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Class 12 Exam 2017 : Mathematics (DAV Public School, Sahibabad)

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Elligibility Test June - 2016 SET A Mathemaitcs Time 3 hrs. Class XII Full Marks 100 The question paper consists of 26questions divided into three sections A, B and C. Section A comprises of 6 questions of 1 mark each Section B comprises of 13 questions of 4 marks each and section C comprises of 7 questions of 6 marks each. SECTION A 1 2 3 4 5 6 Write a skew matrix of order 3 x 3. If A is a 3 3 matrix |3A| = k|A|, then write the value of k. The total revenue in rupees received from the sale of x units of a product is R(x) = 3x2 + 36x + 5.Find the marginal revenue when x = 15. The probability that a person is not a swimmer is 0.3.Find the probability that out of 5 persons 4 are swimmer. The corner points of feasible region are (0,10),(5,5),(15,15)and (0,20). Find the condition so that objective function z = px + qy,( p& q >0) is maximum at (15,15)and (0,20). Give an example of a function which is continuous but not differentiable. SECTION B 7. To raise money for an orphanage, students of three schools A, B and C organized an exhibition in their locality, where they sold paper bags, scrap-books and pastel sheets made by them using recycled paper, at the rate of Rs. 20, Rs. 15 and Rs. 5 per unit respectively. School A sold 25 paper bags, 12 scrap-books and 34 pastel sheets. School B sold 22 paper bags , 15 scrap-books and 28 pastel sheets. While school C sold 26 paper bags, 18 scrap-books and 36 pastel sheets. Using matrices, find the total amount raised by each school. By such exhibition, which values are inculcated in the students? 2 8. Find d y d x2 at t = 4 if y = a cos t and x = b sin t. OR Differentiate with respect to sin x . 9. 10. If X [ ] 1 2 3 4 5 6 = [ ] 7 8 9 2 4 6 , then find the matrix X OR If A = [ ] 2 3 1 2 then prove that A2 4A + 7I = 0.Using this result find A5 x sin x 11. If f(x) = Ix-3I + Ix- 4I , then show that f(x) is not differentiable at x = 3 and x = 4. 12. Out of a group of 8 highly qualified doctors in a hospital, 6 are very kind and cooperative with their patients and so are very popular, while the other two remain reserved. For a health camp, three doctors are selected at random. Find the probability distribution of the number of very popular doctors. What values are expected from the doctors? 13. Using elementary operation find the inverse of matrix A= 14. Using differential, find the approximate value of [ ]. 2 1 4 4 0 2 3 2 7 0.036 OR Find the approximate value of f(2.01), where f(x) = 4x2 + 5x + 2 15. If : [-5,5] R is a differentiable function and if f (x) does not vanish anywhere, then prove that f(-5) f(5) . 16. Find the intervals in which the function f given by f(x) = sinx + cosx , 0 < x < 2 is strictly increasing or strictly decreasing. 17. The two equal sides of an isosceles triangle with fixed base b are decreasing at the rate of 3 cm./sec. How fast is the area decreasing when the two equal sides are equal to the base. OR A man of height 2 meters walks at a uniform speed of 5 Km./hr. away from a lamp post which is 6 meters high. Find the rate at which the length of his shadow increases. 18. Determine the minimum value of Z = - x + 2y (if any),subject to the constraints: x 3 , x + y 5, x + 2y 6 & y 0. 19. Given two events A & B such that P(A) = 0.3 , P(B) = 0.6 . Find the probability of event neither A nor B . SECTION C 20. A given rectangular area is to be fenced off in a field whose length lies along a straight river. If no fencing is needed along the river, show that the least length of fencing will be required when length of the field is twice its breath. 21. 22. Which value do you like most? Explain it. 23. OR 24. Find the equation of all tangents to the curve y = cos (x + y) , - 2 x 2 , that are parallel to the line x + 2y = 0. OR For the curve y = 4x3 2x5, find all the points at which tangent passes through origin. 25.If ab =xx + yx + xy , find derivative of y with respect to x. 26.If y= log { x + x +a 2 2 } , Prove that ( 2 2 x +a 2 d y ) d x2 +x dy dx =0 Solution Section A 1 [ ] 0 2 3 2 0 1 3 1 0 2 3 4 K = 27 126 5 C4(0.7)4(0.3) = 0.36015 5 15p + 15q = 0.p + 20q 6 f(x) = IxI is one of the example of a function which is continuous but not differentiable at x = 0 Section B 7 q = 3p 8 X = b sin t So, dt dx dx dt dy dx sec3 2 d y 2 dx =d2 y 2 dx dy dt = - a sint b cost tan t a b =- sec2 t a 2 b 2 d y d x2 2 d y d x2 at t = 4 3 sec t =- =- a b d dt tan t 1 bcost 4 OR = a b =- a 2 b Y = a cos t 2 2 a 2 b d2 y 2 dx at t = 4 =- x sin x Let u = du dx du dv = & v = sinx sinx xcosx 2 = sin x du dx dv dx = & dv dx = cosx sinx xcosx 2 sin xcosx 9. 10. It is given that: The matrix given on the R.H.S. of the equation is a 2 3 matrix and the one given on the L.H.S. of the equation is a 2 3 matrix. Therefore, X has to be a 2 2 matrix. Now, let Therefore, we have: Equating the corresponding elements of the two matrices, we have: Thus, a = 1, b = 2, c = 2, d = 0 Hence, the required matrix X is OR A2 = -4A = [ ] [ ] [ ] [ ] [ ] [ ] 2 3 1 2 8 12 4 8 2 3 1 2 = &7I= 1 12 4 1 7 0 0 7 Therefore, A2 4A + 7I = 0 0 0 0 =O A2 = 4A - 7I A3 = A A2 = A(4A - 7I) = 4A2 7A = 4(4A - 7I ) 7A = 9A 28I So, A5 = A3 A2 = (9A 28I) (4A - 7I) = 36A2 - 63A 112A + 196I = 36 (4A - 7I ) 175A + 196I [ ] 2 3 = - 31A 56I = -31 1 2 A5 = [ ] 118 93 31 118 - 56 [ ] 1 0 0 1 12. let x be the random variable representing the number of very popular doctors. x 1 2 3 P(x) C (6,1 ) C(2,2) C ( 6,2 ) C(2,1) C ( 6,3 ) C(8,3) C(8,3) C (8,3) P(x) 3 28 15 28 It is expected that a doctor must be Qualified Very kind and cooperative with the patient 13. ] Given that, A= [ 2 1 4 4 0 2 3 2 7 Let A = IA [ ][ ] 2 1 4 1 0 0 4 0 2=0 1 0 A 3 2 7 0 0 1 R3 R3 R1 & R2 R2 2R1 [ ][ ] 2 1 4 1 0 0 0 2 6 = 2 1 0 A 1 1 3 1 0 1 Interchanging R1 & R3 [ ][ ] 1 1 3 1 0 1 0 2 6 = 2 1 0 A 2 1 4 1 0 0 R3 R3 2R1 [ ][ ] 1 1 3 1 0 1 0 2 6 = 2 1 0 A 0 1 2 3 0 2 R1 R1 + R3 & R2 [ ][ ] 1 0 1 2 0 1 0 1 4 = 5 1 2 A 0 1 2 3 0 2 R3 R3 R2 [ ][ ] 1 0 1 2 0 1 0 1 4 = 5 1 2 A 0 0 2 8 1 4 R2 R3 10 28 R3 1 2 R3 [ ][ ] 1 0 1 2 0 1 0 1 4 = 5 1 2 A 0 0 1 4 1/2 2 R2 R2 + 4R3 & R1 R1 - R3 [ ][ ] 1 0 0 2 1/2 1 0 1 0 = 11 1 6 A 0 0 1 4 1/2 2 Thus A = -1 [ ] 2 1/2 1 11 1 6 4 1/2 2 14. Let x = 2 and x = 0.01. Then, we have:f(2.01) = f(x + x) = 4(x + x)2 + 5(x + x) + 2 Now, y = f(x + x) f(x) f(x + x) = f(x) + y Hence, the approximate value of f (2.01) is 28.21. 15. 16. See Example 13 page 204 NCERT 17. Let ABC be isosceles where BC is the base of fixed length b. Let the length of the two equal sides of ABC be a. Draw AD BC. Now, in ADC, by applying the Pythagoras theorem, we have: Area of triangle The rate of change of the area with respect to time (t) is given by, It is given that the two equal sides of the triangle are decreasing at the rate of 3 cm per second. Then, when a = b, we have: Hence, if the two equal sides are equal to the base, then the area of the triangle is decreasing at the rate of . OR See Example 44 page 235 18. The feasible region determined by the constraints, is as follows. It can be seen that the feasible region is unbounded. The values of Z at corner points A (6, 0), B (4, 1), and C (3, 2) are as follows. Corner point Z = x + 2y A(6, 0) Z= 6 B(4, 1) Z= 2 C(3, 2) Z=1 As the feasible region is unbounded, therefore, Z = 1 may or may not be the maximum value. For this, we graph the inequality, x + 2y > 1, and check whether the resulting half plane has points in common with the feasible region or not. The resulting feasible region has points in common with the feasible region. Therefore, Z = 1 is not the maximum value. Z has no maximum value. 19. Section C 20. Second method 21. 22. 23. OR See example 9 page 521 NCERT 24. See example 46 page 237 NCERT OR The equation of the given curve is y = 4x3 2x5. Therefore, the slope of the tangent at a point (x, y) is 12x2 10x4. The equation of the tangent at (x, y) is given by, When the tangent passes through the origin (0, 0), then X = Y = 0. Therefore, equation (1) reduces to: Also, we have For x = 0, y = 0 For x = -1, y = -2 For x= 1, y = 2 So, the required points are (0,0),(1,2) & (-1,-2). 25. See Example 33 page 176 NCERT 26. y= log { x + x2+a2 } dy dx 1 = x x+ x +a 2 2 {1+ x2+a2 dy dx =1 Again, differentiating w.r.t. x x dy x2+a2 d y2 + dx 2 2 dx x +a = 0 2 x ( 2+a2 ) 2 d y d x2 +x dy dx =0 dy dx x +a2 2 = } 1 x +a2 2

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