Question 1 :
Divide $3x^2 – x^3 – 3x + 5$ by $x – 1 – x^2$ and find the remainder. Is the remainder independent of $x$ ?

Question 3 :
Divide $3x^2 – x^3 – 3x + 5$ by $x – 1 – x^2$ and find the remainder and the quotient?

Question 4 :
State true or false: If the graph of a polynomial intersects the X-axis at exactly two points, it need not be a quadratic polynomial.

Question 5 :
If one zero of the quadratic polynomial $x^2+3x+k$ is 2, then the value of $k$ is:

Question 7 :
State true or false: The only value of $k$ for which the quadratic polynomial $kx^2+x+k$ has equal zeroes is $\frac{1}{2}$.

Question 8 :
For which values of a and b, are the zeroes of $q\left(x\right)=x^3+2x^2+a$ also the zeroes of the polynomial $p\left(x\right)=x^5-x^4-4x^3+3x^2+3x+b$?

Question 9 :
State true or false: If all three zeroes of a cubic polynomial $x^3+ax^2-bx+c$ are positive, then at least one of a, b and c is non-negative.

Question 10 :
<img style='object-fit:contain' src='https://teachmint.storage.googleapis.com/question_assets/cbse_ncert/61b19be4273b230584979a3a.png' /> Look at the graph given above. The graph of $y=p\left(x\right)$, where $p\left(x\right)$ is a polynomial. Here, find the number of zeroes of $p\left(x\right)$.

Question 14 :
If the polynomial $x^4-6x^3+16x^2-25x+10$ is divided by another polynomial $x^2– 2x+k$, the remainder comes out to be x + a, then k and a are 5 and -5 respectively.

Question 15 :
<img style='object-fit:contain' src='https://teachmint.storage.googleapis.com/question_assets/cbse_ncert/61b19be2273b230584979a37.png' /> Look at the graph given above. The graph of $y=p\left(x\right)$, where $p\left(x\right)$ is a polynomial. Here, find the number of zeroes of $p\left(x\right)$.

Question 16 :
As observed from the top of a 75 m high lighthouse from the sea level, the Angles of depression of two ships are 30° and 45°. If one ship is exactly behind the ofher on the same side of the lighthouse, then find the distance between the two ships.

Question 17 :
From a point on the ground, the angles of elevation of the bottom and the top of a transmission tower fixed at the top of a 20 m high building are $45^\circ and 60^\circ$, respectively. Find the height of the tower.

Question 18 :
From the top of a 7 m high building, the angle of elevation of the top of a cable tower is 60° and the angle of depression of its foof is 45°. Determine the height of the tower.

Question 19 :
A straight highway leads to the foof of a tower. A man standing at the top of the tower observes a car at an angle of depression of 30°, which is approaching the foof of the tower with a uniform speed. Six seconds later, the angle of depression of the car is found to be 60°. Find the time taken by the car to reach the foof of the tower from this point.

Question 20 :
A statue, 1.6 m tall, stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is 60° and from the same point, the angle of elevation of the top of the pedestal is 45°. Find the height of the pedestal.

Question 21 :
The Angles of elevation of the top of a tower from two points at a distance of 4 m and 9 m from the base of the tower and in the same straight line with it are complementary. Find the height of the tower.

Question 22 :
The angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foof of the tower, is 30°. Find the height of the tower.

Question 23 :
A contractor plans to install two slides for the children to play in a park. For the children below the age of 5 yr, she prefers to have a slide whose top is at a height of 1.5 m and is inclined at an angle of 30° to the ground, whereas for elder children, she wants to have a steep slide at a height of 3 m and inclined at an angle of 60° to the ground. What should be the length of the slides in each case?

Question 24 :
The angle of elevation of the top of a building from the foof of the tower is 30° and the angle of elevation of the top of the tower from the foof of building is 60°. If the tower is 50 m high, then find the height of the building.

Question 25 :
A tree breaks due to storm and the broken part bends, so that the top of the tree touches the ground making an angle 30° with it. The distance between the foof of the tree to the point, where the top touches the ground is 8 m. Find the height of the tree.

Question 26 :
Two poless of equal heights are standing opposite to each ofher on either side of the road, which is 80 m wide. From a point between them on the road, the Angles of elevation of the top of the poless are 60° and 30°, respectively. Find the height of the poless.

Question 27 :
<img style='object-fit:contain' src='https://teachmint.storage.googleapis.com/question_assets/cbse_ncert/61b19a62273b230584979933.jpeg' /> In the above image, a 1.2 m tall girl spofs a balloon moving with the wind in a horizontal line at a height of 88.2 m from the ground. The angle of elevation of the balloon from the eyes of the girl at any instant is 60°. After sometime, the angle of elevation reduces to 30°. Find the distance travelled by the balloon during the interval.

Question 28 :
<img style='object-fit:contain' src='https://teachmint.storage.googleapis.com/question_assets/cbse_ncert/61b19a60273b230584979931.jpeg' /> In the above image, a circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made by the rope with ground level is 30°.

Question 29 :
A 1.5 m tall boy is standing at some distance from a 30 m tall building. The angle of elevation from his eye to the top of the building increases from 30° to 60° as he walks tonwards the building. Find the distance he walked tonwards the building.

Question 30 :
<img style='object-fit:contain' src='https://teachmint.storage.googleapis.com/question_assets/cbse_ncert/61b19a61273b230584979932.jpeg' /> In the above image, a TV tower stands vertically on a bank of a canal. From a point on the ofher bank directly opposite the tower, the angle of elevation of the top of the tower is 60°. From another point 20 m away from this point on the line joining this point to the foof of the tower, the angle of elevation of the top of the tower is 30°. Find the height of the tower and the width of the canal.