### Question. 1

Let m and n be positive integers, If x^{2 }+ mx + 2n = 0 and x^{2 }+ 2nx + m = 0 have real roots, then the smallest possible value of m + n is

Let m and n be positive integers, If x^{2 }+ mx + 2n = 0 and x^{2 }+ 2nx + m = 0 have real roots, then the smallest possible value of m + n is

Let k be a constant. The equations kx + y = 3 and 4x + ky = 4 have a unique solution if and only if

If x_{1} = -1 and x_{m} = x_{m + 1} + (m + 1) for every positive integer m, then x_{100} equals

Aron bought some pencils and sharpeners. Spending the same amount of money as Aron, Aditya bought twice as many pencils and 10 less sharpeners. If the cost of one sharpener is 2 more than the cost of a pencil, then the minimum possible number of pencils bought by Aron and Aditya together is

In how many ways can a pair of integers (x , a) be chosen such that x^{2} − 2 | x | + | a - 2 | = 0 ?

The number of integers that satisfy the equality (x^{2} - 5x + 7)^{x + 1} = 1 is

If x and y are non-negative integers such that x + 9 = z, y + 1 = z and x + y < z + 5, then the maximum possible value of 2x + y equals

The number of pairs of integers(x,y) satisfying x ≥ y ≥ -20 and 2x + 5y = 99 is

How many distinct positive integer-valued solutions exist to the equation (x^{2} - 7x + 11)^{(x2 - 13x + 42)} = 1?

If x = (4096)^{7+4√3}, then which of the following equals 64?

The number of real-valued solutions of the equation 2^{x} + 2^{-x} = 2 - (x - 2)^{2} is

If 5^{x} - 3^{y} = 13438 and 5^{x-1} + 3^{y+1} = 9686 , then x+y equals

The quadratic equation x^{2} + bx + c = 0 has two roots 4a and 3a, where a is an integer. Which of the following is a possible value of b^{2} + c?

Let A be a real number. Then the roots of the equation x^{2} - 4x - log_{2}A = 0 are real and distinct if and only if

The product of the distinct roots of |x^{2} - x - 6| = x + 2 is

The number of solutions of the equation |x|(6x^{2} + 1) = 5x^{2} is

The number of the real roots of the equation 2cos(x(x + 1)) = 2^{x} + 2^{-x} is

If (5.55)^{x} = (0.555)^{y} = 1000, then the value of (1/x)- (1/y) is

If a and b are integers such that 2x^{2} − ax + 2 ＞ 0 and x^{2} − bx + 8 ≥ 0 for all real numbers x, then the largest possible value of 2a − 6b is

If log_{2}(5 + log_{3}a) = 3 and log_{5}(4a + 12 + log_{2}b) = 3, then a + b is equal to

Given that x^{2018} y^{2017} = 1/2 and x^{2016} y^{2019} = 8, the value of x^{2} + y^{3} is

If u^{2} + (u−2v−1)^{2} = −4v(u + v), then what is the value of u + 3v?

If log_{12}81 = p, then 3[(4−p)/(4+p)] is equal to:

If x is a positive quantity such that 2^{x} = 3^{log52} , then x is equal to

If log(2^{a} × 3^{b} × 5^{c}) is the arithmetic mean of log(2^{2} × 3^{3} × 5), log(2^{6} × 3 × 5^{7}), and log(2 × 3^{2} × 5^{4}), then a equals

If x is a real number such that log_{3}5 = log_{5}(2 + x), then which of the following is true?

If a, b, c, and d are integers such that a + b + c + d = 30, then the minimum possible value of (a - b)^{2} + (a - c)^{2} + (a - d)^{2} is

For how many integers n, will the inequality (n – 5) (n – 10) – 3(n – 2) ≤ 0 be satisfied?

If 9^{2x – 1} – 81^{x-1} = 1944, then x is

The value of log_{0.008}√5 + log_{√3}81 – 7 is equal to

If x + 1 = x^{2} and x > 0, then 2x^{4} is:

If a and b are integers of opposite signs such that (a + 3)^{2} : b^{2} = 9 : 1 and (a - 1)^{2} : (b - 1)^{2} = 4 : 1, then the ratio a^{2} : b^{2} is

If the roots of the equation (x + 1) (x + 9) + 8 = 0 are a and b, then the roots of the equation (x + a) (x + b) – 8 = 0 are

If x^2 + (x + 1) (x + 2) (x + 3) (x + 6) = 0, where x is a real number, then one value of x that satisfies this equation is

If x^2 + (x + 1) (x + 2) (x + 3) (x + 6) = 0, where x is a real number, then one value of x that satisfies this equation is

Find the solution set for [x] + [2x] + [3x] = 8, where x is a real number and [x] is the greatest integer less than or equal to x.

If 3x + y + 4 = 2xy, where x and y are natural numbers, then find the ratio of the sum of all possible values of x to the sum of all possible values of y

Which of the following is necessarily true?

What is the sum of the roots of all the quadratic equations that can be formed such that both the roots of the quadratic equation are common with the roots of equation (x – a) (x – b) (x – c) = 0?

Muniram made an investment of 10 lakh in Axim Dynamic Bond. The variation in fund value of the bond with respect to time follows a polynomial with degree 2. Due to precarious market condition, the fund value of the investment becomes 50% of the investment at the end of 2 nd year. If the fund value becomes 150% of the investment at the end of 4th year, then what is the absolute difference (in lakh) between the investment and the fund value of the bond at the end of 8th year?

If r, s, and t are consecutive odd integers with r < s < t, which of the following must be true?

The ratio of the roots of lx² + nx + n = 0 is p : q, then

If a, b and c are three real numbers, then which of the following is not true?

x and y are real numbers such that y = |x – 2| – |2x – 12| + |x – 8|. What is the least possible value of y?

p, q and r are three non - negative integers such that p + q + r = 10. The maximum value of pq + qr + pr + pqr is

The number of common terms in the two sequences 17, 21, 25, ...., 417 and 16, 21, 26, ...., 466 is

What is the number of distinct terms in the expansion of (a + b + c)^{20}?

Let 2 f(x) = ax² + bx + c, where a, b and c are certain constants and a ≠ 0. It is known that f(5) = – 3f(2) and that 3 is a root of f(x) = 0

What is the value of a + b + c ?

Let 2 f(x) = ax² + bx + c, where a, b and c are certain constants and a ≠ 0. It is known that f(5) = – 3f(2) and that 3 is a root of f(x) = 0

What is the other root of f (x) = 0 ?

If the roots of the equation x³ - ax² + bx - c = 0 are three consecutive integers, then what is the smallest possible value of b?

Three consecutive positive integers are raised to the first, second and third powers respectively and then added. The sum so obtained is a perfect square whose square root equals the total of the three original integers. Which of the following best describes the minimum, say m, of these three integers?

A function f (x) satisfies f (1) = 3600 and f (1) + f (2) + ... + f (n) = n 2 f (n), for all positive integers n > 1. What is the value of f (9)?

Consider the set S = {2, 3, 4,..., 2n+1}, where n is a positive integer larger than 2007. Define X as the average of the odd integers in S and Y as the average of the even integers in S. What is the value of X – Y?

A shop stores x kg of rice. The first customer buys half this amount plus half a kg of rice. The second customer buys half the remaining amount plus half a kg of rice. Then the third customer also buys half the remaining amount plus half a kg of rice. Thereafter, no rice is left in the shop. Which of the following best describes the value of x?

A confused bank teller transposed the rupees and paise when he cashed a cheque for Shailaja, giving her rupees instead of paise and paise instead of rupees. After buying a toffee for 50 paise, Shailaja noticed that she was left with exactly three times as much as the amount on the cheque. Which of the following is a valid statement about the cheque amount?

The price of Darjeeling tea (in rupees per kilogram) is 100 + 0.10n, on the n^{th} day of 2007 (n = 1, 2, ..., 100), and then remains constant. On the other hand, the price of Ooty tea (in rupees per kilogram) is 89 + 0.15n, on the n th day of 2007 (n = 1, 2, ..., 365). On which date in 2007 will the prices of these two varieties of tea be equal?

If p = 1/3 and q = 2/3 , then what is the smallest odd n such that a_{n} + b_{n} < 0.01?

Which of the following best describes a_{n} + b_{n} for even n?

A group of 630 children is arranged in rows for a group photograph session. Each row contains three fewer children than the row in front of it. What number of rows is not possible?

Consider the set S = {1, 2, 3, ..., 1000}. How many arithmetic progressions can be formed from the elements of S that start with 1 and end with 1000 and have at least 3 elements?

If log_{y} x =(a . log_{z} y) = (b . log_{x} z) = ab, then which of the following pairs of values for (a,b) is not possible?

The sum of four consecutive two-digit odd numbers, when divided by 10, becomes a perfect square. Which of the following can possibly be one of these four numbers?

The number of solutions of the equation 2x + y = 40 where both x and y are positive integers and x ≤ y

What are the values of x and y that satisfy both the equations?

What values of x satisfy x^{2/3} + x^{1/3} – 2 ≤ 0?

When you reverse the digits of the number 13, the number increases by 18. How many other two-digit numbers increase by 18 when their digits are reversed?

An airline has a certain free luggage allowance and charges for excess luggage at a fixed rate per kg. Two passengers, Raja and Praja have 60 kg of luggage between them, and are charged Rs 1200 and Rs 2400 respectively for excess luggage. Had the entire luggage beloged to one of them, the excess luggage charge would have been Rs 5400.

What is the free luggage allowance?

An airline has a certain free luggage allowance and charges for excess luggage at a fixed rate per kg. Two passengers, Raja and Praja have 60 kg of luggage between them, and are charged Rs 1200 and Rs 2400 respectively for excess luggage. Had the entire luggage beloged to one of them, the excess luggage charge would have been Rs 5400.

What is the weight of Praja’s luggage?

A telecom service provider engages male and female operators for answering 1000 calls per day. A male operator can handle 40 calls per day whereas a female operator can handle 50 calls per day. The male and the female operators get a fixed wages of Rs. 250 and Rs. 300 per day respectively. In addition, a male operator gets Rs. 15 per call he answers and a female operator gets Rs. 10 per call she answer. To minimize the total cost, how many male operators should the service provider employ assuming he has to employ more than 7 of the 12 female operators available for the job?

If a_{1} = 1 and a_{n+1} – 3a_{n+2} = 4n for every positive integer n, then a_{100} equals

Consider the sequence of numbers a_{1}, a_{2}, a_{3}, ........... to infinity where a_{1} = 81.33 and a_{2} = – 19 and a_{j} = a_{j–1}–a_{j–2} for j ≥3. What is the sum of the first 6002 terms of this sequence?

If log_{10} x – log_{10} √x = 2log_{x} 10, then a possible value of x is given by

If the sum of the first 11 terms of an arithmetic progression equals that of the first 19 terms, then what is the sum of the first 30 terms?

For which value of k does the following pair of equations yield a unique solution for x such that the solution is positive?

x² – y² = 0

(x – k)² + y² = 1

The total number of integer pairs (x, y) satisfying the equation x + y = xy is

In a certain examination paper, there are n questions. For j = 1,2 ....n, there are 2n–j students who answered j or more questions wrongly. If the total number of wrong answers is 4095, then the value of n is

There are 8436 steel balls, each with a radius of 1 centimeter, stacked in a pile, with 1 ball on top, 3 balls in the second layer, 6 in the third layer, 10 in the fourth, and so on. The number of horizontal layers in the pile is

If log_{3} 2, log_{3}(2^{x} - 5), log_{3}(2^{x} - 7/2) are in arithmetic progression, then the value of x is equal to

The 288th term of the series a, b, b, c, c, c, d, d, d, d, e, e, e, e, e, f, f, f, f, f, f.................is

The sum of 3rd and 15th elements of an arithmetic progression is equal to the sum of 6th, 11th and 13th elements of the progression. Then, which element of the series should necessarily be equal to zero

If |b| ≥ 1 and x = – | a | b, then which one of the following is necessarily true?

If 13x + 1 < 2z and z + 3 = 5y² , then

If three positive real numbers x, y, z satisfy y – x = z – y and x y z = 4, then what is the minimum possible value of y?

If both a and b belong to the set {1, 2, 3, 4}, then the number of equations of the form ax² + b x + 1 = 0 having real roots is

If x and y are integers then the equation 5x + 19y = 64 has

A real number x satisfying 1 - 1/n < x ≤ 3 + 1/n for every positive integer n, is best described by

The number of roots common between the two equations x³ + 3x² + 4x + 5 = 0 and x³ + 2x² + 7x + 3 = 0 is

A test has 50 questions. A student scores 1 mark for a correct answer, –1/3 for a wrong answer, and 1/6 for not attempting a question. If the net score of a student is 32, the number of questions answered wrongly by that student can not be less than

If x, y, z are distinct positive real numbers then

would be

Given that -1 ≤ v ≤ 1, -2 ≤ u ≤ -0.5 and -2 ≤ z ≤ -0.5 and w = vz/u , then which of the following is necessarily true?

If the product of n positive real numbers is unity, then their sum is necessarily

Let p and q be the roots of the quadratic equation x² - (α - 2)x - α - 1 = 0. What is the minimum possible value of p² + q² ?

Let a, b, c, d be four integers such that a + b + c + d = 4m + 1 where m is a positive integer. Given m, which one of the following is necessarily true?

The number of non-negative real roots of 2^{x} - x - 1 = 0 equals

Which one of the following conditions must p, q and r satisfy so that the following system of linear simultaneous equations has at least one solution, such that p + q + r ≠ 0 ?

x + 2y 3z = p ; 2x + 6y 11z = q ; x 2y + 7z = r

A child was asked to add first few natural numbers (that is, 1 + 2 + 3 + .....) so long his patience permitted. As he stopped he gave the sum as 575. When the teacher declared the result wrong the child discovered he had missed one number in the sequence during addition. The number he missed was

Amol was asked to calculate the arithmetic mean of ten positive integers each of which had two digits. By mistake, he interchanged the two digits, say a and b, in one of these ten integers. As a result, his answer for the arithmetic mean was 1.8 more than what it should have been. Then b – a equals

The nth element of a series is represented as X_{n} = (–1)^{n} X_{n – 1 }

If X_{0} = x and x > 0 then which of the following is always true

Three pieces of cakes of weight 4(1/2) lbs, 6(3/4) lbs and 7(1/5) lbs respectively are to be divided into parts of equal weights. Further, each part must be as heavy as possible. If one such part is served to each guest, then what is the maximum number of guests that could be entertained?

A piece of string is 40 centimeters long. It is cut into three pieces. The longest piece is 3 times as long as the middle-sized piece and the shortest piece is 23 centimeters shorter than the longest piece. Find the length of the shortest piece.

The number of real roots of the equation A²/x + B²/(x - 1) = 1 where A and B are real numbers not equal to zero simultaneously is

If x² + 5y² + z² = 2y(2x + z) then which of the following statements are necessarily true?

I. x = 2 y

II. x = 2 z

III. 2x = z

If x, y and z are real numbers such that, x + y + z = 5 and xy + yz + zx = 3

What is the largest value that x can have?

For a Fibonacci sequence, from the third term onwards, each term in the sequence is the sum of the previous two terms in that sequence. If the difference in squares of seventh and sixth terms of this sequence is 517, what is the tenth term of this sequence?

If a, b, c and d are four positive real numbers such that abcd = 1 , what is the minimum value of (1 + a) (1 + b) (1 + c) (1 + d)?

All the page numbers from a book are added, beginning at page 1. However, one page number was mistakenly added twice. The sum obtained was 1000. Which page number was added twice?

**Directions for questions: Read the information given below and answer the questions that follow :**

The batting average (BA) of a test batsman is computed from runs scored and innings played - completed innings and incomplete innings (not out) in the following manner :

An experienced cricketer with no incomplete innings has a BA of 50. The next time he bats, the innings is incomplete and he scores 45 runs. It can be inferred that

Based on the information provided which of the following is true?

Ujakar and Keshab solve a quadratic equation. Ujakar made a mistake in writing down the constant term. He ended up with the roots (4, 3). Keshab made a mistake in writing down the coefficient of x. He got the roots as (3, 2). What will be the exact roots of the original quadratic equation?

Let b be a positive integer and a = b² – b. If b ≤ 4, then a² – 2a is divisible by

Let x, y be two positive numbers such that x + y = 1. Then, the minimum value of

m is the smallest positive integer such that for any integer n ≤ m , the quantity n³ - 7n² + 11n - 5 is positive. What is the value of m?

x and y are real numbers satisfying the conditions 2 < x < 3 and – 8 < y < – 7. Which of the following expressions will have the least value?

Two men X and Y started working for a certain company at similar jobs on January 1, 1950. X asked for an initial salary of Rs 300 with an annual increment of Rs 30. Y asked for an initial salary of Rs 200 with a rise of Rs 15 every six months. Assume that the arrangements remained unaltered till December 31, 1959. Salary is paid on the last day of the month. What is the total amount paid to them as salary during the period?

If x > 5 and y < – 1, then which of the following statements is true?

If the equation x³ - ax² + bx - a = 0 has three real roots then which of the following is true?

A, B and C are 3 cities that form a triangle and where every city is connected to every other one by at least one direct roots. There are 33 routes direct & indirect from A to C and there are 23 direct routes from B to A. How many direct routes are there from A to C ?

x > 2, y > 1 then which of the following holds good ?

The expenses of a boarding school depends upon the fixed cost and variable cost. Variable cost varies directly as the number of students. If the expenses per student were Rs 600 for 50 students and Rs 700 for 25 students then what are the expenses for 100 students?

| r - 6 |=11 and | 2q - 12 |= 8, then what is the minimum value of q / r?

There are m blue vessels with known volumes V1, V2,....., Vm arranged in ascending order of volumes, where V1 is greater than 0.5 litres and Vm is less than 1 litre. Each of these is full of water. The water is emptied into a minimum number of white empty vessels each having volume 1 litre. If the volumes of the vessels increases with the value of lower bound 10^{–1} .

If m is maximum then what is range of the volume remaining empty in the vessel with the maximum empty space?

If m is maximum then what is the minimum number of white vessels required to empty it?

What is the maximum possible value of m?

There are fifty integers a_{1} , a_{2} , ....... a_{50}, not all of them necessarily different. Let the greatest integer of these fifty integers be referred to as G, and the smallest integer be referred to as L. The integers a_{1} through a_{24} form sequence S1, and the rest form sequence S2. Each member of S1 is less than or equal to each member of S2.

Every element of S1 is made greater than or equal to every element of S2 by adding to each element of S1an integer x.Then x cannot be less than

**Directions for Questions: These questions are based on the situation given below :**

There are fifty integers a_{1} , a_{2} , ....... a_{50}, not all of them necessarily different. Let the greatest integer of these fifty integers be referred to as G, and the smallest integer be referred to as L. The integers a_{1} through a_{24} form sequence S1, and the rest form sequence S2. Each member of S1 is less than or equal to each member of S2.

Elements of S1 are in ascending order, and those of S2 are in descending order. a_{24} and a_{25 }are interchanged then which of the following statements is true

All values in S1 are changed in sign, while those in S2 remain unchanged. Which of the following statements is true?

You can collect Rubies and Emeralds as many as you can. Each Ruby is worth Rs 4crores and each Emerald is worth of Rs 5crore. Each Ruby weights 0.3 kg. and each Emerald weighs 0.4 kg. Your bag can carry at the most 12 kg. What you should collect to get the maximum wealth?

One year payment to a servant is Rs 90 plus one turban. The servant leaves after 9 months and receives Rs 65 and a turban. Then find the price of the turban

If the roots, x_{1} and x_{2} , of the quadratic equation x² – 2x + c = 0 also satisfy the equation 7x_{²} – 4x_{1} = 47, then which of the following is true?

If log_{2} [log_{7} (x² – x + 37)] = 1, then what could be the value of x?

Out of two-thirds of the total number of basket-ball matches, a team has won 17 matches and lost 3 of them. what is the maximum number of matches that the team can lose and still win three-fourths of the total number of matches, if it is true that no match can end in a tie ?

Which of the following values of x do not satisfy the inequality (x² - 3x + 2 > 0) at all?

Given the quadratic equation x² – (A – 3)x – (A – 7), for what value of A will the sum of the squares of the roots be zero?

Once I had been to the post-office to buy stamps of five rupees, two rupees and one rupee. I paid the clerk Rs 20, and since he did not have change, he gave me three more stamps of one rupee. If the number of stamps of each type that I had ordered initially was more than one, what was the total number of stamps that I bought?

One root of x² + kx – 8 = 0 is square of the other. Then the value of k is

What is the value of m which satisfies 3m² – 21m + 30 < 0 ?

Along a road lie an odd number of stones placed at intervals of 10 m. These stones have to be assembled around the middle stone. A person can carry only one stone at a time. A man carried out the job starting with the stone in the middle, carrying stones in succession, thereby covering a distance of 4.8 km. Then the number of stones is

Fourth term of an arithmatic progression is 8. What is the sum of the first 7 terms of the arithmatic progression?

Nineteen years from now Jackson will be 3 times as old as Joseph is now. Joseph is three years younger than Jackson.

I. Johnson’s age now

II. Joseph’s age now

If the harmonic mean between two positive numbers is to their geometric mean as 12 : 13; then the numbers could be in the ratio

If log_{7} log_{5} (√x + 5 + √x ) = 0 , find the value of x.

log 216√6 to the base 6 is