Page 1 CBSE XII | Mathematics Sample Paper – 3 Mathematics Class XII Sample Paper 3 Time: 3 hour 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 – A 1. This graph does not represent a function. Make the required changes in this graph, and draw the graph, so that it represents a function. 2. Find the value of ‘ ?’ for which ?(î + j + ˆ k ) is a unit vector. 3. Find the slope of the tangent to the curve y = x 3 – x + 1 at the point where the curve cuts the y-axis. 4. This 3 x 2 matrix gives information about the number of men and women workers in three factories I, II and III who lost their jobs in the last 2 months. What do you infer from the entry in the third row and second column of this matrix? Men workers Women workers Factory I 40 15 Factory II 35 40 Factory III 72 64 Page 2 CBSE XII | Mathematics Sample Paper – 3 Mathematics Class XII Sample Paper 3 Time: 3 hour 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 – A 1. This graph does not represent a function. Make the required changes in this graph, and draw the graph, so that it represents a function. 2. Find the value of ‘ ?’ for which ?(î + j + ˆ k ) is a unit vector. 3. Find the slope of the tangent to the curve y = x 3 – x + 1 at the point where the curve cuts the y-axis. 4. This 3 x 2 matrix gives information about the number of men and women workers in three factories I, II and III who lost their jobs in the last 2 months. What do you infer from the entry in the third row and second column of this matrix? Men workers Women workers Factory I 40 15 Factory II 35 40 Factory III 72 64 CBSE XII | Mathematics Sample Paper – 3 OR If for any 2 x 2 square matrix A, A (adj A) = 80 80 ?? ?? ?? , then write the value of |A|. SECTION – B 5. Evaluate: 1 1 sin sin 32 ? ?? ? ?? ?? ?? ?? ?? ?? 6. Evaluate: 1 1 2x log dx 2x ? ? ?? ?? ? ?? ? 7. ? ? ? ? For what value of 'a' the vectors 2i 3j 4k and ai 6j 8k are collinear? 8. ? ?? ?? ?? 1 25 Write A for A = . 13 OR If A is a skew-symmetric matrix of order 3, then prove that det A = 0. 9. ? ? ? ? 1 2 1 Simplify cot for x < 1. x1 10. ? ? ? ? ? ? ? ? 1 2 2 2 5x 1 1 dy 2 3 If y = tan , x < , then prove that . dx 66 1 6x 1 4x 1 9x OR The volume of a cube is increasing at the rate of 9 cm 3 /s. How fast is its surface area increasing when the length of an edge is 10 cm ? Page 3 CBSE XII | Mathematics Sample Paper – 3 Mathematics Class XII Sample Paper 3 Time: 3 hour 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 – A 1. This graph does not represent a function. Make the required changes in this graph, and draw the graph, so that it represents a function. 2. Find the value of ‘ ?’ for which ?(î + j + ˆ k ) is a unit vector. 3. Find the slope of the tangent to the curve y = x 3 – x + 1 at the point where the curve cuts the y-axis. 4. This 3 x 2 matrix gives information about the number of men and women workers in three factories I, II and III who lost their jobs in the last 2 months. What do you infer from the entry in the third row and second column of this matrix? Men workers Women workers Factory I 40 15 Factory II 35 40 Factory III 72 64 CBSE XII | Mathematics Sample Paper – 3 OR If for any 2 x 2 square matrix A, A (adj A) = 80 80 ?? ?? ?? , then write the value of |A|. SECTION – B 5. Evaluate: 1 1 sin sin 32 ? ?? ? ?? ?? ?? ?? ?? ?? 6. Evaluate: 1 1 2x log dx 2x ? ? ?? ?? ? ?? ? 7. ? ? ? ? For what value of 'a' the vectors 2i 3j 4k and ai 6j 8k are collinear? 8. ? ?? ?? ?? 1 25 Write A for A = . 13 OR If A is a skew-symmetric matrix of order 3, then prove that det A = 0. 9. ? ? ? ? 1 2 1 Simplify cot for x < 1. x1 10. ? ? ? ? ? ? ? ? 1 2 2 2 5x 1 1 dy 2 3 If y = tan , x < , then prove that . dx 66 1 6x 1 4x 1 9x OR The volume of a cube is increasing at the rate of 9 cm 3 /s. How fast is its surface area increasing when the length of an edge is 10 cm ? CBSE XII | Mathematics Sample Paper – 3 11. Obtain the differential equation of the family of circles passing through the points (a,0) and (-a,0). 12. ? 2 1 1 If P(A)= , P(B)= , P(A B)= , then find P(A /B). 5 3 5 OR A die, whose faces are marked 1, 2, 3 in red and 4, 5, 6 in green, is tossed. Let A be the event “number obtained is even” and B be the event “Number obtained is red.” Find if A and B are independent events. SECTION – C 13. ? ? ? ?? 3 5 3 x 5 x Differentiate , wrt x 7 3x 8 5x OR ? ? ? 2 '' ' If y = acos(log x) + bsin(log x), prove that x y xy y 0, 14. ? ? ? ? Let f(x) x 3, g(x) = x 3; x N, Show that (i) f is not an onto function (ii) gof is an onto function 15. Find the distance between the parallel planes r. 2i 1j 3k 4 and r. 6i 3j 9k 13 0 16. 2 2 2 2 A plane is at a distance of p units from the origin. It makes an intercept of a,b,c with the x , y and z axis repectively. Show that it satisfies the equation: 1 1 1 1 a b c p ? ? ? Page 4 CBSE XII | Mathematics Sample Paper – 3 Mathematics Class XII Sample Paper 3 Time: 3 hour 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 – A 1. This graph does not represent a function. Make the required changes in this graph, and draw the graph, so that it represents a function. 2. Find the value of ‘ ?’ for which ?(î + j + ˆ k ) is a unit vector. 3. Find the slope of the tangent to the curve y = x 3 – x + 1 at the point where the curve cuts the y-axis. 4. This 3 x 2 matrix gives information about the number of men and women workers in three factories I, II and III who lost their jobs in the last 2 months. What do you infer from the entry in the third row and second column of this matrix? Men workers Women workers Factory I 40 15 Factory II 35 40 Factory III 72 64 CBSE XII | Mathematics Sample Paper – 3 OR If for any 2 x 2 square matrix A, A (adj A) = 80 80 ?? ?? ?? , then write the value of |A|. SECTION – B 5. Evaluate: 1 1 sin sin 32 ? ?? ? ?? ?? ?? ?? ?? ?? 6. Evaluate: 1 1 2x log dx 2x ? ? ?? ?? ? ?? ? 7. ? ? ? ? For what value of 'a' the vectors 2i 3j 4k and ai 6j 8k are collinear? 8. ? ?? ?? ?? 1 25 Write A for A = . 13 OR If A is a skew-symmetric matrix of order 3, then prove that det A = 0. 9. ? ? ? ? 1 2 1 Simplify cot for x < 1. x1 10. ? ? ? ? ? ? ? ? 1 2 2 2 5x 1 1 dy 2 3 If y = tan , x < , then prove that . dx 66 1 6x 1 4x 1 9x OR The volume of a cube is increasing at the rate of 9 cm 3 /s. How fast is its surface area increasing when the length of an edge is 10 cm ? CBSE XII | Mathematics Sample Paper – 3 11. Obtain the differential equation of the family of circles passing through the points (a,0) and (-a,0). 12. ? 2 1 1 If P(A)= , P(B)= , P(A B)= , then find P(A /B). 5 3 5 OR A die, whose faces are marked 1, 2, 3 in red and 4, 5, 6 in green, is tossed. Let A be the event “number obtained is even” and B be the event “Number obtained is red.” Find if A and B are independent events. SECTION – C 13. ? ? ? ?? 3 5 3 x 5 x Differentiate , wrt x 7 3x 8 5x OR ? ? ? 2 '' ' If y = acos(log x) + bsin(log x), prove that x y xy y 0, 14. ? ? ? ? Let f(x) x 3, g(x) = x 3; x N, Show that (i) f is not an onto function (ii) gof is an onto function 15. Find the distance between the parallel planes r. 2i 1j 3k 4 and r. 6i 3j 9k 13 0 16. 2 2 2 2 A plane is at a distance of p units from the origin. It makes an intercept of a,b,c with the x , y and z axis repectively. Show that it satisfies the equation: 1 1 1 1 a b c p ? ? ? CBSE XII | Mathematics Sample Paper – 3 17. 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? 18. 2 1 2 1 2 5 Solve the equation: (tan x) (cot x) 8 ?? ? ?? 19. Find the equation of a tangent to the curve given by 33 x asin t , y bcos t ?? at a point, where t 2 ? ? . 20. Evaluate: 4 0 sinx cosx dx 9 16sin2x ? ? ? ? 21. If cos sin A sin cos ?? ?? ? ?? ? ? ? ?? then prove that n cosn sinn A ,n N sinn cosn ?? ?? ?? ?? ? ? ? ?? OR ? ?? ? ? ? ? ?? 2 2 2 If is one of the cube roots of unity, evaluate the given determinant 1 1 1 22. ? Show that the function f defined by f(x) = 1-x + x , x R is continuous. OR Show that a logarithmic function is continuous at every point in its domain. 23. ? ? ? ? ? ? ? ? ? 2 (3 sin 2)cos Evaluate: d 5 cos 4sin Page 5 CBSE XII | Mathematics Sample Paper – 3 Mathematics Class XII Sample Paper 3 Time: 3 hour 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 – A 1. This graph does not represent a function. Make the required changes in this graph, and draw the graph, so that it represents a function. 2. Find the value of ‘ ?’ for which ?(î + j + ˆ k ) is a unit vector. 3. Find the slope of the tangent to the curve y = x 3 – x + 1 at the point where the curve cuts the y-axis. 4. This 3 x 2 matrix gives information about the number of men and women workers in three factories I, II and III who lost their jobs in the last 2 months. What do you infer from the entry in the third row and second column of this matrix? Men workers Women workers Factory I 40 15 Factory II 35 40 Factory III 72 64 CBSE XII | Mathematics Sample Paper – 3 OR If for any 2 x 2 square matrix A, A (adj A) = 80 80 ?? ?? ?? , then write the value of |A|. SECTION – B 5. Evaluate: 1 1 sin sin 32 ? ?? ? ?? ?? ?? ?? ?? ?? 6. Evaluate: 1 1 2x log dx 2x ? ? ?? ?? ? ?? ? 7. ? ? ? ? For what value of 'a' the vectors 2i 3j 4k and ai 6j 8k are collinear? 8. ? ?? ?? ?? 1 25 Write A for A = . 13 OR If A is a skew-symmetric matrix of order 3, then prove that det A = 0. 9. ? ? ? ? 1 2 1 Simplify cot for x < 1. x1 10. ? ? ? ? ? ? ? ? 1 2 2 2 5x 1 1 dy 2 3 If y = tan , x < , then prove that . dx 66 1 6x 1 4x 1 9x OR The volume of a cube is increasing at the rate of 9 cm 3 /s. How fast is its surface area increasing when the length of an edge is 10 cm ? CBSE XII | Mathematics Sample Paper – 3 11. Obtain the differential equation of the family of circles passing through the points (a,0) and (-a,0). 12. ? 2 1 1 If P(A)= , P(B)= , P(A B)= , then find P(A /B). 5 3 5 OR A die, whose faces are marked 1, 2, 3 in red and 4, 5, 6 in green, is tossed. Let A be the event “number obtained is even” and B be the event “Number obtained is red.” Find if A and B are independent events. SECTION – C 13. ? ? ? ?? 3 5 3 x 5 x Differentiate , wrt x 7 3x 8 5x OR ? ? ? 2 '' ' If y = acos(log x) + bsin(log x), prove that x y xy y 0, 14. ? ? ? ? Let f(x) x 3, g(x) = x 3; x N, Show that (i) f is not an onto function (ii) gof is an onto function 15. Find the distance between the parallel planes r. 2i 1j 3k 4 and r. 6i 3j 9k 13 0 16. 2 2 2 2 A plane is at a distance of p units from the origin. It makes an intercept of a,b,c with the x , y and z axis repectively. Show that it satisfies the equation: 1 1 1 1 a b c p ? ? ? CBSE XII | Mathematics Sample Paper – 3 17. 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? 18. 2 1 2 1 2 5 Solve the equation: (tan x) (cot x) 8 ?? ? ?? 19. Find the equation of a tangent to the curve given by 33 x asin t , y bcos t ?? at a point, where t 2 ? ? . 20. Evaluate: 4 0 sinx cosx dx 9 16sin2x ? ? ? ? 21. If cos sin A sin cos ?? ?? ? ?? ? ? ? ?? then prove that n cosn sinn A ,n N sinn cosn ?? ?? ?? ?? ? ? ? ?? OR ? ?? ? ? ? ? ?? 2 2 2 If is one of the cube roots of unity, evaluate the given determinant 1 1 1 22. ? Show that the function f defined by f(x) = 1-x + x , x R is continuous. OR Show that a logarithmic function is continuous at every point in its domain. 23. ? ? ? ? ? ? ? ? ? 2 (3 sin 2)cos Evaluate: d 5 cos 4sin CBSE XII | Mathematics Sample Paper – 3 SECTION - D 24. Show that the right circular cone of least curved surface and given volume has an altitude equal to 2 times the radius of the base. OR 43 Find the points at which the function f given by f(x) = (x - 2 ) (x 1) is minimum. ? 25. Obtain the inverse of the following matrix using elementary operations. 0 1 2 A= 1 2 3 3 1 1 ?? ?? ?? ?? ?? 26. ? ? 22 22 22 22 Calculate the area xy (i) between the curves + 1,and the x-axis between x = 0 to x = a ab xy (ii) Triangle AOB is in the first quadrant of the ellipse + 1,where OA = a and OB = b. ab Find the area enclosed between the chord AB and the arc AB of the ellipse (iii) Find the ratio of the two areas found. OR ?? ?? 2 2 2 2 Find the smaller of the two areas in which the circle x y 2a is divided by the parabola y ax, a 0 27. Find the equation of a plane that is parallel to the x-axis and passes through the line common to two intersectiing planes r. i+j+k 1 0 and r. 2i+3j-k 4 28. Two trainee carpenters A and B earn Rs. 150 and Rs. 200 per day respectively. A can make 6 frames and 4 stools per day while B can make 10 frames and 4 stools per day. How many days shall each work, if it is desired to produce atleast 60 frames and 32 stools at a minimum labour cost? Solve the problem graphically.Read More

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