Page 1 CBSE XII | Mathematics Board Paper 2018 â€“ Set 3 CBSE Board Class XII Mathematics Board Paper 2018 Time: 3 hrs Total Marks: 100 General Instructions: 1. All questions are compulsory. 2. The question paper consists of 29 questions divided into four Section A, B, C and Ds. Section A comprises of 4 questions of one mark each, Section B comprises of 8 questions of two marks each, Section C comprises of 11 questions of four marks each and Section D comprises of 6 questions of six marks each. 3. All questions in section A are to be answered in one word, one sentence or as per the exact requirement of the question. 4. There is no overall choice. However, internal choice has been provided in 3 questions of four marks each and 3 questions of six marks each. You have to attempt only one of the alternatives in all such questions. 5. Use of calculators is not permitted. You may ask for logarithmic tables, if required. SECTION â€“ A 1. o Find the magnitude of each of two vectors a and b, having the same magnitude 9 such that the angle between them is 60 and their scalar product is . 2 2. -1 1 Find the value of tan 3 cot ( 3). ? ?? 3. if a b denotes the larger of 'a' and 'b' and if a o b = (a b)+3, then write the value of (5) o (10) where and o are binary operations. ?? ? 4. 0 a 3 If the matrix A = 2 0 1 is skew symmetric, find the values of 'a' and 'b' b 1 0 ? ?? ?? ? ?? ?? ?? Page 2 CBSE XII | Mathematics Board Paper 2018 â€“ Set 3 CBSE Board Class XII Mathematics Board Paper 2018 Time: 3 hrs Total Marks: 100 General Instructions: 1. All questions are compulsory. 2. The question paper consists of 29 questions divided into four Section A, B, C and Ds. Section A comprises of 4 questions of one mark each, Section B comprises of 8 questions of two marks each, Section C comprises of 11 questions of four marks each and Section D comprises of 6 questions of six marks each. 3. All questions in section A are to be answered in one word, one sentence or as per the exact requirement of the question. 4. There is no overall choice. However, internal choice has been provided in 3 questions of four marks each and 3 questions of six marks each. You have to attempt only one of the alternatives in all such questions. 5. Use of calculators is not permitted. You may ask for logarithmic tables, if required. SECTION â€“ A 1. o Find the magnitude of each of two vectors a and b, having the same magnitude 9 such that the angle between them is 60 and their scalar product is . 2 2. -1 1 Find the value of tan 3 cot ( 3). ? ?? 3. if a b denotes the larger of 'a' and 'b' and if a o b = (a b)+3, then write the value of (5) o (10) where and o are binary operations. ?? ? 4. 0 a 3 If the matrix A = 2 0 1 is skew symmetric, find the values of 'a' and 'b' b 1 0 ? ?? ?? ? ?? ?? ?? CBSE XII | Mathematics Board Paper 2018 â€“ Set 3 SECTION B 5. A black and red die are rolled together. Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4. 6. If is the angle between two vectors i 2j k and 3i-2j+k , find sin ?? ? ? ? ? 7. Find the differential equation representing the family of curves y = ae bx+5 , where a and b are arbitrary constants. 8. 2 2 Evaluate : cos 2x + 2 sin x dx cos x ? 9. The total cost C(x) associated with the production of x units of an item is given by C(x) = 0.005x 3 â€“ 0.02x 2 + 30x + 5000. Find the marginal cost when 3 units are produced, where by marginal cost we mean the instantaneous rate of change of total cost at any level of output. 10. -1 1 cos x Differentiate tan with respect to x. sin x ??? ?? ?? 11. -1 -1 2 -3 Given A = , compute A and show that 2A 9I A -4 7 ?? ?? ?? ?? 12. Prove that : -1 -1 3 11 3 sin x sin (3x 4x ),x , 22 Page 3 CBSE XII | Mathematics Board Paper 2018 â€“ Set 3 CBSE Board Class XII Mathematics Board Paper 2018 Time: 3 hrs Total Marks: 100 General Instructions: 1. All questions are compulsory. 2. The question paper consists of 29 questions divided into four Section A, B, C and Ds. Section A comprises of 4 questions of one mark each, Section B comprises of 8 questions of two marks each, Section C comprises of 11 questions of four marks each and Section D comprises of 6 questions of six marks each. 3. All questions in section A are to be answered in one word, one sentence or as per the exact requirement of the question. 4. There is no overall choice. However, internal choice has been provided in 3 questions of four marks each and 3 questions of six marks each. You have to attempt only one of the alternatives in all such questions. 5. Use of calculators is not permitted. You may ask for logarithmic tables, if required. SECTION â€“ A 1. o Find the magnitude of each of two vectors a and b, having the same magnitude 9 such that the angle between them is 60 and their scalar product is . 2 2. -1 1 Find the value of tan 3 cot ( 3). ? ?? 3. if a b denotes the larger of 'a' and 'b' and if a o b = (a b)+3, then write the value of (5) o (10) where and o are binary operations. ?? ? 4. 0 a 3 If the matrix A = 2 0 1 is skew symmetric, find the values of 'a' and 'b' b 1 0 ? ?? ?? ? ?? ?? ?? CBSE XII | Mathematics Board Paper 2018 â€“ Set 3 SECTION B 5. A black and red die are rolled together. Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4. 6. If is the angle between two vectors i 2j k and 3i-2j+k , find sin ?? ? ? ? ? 7. Find the differential equation representing the family of curves y = ae bx+5 , where a and b are arbitrary constants. 8. 2 2 Evaluate : cos 2x + 2 sin x dx cos x ? 9. The total cost C(x) associated with the production of x units of an item is given by C(x) = 0.005x 3 â€“ 0.02x 2 + 30x + 5000. Find the marginal cost when 3 units are produced, where by marginal cost we mean the instantaneous rate of change of total cost at any level of output. 10. -1 1 cos x Differentiate tan with respect to x. sin x ??? ?? ?? 11. -1 -1 2 -3 Given A = , compute A and show that 2A 9I A -4 7 ?? ?? ?? ?? 12. Prove that : -1 -1 3 11 3 sin x sin (3x 4x ),x , 22 CBSE XII | Mathematics Board Paper 2018 â€“ Set 3 SECTION C 13. Two numbers are selected at random (without replacement) from the first five positive integers. Let X denote the larger of the two numbers obtained. Find the mean and variance of X. 14. An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when depth of the tank is half of its width. If the cost is to be borne by nearby settled lower income families, for whom water will be provided, what kind of value is hidden in this question? 15. Find the equations of the tangent and the normal, to the curve 16x 2 + 9y 2 = 145 at the point (x1, y1), where x1 = 2 and y1 > 0. OR 4 32 x Find the intervals in which the function f(x) = x 5x 24x 12 is 4 (a) strictly increasing, (b) strictly decreasing. ? ? ? ? 16. 2 2 2 dy If (x +y ) = xy, find . dx OR dy If x = a (2 - sin2 ) and y = a (1- cos2 ), find when = . dx 3 ? ? ? ? ? 17. 2 2 2 d y dy if y = sin (sin x), prove that tan x +y cos x 0 dx dx ?? 18. x x 2 Find the particular solution of the differential equation e tan y dx + (2 - e ) sec y dy = 0, given that y = when x = 0. 4 OR Find the particula ? dy r solution of the differential equation 2y tan x = sin x, given that dx y = 0 when x = 3 ? ? Page 4 CBSE XII | Mathematics Board Paper 2018 â€“ Set 3 CBSE Board Class XII Mathematics Board Paper 2018 Time: 3 hrs Total Marks: 100 General Instructions: 1. All questions are compulsory. 2. The question paper consists of 29 questions divided into four Section A, B, C and Ds. Section A comprises of 4 questions of one mark each, Section B comprises of 8 questions of two marks each, Section C comprises of 11 questions of four marks each and Section D comprises of 6 questions of six marks each. 3. All questions in section A are to be answered in one word, one sentence or as per the exact requirement of the question. 4. There is no overall choice. However, internal choice has been provided in 3 questions of four marks each and 3 questions of six marks each. You have to attempt only one of the alternatives in all such questions. 5. Use of calculators is not permitted. You may ask for logarithmic tables, if required. SECTION â€“ A 1. o Find the magnitude of each of two vectors a and b, having the same magnitude 9 such that the angle between them is 60 and their scalar product is . 2 2. -1 1 Find the value of tan 3 cot ( 3). ? ?? 3. if a b denotes the larger of 'a' and 'b' and if a o b = (a b)+3, then write the value of (5) o (10) where and o are binary operations. ?? ? 4. 0 a 3 If the matrix A = 2 0 1 is skew symmetric, find the values of 'a' and 'b' b 1 0 ? ?? ?? ? ?? ?? ?? CBSE XII | Mathematics Board Paper 2018 â€“ Set 3 SECTION B 5. A black and red die are rolled together. Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4. 6. If is the angle between two vectors i 2j k and 3i-2j+k , find sin ?? ? ? ? ? 7. Find the differential equation representing the family of curves y = ae bx+5 , where a and b are arbitrary constants. 8. 2 2 Evaluate : cos 2x + 2 sin x dx cos x ? 9. The total cost C(x) associated with the production of x units of an item is given by C(x) = 0.005x 3 â€“ 0.02x 2 + 30x + 5000. Find the marginal cost when 3 units are produced, where by marginal cost we mean the instantaneous rate of change of total cost at any level of output. 10. -1 1 cos x Differentiate tan with respect to x. sin x ??? ?? ?? 11. -1 -1 2 -3 Given A = , compute A and show that 2A 9I A -4 7 ?? ?? ?? ?? 12. Prove that : -1 -1 3 11 3 sin x sin (3x 4x ),x , 22 CBSE XII | Mathematics Board Paper 2018 â€“ Set 3 SECTION C 13. Two numbers are selected at random (without replacement) from the first five positive integers. Let X denote the larger of the two numbers obtained. Find the mean and variance of X. 14. An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when depth of the tank is half of its width. If the cost is to be borne by nearby settled lower income families, for whom water will be provided, what kind of value is hidden in this question? 15. Find the equations of the tangent and the normal, to the curve 16x 2 + 9y 2 = 145 at the point (x1, y1), where x1 = 2 and y1 > 0. OR 4 32 x Find the intervals in which the function f(x) = x 5x 24x 12 is 4 (a) strictly increasing, (b) strictly decreasing. ? ? ? ? 16. 2 2 2 dy If (x +y ) = xy, find . dx OR dy If x = a (2 - sin2 ) and y = a (1- cos2 ), find when = . dx 3 ? ? ? ? ? 17. 2 2 2 d y dy if y = sin (sin x), prove that tan x +y cos x 0 dx dx ?? 18. x x 2 Find the particular solution of the differential equation e tan y dx + (2 - e ) sec y dy = 0, given that y = when x = 0. 4 OR Find the particula ? dy r solution of the differential equation 2y tan x = sin x, given that dx y = 0 when x = 3 ? ? CBSE XII | Mathematics Board Paper 2018 â€“ Set 3 19. Find the shortest distance between the lines r (4i j) (i 2j 3k) and r (i j 2k) (2i 4j 5k) ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 20. 2 Find : 2 cos x dx (1 sinx)(1 sin x) ?? ? 21. Suppose a girl throws a die. If she gets 1 or 2, she tosses a coin three times and notes the number of tails. If she gets 3,4,5 or 6, she tosses a coin once and notes whether a â€˜headâ€™ or â€˜tailâ€™ is obtained. If she obtained exactly one â€˜tailâ€™. What is the probability that she threw 3, 4, 5 sor 6 with the die? 22. Let a 4i 5j k, b i - 4j 5k and c 3i + j k. Find a vector d which is perpendicular to both c and b and d . a = 21. ? ? ? ? ? ? ? ? ? ? 23. Using properties of determinants, prove that 1 1 1+3x 1+3y 1 1 9(3xyz+xy+yz+zx) 1 1+3z 1 ? 24. Using integration, find the area of the region in the first quadrant enclosed by the x- axis, the line y = x and the circle x 2 + y 2 = 32 25. Let A = {x Z :0 x 12). Show that R = {(a,b): a,b A, a b is divisible by 4} is an equivalence relation. Find the set of all elements related to 1. Also write the equivalence class {2} ? ? ? ?? 2 OR x Show that function f : R R defined by f(x) = , x R is neither one-one nor onto. x1 Also, if g : R R is defined as g(x) = 2x - 1, find fog (x) ? ? ? ? ? Page 5 CBSE XII | Mathematics Board Paper 2018 â€“ Set 3 CBSE Board Class XII Mathematics Board Paper 2018 Time: 3 hrs Total Marks: 100 General Instructions: 1. All questions are compulsory. 2. The question paper consists of 29 questions divided into four Section A, B, C and Ds. Section A comprises of 4 questions of one mark each, Section B comprises of 8 questions of two marks each, Section C comprises of 11 questions of four marks each and Section D comprises of 6 questions of six marks each. 3. All questions in section A are to be answered in one word, one sentence or as per the exact requirement of the question. 4. There is no overall choice. However, internal choice has been provided in 3 questions of four marks each and 3 questions of six marks each. You have to attempt only one of the alternatives in all such questions. 5. Use of calculators is not permitted. You may ask for logarithmic tables, if required. SECTION â€“ A 1. o Find the magnitude of each of two vectors a and b, having the same magnitude 9 such that the angle between them is 60 and their scalar product is . 2 2. -1 1 Find the value of tan 3 cot ( 3). ? ?? 3. if a b denotes the larger of 'a' and 'b' and if a o b = (a b)+3, then write the value of (5) o (10) where and o are binary operations. ?? ? 4. 0 a 3 If the matrix A = 2 0 1 is skew symmetric, find the values of 'a' and 'b' b 1 0 ? ?? ?? ? ?? ?? ?? CBSE XII | Mathematics Board Paper 2018 â€“ Set 3 SECTION B 5. A black and red die are rolled together. Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4. 6. If is the angle between two vectors i 2j k and 3i-2j+k , find sin ?? ? ? ? ? 7. Find the differential equation representing the family of curves y = ae bx+5 , where a and b are arbitrary constants. 8. 2 2 Evaluate : cos 2x + 2 sin x dx cos x ? 9. The total cost C(x) associated with the production of x units of an item is given by C(x) = 0.005x 3 â€“ 0.02x 2 + 30x + 5000. Find the marginal cost when 3 units are produced, where by marginal cost we mean the instantaneous rate of change of total cost at any level of output. 10. -1 1 cos x Differentiate tan with respect to x. sin x ??? ?? ?? 11. -1 -1 2 -3 Given A = , compute A and show that 2A 9I A -4 7 ?? ?? ?? ?? 12. Prove that : -1 -1 3 11 3 sin x sin (3x 4x ),x , 22 CBSE XII | Mathematics Board Paper 2018 â€“ Set 3 SECTION C 13. Two numbers are selected at random (without replacement) from the first five positive integers. Let X denote the larger of the two numbers obtained. Find the mean and variance of X. 14. An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when depth of the tank is half of its width. If the cost is to be borne by nearby settled lower income families, for whom water will be provided, what kind of value is hidden in this question? 15. Find the equations of the tangent and the normal, to the curve 16x 2 + 9y 2 = 145 at the point (x1, y1), where x1 = 2 and y1 > 0. OR 4 32 x Find the intervals in which the function f(x) = x 5x 24x 12 is 4 (a) strictly increasing, (b) strictly decreasing. ? ? ? ? 16. 2 2 2 dy If (x +y ) = xy, find . dx OR dy If x = a (2 - sin2 ) and y = a (1- cos2 ), find when = . dx 3 ? ? ? ? ? 17. 2 2 2 d y dy if y = sin (sin x), prove that tan x +y cos x 0 dx dx ?? 18. x x 2 Find the particular solution of the differential equation e tan y dx + (2 - e ) sec y dy = 0, given that y = when x = 0. 4 OR Find the particula ? dy r solution of the differential equation 2y tan x = sin x, given that dx y = 0 when x = 3 ? ? CBSE XII | Mathematics Board Paper 2018 â€“ Set 3 19. Find the shortest distance between the lines r (4i j) (i 2j 3k) and r (i j 2k) (2i 4j 5k) ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 20. 2 Find : 2 cos x dx (1 sinx)(1 sin x) ?? ? 21. Suppose a girl throws a die. If she gets 1 or 2, she tosses a coin three times and notes the number of tails. If she gets 3,4,5 or 6, she tosses a coin once and notes whether a â€˜headâ€™ or â€˜tailâ€™ is obtained. If she obtained exactly one â€˜tailâ€™. What is the probability that she threw 3, 4, 5 sor 6 with the die? 22. Let a 4i 5j k, b i - 4j 5k and c 3i + j k. Find a vector d which is perpendicular to both c and b and d . a = 21. ? ? ? ? ? ? ? ? ? ? 23. Using properties of determinants, prove that 1 1 1+3x 1+3y 1 1 9(3xyz+xy+yz+zx) 1 1+3z 1 ? 24. Using integration, find the area of the region in the first quadrant enclosed by the x- axis, the line y = x and the circle x 2 + y 2 = 32 25. Let A = {x Z :0 x 12). Show that R = {(a,b): a,b A, a b is divisible by 4} is an equivalence relation. Find the set of all elements related to 1. Also write the equivalence class {2} ? ? ? ?? 2 OR x Show that function f : R R defined by f(x) = , x R is neither one-one nor onto. x1 Also, if g : R R is defined as g(x) = 2x - 1, find fog (x) ? ? ? ? ? CBSE XII | Mathematics Board Paper 2018 â€“ Set 3 26. Find the distance of the point (-1,-5,-10) from the point of intersection of the line r = 2i - j + 2k + (3i 4j + 2k) and the plane r = (i - j + k) 5. ? ? ? ? ? ? 27. A factory manufactures two types of screws A and B, each type requiring the use of two machines, an automatic and a hand-operated. It takes 4 minutes on the automatic and 6 minutes on the hand-operated machines to manufacture a packet of screws â€˜Aâ€™ while it takes 6 minutes on the automatic and 3 minutes on the hand-operated machine to manufacture packet of screws â€˜Bâ€™. Each machine is available for at most 4 hours on any day. The manufacturer can sell a packet of screws â€˜Aâ€™ at a profit of 70 paise and screws â€˜Bâ€™ at a profit of 1. Assuming that he can sell all the screws he manufactures how many packets of each type should the factory owner produce in a day in order to maximize his profit? Formulate the above LPP and solve it graphically and find the maximum profit. 28. /4 0 3 2x 1 Evaluate : sin x + cos x dx 16 9 sin 2x OR Evaluate : (x 3x e ) dx as the limit of the sum ? ? ?? ? ? 29. -1 2 3 5 If A = 3 2 4 , find A Use it to solve the system of equations 1 1 2 2x -3y +5z = 11 3x + 2y -4z = -5 x + y - 2z = -3 OR Using el ? ?? ?? ? ?? ?? ? ?? ementary row transformations, find the inverse of the matrix 1 2 3 A = 2 5 7 2 4 5 ?? ?? ?? ?? ? ? ? ??Read More

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