Page 1 CBSE XII | Mathematics Board Paper 2019 â€“ All India Set â€“ 1 CBSE Board Class XII Mathematics Board Paper â€“ 2019 All India Set â€“ 1 Time: 3 hrs Total Marks: 100 General Instructions: i. All questions are compulsory. ii. The question paper consists of 29 questions divided into four sections: A, B, C and D. 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. iii. All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of the question. iv. There is no overall choice. However, internal choice has been provided in 1 question of Section A, 3 questions of Section B, 3 questions of Section C and 3 questions of Section D. You have to attempt only one of the alternatives in all such questions. v. Use of calculators is not permitted. You may ask for logarithmic tables, if required. SECTION A Questions number 1 to 4 carry 1 mark each. 1. If A is a square matrix of order 3 with |A| = 4, then write the value of |-2A|. 2. If y = sin -1 x + cos -1 x, find dy dx . 3. Write the order and the degree of the differential equation ?? ?? ?? ???? ?? ?? ?? ?? ?? ?? 3 2 2 4 4 d y dy x dx dx . 4. If a line has the direction ratios -18, 12, -4, then what are its direction cosines? OR Find the cartesian equation of the line which passes through the point (-2, 4, -5) and is parallel to the line ? ? ? ?? x 3 4 y z 8 . 3 5 6 Page 2 CBSE XII | Mathematics Board Paper 2019 â€“ All India Set â€“ 1 CBSE Board Class XII Mathematics Board Paper â€“ 2019 All India Set â€“ 1 Time: 3 hrs Total Marks: 100 General Instructions: i. All questions are compulsory. ii. The question paper consists of 29 questions divided into four sections: A, B, C and D. 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. iii. All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of the question. iv. There is no overall choice. However, internal choice has been provided in 1 question of Section A, 3 questions of Section B, 3 questions of Section C and 3 questions of Section D. You have to attempt only one of the alternatives in all such questions. v. Use of calculators is not permitted. You may ask for logarithmic tables, if required. SECTION A Questions number 1 to 4 carry 1 mark each. 1. If A is a square matrix of order 3 with |A| = 4, then write the value of |-2A|. 2. If y = sin -1 x + cos -1 x, find dy dx . 3. Write the order and the degree of the differential equation ?? ?? ?? ???? ?? ?? ?? ?? ?? ?? 3 2 2 4 4 d y dy x dx dx . 4. If a line has the direction ratios -18, 12, -4, then what are its direction cosines? OR Find the cartesian equation of the line which passes through the point (-2, 4, -5) and is parallel to the line ? ? ? ?? x 3 4 y z 8 . 3 5 6 CBSE XII | Mathematics Board Paper 2019 â€“ All India Set â€“ 1 SECTION B Questions number 5 to 12 carry 2 marks each. 5. If * is defined on the set R of all real numbers by * : a * b = ? 22 a b , find the identity element, if it exists in R with respect to *. 6. ? ? ? ? ? ? ? ? ? ? ? ? 0 2 0 3a If A = and kA = , then find the values of k, a and b. 3 -4 2b 24 7. Find: ? ? ? sinx - cosx dx, 0 < x < /2 1 sin2x 8. Find: ? ? ? ? ? ? ? sin x a dx sin x a OR Find: ?(logx) 2 dx 9. Form the differential equation representing the family of curves y 2 = m(a 2 â€“ x 2 ) by eliminating the arbitrary constants â€˜mâ€™ and â€˜aâ€™. 10. Find a unit vector perpendicular to both the vectors a and b, where ?? a i - 7j + 7k and b 3i - 2j + 2k . OR Show that the vectors i - 2j + 3k, -2i + 3j - 4k and i - 3j + 5k are coplanar. 11. Mother, father and son line up at random for a family photo. If A and B are two events given by A = Son on one end, B = Father in the middle, find P(B/A). 12. Let X be a random variable which assumes values x1, x2, x3, x4 such that 2P(X = x1) = 3P(X = x2) = P(X = x3) = 5P(X = x4). Find the probability distribution of X. Page 3 CBSE XII | Mathematics Board Paper 2019 â€“ All India Set â€“ 1 CBSE Board Class XII Mathematics Board Paper â€“ 2019 All India Set â€“ 1 Time: 3 hrs Total Marks: 100 General Instructions: i. All questions are compulsory. ii. The question paper consists of 29 questions divided into four sections: A, B, C and D. 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. iii. All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of the question. iv. There is no overall choice. However, internal choice has been provided in 1 question of Section A, 3 questions of Section B, 3 questions of Section C and 3 questions of Section D. You have to attempt only one of the alternatives in all such questions. v. Use of calculators is not permitted. You may ask for logarithmic tables, if required. SECTION A Questions number 1 to 4 carry 1 mark each. 1. If A is a square matrix of order 3 with |A| = 4, then write the value of |-2A|. 2. If y = sin -1 x + cos -1 x, find dy dx . 3. Write the order and the degree of the differential equation ?? ?? ?? ???? ?? ?? ?? ?? ?? ?? 3 2 2 4 4 d y dy x dx dx . 4. If a line has the direction ratios -18, 12, -4, then what are its direction cosines? OR Find the cartesian equation of the line which passes through the point (-2, 4, -5) and is parallel to the line ? ? ? ?? x 3 4 y z 8 . 3 5 6 CBSE XII | Mathematics Board Paper 2019 â€“ All India Set â€“ 1 SECTION B Questions number 5 to 12 carry 2 marks each. 5. If * is defined on the set R of all real numbers by * : a * b = ? 22 a b , find the identity element, if it exists in R with respect to *. 6. ? ? ? ? ? ? ? ? ? ? ? ? 0 2 0 3a If A = and kA = , then find the values of k, a and b. 3 -4 2b 24 7. Find: ? ? ? sinx - cosx dx, 0 < x < /2 1 sin2x 8. Find: ? ? ? ? ? ? ? sin x a dx sin x a OR Find: ?(logx) 2 dx 9. Form the differential equation representing the family of curves y 2 = m(a 2 â€“ x 2 ) by eliminating the arbitrary constants â€˜mâ€™ and â€˜aâ€™. 10. Find a unit vector perpendicular to both the vectors a and b, where ?? a i - 7j + 7k and b 3i - 2j + 2k . OR Show that the vectors i - 2j + 3k, -2i + 3j - 4k and i - 3j + 5k are coplanar. 11. Mother, father and son line up at random for a family photo. If A and B are two events given by A = Son on one end, B = Father in the middle, find P(B/A). 12. Let X be a random variable which assumes values x1, x2, x3, x4 such that 2P(X = x1) = 3P(X = x2) = P(X = x3) = 5P(X = x4). Find the probability distribution of X. CBSE XII | Mathematics Board Paper 2019 â€“ All India Set â€“ 1 OR A coin is tossed 5 times, Find the probability of getting (i) at least 4 heads, and (ii) at most 4 heads. SECTION C Questions number 13 to 23 carry 4 marks each. 13. Show that the relation R on the set Z of all integers, given by R = [(a,b) : 2 divides (a - b)] is an equivalence relation. OR If f(x) = ? ? ? 4x 3 2 , x 6x 4 3 , show that fof(x) = x for all ? 2 x 3 . Also, find the inverse of f. 14. If tan -1 x â€“ cot -1 x = tan -1 ?? ?? ?? 1 3 , x > 0, find the value of x and hence find the value of sec -1 2 x ?? ?? ?? . 15. Using properties of determinants, prove that b c a a b c a b = 4abc c c a b ? ? ? 16. If sin y = x sin (a + y), prove that 2 dy sin (a + y) dx sina ? OR If (sin x) y = x + y, find dy dx . 17. If y = (sec -1 x) 2 , x > 0, show that ? ? ? ? 2 2 2 3 2 d y dy x x 1 2x x 2 0 dx dx ? ? ? ? ? 18. Find the equation of the tangent and the normal to the curve ? ? ? ? x - 7 y x 2 x 3 ? ?? at the point where it cuts the x-axis. Page 4 CBSE XII | Mathematics Board Paper 2019 â€“ All India Set â€“ 1 CBSE Board Class XII Mathematics Board Paper â€“ 2019 All India Set â€“ 1 Time: 3 hrs Total Marks: 100 General Instructions: i. All questions are compulsory. ii. The question paper consists of 29 questions divided into four sections: A, B, C and D. 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. iii. All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of the question. iv. There is no overall choice. However, internal choice has been provided in 1 question of Section A, 3 questions of Section B, 3 questions of Section C and 3 questions of Section D. You have to attempt only one of the alternatives in all such questions. v. Use of calculators is not permitted. You may ask for logarithmic tables, if required. SECTION A Questions number 1 to 4 carry 1 mark each. 1. If A is a square matrix of order 3 with |A| = 4, then write the value of |-2A|. 2. If y = sin -1 x + cos -1 x, find dy dx . 3. Write the order and the degree of the differential equation ?? ?? ?? ???? ?? ?? ?? ?? ?? ?? 3 2 2 4 4 d y dy x dx dx . 4. If a line has the direction ratios -18, 12, -4, then what are its direction cosines? OR Find the cartesian equation of the line which passes through the point (-2, 4, -5) and is parallel to the line ? ? ? ?? x 3 4 y z 8 . 3 5 6 CBSE XII | Mathematics Board Paper 2019 â€“ All India Set â€“ 1 SECTION B Questions number 5 to 12 carry 2 marks each. 5. If * is defined on the set R of all real numbers by * : a * b = ? 22 a b , find the identity element, if it exists in R with respect to *. 6. ? ? ? ? ? ? ? ? ? ? ? ? 0 2 0 3a If A = and kA = , then find the values of k, a and b. 3 -4 2b 24 7. Find: ? ? ? sinx - cosx dx, 0 < x < /2 1 sin2x 8. Find: ? ? ? ? ? ? ? sin x a dx sin x a OR Find: ?(logx) 2 dx 9. Form the differential equation representing the family of curves y 2 = m(a 2 â€“ x 2 ) by eliminating the arbitrary constants â€˜mâ€™ and â€˜aâ€™. 10. Find a unit vector perpendicular to both the vectors a and b, where ?? a i - 7j + 7k and b 3i - 2j + 2k . OR Show that the vectors i - 2j + 3k, -2i + 3j - 4k and i - 3j + 5k are coplanar. 11. Mother, father and son line up at random for a family photo. If A and B are two events given by A = Son on one end, B = Father in the middle, find P(B/A). 12. Let X be a random variable which assumes values x1, x2, x3, x4 such that 2P(X = x1) = 3P(X = x2) = P(X = x3) = 5P(X = x4). Find the probability distribution of X. CBSE XII | Mathematics Board Paper 2019 â€“ All India Set â€“ 1 OR A coin is tossed 5 times, Find the probability of getting (i) at least 4 heads, and (ii) at most 4 heads. SECTION C Questions number 13 to 23 carry 4 marks each. 13. Show that the relation R on the set Z of all integers, given by R = [(a,b) : 2 divides (a - b)] is an equivalence relation. OR If f(x) = ? ? ? 4x 3 2 , x 6x 4 3 , show that fof(x) = x for all ? 2 x 3 . Also, find the inverse of f. 14. If tan -1 x â€“ cot -1 x = tan -1 ?? ?? ?? 1 3 , x > 0, find the value of x and hence find the value of sec -1 2 x ?? ?? ?? . 15. Using properties of determinants, prove that b c a a b c a b = 4abc c c a b ? ? ? 16. If sin y = x sin (a + y), prove that 2 dy sin (a + y) dx sina ? OR If (sin x) y = x + y, find dy dx . 17. If y = (sec -1 x) 2 , x > 0, show that ? ? ? ? 2 2 2 3 2 d y dy x x 1 2x x 2 0 dx dx ? ? ? ? ? 18. Find the equation of the tangent and the normal to the curve ? ? ? ? x - 7 y x 2 x 3 ? ?? at the point where it cuts the x-axis. CBSE XII | Mathematics Board Paper 2019 â€“ All India Set â€“ 1 19. Find: ? ? ? ? 22 sin2x dx sin x 1 sin x 3 ?? ? 20. Prove that ? ? ? ? bb aa /3 /6 f x dx = f a b x dx and hence evaluate dx . 1 tanx ? ? ?? ? ?? ? 21. Solve the differential equation: dy x y dx x y ? ? ? OR Solve the differential equation: (1 + x 2 )dy + 2xy dx = cot x dx 22. Let a, b and c be three vectors such that a 1, b 2 and c 3. ? ? ? If the projection of b along a is equal to the projection of c along a; and b, c are perpendicular to each other, then find 3a - 2b + 2c . 23. Find the value of ? for which the following lines are perpendicular to each other: 1 y x 5 2 y 1 z x z 1 2 ; 5 2 5 1 1 2 3 ? ? ? ? ? ? ? ? ? ? ? ? ? Hence, find whether the lines intersect or not. Page 5 CBSE XII | Mathematics Board Paper 2019 â€“ All India Set â€“ 1 CBSE Board Class XII Mathematics Board Paper â€“ 2019 All India Set â€“ 1 Time: 3 hrs Total Marks: 100 General Instructions: i. All questions are compulsory. ii. The question paper consists of 29 questions divided into four sections: A, B, C and D. 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. iii. All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of the question. iv. There is no overall choice. However, internal choice has been provided in 1 question of Section A, 3 questions of Section B, 3 questions of Section C and 3 questions of Section D. You have to attempt only one of the alternatives in all such questions. v. Use of calculators is not permitted. You may ask for logarithmic tables, if required. SECTION A Questions number 1 to 4 carry 1 mark each. 1. If A is a square matrix of order 3 with |A| = 4, then write the value of |-2A|. 2. If y = sin -1 x + cos -1 x, find dy dx . 3. Write the order and the degree of the differential equation ?? ?? ?? ???? ?? ?? ?? ?? ?? ?? 3 2 2 4 4 d y dy x dx dx . 4. If a line has the direction ratios -18, 12, -4, then what are its direction cosines? OR Find the cartesian equation of the line which passes through the point (-2, 4, -5) and is parallel to the line ? ? ? ?? x 3 4 y z 8 . 3 5 6 CBSE XII | Mathematics Board Paper 2019 â€“ All India Set â€“ 1 SECTION B Questions number 5 to 12 carry 2 marks each. 5. If * is defined on the set R of all real numbers by * : a * b = ? 22 a b , find the identity element, if it exists in R with respect to *. 6. ? ? ? ? ? ? ? ? ? ? ? ? 0 2 0 3a If A = and kA = , then find the values of k, a and b. 3 -4 2b 24 7. Find: ? ? ? sinx - cosx dx, 0 < x < /2 1 sin2x 8. Find: ? ? ? ? ? ? ? sin x a dx sin x a OR Find: ?(logx) 2 dx 9. Form the differential equation representing the family of curves y 2 = m(a 2 â€“ x 2 ) by eliminating the arbitrary constants â€˜mâ€™ and â€˜aâ€™. 10. Find a unit vector perpendicular to both the vectors a and b, where ?? a i - 7j + 7k and b 3i - 2j + 2k . OR Show that the vectors i - 2j + 3k, -2i + 3j - 4k and i - 3j + 5k are coplanar. 11. Mother, father and son line up at random for a family photo. If A and B are two events given by A = Son on one end, B = Father in the middle, find P(B/A). 12. Let X be a random variable which assumes values x1, x2, x3, x4 such that 2P(X = x1) = 3P(X = x2) = P(X = x3) = 5P(X = x4). Find the probability distribution of X. CBSE XII | Mathematics Board Paper 2019 â€“ All India Set â€“ 1 OR A coin is tossed 5 times, Find the probability of getting (i) at least 4 heads, and (ii) at most 4 heads. SECTION C Questions number 13 to 23 carry 4 marks each. 13. Show that the relation R on the set Z of all integers, given by R = [(a,b) : 2 divides (a - b)] is an equivalence relation. OR If f(x) = ? ? ? 4x 3 2 , x 6x 4 3 , show that fof(x) = x for all ? 2 x 3 . Also, find the inverse of f. 14. If tan -1 x â€“ cot -1 x = tan -1 ?? ?? ?? 1 3 , x > 0, find the value of x and hence find the value of sec -1 2 x ?? ?? ?? . 15. Using properties of determinants, prove that b c a a b c a b = 4abc c c a b ? ? ? 16. If sin y = x sin (a + y), prove that 2 dy sin (a + y) dx sina ? OR If (sin x) y = x + y, find dy dx . 17. If y = (sec -1 x) 2 , x > 0, show that ? ? ? ? 2 2 2 3 2 d y dy x x 1 2x x 2 0 dx dx ? ? ? ? ? 18. Find the equation of the tangent and the normal to the curve ? ? ? ? x - 7 y x 2 x 3 ? ?? at the point where it cuts the x-axis. CBSE XII | Mathematics Board Paper 2019 â€“ All India Set â€“ 1 19. Find: ? ? ? ? 22 sin2x dx sin x 1 sin x 3 ?? ? 20. Prove that ? ? ? ? bb aa /3 /6 f x dx = f a b x dx and hence evaluate dx . 1 tanx ? ? ?? ? ?? ? 21. Solve the differential equation: dy x y dx x y ? ? ? OR Solve the differential equation: (1 + x 2 )dy + 2xy dx = cot x dx 22. Let a, b and c be three vectors such that a 1, b 2 and c 3. ? ? ? If the projection of b along a is equal to the projection of c along a; and b, c are perpendicular to each other, then find 3a - 2b + 2c . 23. Find the value of ? for which the following lines are perpendicular to each other: 1 y x 5 2 y 1 z x z 1 2 ; 5 2 5 1 1 2 3 ? ? ? ? ? ? ? ? ? ? ? ? ? Hence, find whether the lines intersect or not. CBSE XII | Mathematics Board Paper 2019 â€“ All India Set â€“ 1 SECTION D Questions number 24 to 29 carry 6 marks each. 24. If A = 1 1 1 1 0 1 3 ' find A 1 2 1 ? ?? ?? ?? ?? ? ?? Hence, solve the following system of equations: x + y + z = 6, y + 3z = 11 and x â€“ 2y + z = 0 OR Find the inverse of the following matrix, using elementary transformations: 2 3 1 A = 2 4 1 3 7 2 ?? ?? ?? ?? ?? 25. Show that the height of a cylinder, which is open at the top, having a given surface area and greatest volume, is equal to the radius of its base. 26. Find the area of the triangle whose vertices are (-1, 1), (0, 5) and (3, 2), using intergration. OR Find the area of the region bounded by the curves (x - 1) 2 + y 2 = 1 and x 2 + y 2 = 1, using integration. 27. Find the vector and cartesian equations of the plane passing through the points (2, 5, -3), (-2, -3, 5) and (5, 3, -3). Also, find the point of intersection of this plane with the line passing through points (3, 1, 5) and (-1, -3, -1). OR Find the equation of the plane passing through the intersection of the planes ? ? ? ? r. i + j + k 1 and r. 2i + 3j - k + 4 0 ?? and parallel to x-axis. Hence, find the distance of the plane from x-axis.Read More

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