Quant Boosters  Kamal Lohia  Set 7

Q3) Find pair of positive integers (m, n) such that the quadratic equation 4x²  2mx + n = 0 has two real roots both of which are between 0 and 1.

Q4) Solve the equation {√(2 + √3)}ⁿ + {(2  √3)}ⁿ = 4 for real n.

Q5) Let S be the sum of reciprocals of two real roots of the equation (a²  4)x² + (2a  1)x + 1 = 0 where a is a real number, find the range of S.

Q6) Let a, b be two real number, a > 0 and the equation x  a  b = 5 has three distinct roots, find the value of b

Q7) Find f(x) such that 2f(1  x) + 1 = xf(x).

Q8) Find sum of all real x such that (x  1)^4 + (x + 3)^4 = 82.

Q9) Let x₁, x₂ be the two real roots of the quadratic equation x² + x  3 = 0, find the value of x₁³  4x₂² + 19.

Q10) If x₁, x₂ be the two real roots of the quadratic equation x² + ax + a  1/2 = 0, find the value of a such that (x₁  3x₂)(x₂  3x₁) reaches the maximum value.

Q11) Find the number of positive integral solutions (x, y) such that:
x√(y) + y√(x) = √(2013x)  √(2013y) + √(2013xy) = 2013.

Q12) Given f(1) = 1/5 and when n > 1, {f(n  1)}/f(n) = {2nf(n  1) + 1}/{1  2f(n)}, find f(n).

Q13) Given 2x + 6y ≤ 15, x ≥ 0, y ≥ 0, find the maximum value of 4x + 3y.

Q14) Find the product of maximum and minimum possible value of real y such that y = (x²  x + 1)/(x² + x + 1) where x is a real number.

Q15) Hexagon ABCDEF is inscribed in a circle. The sides AB, CD, and EF are each x units in length whereas the sides BC, DE, and FA are each y units in length. Then, the radius of the circle is
(a) rt[(x² + y² + xy)/3]
(b) rt[(x² + y² + xy)/2]
(c) rt[(x² + y² – xy)/3]
(d) rt[(x² + y² – xy)/2]

Q16) ABC is an isosceles triangle with AB= AC = 4. O is a point inside the triangle ABC.
Find the area of triangle OBC.
I. ∠BAC = 45˚.
II. ∠BOC = 90˚.(a) question can be answered by using one statement alone but not by other statement alone.
(b) question can be answered by using either statement alone.
(c) question can be answered by using both statements together but not by either alone.
(d) question cannot be answered even by using both statements together.

Q17) In figure, CD = BC, ∠BAD = 66˚, AB is the diameter and O the center of the semicircle.
Measure the angle ∠DEC
(a) 48˚
(b) 54˚
(c) 57˚
(d) 63˚

Q18) ABCDHFEG is a cuboid and EFHGJ is a right pyramid with base EFHG and J lying on plane ABCD. If volume inside cuboid but outside pyramid is 480cm³, find h.
(a) 8
(b) 10
(c) 12
(d) 16

Q19) ABCD is a square with side length 2 cm. It is divided into five rectangles of equal areas, as shown in the figure. The perimeter of the rectangle BEFG is
(a) 51/16
(b) 36/11
(c) 58/15
(d) 47/13

Q20) Three circles are in a row touching each other such that all three of them have two common tangents. The radii of the largest and the smallest circle are 9 and 4 respectively. Line segment AB passes through the centres of the circles and lies on the two outer circles. What is the length of AB?
(a) 38
(b) 26 + 3rt(32)
(c) 26 + 2rt(38)
(d) 19

Q21) A regular hexagon with a perimeter of 12rt(2) units lies in the Cartesian plane in such a way that its center is on the origin, two of the vertices lie on the xaxis, and the midpoints of two of its sides lie on the yaxis. If the portion of the hexagon that lies in Quadrant I is completely revolved around the xaxis, a solid whose volume is X cubic units results. If the same portion is completely revolved around the yaxis, a solid with a volume of Y cubic units results. Evaluate (X/Y)² .
(a) 48/49
(b) 1
(c) 4/3
(d) 16/9

Q22) One angle of a regular polygon measures 179˚. How many sides does it have?
(a) 179
(b) 180
(c) 280
(d) 360