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MATHMAA

# KVPY2013

This page contains KVPY2013 questions of Part I and Part II. The solutions are continued in next pages to get clear information.

## KVPY2013

1) Let x,y,z be three non negative integers such that x+y+z=10. The maximum possible value of xyz+xy+yz+zx is
A) 52           B)64           C) 69        D) 73

2) If a,b are natural numbers such that 2013+a² = b², then the minimum possible value of ab is
a) 671        b) 668        c) 658      d) 645

3) The number of values of b for which there is an isosceles triangle with side lengths b+5, 3b-2 and 6-b is
a) 0            b) 1          c) 2           d)3

4) Let a,b be non-zero real numbers. Which of the following statements about the following quadratic equation
ax² + (a+b)x + b = 0
is necessarily true ?
i) It has at least one negative root.
ii) It has at least one positive root.
iii) Both its roots are real.
(A) (i) and (ii) only              (B) (i) and (iii) only
(C) (ii) and (iii) only             (D) All of them

5) Let x,y,z be non-zero real numbers such that $$\frac{x}{y} + \frac{y}{z} + \frac{z}{x} = 7$$ and $$\frac{y}{z} + \frac{z}{x} + \frac{x}{z} = 9$$ then $$\frac{x³}{y³} + \frac{y³}{z³} + \frac{z³}{x³}$$-3 is equal to
(A) 152         (B) 153      (C) 154       (D) 155

6) In a triangle ABC with $$\angle A$$ < $$\angle B$$ < $$\angle C$$, points D,E,F are the interior of segments BC,CA,AB respectively. Which of the following triangles CANNOT BE similar to ABC.
A) Triangle ABD              B) Triangle BCE
C) Triangle CAF              D) Triangle DEF

7) Tangents to a circle at points P and Q on the circle intersect at a point R. If PQ=6 and PR=5 then the radius of the circle is
A) $$\frac{13}{3}$$       B)4
C) $$\frac{15}{4}$$       D)$$\frac{16}{5}$$

8) In an acute-angled triangle ABC, the altitudes from A, B, C when extended intersect the circumcircle again at points A1, B1, C1 respectively. If $$\angle ABC$$=45° , then $$\angle A1B1C1$$ equals
A) 45°    B)60°     C)90°     D)135°

9) In a rectangle ABCD, points X and Y are the midpoints of AD and DC, respectively. Lines BX and CD when extended intersect at E, lines BY and AD when extended intersect at F. If the area of ABCD is 60 then the area of BEF is
A) 60      B)80   C)90       D)120

10) In the figure given below, ABCDEF is a regular hexagon of side length 1, AFPS and ABQR are squares . Then the ratio Area(APQ)/Area(SRP) equals

A) $$\frac{\sqrt{2}+1}{2}$$  B) $$\sqrt{2}$$    C)$$\frac{3\sqrt{3}}{4}$$  D) 2

11) A person X is running around a circular track completing one round every 40 seconds. Another person Y running in the opposite direction meets X every 15 seconds. The time, expressed in seconds, taken by Y to complete one round is
A) 12.5     B) 24       C) 25      D) 55

12) The least positive integer n for which
$$\sqrt{n+1}$$ - $$\sqrt{n-1}$$ < 0.2 is
A)  24    B) 25    C)  26    D)  27

13) How many natural numbers n are there such that n! + 10 is perfect square?
A) 1      B)  2      C)  4      D)  infinitely many

14) Ten points lie in a plane so that no three of them are collinear. The number of lines passing through exactly two of these points and dividing the plane into two regions each containing four of the remaining points is
A)  1         B)  5
C) 10         D) dependent on the configuration of points

15)In a city, the total income of all people with salary below Rs. 10000 per annum is less than the total income of all people with salary above Rs. 10000 per annum. If the salaries of people in the first group increases by 5% and salaries of the people in the second group decreases by 5% then the average income of all people
A) increases           B) decreases
C)remains the same    D) cannot be determined from the data

This completes the part I KVPY2013 questions which carry 1 mark each .

To go through the Part II questions please click on Next  to go back click BACK.