# Model Question Paper Mathematics Class XII

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1 Model Question Pape Mathematics Class XII Time Allowed : 3 hous Maks: 100 Ma: Geneal Instuctions (i) The question pape consists of thee pats A, B and C. Each question of each pat is compulsoy. (ii) Pat A Question numbe 1 to 0 ae of 1 mak each. (iii) Pat B Question numbe 1 to 31 ae of 4 maks each (iv) Question numbe 3 to 37 ae of 6 maks each Pat A Choose the coect answe in each of the questions fom 1 to 0. Each of these question contain 4 options with just one coect option. 1. Let N be the set of natual numbes and R be the elation in N defined as R = {(a, b) : a = b, b > 6}. Then (A) (, 4) R (B) (3, 8) R (C) (6, 8) R (D) (8, 7) R.. If sin 1 = y, then (A) 0 y (C) 0 < y < (B) (D) y < y < sin sin ( ) 3 is equal to (A) 1 (B) 1 3

2 (C) 1 4 (D) 1 4. The numbe of all possible matices of ode 3 3 with each enty 0 o 1 is (A) 7 (B) 18 (C) 81 (D) Let A be a squae mati of ode 3 3, then k A is equal to (A) k A (B) k A (C) k 3 A (D) 3k A , when If f ( ) =, then 5, when 3 (A) f is continuous at = (B) f is discontinuous at = (C) f is continuous at = 3 (D) f is continuous at = 3 and discontinuous at = d 7. (log tan ) is equal to d (A) sec (C) sec (B) cosec (D) cosec 8. The ate of change of the aea of a cicle with espect to its adius when = 6 is (A) 10 (B) 1 (C) 8 (D) The inteval in which y = e is inceasing is (A) (, ) (B) (, 0) (C) (, ) (D) (0, ) 10. sec d is equal to (A) tan + C (C) tan log sin (B) tan + C (D) tan + log cos + C

3 1 1 d 11. e e (A) is equal to e + C (B) + C e (C) (D) e 1. Aea bounded by the cuve y = sin between = 0 and = is (A) sq. units (B) 4 sq. units (C) 8sq. units (D) 16sq. units 13. The geneal solution of the diffeential equation dy d (A) e + e y = C (B) e + e y = C (C) e + e y = C (D) e + e y = C + y = e is 14. The integating facto of the diffeential equation (A) e (B) e y dy y d = is (C) 1 (D) 15. The vecto î + α ĵ + ˆk is pependicula to the vecto î ĵ ˆk if (A) α = 5 (B) α = 5 (C) α = 3 (D) α = 3 u= i + j k v is 16. If ˆ ˆ ˆ and v= 6 iˆ 3ˆj+ kˆ, then a unit vecto pependicula to both u and (A) iˆ 10 ˆj 18k ˆ (B) 1 1 ˆ ˆ 18 ˆ i j k (C) 1 ( 7 i ˆ 10 ˆj 18k ˆ ) (D) 1 ( ˆ 10 ˆ 18 ˆ ) i j k

4 17. If the given lines is 1 y z and y z = = = = 3 k 3k 1 5 ae pependicula, then k (A) 10 (B) 10 7 (C) 10 7 (D) 7 10.(3i 4 j+ 5 k).(i j k) 18. The angle between the planes ˆ ˆ ˆ = 0 and ˆ ˆ ˆ is (A) 3 (C) 6 (B) (D) The pobability of obtaining an even pime numb on each die, when a pai of dice is olled is (A) 0 (B) 1 3 (C) 1 1 (D) If R is the set of eal numbes and f : R R be the function defined by f () = (3 ), then (fof) () is equal to (A) 1 3 (B) 3 (C) (D) (3 3 ) Pat B 1. Let R be the set of eal numbes, f : R R be the function defined by f () = 4 + 3, R. Show that f is invetible. Find the invese of f.. Pove that tan = cos, 1

5 Δ= y y 1+ y = 0 3. If, y, z ae diffeent numbes and, then show that 1 + y z 3 z z 1+ z = 0 4. Is the function f defined by f () = you answe. + 5,if 1 5,if > 1 a continuous function? Justify 5. If = a (cos t + t sin t), y = a (sin t t cos t). Find d y d 6. Find Find 1 (sin ) d + 1 d Evaluate 1- d 8. Find the solution of the diffeential equation ( + y ) d = y dy Find the equation of the cuve passing though the oigin and satisfying the diffeential equation (1 + ) dy + y = 4 d 9. If the magnitude of the vectos, a, b, c ae 3, 4, 5 espectively and a and b + c, b and ( c + a ), c and ( a + b ) ae mutually pependicula, then find the magnitude of ( a + b + c ) 30. Find the equation of the staight line passing though (1,, 3) and pependicula to the plane + y 5z + 9 = 0 Find the equation of the plane passing though the point ( 1, 3, ) and pependicula to each of the planes + y + 3z = 5 and 3 + 3y + z = A man and his wife appea fo an inteview fo two posts. The pobability of

6 husband s selection is 1 7 and that of the wife s selection is 1. Find the pobability that only one of them will be 5 selected. Pat-C 3. Obtain the invese of the following mati using elementay opeations. 0 1 A= If 3 5 A= 3 4, find A 1. Using A 1, solve the system of equations: 1 1 3y + 5z = y 4z = 5 + y = Show that the function f given by f () = tan 1 (sin + cos), > 0 is always an stictly inceasing function in 0, 4 An open bo is to be constucted by emoving equal squaes fom each cone of a 3 mete by 8 mete ectangula sheet of aluminium and folding up the sides. Find the volume of the lagest such bo. 34. Find the aea of the egion enclosed between the two cicles + y = 4 and ( ) + y = 4. Pove that the cuves y = 4 and = 4y divide the aea of the squae bounded by = 0, = 4, y = 4 and y = 0 into thee equal pats. 35. A dietician has to develop a special diet using two foods P and Q. Each packet (containing 30g) of food P contains 1 units of calcium, 4 units of ion, 6 units of cholesteol and 6 units of vitamin A. Each packet of the same quantity of food Q

7 contains 3 units of calcium, 0 units of ion, 4 units of cholesteol, and 3 units of vitamin A. The diet equies atleast 40 units of calcium, atleast 460 units of ion and atmost 300 units of cholesteol. How many packets of each food should be used to minimize the amount of vitamin A in the diet? What is the minimum amount of vitamin A? 36. A docto is to visit a patient. Fom the past epeience, it is known that the pobabilities that he will come by tain, bus, scoote o by any othe means of tanspot ae espectively 3, 1, 1 and. The pobabilities that he will be late ae 1, 1 and 1 if he comes by tain, bus and scoote espectively, but if he comes by othe means of tanspot, then he will not be late. When he aives, he is late. What is the pobability that he comes by tain. Let a pai of dice be thown and the andom vaiable X be the sum of the numbes that appea on the two dice. Find the mean o epectation of X. 37. Find the distance between the point P (6, 5, 9) and the plane detemined by points A (3, 1, ), B (5,, 4) and C ( 1, 1, 6).

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