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### Mathematics 2008

 1.  Let p be the statement “x is an irrational number”, q be the statement “y is a transcendental number”, and r be the statement “x is a rational number if y isa transcendental number”.  Statement-1:  r is equivalent to either q or p .  Statement 2: r is equivalent to  ~(p ~q)   (1)   Statement -1 is true , Statement -2 is false (2)   Statement -1 is false, Statement -2 is true (3)   Statement -1 is true, Statement -2 is true:        Statement -2 is a correct explanation for        Statement -1 (4)  Statement -1 is true, Statement -2 is true:       Statement -2 is not a correct explanation for                Statement -1   2.  In  a shop there are five types of ice-creams available. A child buys six ice-creams. Statement -1 : The number of different ways the child can buy the six ice-creams is  . Statement -2 : The number of different ways the child can buy the six ice-creams is equal of different ways of arranging 6 A’s and 4 B’s in a row   (1)        Statement -1 is true, Statement -2 is false (2)        Statement -1 is false, Statement -2 is true (3)        Statement -1 is true , Statement  -2 is true           Statement -2 is a correct explanation for Statement -1 (4)        Statement -1 is true , Statement-2 is true :               Statement -2 is not  a correct explanation for Statement -1   Ans.  x1, x2, x3, x4, x5 be the number of icecreams selected from 5 types of ice-creams x1+ x2+ x3+ x4+ x5 = 6 i = 1, 2, 3, 4, 5, 6 solve to get number of ways => statement (1) is false but statement (2) is correct.     3.  Statement-1:   Statement-2:     (1)       Statement -1 is true, Statement -2 is false   (2)       Statement -1 is false, Statement -2 is true (3)       Statement -1 is true, Statement -2 is true              Statement -2 is a correct explanation for Statement -1   (4)       Statement -1 is true, Statement-2 is true :                  Statement -2 is not  a correct explanation for Statement -1       Ans.   4.  Statement-1: For every natural number Statement-2: For every natural number   (1)       Statement -1 is true, Statement -2 is false (2)       Statement -1 is false, Statement  -2 is true (3)       Statement -1 is true, Statement -2 is true              Statement -2 is a correct explanation for Statement -1   (4        Statement -1 is true, Statement-2 is true :                  Statement -2 is not  a correct explanation for Statement -1       Ans. statement 1 is correct statement 1 is correct using statement 2   5. Let A be a  2 x 2 matrix with real entries. Let I be the 2 x 2 identity matrix. Denote by tr(A), the sum of diagonal entries of A. Assume that A2  = 1       Statement -1 :  If A 1 and A- 1 , then A = - 1           Statement- 2: If   A 1 and A  - 1, then tr (A)  0.     (1)   Statement -1 is true, Statement -2 is false (2)    Statement -1 is false, Statement  -2 is true (3)   Statement -1 is true, Statement -2 is true          Statement -2 is a correct explanation for Statement -1   (4)   Statement -1 is true, Statement-2 is true :        Statement -2 is not  a correct explanation for Statement -1   Ans.   6.  The statement p (q  p ) is equivalent to   (1)          p  (p  q) (2)          p  (p  q) (3)          p   (4)          p      7.  The value of cost   Ans.    8.  The differential equation of the family of circles with fixed radius 5 units and centre on the line y = 2 is   Ans.    9.   Then which one of the following is true?   Ans.    10.  The area of the plane region bounded by the curves x + 2y2 = 0 and x + 3y2 = 1 is equal to   Ans.        Ans.    12.  AB is a vertical pole with B at the ground level and A at the top. A man finds that the angle of elevation of the point A from a certain point C on the ground is 60o. He moves away from the pole along the line BC to a point D such that CD = 7 m. From D the angle of elevation of the point A is 45o. Then the height of the pole is   Ans.    13.  How many real solutions does the equation x7 + 14x5 + 16x3 + 30x - 560 = 0 have ?   (1)        5          (2)        7          (3)        1          (4)        3   Ans.    14.  Then which one of the following is true ?   (1)   f is differentiable at x = 1 but not at x = 0 (2)   f is neither differentiable at x = 0 nor at x = 1 (3)   f is differentiable at x = 0 and x = 1 (4)   f is differentiable at x = 0 but not at x = 1   Ans.  = Oscillating between -1 & 1 => RHD does not exists Non diff. At x=1 At x=0, f(x) is continuous & diff. As both (x-1) and sin(1/x-1) are continuous & diff. At x=0   15.  The first two terms of a geometric progression add up to 12. The sum if the third and the fourth terms is 48. If the terms of the geometric progression ate alternately positive and negative, then the first term is   (1)               4 (2)               –4 (3)               –12 (4)               12   Ans.  Given  b+br = 12 ---(1) br2 + br3 = 48    r<0 b(1+r) = 12 & br2 (1+r) = 48 Divide => r2 = 4 => As r<0, we take r = -2 Replace r in (1) to get b(1-2) = 12 => b = -12   16.  It is given that the events A and B are such that P(A) =, P(A | B) = and P(B | A) = . Then P(B) is   Ans.   17.  A die is thrown. Let A be the event that the number obtained is greater than 3. Let B be the event that the number obtained is less than 5. Then is   Ans.    18.  Suppose the cubic x3 – px + q has three distinet real roots where p> 0 and q > 0. Then which one of the following holds ?   (1)               The cubic has maxima at both and (2)               The cubic has minima at and maxima at (3)               The cubic has minima at and maxima at (4)               The cubic has minima at both and     Ans.  let f(x) = x3-px + q f ’(x) = 3x2 - p     19.  How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two S are adjacent ?     Ans.  M|||PP|  SSSS Arrange M|||PP|  in  Between 7 letter there are 8 possibilities for 4 select 4 position out of 8 in  ways   20.  The perpendicular bisector of the line segment joining P(1, 4) and Q(k, 3) has y-intercept –4. Then a possible value of k is   (1)   -4 (2)   1 (3)   2 (4)   –2     Ans.    21.  A parabola has the origin as its focus and the line x = 2 as the directrix. Then the vertex of the parabola is at   (1)   (2, 0) (2)   (0, 2) (3)   (1, 0) (4)   (0, 1)   Ans.  distance of vertex from directrix = distance from focus => V=(1, 0)   22.  The point diametrically opposite to the point P(1, 0) on the circle x2 + y2 + 2x + 4y – 3 = 0 is   (1)   (3, 4) (2)   (3, -4) (3)   (-3, 4) (4)   (-3, -4)     23.  A focus of an ellipse is at the origin. The directrix is the line x = 4 and the eccentricity is . Then the length of the semi-major axis is   Ans.    24.  The solution of the differential equation               satisfying the condition y(1) = 1 is   (1)               y = x ln x + x (2)               y = ln x + x (3)               y = x ln x x + x2 (4)               y = x e(x-1)    Ans.    25.  Let a, b, c be any real numbers. Suppose that there are real numbers x, y, z not all zero such that x = cy + bz, y = az + cz, and z = bx +ay. Then a2 + b2 + c2 + 2abc is equal to   (1)   1 (2)   2 (3)   –1 (4)   0   Ans.  x - cy - bz = 0 cx - y + az = 0 bx + ay - z = 0 It is given that system of equations is consistent. i.e. posses a solution. It is given that not all x, y, z are zero. => Non trivial solution exist => D = 0 Evalute  1(-1 - a2) + c(-c - ab) - b(ac + b) = 0 =>-1 - a2 - c2 - abc - abc - b2 = 0 => a2 + b2 + c2 + 2abc = -1   26.  Let A be a square matrix all of whose entries are integers. Then which one of the following is true ?   (1)   If det A = , then A-1 need not exist (2)   If det A = , then A-1 exists but all its entries are not necessarily integers (3)   If det A , then A-1 exists and all its entries are non-integers. (4)   If det A = , then A-1 exists and all its entries  are integers    Ans.  For example, let  det(A) = 1   A-1 exists & all entries are integers.   27.  The quadratic equations x2 – 6x + a = 0 and x2 – cx + 6 = 0have one root in common. The other roots of the first and second equations are integers in the ration 4: 3. Then the common root is   (1)        2          (2)        1          (3)        4          (4)        3   Ans.    28.  The mean of the numbers a, b, 8, 5, 10 is 6 and the variance is 6.80. Then which one of the following gives possible values of a and b ?   (1)               a = 3, b = 4 (2)               a = 0, b = 7 (3)               a = 5, b = 2 (4)               a = 1, b = 6   Ans.  simplify (1) to get: 23 + a + b=30 => a + b = 7 -----(3) simplify (2) to get (a - 6)2 + (b - 6)2 + 4 + 1 + 16=34 => (a - 6)2 + (7- a - 6)2 = 13 (a2-12a + 36) + (a2 +1 - 2a) = 13 2a2-14a + 24 = 0 a2 - 7a +12 = 0 => (a-4) (a-3) = 0 => b = 3, or 4   29.  The vector lies in the plane of the vectors and and bisects the angle between  Then which one of the following gives possible values of   Ans.    30.  The non-zero vectors are related by   Then the angle between is   Ans.   31.  The line passing through the points (5, 1, a) and (3, b, 1) crosses the yz-plane at the point Then   (1)               a = 8, b = 2 (2)               a = 2, b = 8 (3)               a = 4, b = 6 (4)               a = 6, b = 4   Ans.  Equation of line is let coordinates of intersection of l with yz plane be [2k+5, (1-b)k+1, (a-1)k+a] it lies on yz plane, 2k+5 = 0 => k = -5/2 Also (1-b)k+1=17/2 => 1-b = 15/2(-2/5) = -3 => b =4 Also (a-1)k+a =-13/2 A(1+k)-k = -13/2 => a(1-5/2) = -13/2-5/2 = -9 => -3/2 a = -9 => a = 6   32.  If the straight lines   intersect at a point, then the integer k is equal to     (1)               -2 (2)               –5 (3)               5 (4)               2   Ans.     33.  The conjugate of a complex number is  Then that complex number is   Ans.    34.  Let R be the real line. Consider the following subsets of the plane R R:                         S = {(x, y) : y = x + 1 and 0 < x < 2}                         T = {(x, y) : x – y is an integer}.         Which one of the following is true ?   (1)               T is an equivalence relation on R but S is not (2)               Neither S nor T is an equivalence relations on R (3)               Both S and T are equivalence relation on R (4)               S is an equivalence relation on R but T is not    35.   Let f : N  Y be a function defined as f(x) =4x + 3 where Y = {y N : y = 4x + 3 for some x N}. Show that f is invertible and its inverse is   Ans.  To find inverse of  f, replace y by x and x by y. i.e. x = 4f -1(x)+3 => f –1(x) = g(x) = x-3/4 Interims of y g(y) = y-3/4