Practice Problems (MCQ Questions may have more than 1 correct answer)
Q-1) Solve the equation in reals \(|x^2+x-4|=|x^2-4|+|x|\).
Q-2) If \(\alpha\), \(\beta\) be the roots of the equation \(x^2-p(x+1)-q=0\), then the value of \(\frac{\alpha^2+2\alpha+1}{\alpha^2+2\alpha+q}+\frac{\beta^2+2\beta+1}{\beta^2+2\beta+q}\) is
a) \(2\) b) \(1\) c) \(0\) d) None of these
Q-3) Let \(f(x)=ax^2+bx+c\) be such that \(f(x)=x\) has no real roots. Show that the equation \(f(f(x))=x\) does not have any real roots.
Q-4) Find \(a\in\mathbb{R}\) such that the sum of the squares of the roots of the equation \(x^2-(a-2)x-a-1=0\) is minimal.
Q-5) Let \(a,b,c\) be distinct numbers. Show that for all \(x\), \(\frac{(x-a)(x-b)}{(c-a)(c-b)}+\frac{(x-b)(x-c)}{(a-b)(a-c)}+\frac{(x-c)(x-a)}{(b-c)(b-a)}=1\).
(You may use the fact that if a polynomial of degree n has n+1 roots, then the value of the poynomial is 0 for all real numbers)
Q-6) The number of values of \(a\) for which \((a^2-3a-2)x^2+(a^2-5a+6)x+a^2-4=0\) is an identity in x is
a) \(0\) b) \(1\) c) \(2\) d) None of these
Q-7) Let \(f(x)=x-[x]\), where \([x]\) denotes the greatest integer less than or equal to \(x\). The number of solutions of \(f\left(x \right)+f\left(\frac{1}{x} \right)=1\) is/are
a) \(0\) b) infinitely many c) \(1\) d) None of these
Q-8) The equation \(ax^2-bx+c=0\) where \(a,b,c \in \mathbb{N}\) has 2 distinct real roots in the interval \((1,2)\). Find the maximum values of \(a,b,c\).
Q-9) [Integer Answer Type] If the roots of the cubic equation \(x^3+ax^2+bx+c=0\) are 3 consecutive positive integers, find the value of \(\frac{a^2}{b+1}\).
Q-10) Find the solution set of the system
\(x+2y+z=1\),
\(2x-3y-w=2\),
\(x \ge 0, y\ge 0, z\ge 0, w \ge 0\)
Q-11) Let \(\alpha, \beta, \gamma\) be the roots of the equation \((x-a)(x-b)(x-c)=d\), where \(d \ne 0\). The roots of the equation \((x-\alpha)(x-\beta)(x-\gamma)+d=0\) are
a) \(a,b,d\) b) \(b,c,d\) c) \(a,b,c\) d) None of these
Q-12) The number of values \(a\) for which the equations \(x^3+ax+1=0\) & \(x^4+ax^2+1=0\) have a common root are:
a) \(o\) b) \(1\) c) \(2\) d) infinitely many
Q-13) The number of solutions of the equation \(|[x]-2x|=4\), where \([x]\) denotes the greatest integer less than or equal to \(x\) is:
a) \(2\) b) \(3\) c) \(4\) d) None of these
Q-14) Find all real values of \(a\) such that \(2\) lies between the roots of the equation \(x^2+(a+2)x-(a+3)=0\)
Q-15) Consider the equation \(ax^2+bx+c=0\) with \(a,b,c\) as odd integers. Can this equation have rational roots?
<<In case there is any confusion over any example (solved or not), problem or concept, please feel free to create a thread in the appropriate forum. Our memebrs/experts will try their best to help solve the problem for you>>