PUC Science इयत्ता ११ - Karnataka Board PUC Question Bank Solutions

Write the sum of the coefficients in the expansion of \[\left( 1 - 3x + x^2 \right)^{111}\]

Find the equation of the set of the points P such that its distances from the points A(3, 4, –5) and B(–2, 1, 4) are equal.

If a and b denote respectively the coefficients of x^m and xⁿ in the expansion of \[\left( 1 + x \right)^{m + n}\], then write the relation between a and b.

If a and b are coefficients of xⁿ in the expansions of \[\left( 1 + x \right)^{2n} \text{ and } \left( 1 + x \right)^{2n - 1}\] respectively, then write the relation between a and b.

If a and b denote the sum of the coefficients in the expansions of \[\left( 1 - 3x + 10 x^2 \right)^n\]  and \[\left( 1 + x^2 \right)^n\]  respectively, then write the relation between a and b.

The term without x in the expansion of \[\left( 2x - \frac{1}{2 x^2} \right)^{12}\] is 

Find the ratio in which the sphere x² + y² + z² = 504 divides the line joining the points (12, –4, 8) and (27, –9, 18).

Show that the plane ax + by + cz + d = 0 divides the line joining the points (x₁, y₁, z₁) and (x₂, y₂, z₂) in the ratio \[- \frac{a x_1 + b y_1 + c z_1 + d}{a x_2 + b y_2 + c z_2 + d}\]

Write the distance of the point P (2, 3,5) from the xy-plane.

Write the distance of the point P(3, 4, 5) from z-axis.

The coordinates of the mid-points of sides AB, BC and CA of  △ABC are D(1, 2, −3), E(3, 0,1) and F(−1, 1, −4) respectively. Write the coordinates of its centroid.

Write the coordinates of the foot of the perpendicular from the point (1, 2, 3) on y-axis.

Write the length of the perpendicular drawn from the point P(3, 5, 12) on x-axis.

What is the locus of a point for which y = 0, z = 0?

Find the ratio in which the line segment joining the points (2, 4,5) and (3, −5, 4) is divided by the yz-plane.

Find the point on y-axis which is at a distance of  \[\sqrt{10}\] units from the point (1, 2, 3).

Find the point on x-axis which is equidistant from the points A (3, 2, 2) and B (5, 5, 4).

The ratio in which the line joining (2, 4, 5) and (3, 5, –9) is divided by the yz-plane is

The ratio in which the line joining the points (a, b, c) and (–a, –c, –b) is divided by the xy-plane is

If the coefficient of x in \[\left( x^2 + \frac{\lambda}{x} \right)^5\]  is 270, then \[\lambda =\]