List of practice Questions

If \( \alpha \) and \( \beta \) are the zeroes of the polynomial \( p(x) = x^2 + 3x - 4 \), then the value of \( \frac{\alpha \beta}{4} \) is
If one zero of the polynomial \( q(x) \) is -3, then one factor of \( q(x) \) is
If \( f(x) = x^4 - 2x^3 - x + 2 \) is divided by \( g(x) = x^2 - 3x + 2 \), the degree of the quotient is
If \( \alpha \) and \( \beta \) are the zeroes of the polynomial \( x^2 - 3(x+1) - 5 \), then the value of \( (\alpha + 1)(\beta + 1) \) is
If \( 2a + 3b = 8 \) and \( 3a - 4b = -5 \), then
The angles of elevation of the top of a tower from two points at distances of 4 m and 9 m from the base of the tower, which are in the same straight line, are complementary. Prove that the height of the tower is 6 m.
Draw a circle of radius 6 cm. From a point 10 cm away from its center, construct the pair of tangents to the circle and measure their lengths.
Find the median of the following data:
\[ \begin{array}{|c|c|} \hline \text{Class Interval} & \text{Frequency} \\ \hline 40-45 & 2 \\ 45-50 & 3 \\ 50-55 & 8 \\ 55-60 & 6 \\ 60-65 & 6 \\ 65-70 & 3 \\ 70-75 & 2 \\ \hline \end{array} \]
Which of the following is not a polynomial?
The degree of the polynomial \( (x^5 + x^2 + 3x) (x^6 + x^5 + x^2 + 1) \) is:
The zeroes of the polynomial \( x^2 - 11 \) are:
If -2 and -3 are the zeroes of the quadratic polynomial \( x^2 + (a+1)x + b \), then
Divide polynomial \( 2x^4 + 3x^3 - 2x^2 - 9x - 12 \) by polynomial \( x^2 - 3 \).
Find the sum of 10 terms of A.P. 9, 17, 25, ...
Find the sum of the first 15 multiples of 8.
In triangle \( \triangle PQR \), two points \( S \) and \( T \) are on sides \( PQ \) and \( PR \) such that \( \frac{PS}{SQ} = \frac{PT}{TR} \) and \( \angle PST = \angle PRQ \). Prove that \( \triangle PQR \) is an isosceles triangle.
If one zero of the polynomial \( (a^2 + 9)x^2 + 13x + 6a \) is reciprocal of the other, find \( a \).
On adding 1 to the numerator of a fraction it becomes \( \frac{1}{2} \) and on adding 1 to the denominator it becomes \( \frac{1}{3} \). Write the equation for this statement.
Solve the pair of equations \( 8x + 5y = 9 \) and \( 3x + 2y = 4 \) by substitution method.
Prove that
\[ \left( \frac{\sin A}{1+\cos A} + \frac{1+\cos A}{\sin A} \right) \left( \frac{\sin A}{1-\cos A} + \frac{1-\cos A}{\sin A} \right) = 4 \csc A \cot A. \]
Metallic spheres of radii 6 cm, 8 cm, and 10 cm respectively, are melted to form a single solid sphere. Find the radius of the resulting sphere.
Find the area of the triangle whose vertices are \( A(2,1) \), \( B(4,5) \) and \( C(0,3) \).
Prove that \( \frac{\sec\theta + \tan\theta}{\sec\theta - \tan\theta} = \left(\frac{1 + \sin\theta}{\cos\theta}\right)^2 \).
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?