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True/False: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

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Q.1. For every integer n > 1, the inequality True/False: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE holds.                (1981 - 2 Marks)

Ans. T

Sol. Consider n numbers, namely 1, 2, 3, 4, ....n.
KEY CONCEPT : Now using  A.M. > G.M. for distinct numbers, we get

True/False: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

True/False: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEETrue/False: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

 

 

Q.2. The equation 2x2 + 3x + 1 = 0 has an irrational root. (1983 - 1 Mark)

Ans. F

Sol. 2x2 + 3x + 1 = 0 ⇒ x = -1,-1 /2 both are rational

∴ Statement is  FALSE.

 

 

Q.3. If a < b < c < d, then the r oots of the equation

(x – a) (x – c) + 2(x – b) (x – d) = 0 are real and distinct. (1984 - 1 Mark)

Ans. T

Sol. f(x) = (x – a) (x – c) + 2 (x – b) (x – d).

f (a) = + ve; f (b) = – ve ; f(c) = – ve ;  

f (d) = + ve

∴ There exists two real and distinct roots one in the interval (a, b) and other in (c, d). Hence, (True).

 

 

Q.4. If n1, n2, ......np are p positive integers, whose sum is an even number, then the number of odd integers among them is odd. (1985 - 1 Mark)

Ans. F

Sol.Consider N = n1 + n+ n3+....+ np, where N is an even number.
Let k numbers among these p numbers be odd, then p – k are  even numbers.
Now sum of (p – k) even numbers is even and for N to be an even number, sum of k odd numbers must be even which is possible only when k is even.
∴ The given statement is false.

 

Q.5. If P(x) = ax2 + bx + c and Q(x) = –ax2 + dx + c, where ac ≠ 0 , then P(x)Q(x)=0 has at least two real roots. (1985 - 1 Mark)

 

Ans. T

Sol. P(x).Q (x) = (ax2 + bx + c) (–ax2 + bx + c)

⇒ D1 = b2 – 4ac and D2 = b2 + 4ac

clearly, D1 + D2 = 2b2 ≥ 0

∴ atleast one of D1 and D2 is (+ ve). Hence, atleast two real roots.
Thus, (True)

Q.6. If x and y are positive real numbers and m, n are any positive integers, then True/False: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE (1989 - 1 Mark)

Ans.  F

Sol. As x and y are positive real numbers and m and n are positive integers

True/False: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEand True/False: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

{For two +ve numbers A.M. ≥ G.M.}

True/False: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE....(1)

and True/False: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE....(2)

Multiplying (1) and (2), we get

True/False: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEETrue/False: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Hence the statement is false.

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