Mathem,q,ties PART -2 SECTION -A Single Correct Choice Type This section contains 8 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D)for its answer, out which ONLY ONE is correct. 1. ,, ;e" (1 + looxee -x2oo)o^ , (r_xloo)J1_x2oo (A)k=100,m=1 /u \lTl = "* I 1-^, I +c. then f 1+x* ) 2. 3. Two systems of rectangular axes have same origin. lf a plane cuts them at distance 2, 3,7 and -3, -1, l, then 1"" is equal to (B)k=99,m=l (D) k=100,m=-) (C)k=99,m-'t (A) I (c) 1o (A)-1 (c) 1 The vertices of a triangle ABC lie on the hyperbola xy = 16. lf a tangent to the hyperbola through A is perpendicular to the side BC and r and R respectively denote the inradius and circumradius of the triangle ABC, then (A)r=zRsinP sin9 22 BC (C) r = 2V2 Rsin: sin-(B) e (D) none of these (B) 0 (D)2 (B) r= nsin9 sin9 (D) r = .tr nsin| "inf, 4. Thevalueoftheparametermsuchthattheareaboundedbyy=xt-1,andtheliney=mx+c attains its minimum value (1, and c are positive real values) is Space for rough work5. The equation of the side BC of an equilateral triangle ABC is 4x -3y + 6 = 0, the coordinates of A are (2, 1). A point p | ?, f, ) ties insiOe the triangle ABC. lf D, E, F are the feet of perpendicutar ' \2') from P on BC, CA, AB respectively. Then fiPDl + lPEl + lPFll (where [.] denotes the greatest integer function) is equal to (A) 1 (B) 2 (c) 3 (D) 4 6. The probability of the product of 3 numbers chosen from the set A = {1,2,3, 4, ..., ZO.r is divisible by 4 is equalto (A) 1 (B) 99 ' ' 76 '-' 76 (c) -(D) 9 76 "4 + ?rX e2x2 + ....i ezox21, where p, g, or, ?2, ... e R and €lo + elr + dz*0, 19 (B)p= 10,q= 2oo (D) none of these 8. lf 'f isadifferentiablefunctionand0 81 (c) > 83 (B) ' et (D) > 83 Space for rough workMultiple Correct Ghoice Type This section contains 5 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D)for its answer, out which ONE OR MORE is/are correct 5. of the determinant chosen at random from all the only is positive or negative respectively. Then (B) P(A) < P(B) (D) none of these (B) cotlcotixl = x (D) sinlsin-'xl= lxl 7. Let the equationsof a circle and a parabola be ^'* y" -4x-6 = 0and y= 9x respectivelythen (A) (1, -1) is a point on the common chord of contact (B) the equation of the common chord is y + 1 -9 (C) the length of the common chord is 6 (D) none of these 8. ff ir(r +1)(2r+3)=6na +bn3 +cn2 +dn+e, then lf A and B are two events that the value determinants of order 2 with entries O and 1 (A) P(A)> P(B) (c) P(A) = P(B) = 1 t 6. Which of the following is /are true? (A) tanltan-1xl = lxl (C)tan-'ltanxl= lxl (A)a+c=b+d ) (C) a, b, i c -1 are in A.P. 5 (B)e=0 (D) : is an inteser t1 ll(s+lt-tl)dt, x>2 9. Letf(x)=lJ ,then l0 Il5x+1, x<2 (A) f(x) is differentiable at x = 2 (B) f(x) is continuous but not differentiable at x = 2 (C) f(x) is differentiable everywhere (D) the right derivative of f(x) atx= 2 does not exist Space for rough workSECTION -B Matrix -Match Type This section contains 2 questions, Each question contains statements given in two columns, which have to be matched. The statements in Column I are labelled A, B, C and D, while the statements in Column ll are labelled p, q, r, s and t. Any given statement in Column I can have correct matching with ONE OR MORE statement(s) in Column ll. The appropriate bubbles corresponding to the answers to these questions have to be darkened as illustrated in the following example: lf the correct matches are A-p, s and t; B -q and r; C -p and q; and D -s and t; then the correct darkening of bubbles will look like the following: ABcD @e@@@c @oc oe@1. Match the following: Column -| (A) lf f,(x) = logloglog"'logx ' Then -r times F, ' t | {4 (x)r, (x)...f,00 (x)}-'ox = fr (x) + c , where k = (B) f(x) is a differentiable function satisfying f'(x) + 100f(x) < 1 V x /. t \ e Dt. Then I f (x)-, I is a non-decreasing function of x where \ K/k-(C) f(x) = p"*1x + [xJ, x integer function), then Column -ll 106 -txl) (where [.] denotes the greatest 10 I f(x)dx = io (p) (q) (r) (s) (t) 105 101 100 102 2. (D) rhe vatue "f J#*#fudx=lbslsinkx+coskxl+c, where k = LeJJ(x +y) = f(x)f(y) Vx, y e R and f(t)=2.11intriangleABC, a = f(3), b = f(1) + f(3)and c=f(2) + f(3); r, R denote the inradius and circumradius of triangle ABC, then Column -| R (A) r (B) tan2A > Column -ll 10 I 7I16 7J7 2 (p) (q) (r) (s) (t) (c) (D) sec'A < r< Space for rough workSECTION . C lnteger Answer Type This section contains 8 questions. The answerto each of the questions is a single digit ,x y z w integer, ranging from 0 to 9. The appropriate bubbles below the respective {'uesti6n lO@6G numbers in the ORS have to be darkened. For example, if the correct answers to question numbers x, Y, z and w (say) are 6, 0, 9 and 2, respectively, then the correct darkening of bubbles will look like the following: ;; "tan-'(x'-tex2+36) -1, then sum of the roots of the given equation is equal to (where [.] denotes the greatest integer function), then l, = 3' Let f: R -+ n nae,lg{ned by f(x) = 1 -xt, then the number of points of intersection for the curues y = f(x) and y = f-'(x) is 4. A family of circles touches the straight lines xy + 4 = 2(x+ y). The areas of the all circles are 4n each. lf the area of the polygon formed by joining the centies of all circles be bc, where bc is a Space for rough work 1. 2. rr ri' l[3lo* =,n^ *-'_dLe*-J5. The probability of India wining a test match against Australia is I. Assuming that thg outcomes 2 are independent and the matches end in a win or loss. The probability that in a 5 match ""rLs India's second with at 3'd test is I , then p = _ p lz -11 6,trn=|,-.|andiftheva|ueofdet(|+2A+3A2+4A3+5Aa+.'.tooo) LT -' ) 7. lf [1 +tan(k+ 13)'][1 +tan(k+ 14)"J[1 +tan(k* 1S).] ...[1 +tan(31 _k)"I1 +tan(32_k)ol=oF, where o, and B are prime numbers then /ffi-fg$l is equat to. \5 ) lf A = -z\+zj+S[ and E =l(ei++i-ti), where t(*l = ]1-r)'ol, " *integer. I is siven that f(x) ll, x e integer is continuous at x = 0. Then A 'E = -(where [.] denotes greatest integer function) Space for rough work