List of practice Questions

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?

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.

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 \( B(3,-8) \)?