Page 1 CBSE XII | Mathematics Sample Paper â€“ 8 Mathematics Class XII Sample Paper â€“ 8 Time: 3 hours Total Marks: 100 1. All questions are compulsory. 2. The question paper consist of 29 questions divided into three 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. 3. Use of calculators is not permitted. SECTION â€“ A 1. Write the position of the element 6 in the given matrix, and denote it as aij. 1 16 8 9 7 5 3 2 4 10 6 11 ?? ?? ?? ?? ?? 2. Find dy dx , if 2x + 3y = cos x 3. Is the differential equation given by 2 2 2 d y dy s sy s dx dx ?? , linear or nonlinear. Give reason. 4. The Cartesian equations of a line are x 5 y 4 z 6 3 7 2 ? ? ? ?? . Find a vector equation for the line. OR Find the angle between following pairs of line x 4 y 1 z 3 x 1 y 4 z 5 and 3 5 4 1 1 2 ? ? ? ? ? ? ? ? ? ? SECTION â€“ B 5. Consider the binary operation * on the set {1, 2, 3, 4, 5} defined by a * b = min {a, b}. Write the operation table of the operation *. 6. Solve the matrix equation 2 2 x2 x 3 2y 9 y ? ?? ? ? ? ? ?? ?? ? ? ? ? ? ? ? ? ?? 7. Evaluate: 2 5x 2 dx 1 2x 3x Page 2 CBSE XII | Mathematics Sample Paper â€“ 8 Mathematics Class XII Sample Paper â€“ 8 Time: 3 hours Total Marks: 100 1. All questions are compulsory. 2. The question paper consist of 29 questions divided into three 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. 3. Use of calculators is not permitted. SECTION â€“ A 1. Write the position of the element 6 in the given matrix, and denote it as aij. 1 16 8 9 7 5 3 2 4 10 6 11 ?? ?? ?? ?? ?? 2. Find dy dx , if 2x + 3y = cos x 3. Is the differential equation given by 2 2 2 d y dy s sy s dx dx ?? , linear or nonlinear. Give reason. 4. The Cartesian equations of a line are x 5 y 4 z 6 3 7 2 ? ? ? ?? . Find a vector equation for the line. OR Find the angle between following pairs of line x 4 y 1 z 3 x 1 y 4 z 5 and 3 5 4 1 1 2 ? ? ? ? ? ? ? ? ? ? SECTION â€“ B 5. Consider the binary operation * on the set {1, 2, 3, 4, 5} defined by a * b = min {a, b}. Write the operation table of the operation *. 6. Solve the matrix equation 2 2 x2 x 3 2y 9 y ? ?? ? ? ? ? ?? ?? ? ? ? ? ? ? ? ? ?? 7. Evaluate: 2 5x 2 dx 1 2x 3x CBSE XII | Mathematics Sample Paper â€“ 8 8. Evaluate: 2 22 x dx x 4 x 9 OR Evaluate: ? ? ? ? ? ? ? x 3 x 3 e dx x5 9. Form differential equation of the family of curves y = a sin (bx + c), a and c being parameters. 10. ABCD is a parallelogram with AB 2i 4j 5k;AD i 2j 3k Find a unit vector parallel to its diagonal AC. Also, find the area of the parallelogram ABCD OR Find the projection of (b c) ? on a , where a 2i 2j k, b i 2j 2k and c 2i j 4k . 11. Four cards are drawn successively with replacement from a well shuffled deck of 52 cards. What is the probability that (i) All the four cards are spades? (ii) Only 3 cards are spades? (iii) None is a spade? 12. A random variable X has the following probability distribution : X 0 1 2 3 4 5 6 7 P(X) 0 k 2k 2k 3k k 2 2k 2 7k 2 +k Determine: (i) k (ii) P(X < 3) (iii) P(X > 6) (iv) P(1 ? X < 3) OR A and B throw a dice alternatively till one of them gets a â€˜6â€™ and wins the game. Find their respective probabilities of winning, if A starts first. SECTION â€“ C 13. Let A = Q x Q and let * be a binary operation on A defined by (a, b)*(c, d) = (ac, b + ad) for (a, b), (c, d) ? A. Determine whether * is Commutative and associative. Then, with respect to * on A (i) Find the identify element in A. (ii) Find the invertible elements of A. Page 3 CBSE XII | Mathematics Sample Paper â€“ 8 Mathematics Class XII Sample Paper â€“ 8 Time: 3 hours Total Marks: 100 1. All questions are compulsory. 2. The question paper consist of 29 questions divided into three 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. 3. Use of calculators is not permitted. SECTION â€“ A 1. Write the position of the element 6 in the given matrix, and denote it as aij. 1 16 8 9 7 5 3 2 4 10 6 11 ?? ?? ?? ?? ?? 2. Find dy dx , if 2x + 3y = cos x 3. Is the differential equation given by 2 2 2 d y dy s sy s dx dx ?? , linear or nonlinear. Give reason. 4. The Cartesian equations of a line are x 5 y 4 z 6 3 7 2 ? ? ? ?? . Find a vector equation for the line. OR Find the angle between following pairs of line x 4 y 1 z 3 x 1 y 4 z 5 and 3 5 4 1 1 2 ? ? ? ? ? ? ? ? ? ? SECTION â€“ B 5. Consider the binary operation * on the set {1, 2, 3, 4, 5} defined by a * b = min {a, b}. Write the operation table of the operation *. 6. Solve the matrix equation 2 2 x2 x 3 2y 9 y ? ?? ? ? ? ? ?? ?? ? ? ? ? ? ? ? ? ?? 7. Evaluate: 2 5x 2 dx 1 2x 3x CBSE XII | Mathematics Sample Paper â€“ 8 8. Evaluate: 2 22 x dx x 4 x 9 OR Evaluate: ? ? ? ? ? ? ? x 3 x 3 e dx x5 9. Form differential equation of the family of curves y = a sin (bx + c), a and c being parameters. 10. ABCD is a parallelogram with AB 2i 4j 5k;AD i 2j 3k Find a unit vector parallel to its diagonal AC. Also, find the area of the parallelogram ABCD OR Find the projection of (b c) ? on a , where a 2i 2j k, b i 2j 2k and c 2i j 4k . 11. Four cards are drawn successively with replacement from a well shuffled deck of 52 cards. What is the probability that (i) All the four cards are spades? (ii) Only 3 cards are spades? (iii) None is a spade? 12. A random variable X has the following probability distribution : X 0 1 2 3 4 5 6 7 P(X) 0 k 2k 2k 3k k 2 2k 2 7k 2 +k Determine: (i) k (ii) P(X < 3) (iii) P(X > 6) (iv) P(1 ? X < 3) OR A and B throw a dice alternatively till one of them gets a â€˜6â€™ and wins the game. Find their respective probabilities of winning, if A starts first. SECTION â€“ C 13. Let A = Q x Q and let * be a binary operation on A defined by (a, b)*(c, d) = (ac, b + ad) for (a, b), (c, d) ? A. Determine whether * is Commutative and associative. Then, with respect to * on A (i) Find the identify element in A. (ii) Find the invertible elements of A. CBSE XII | Mathematics Sample Paper â€“ 8 OR Let A = Q × Q, Q being the set of rational. Let â€˜*â€™ be a binary operation on A, defined by (a, b) * (c, d) = (ac, ad + b). Show that (i) â€˜*â€™ is not commutative (ii) â€˜*â€™ is associative (iii) The identity element with respect to â€˜*â€™ is (1, 0) 14. Write in the simplest form: 12 y cot 1 x x ? ? ? ? ?? ?? ?? 15. Let a, b, and c be positive numbers , but not equal and not all are zero. a b c Show that the value of the determinant b c a is negative c a b 16. If sin y = x sin (a + y), prove that ? ? 2 sin a y dy dx sina ? ? OR If 11 x y dy y b tan tan , find a x dx ?? ?? ?? ?? ?? 17. If 1 3 cos3x dy 3 y cos , then show that cos x dx cosx cos3x ? ?? 18. A man of height 2 m walks at a uniform speed of 5 km/h away from a lamp post which is 6 m high. Find the rate at which the length of his shadow increases. 19. Evaluate: 4 2 2 4 1 dx sin x+sin xcos x+cos x ? 20. Evaluate: ? ? 2 1 7x 5 dx, ? ? ? as a limit of sums. 21. Solve the initial value problem: cos (x + y)dy = dx, y(0) = 0. OR Solve: ? ? 2 2 dy x y a dx ?? Page 4 CBSE XII | Mathematics Sample Paper â€“ 8 Mathematics Class XII Sample Paper â€“ 8 Time: 3 hours Total Marks: 100 1. All questions are compulsory. 2. The question paper consist of 29 questions divided into three 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. 3. Use of calculators is not permitted. SECTION â€“ A 1. Write the position of the element 6 in the given matrix, and denote it as aij. 1 16 8 9 7 5 3 2 4 10 6 11 ?? ?? ?? ?? ?? 2. Find dy dx , if 2x + 3y = cos x 3. Is the differential equation given by 2 2 2 d y dy s sy s dx dx ?? , linear or nonlinear. Give reason. 4. The Cartesian equations of a line are x 5 y 4 z 6 3 7 2 ? ? ? ?? . Find a vector equation for the line. OR Find the angle between following pairs of line x 4 y 1 z 3 x 1 y 4 z 5 and 3 5 4 1 1 2 ? ? ? ? ? ? ? ? ? ? SECTION â€“ B 5. Consider the binary operation * on the set {1, 2, 3, 4, 5} defined by a * b = min {a, b}. Write the operation table of the operation *. 6. Solve the matrix equation 2 2 x2 x 3 2y 9 y ? ?? ? ? ? ? ?? ?? ? ? ? ? ? ? ? ? ?? 7. Evaluate: 2 5x 2 dx 1 2x 3x CBSE XII | Mathematics Sample Paper â€“ 8 8. Evaluate: 2 22 x dx x 4 x 9 OR Evaluate: ? ? ? ? ? ? ? x 3 x 3 e dx x5 9. Form differential equation of the family of curves y = a sin (bx + c), a and c being parameters. 10. ABCD is a parallelogram with AB 2i 4j 5k;AD i 2j 3k Find a unit vector parallel to its diagonal AC. Also, find the area of the parallelogram ABCD OR Find the projection of (b c) ? on a , where a 2i 2j k, b i 2j 2k and c 2i j 4k . 11. Four cards are drawn successively with replacement from a well shuffled deck of 52 cards. What is the probability that (i) All the four cards are spades? (ii) Only 3 cards are spades? (iii) None is a spade? 12. A random variable X has the following probability distribution : X 0 1 2 3 4 5 6 7 P(X) 0 k 2k 2k 3k k 2 2k 2 7k 2 +k Determine: (i) k (ii) P(X < 3) (iii) P(X > 6) (iv) P(1 ? X < 3) OR A and B throw a dice alternatively till one of them gets a â€˜6â€™ and wins the game. Find their respective probabilities of winning, if A starts first. SECTION â€“ C 13. Let A = Q x Q and let * be a binary operation on A defined by (a, b)*(c, d) = (ac, b + ad) for (a, b), (c, d) ? A. Determine whether * is Commutative and associative. Then, with respect to * on A (i) Find the identify element in A. (ii) Find the invertible elements of A. CBSE XII | Mathematics Sample Paper â€“ 8 OR Let A = Q × Q, Q being the set of rational. Let â€˜*â€™ be a binary operation on A, defined by (a, b) * (c, d) = (ac, ad + b). Show that (i) â€˜*â€™ is not commutative (ii) â€˜*â€™ is associative (iii) The identity element with respect to â€˜*â€™ is (1, 0) 14. Write in the simplest form: 12 y cot 1 x x ? ? ? ? ?? ?? ?? 15. Let a, b, and c be positive numbers , but not equal and not all are zero. a b c Show that the value of the determinant b c a is negative c a b 16. If sin y = x sin (a + y), prove that ? ? 2 sin a y dy dx sina ? ? OR If 11 x y dy y b tan tan , find a x dx ?? ?? ?? ?? ?? 17. If 1 3 cos3x dy 3 y cos , then show that cos x dx cosx cos3x ? ?? 18. A man of height 2 m walks at a uniform speed of 5 km/h away from a lamp post which is 6 m high. Find the rate at which the length of his shadow increases. 19. Evaluate: 4 2 2 4 1 dx sin x+sin xcos x+cos x ? 20. Evaluate: ? ? 2 1 7x 5 dx, ? ? ? as a limit of sums. 21. Solve the initial value problem: cos (x + y)dy = dx, y(0) = 0. OR Solve: ? ? 2 2 dy x y a dx ?? CBSE XII | Mathematics Sample Paper â€“ 8 22. a) If Ë† Ë†Ë† i . j.k represents the right handed system of mutually perpendicular vectors and Ë† Ë† Ë† Ë† Ë† 3i j; 2i j 3k ? ? ? ? ? ? ? , then express ? in the form of 12 ? ? ? ? ? where 1 ? is parallel to 2 and ?? is perpendicular to ? . b) Let a, b and c be three vectors of magnitude 3, 4 and 5 units respectively. If each of these is perpendicular to the sum of the other two vectors, find a b c ?? . 23. Find the coordinates of the point, where the line x 2 y 1 z 2 3 4 2 intersects the plane x â€“ y + z â€“ 5 = 0. Also find the angle between the line and the plane. SECTION â€“ D 24. If 1 0 2 A 0 2 2 203 ?? ?? ? ?? ?? ?? , then show that A is a root of polynomials f(x) = x 3 â€“ 6x 2 + 7x + 2 OR If 1 1 0 A 0 1 1 2 3 4 ?? ?? ?? ?? ?? ?? ?? and 1 2 3 B 0 1 0 1 1 0 ?? ?? ? ?? ?? ?? , show that AB?BA 25. Find the volume of the largest cylinder which can be inscribed in a sphere of radius r. 26. Find the area of the region {(x, y):0 = y = x² + 1, 0 = y = x + 1, 0 = x = 2} OR Calculate the area ? 22 22 xy (i) between the curves + 1,and the x-axis between x = 0 to x = a ab 22 22 xy (ii) Triangle AOB is in the first quadrant of the ellipse + 1, ab Where OA = a and OB = b. Find the area enclosed between the chord AB and the arc AB of the ellipse ? (iii) Find the ratio of the two areas found. Page 5 CBSE XII | Mathematics Sample Paper â€“ 8 Mathematics Class XII Sample Paper â€“ 8 Time: 3 hours Total Marks: 100 1. All questions are compulsory. 2. The question paper consist of 29 questions divided into three 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. 3. Use of calculators is not permitted. SECTION â€“ A 1. Write the position of the element 6 in the given matrix, and denote it as aij. 1 16 8 9 7 5 3 2 4 10 6 11 ?? ?? ?? ?? ?? 2. Find dy dx , if 2x + 3y = cos x 3. Is the differential equation given by 2 2 2 d y dy s sy s dx dx ?? , linear or nonlinear. Give reason. 4. The Cartesian equations of a line are x 5 y 4 z 6 3 7 2 ? ? ? ?? . Find a vector equation for the line. OR Find the angle between following pairs of line x 4 y 1 z 3 x 1 y 4 z 5 and 3 5 4 1 1 2 ? ? ? ? ? ? ? ? ? ? SECTION â€“ B 5. Consider the binary operation * on the set {1, 2, 3, 4, 5} defined by a * b = min {a, b}. Write the operation table of the operation *. 6. Solve the matrix equation 2 2 x2 x 3 2y 9 y ? ?? ? ? ? ? ?? ?? ? ? ? ? ? ? ? ? ?? 7. Evaluate: 2 5x 2 dx 1 2x 3x CBSE XII | Mathematics Sample Paper â€“ 8 8. Evaluate: 2 22 x dx x 4 x 9 OR Evaluate: ? ? ? ? ? ? ? x 3 x 3 e dx x5 9. Form differential equation of the family of curves y = a sin (bx + c), a and c being parameters. 10. ABCD is a parallelogram with AB 2i 4j 5k;AD i 2j 3k Find a unit vector parallel to its diagonal AC. Also, find the area of the parallelogram ABCD OR Find the projection of (b c) ? on a , where a 2i 2j k, b i 2j 2k and c 2i j 4k . 11. Four cards are drawn successively with replacement from a well shuffled deck of 52 cards. What is the probability that (i) All the four cards are spades? (ii) Only 3 cards are spades? (iii) None is a spade? 12. A random variable X has the following probability distribution : X 0 1 2 3 4 5 6 7 P(X) 0 k 2k 2k 3k k 2 2k 2 7k 2 +k Determine: (i) k (ii) P(X < 3) (iii) P(X > 6) (iv) P(1 ? X < 3) OR A and B throw a dice alternatively till one of them gets a â€˜6â€™ and wins the game. Find their respective probabilities of winning, if A starts first. SECTION â€“ C 13. Let A = Q x Q and let * be a binary operation on A defined by (a, b)*(c, d) = (ac, b + ad) for (a, b), (c, d) ? A. Determine whether * is Commutative and associative. Then, with respect to * on A (i) Find the identify element in A. (ii) Find the invertible elements of A. CBSE XII | Mathematics Sample Paper â€“ 8 OR Let A = Q × Q, Q being the set of rational. Let â€˜*â€™ be a binary operation on A, defined by (a, b) * (c, d) = (ac, ad + b). Show that (i) â€˜*â€™ is not commutative (ii) â€˜*â€™ is associative (iii) The identity element with respect to â€˜*â€™ is (1, 0) 14. Write in the simplest form: 12 y cot 1 x x ? ? ? ? ?? ?? ?? 15. Let a, b, and c be positive numbers , but not equal and not all are zero. a b c Show that the value of the determinant b c a is negative c a b 16. If sin y = x sin (a + y), prove that ? ? 2 sin a y dy dx sina ? ? OR If 11 x y dy y b tan tan , find a x dx ?? ?? ?? ?? ?? 17. If 1 3 cos3x dy 3 y cos , then show that cos x dx cosx cos3x ? ?? 18. A man of height 2 m walks at a uniform speed of 5 km/h away from a lamp post which is 6 m high. Find the rate at which the length of his shadow increases. 19. Evaluate: 4 2 2 4 1 dx sin x+sin xcos x+cos x ? 20. Evaluate: ? ? 2 1 7x 5 dx, ? ? ? as a limit of sums. 21. Solve the initial value problem: cos (x + y)dy = dx, y(0) = 0. OR Solve: ? ? 2 2 dy x y a dx ?? CBSE XII | Mathematics Sample Paper â€“ 8 22. a) If Ë† Ë†Ë† i . j.k represents the right handed system of mutually perpendicular vectors and Ë† Ë† Ë† Ë† Ë† 3i j; 2i j 3k ? ? ? ? ? ? ? , then express ? in the form of 12 ? ? ? ? ? where 1 ? is parallel to 2 and ?? is perpendicular to ? . b) Let a, b and c be three vectors of magnitude 3, 4 and 5 units respectively. If each of these is perpendicular to the sum of the other two vectors, find a b c ?? . 23. Find the coordinates of the point, where the line x 2 y 1 z 2 3 4 2 intersects the plane x â€“ y + z â€“ 5 = 0. Also find the angle between the line and the plane. SECTION â€“ D 24. If 1 0 2 A 0 2 2 203 ?? ?? ? ?? ?? ?? , then show that A is a root of polynomials f(x) = x 3 â€“ 6x 2 + 7x + 2 OR If 1 1 0 A 0 1 1 2 3 4 ?? ?? ?? ?? ?? ?? ?? and 1 2 3 B 0 1 0 1 1 0 ?? ?? ? ?? ?? ?? , show that AB?BA 25. Find the volume of the largest cylinder which can be inscribed in a sphere of radius r. 26. Find the area of the region {(x, y):0 = y = x² + 1, 0 = y = x + 1, 0 = x = 2} OR Calculate the area ? 22 22 xy (i) between the curves + 1,and the x-axis between x = 0 to x = a ab 22 22 xy (ii) Triangle AOB is in the first quadrant of the ellipse + 1, ab Where OA = a and OB = b. Find the area enclosed between the chord AB and the arc AB of the ellipse ? (iii) Find the ratio of the two areas found. CBSE XII | Mathematics Sample Paper â€“ 8 27. Find the Cartesian equation of the plane passing through the points A(0, 0, 0) and B(3, -1, 2) and parallel to the line x 4 y 3 z 1 1 4 7 ? ? ? ?? ? OR Find the equation of the line passing through the point (-1,3,-2) and perpendicular to the lines x y z x 2 y 1 z 1 and 1 2 3 3 2 5 . 28. A manufacturing company makes two models A and B of a product. Each piece of Model A requires 9 labour hours for fabricating and 1 labour hour for finishing. Each piece of Model B requires 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available are 180 and 30 respectively. The company makes a profit of Rs. 8000 on each piece of model A and Rs. 12000 on each piece of Model B. How many pieces of Model A and Model B should be manufactured per week to realize a maximum profit? What is the maximum profit per week? 29. Two bags A and B contain 3 red and 4 black balls, and 4 red and 5 black balls respectively. From bag A, one ball is transferred to bag B and then a ball is drawn from bag B. The ball is found to be red in colour. Find the probability that (a)The transferred ball is black ? (b) The transferred ball is red ?Read More

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