Unit-VI (8) 1. Probability, 1+3 +3 (C) =8 Q.No. Marks Total 1-10 11-14 15-26 27-31 1 2 3 5 10 08 42 25 85 Choice : Q.No. : 11, Marks : 2, 6 8 11 5 6 16, 18, 22, 26, 28, 29, 30 3 , 3 , 3 , 3 , 5, 5, 5, (12) = 31 Downloaded From: http://www.cbseportal.com MATHEMATICS Time : 3 Hrs. Max. Marks 85 Special Instructions : 1. You must write Question Paper Series in the circle at top left side of title page of your answer book. 2. While answering your questions you must indicate on your answer book the same question number as appears in your question paper. 3. Do not leave blank page(s) in your answer book. 4. Q.No. 1 to 10 multiple choice questions are of 1 mark each. Questions 11 to 14 are of 2 marks each, Q.Nos. 15 to 26 are of 3 marks each and Q.Nos. 27 to 31 are of 5 marks each. 5. All questions are compulsory. 6. Internal choices have been provided in some questions. You have to attempt only one of the choices in such questions. 7. Use of calculator is not permitted, however, you ask for logarithmic tables, if required from the superintendent of examinations. 8. Try to answer the questions in serial order as far as possible. Q1. If sin 1 x = y then (a) 0 y (c) 0<y< Q2. A = [aij] (a) (b) m n (d) 2 y 2< y < 2 2 is a square matrix if m<n (b) m>n (c) m=n (d) None Q3. Derivative of (ax + b)n is : (a) n (ax + b)n 1 (b) (ax + b)n 1 (c) n(ax + b)n 1 (d) na(ax + b)n 1 a a Q4. The rate of change of the area of a circle with respect to its radius r at r = 6cm is: (a) 10 (b) 12 (c) 8 (d) 11 3 dx 1+x Q5. 1 (a) 3 equals : (b) 2 3 6 (c) (13) (d) 12 Downloaded From: http://www.cbseportal.com Q6. The order of differential equation 2x2y11 3y1 + y = 0 is (a) 2 (b) 1 (c) t t 0 (d) t t t not defined t Q7. If is angle between a and b, then xa . b x = x a b x when is equal to : (a) 0 (b) (c) (d) 2 4 t t Q8. If a is a non zero vector of magnitude 'a' and a non zero scalar, then a is a unit vector if : (a) = 1 (b) = 1 (c) a = x x (d) a = 1 x x Q9. The planes : 2x y + 4z = 5 and 5x 2.5y + 10z = 6 are : (a) Perpendicular (b) Parallel (c) Intersect y-axis (d) passes through 0, 0, 4 Q10. If P(A) = 1 , P(B) = 0, then P (A/B) is 2 1 (a) 0 (b) (c) not defined 2 Q11. Find x and y if 2 1 0 3 x + y 1 0 2 = (d) 5 1 1 6 8 OR By using elementary operations, find the inverse of matrix A = Q12. Examine the function : 1 f(x) = x 5 1 2 2 1 for continuity : Q13. Find the least value of a such that the function f given by f(x) = x + ax + 1 is strictly increasing on (1, 2) Q14. Solve the differential equation : y' + y = 1 Q15. Find gof and fog if f(x) = xxxand g(x) = x5x 2x Q16. Show that : 1 sin ( 2x 1 x ) = 2sin 1 x, 1 x 1 2 2 OR Prove that : (14) Downloaded From: http://www.cbseportal.com 1 1 tan x + tan 1 2x = tan 1 x ( 3x x 1 3x ) , xxx< 1 3 Q17. By using properties of determinant prove that : b +c c+a a+b q+r r+p p+q y+z z+x x+y a b c =2 p q r x y z Q18. Is the function defined by : f(x) = x + 5, if x 1 x 5, if x >1 a continuous function ? OR Differentiate sin cos (x ) w.r.t. x ( Q19. Integrate ) x sec x dx 2 Q20. Find (x +1) dx as the limit of a sum. 0 Q21. By using properties of definite integrals, evaluate : J@2 (2 log sinx log sin2x) dx 0 Q22. Solve the differential equation : 1 (tan y x) dy = (1 + y ) dx OR Solve the differential equation (x + xy) dy = (x + y ) dx Q23. Find the area of a triangle having the points A (1, 1, 1), B(1, 2, 3) and C (2, 3, 1) as its vertices. Q24. Show that the lines y 1 x+3 = 1 3 are coplanar = z 5 5 and x+1 1 (15) = y 2 2 = z 5 5 Downloaded From: http://www.cbseportal.com Q25. Find (i) (ii) (iii) the probability distribution of Number of heads in two tosses of a coin. Number of tails in the simultaneous tosses of three coins. Number of heads in four tosses of a coin. Q26. Find the probability of getting 5 exactly twice in 7 throws of a dice OR Let E and F be events with P(E) = 3 5 P(F) = 3 10 and P(E3F) = 1 . 5 Are E and F independent ? Q27. Solve the following system of equations by matrix method : 3x 2y + 3z = 8 2x + y z = 1 4x 3y + 2z = 4 Q28. Find the local maxima and local minima if any of the function : f(x) = x 6x + 9x + 15 OR A wire of length 28m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of two pieces, so that the combined area of the square and the circle is minimum. Q29. Find the area of the region in the first quadrant enclosed by the x-axis, the line y=x and the circle x + y =32 OR Find the area under the given curves and the given lines. y = x , x =1, x = 2 and x axis Q30. Find the shortest distance between the lines x 3 y +1 z +1 x +1 = = and 1 6 1 7 OR = y 5 z 7 = 2 1 Find the equations of the planes that passes through three points (1, 1, 1), (6, 4, 5), ( 4, 2, 3) Q31. Minimise Z = 3x + 4y subject to x + 2y 8, 3x + 2y 12, x 0, y 0 (16)

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