CAT Algebra

Reading Test 1

Question No : 1

Q.1 If x is a positive real number such that x⁸ + (1/x)⁸ = 47, then the value of x⁹ + (1/x)⁹ is:

Question No : 2

Q.2 Let n and m be two positive integers such that m < n, and there are exactly 41 integers greater Than x^m and less than 8ⁿ, which can be expressed as powers of 2. Then, the smallest possible value of n + m is

Question No : 3

Q.3 For some real numbers a and b, the system of equations x + y = 4 and (a + 5)x + (b² -15)y = 8b has infinite many solutions for x and y. Then, the maximum possible value of ab is

Question No : 4

Q.4 Let n be any natural number such that 5^n-1<3ⁿ⁺¹. Then the least integer value of m that satisfies 3ⁿ⁺¹<2^n+mfor each such n, is

Question No : 5

Q.5 A quadratic equation x²+ bx + c = 0 has two real roots. If the difference between the reciprocals of the roots is 1/3, and the sum of the reciprocals of the squares of the roots is 5/9, then the largest possible value of (b + c) is

Question No : 6

Q.6 The population of a town in 2020 was 100000. The population decreased by y% from the year 2020 to 2021, and increased by x% from the year 2021 to 2022, where x and y are two natural numbers. If the population in 2022 was greater than the population in 2020 and the difference between x and y is 10, then the lowest possible population of the town in 2021 was

Question No : 7

Q.7 The product of the roots of the equation mx² + 6x + (2m -1) = 0 is -1. Then m is:

Question No : 8

Q.8 If 13x + 17y = 643, where x and y are natural numbers, what is the value of two times the product of x and y?

Question No : 9

Q.9 Find the value of √6 + √6 + √6 + ...

Question No : 10

Q.10 One of the two students, while solving a quadratic equation in x, copied the constant term incorrectly and got the roots 3 and 2. The other copied the constant term and coefficient of x² correctly and got his roots as -6 and 1 respectively. The correct roots are:

Question No : 11

Q.11 If p and q are the roots of the equation x²+px + q = 0, then the value of p is:

Question No : 12

Q.12 If the roots of the equation a(b -c)x² + b(c - a)x + c(a - b) = 0 are equal, then a,b,c are in

Question No : 13

Q.13 The number of roots of the equation x - 2/(x-1) = 1 - 2/(x - 1) is

Question No : 14

Q.14 If the roots of the equation a₁x²+a₂x + a₃ = 0 are in the ratio r₁ : r₂ then

Question No : 15

Q.15 For what value of a do the roots of the equation 2x² + 6x + a = 0, satisfy the conditions (α/β) + (β/α) < 2?

Question No : 16

Q.16 For what values of b and c would the equaton x² + bx + c = 0 have roots equal to b and c?