CLASS XII (STREAM SX) CAREER POINT

1 Kota : CAREER POINT Ltd., CP Tower, IPIA, Road , Kota (Raj.), Ph: 0744-5151200 1 KVPY EXAMINATION 2017 CAREER POINT CLASS XII (STREAM SX) KVPY QUESTION PAPER-2017 (STREAM SX) Date : 05 /11/2017 Part A-Mathematics 1. Let BC be a fixed line segment in the plane. The locus of a POINT A such that the triangle ABC is isosceles, is (with finitely many possible exceptional points) [2017] (A) a line (B) a circle (C) the union of a circle and a line (D) the union of two circles and a line Sol. [D] Case (i) : AB C If B = C locus of A is bisector of BC So it is straight line Case (ii) : AB C If A = C BC fixed B(a, 0), C(0, a) BC = AB So, (x a)2 + y2 = 2a2 Circle Case (iii) : A = B AC = BC 22)ak(h = 2a2 x2 + (y a)2 = 2a2 also a circle So union of two circle and a line.

2 2. The number of solution pairs (x, y) of the simultaneous equations log1/3 (x + y) + log3 (x y) = 2 and 2y2 = 512x+1 is [2017] (A) 0 (B) 1 (C) 2 (D) 3 Sol. [B] )yx(log13 + log3(x y) = 2 log3(x + y) + log3(x y) = 2 log3 yxyx = 2 yxyx = 9 1x9y)2(22 )1x(9y222 y2 = 9(x + 1) Solve eliminate y 16x2 225x 225 = 0 x = 15, 1615 At x = 15, y = 12 x = 1615 , y = 43 (not possible) only sol. x = 15, y = 12 only one sol. 3. The value of the limit xlim x2x x42 is [2017] (A) (B) 41 (C) 0 (D) 41 CAREER POINT Kota : CAREER POINT Ltd., CP Tower, IPIA, Road , Kota (Raj.), Ph: 0744-5151200 2 KVPY EXAMINATION 2017 CAREER POINT CLASS XII (STREAM SX) Sol.

3 [D] Rationalise x2xx4x2xx4)x2xx4(lim222x x2x14|x|xlimx at x |x| = x x2x14xxlimx = 221 = 41 4. Let R be a relation on the set of all natural numbers given by a R b a divides b2. Which of the following properties does R satisfy ? I. Reflexivity II. Symmetry III. Transitivity [2017] (A) I only (B) III only (C) I and III only (D) I and II only Sol. [A] (I) This relation is reflexive relation because every natural no. divides square of itself a R a a divides a2 (II) not symmetric eg. 5 R 10 5 Divide 100 But 10 R 5 10 Divide 25 (III) Not transitivity for example if 8 R 4 & 4 R 2 8 R 2 only (I) Option 5. The fractional part of a real number x is x [x], where [x] is the greatest integer less than or equal to x.

4 Let F1 and F2 be the fractional parts of (44 2017)2017 and (44 + 2017)2017 respectively. Then F1 + F2 lies between the numbers [2017] (A) 0 and (B) and (C) and (D) and Sol. [C] I + F2 = 2017442017 F2 = 2017442017 ; 0 < F2 < 1 I + F2 F2 = 2 ..)44(2017C201612017 F2 = F2 F2 = ( )2017 Now, F1 = 2017)201744( = 2017) ( Fractional part can not ve. So, F1 = 1 ( )2017 So, F1 + F2 = 1 1 lie Between & 6. The number of real solutions of the equation 2sin 3x + sin 7x 3 = 0 which lie in the interval [ 2 , 2 ] is [2017] (A) 1 (B) 2 (C) 3 (D) 4 Sol. [B] only possible when sin 3x = 1 & sin 7x = 1 sin 3x = 1 sin 3x = sin (4n + 1)2 , n I 3x = (4n + 1)2 x = (4n + 1)6 sin 7x = sin(4m + 1)2 , m I x = (4m + 1)14 for common solution (4n + 1)6 = (4m + 1)14 Solving these 1 = 3m 7n First solution is m = 5, n = 2 Second solution is m = 12, n = 5 So two solutions are possible 7.

5 Suppose p, q, r are real numbers such that q = p (4 p), r = q (4 q), p = r (4 r). The maximum possible value of p + q + r is [2017] (A) 0 (B) 3 (C) 9 (D) 27 Sol. [C] Add all these p + q + r = 3rqp222 for maximum value p = 3, q = 3, r = 3 Answer is 9. 8. The parabola y2 = 4x + 1 divides the disc x2 + y2 1 into two regions with areas A1 and A2. Then | A1 A2 | equals [2017] (A) 31 (B) 32 (C) 4 (D) 3 Kota : CAREER POINT Ltd., CP Tower, IPIA, Road , Kota (Raj.), Ph: 0744-5151200 3 KVPY EXAMINATION 2017 CAREER POINT CLASS XII (STREAM SX) Sol. [B] A1 = dx1x4204/1 + dxx12102 A1 Solve A1 = 31 + 2 A2 = (1)2 A1 = 2 31 |A1 A2| = 32 9.

6 A shooter can hit a given target with probability 41. She keeps firing a bullet at the target until she hits it successfully three times and then she stops firing. The probability that she fires exactly six bullets lies in the interval [2017] (A) ( , ) (B) ( , ) (C) ( , ) (D) ( , ) Sol. [D] 3rd time target will hit in sixth time So, In first 5 attempt these will be 3L, 2W and at 6th attempt shot will be hit So, 5C3343 241 41 = 4096270 = 10. Consider the following events : E1 : Six fair dice are rolled and at least one die shows six. E2 : Twelve fair dice are rolled and at least two dice show six. Let p1 be the probability of E1 and p2 be the probability of E2.

7 Which of the following is true ? [2017] (A) p1 > p2 (B) p1 = p2 = (C) p1 < p2 (D) p1 = p2 = Sol. [A] p1 = 1 (no die show six) 1 665 = p2 = 1 (no die shown two + one die shown two) p2 = 1 111112126165C65 = p1 > p2 11. For how many different values of a does the following system have at least two distinct solutions ? ax + y = 0 x + (a + 10) y = 0 [2017] (A) 0 (B) 1 (C) 2 (D) Infinitely many Sol. [C] 1a = )10a(1 a2 + 10a 1 = 0 two value of a 12. Let R be the set of real numbers and f : R R be defined by f(x) = 2]x[1}x{ , where [x] is the greatest integer less than or equal to x, and {x} = x [x].

8 Which of the following statements are true ? I. The range of f is a closed interval II. f is continuous on R. III. f is one-one on R. [2017] (A) I only (B) II only (C) III only (D) None of I, II and III Sol. [D] f(x) = 2]x[1}x{ f(x) = onSo3x2;52x2x1;21x1x0;x0x1;21x Now check accordingly Kota : CAREER POINT Ltd., CP Tower, IPIA, Road , Kota (Raj.), Ph: 0744-5151200 4 KVPY EXAMINATION 2017 CAREER POINT CLASS XII (STREAM SX) 13. Let xn = (2n + 3n)1/2n for all natural numbers n. Then [2017] (A) nlimxn = (B) nlimxn =3 (C) nlimxn =3 + 2 (D) nlimxn =5 Sol. [B] n2/1nn2/1nn132)3(lim Put 3limn 14.

9 One of the solutions of the equation 8 sin3 7 sin +3cos = 0 lies in the interval [2017] (A) (0, 10 ] (B) (10 , 20 ] (C) (20 , 30 ] (D) (30 , 40 ] Sol. [B] 6 sin 2 sin 3 7 sin + 3cos = 0 3cos sin = 2 sin 3 It can be written as 2 (sin (60 )) = 2 sin 3 sin (60 ) = sin 3 60 = 4 = 15 is one of the value 15. Let a, b, c, d, e, be real numbers such that a + b < c + d, b + c < d + e, c + d < e + a, d + e < a + b. Then [2017] (A) The largest is a and the smallest is b (B) The largest is a and the smallest is c (C) The largest is c and the smallest is e (D) The largest is c and the smallest is b Sol. [A] (i) a + b < c + d (ii) b + c < d + e (iii) c + d < e + a (iv) d + e < a + b from (i) & (iii) a + b < e + a b < e from (ii) & (iv) b + c < a + b c < a (i) (ii) a c < c e c > e (i) (iv) (a e) + (b d) < (c a) + (d b) from thus d > b (i) + (iii) (ii) c > d overall a is greatest, b is least 16.))))

10 If a fair coin is tossed 5 times, the probability that heads does not occur two or more times in a row is [2017] (A) 5212 (B) 5213 (C) 5214 (D) 5215 Sol. [B] Case (1) : All tail 521 Case (2) : 4T, 1H 5C4421 .521 = 525 Case (3) : T T T 321 4C2 221 = 521 6 = 526 Case (4) : T T 221 321 = 521 overall 5213 17. Consider the following parametric equation of a curve : x ( ) = | cos 4 | cos y( ) = | cos 4 | sin for 0 2 Which one of the following graphs represents the curve ? [2017] (A) y x (B) y x (C) y x (D) x Sol. [A] Make graph and observe yourself Kota : CAREER POINT Ltd.