JEE Advanced Maths: Algebra questions with solutions

Q1. For real numbers a, b, c (all non-zero), suppose a² + b² + c² = ab + bc + ca. If the value of (b+c)/a + (a+c)/b + (a+b)/c is denoted by lambda, find the value of lambda.

1. 2
2. 4
3. 6
4. 8

Answer: 6

The condition implies a = b = c (since each squared difference must be zero); substituting gives each term (b+c)/a = 2, so lambda = 2 + 2 + 2 = 6.

Q2. If x is a real number satisfying x + 1/x = 3, find all possible values of x² - 1/x².

1. 3*sqrt(5)
2. -5*sqrt(3)
3. -3*sqrt(5)
4. 5*sqrt(3)

Answer: 3*sqrt(5)

x² - 1/x² = (x-1/x)(x+1/x) = (+/-sqrt(5))(3) = +/-3*sqrt(5). Since x>0 and x+1/x=3, x could be either root giving +3*sqrt(5) or -3*sqrt(5) depending on which root is chosen.

Q3. The equation sqrt(x + 1) - sqrt(x - 1) = sqrt(4x - 1) has how many real solutions?

1. No solution
2. One solution
3. Two solution
4. More than two solutions

Answer: No solution

After squaring, we get 2x - 2*sqrt(x²-1) = 4x - 1, i.e., 1 - 2x = 2*sqrt(x²-1). Since x >= 1, left side 1 - 2x <= -1 < 0, while right side >= 0. No solution exists.