List of practice Questions

Consider the system of linear equations given below.
\[ ax + y = b \] \[ 16x + ay = 24 \] Suppose the values of \(a\) and \(b\) are chosen such that the system of linear equations produce multiple solutions. Then the product of \(a\) and \(b\) is _____. (answer in integer)

A quadratic equation \[ x^2-ax+b=0,\qquad a>0 \] has real roots. If the sum of the roots is less than the product of the roots, then \(b\) lies in the interval

If \(\alpha,\beta,5\) are the roots of the equation \[ x^3-ax+a= \frac{\sin^2x+\cos^4x} {\cos^2x+\sin^4x}, \] then \(a(\alpha+\beta)=\)

When each root of the equation \[ x^3-ax^2+bx+1=0 \] is diminished by \(h\), if the transformed equation takes the form \[ 9x^3+\left(9+\frac{a^3}{3}\right)=0, \] then \(ah=\)

If the equations \[ ax^2+2bx+3c=0,\qquad 3x^2+8x+15=0 \] have a common root, where \(a,b,c\) are sides of triangle \(ABC\), then \[ \cos2A+\cos2B+\cos2C= \]

If \( \theta_1 \) and \( \theta_2 \) are the values of \( \theta \in (0,\pi) \) for which the system of linear equations \[ x+3y+7z=0, \] \[ -x+4y+7z=0, \] \[ (\sin 3\theta)x+(\cos 2\theta)y+2z=0 \] has a non-trivial solution, then \( |\theta_1-\theta_2| \) is equal to

On the inauguration day of a new showroom, a lucky draw was organized and some vouchers of ₹ 1,000 and ₹ 500 were given to the lucky draw winners. A total of 60 vouchers were given on the day. The number of ₹ 1,000 vouchers added to 3 times the number of ₹ 500 vouchers, gives 100. Express the given information as a system of linear equations in two variables. Hence, find the number of vouchers of each type by matrix method.

A system of equations is given in the matrix form as $\begin{bmatrix} \alpha & 2 & 3 \\ 2 & 3 & -\alpha \\ 3 & 5 & \alpha+1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 2 \\ 3 \\ 5 \end{bmatrix}$ where $\alpha$ is an integer. If the system of equations does not have a unique solution, then the value of $\alpha$ is equal to

If the system \[ x+y-z=\lambda, \] \[ 2x-y+\mu z=2, \] \[ x-y+3z=1 \] is inconsistent, then

If \(a\in \mathbb{Z}\) and the equation \[ (x-a)(x-10)+1=0 \] has integral roots, then the values of \(a\) are:

If \(\alpha,\beta\) are the rational roots and \(l,m\) are the irrational roots of \[ (x^2-9x+11)^2-(x-4)(x-5)=3, \] then \(\alpha+\beta+lm=\)

Let \(\alpha,\beta,\gamma,\delta\) be the roots of \[ 4x^4+8x^3-17x^2-12x+9=0. \] If \[ 4(\alpha+4)(\beta+4)(\gamma+4)(\delta+4)=k, \] then \(k=\)

The number of real roots of the equation \[ x^7+3x^5-13x^3-15x=0 \] is:

If the minimum value of $ax^2-7x+3a$ is $-\frac{1}{8}$, find sum of roots of $ax^2-7x+3a=0$.}

If $\alpha,\beta$ are roots of $2x^2-x-3\lambda=0$ and $\alpha,\gamma$ are roots of $2x^2+9x+2\lambda=0$, find equation of roots $2\alpha+\beta$ and $\beta+\gamma$.

If $\alpha,\beta,\gamma$ are roots of $x^3-ax^2-4x+4a=0$ with $\alpha+\beta=0$, $\beta+\gamma=5$, find sum of all possible $a$.

If $2x^5+ax^4-12x^3+bx^2+x+c=0$ is reciprocal equation of class one, find sum of rational roots.

If the system of equations \[ ax+y-2z=3,\qquad 2x-y+3z=b,\qquad x+2y-z=3 \] has infinitely many solutions, then \(3a-2b=\)

Consider the system of linear equations \[ x+y+z=6, \] \[ x+2y+3z=10, \] \[ 3x+2y+\lambda z=\mu. \] If the system has infinitely many solutions, then the value of \(\mu+\lambda\) is:

If the system of equations:
\(x + y + z = 5\)
\(x + 2y + 3z = 9\)
\(x + 3y + \lambda z = \mu\)
has infinitely many solutions, then the value of \(\lambda + \mu\) is:

Let \( \alpha, \beta \in \mathbb{R} \) be such that the system of linear equations} \[ x + 2y + z = 5 \] \[ 2x + y + \alpha z = 5 \] \[ 8x + 4y + \beta z = 18 \] has no solution. Then \( \frac{\beta}{\alpha} \) is equal to:

If the system of equations} \[ x + 5y + 6z = 4 \] \[ 2x + 3y + 4z = 7 \] \[ x + 6y + az = b \] has infinitely many solutions, then the point \( (a, b) \) lies on the line}

If the system of linear equations $x + y + z = 1$, $x + 2y + 4z = \eta$, $x + 4y + 10z = \eta^2$ has a solution, then the value of $\eta$ is:

If the system of equations \[ x+y+z=6 \] \[ x+2y+3z=14 \] \[ 2x+y+z=7 \] has solution $(x,y,z)$, then the value of $x+y+z$ is: