List of practice Questions

Find the discriminant of the quadratic equation \( \sqrt{2}x^2 - x - \sqrt{2} = 0 \) and hence find the nature of the roots.
For what value of \( k \) will the equation \( kx(x-2) + 6 = 0 \) have equal roots?
Check whether 301 is a term of the sequence \( 5, 11, 17, 23, \dots \).
Prove that \( (2+\sqrt{3})^2 \) is not a rational number.
Find the zeroes of the quadratic polynomial \( 3x^2 - x - 4 \) and verify the relationship between the zeroes and the coefficients.
Using Euclid division algorithm find the HCF of 148 and 185.
Find the mean of the following data: \ \[ \begin{array}{|c|c|c|c|c|c|} \hline \text{Variable} & 2 & 4 & 6 & 10 & 12 \\ \hline \text{Frequency} & 3 & 2 & 3 & 1 & 2 \\ \hline \end{array} \]
If the angle between two tangents drawn from an external point \( P \) to a circle of radius 3 cm and centre \( O \) is \( 60^\circ \), find the length of \( OP \).
For what value of \( k \) equations \( kx + y = 1 \) and \( (k-1)x + 2y = 3 \) have no solution?
Write the solution of the equation \( 3x + y = 11 \) in natural numbers.
Find the roots of the equation \( \frac{1}{x} - \frac{1}{x-2} = 3 \), \( x \neq 0,2 \).
In \( \triangle ABC \), \( \angle C = 90^\circ \) and \( P, Q \) are midpoints of \( CA \) and \( CB \) respectively. Prove that \( 4AQ^2 = 4AC^2 + BC^2 \).
If point \( A(x,2) \) is equidistant from the points \( B(8,-2) \) and \( C(2,-2) \), find the value of \( x \).
In what ratio does the point \( (-4,6) \) divide the line segment joining \( A(-6,10) \) and \( B(3,-8) \)?
For what value of \( m \), \( -4 \) is one of the zeros of the polynomial \( x^2 - x - (2m+2) \)?
If 1 is one zero of the polynomial \( p(x) = ax^2 - 3(a-1)x - 1 \), then the value of \( a \) is:
Which of the following quadratic polynomials has zeros \( \frac{3}{5} \) and \( -\frac{1}{2} \)?
If \( \alpha \) and \( \beta \) are the zeros of the polynomial \( p(x) = x^2 - 3x - 4 \), then the value of \( \frac{4}{3} (\alpha + \beta) \) is:
If one zero of the polynomial \( p(x) \) is 5, then one factor of \( p(x) \) is:
If \( p(x) = x^4 + 2x^3 - 17x^2 - 4x + 30 \) is divided by \( q(x) = x^2 + 2x - 15 \), then the degree of the quotient is:
If \( \alpha \) and \( \beta \) are the zeros of the polynomial \( x^2 + 5x + 8 \), then the value of \( \alpha^2 + \beta^2 + 2\alpha\beta \) is:
If \( TP \) and \( TQ \) are two tangents drawn from an external point \( T \) to a circle whose centre is \( O \) such that \( \angle POQ = 120^\circ \), then the value of \( \angle OTP \) is:
If \( \tan 2A = \cot (A - 18^\circ) \) where \( 2A \) is an acute angle, then the value of \( A \) is:
If \( \sin \theta = \frac{a}{b} \), then the value of \( \cos \theta \) is:
Which of the following is not a polynomial