Question 1 :
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 2 :
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 3 :
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 4 : <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.
Question 5 :
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 6 :
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 7 :
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 distances of the point from the poless.
Question 8 :
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 9 : <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 10 :
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 11 :
State True or False whether the following quadratic equation has two distinct real roots: $\left(x+1\right)\left(x-2\right)+x=0$
Question 12 :
Justify why the following quadratic equation has two distinct real roots: $3x^2-4x+1=0$
Question 13 :
Which constant must be added and subtracted to solve the quadratic equation $9x^2+\frac{3}{4}x-\sqrt{2}=0$ by the method of completing the square?
Question 14 :
Find the values of k for each of the following quadratic equations, so that they have two equal roots: $kx (x – 2) + 6 = 0$
Question 16 :
Find the roots of the following quadratic equation by factorisation: $\sqrt{2}x^2+7x+5\sqrt{2}=0$
Question 19 :
Check whether the following is a quadratic equation: $(2x – 1)(x – 3) = (x + 5)(x – 1)$
Question 20 :
Check whether the following is quadratic equation : (2x- 1)(x -3)=(x +5)(x -1)
Question 21 :
State True or False: Every quadratic equation has at least one real root.
Question 24 :
Using method of completing the square , solve for x: $4x^2+3x+5=0$
Question 27 :
Check whether the following is a quadratic equation: $x^2 – 2x = (–2) (3 – x)$
Question 28 :
Does the following equation has the sum of its roots as 3? $\sqrt{2}x^2-\frac{3}{\sqrt{2}}x+1=0$
Question 29 :
What is the general expression (standard form) for quadratic equations ?
Question 30 :
Represent the following situation in the form of a quadratic equation : The product of two consecutive positive integers is 306. We need to find the integers.
Question 31 :
A tent is in the shape of a cylinder surmounted by a conical top. If the height and diameter of the cylindrical part are 2.1 m and 4 m respectively, and the slant height of the top is 2.8 m, find the area of the canvas used for making the tent. (Note that the base of the tent will not be covered with canvas.)
Question 32 :
The slant height of a frustum of a cone is 4 cm and the perimeters (circumference) of its circular ends are 18 cm and 6 cm. Find the curved surface area of the frustum.
Question 33 :
In one fortnight of a given month, there was a rainfall of 10 cm in a river valley. If the area of the valley is 7280 $km^2$, check whether the total rainfall is approximately equivalent to the addition to the the normal water of three rivers each 1072 km long, 75 m wide and 3 m deep .
Question 34 :
A shuttle cock used for playing badminton has the shape of the combination of
Question 35 :
A copper wire, 3 mm in diameter, is wound about a cylinder whose length is 12 cm, and diameter 10 cm, so as to cover the curved surface of the cylinder. Find the length of the wire, assuming the density of copper to be 8.88 g per $cm^3$ .
Question 36 :
A spherical steel ball is melted to make eight new identical balls.Then, the radius of each new ball be $\frac{1}{8}$th the radius of the original ball.
Question 37 :
The diameters of the two circular ends of the bucket are 44 cm and 24 cm. The height of the bucket is 35 cm. The capacity of the bucket is
Question 38 :
A hemispherical bowl of internal radius 9 cm is full of liquid. The liquid is to be filled into cylindrical shaped bottles each of radius 1.5 cm and height 4 cm. How many bottles are needed to empty the bowl?
Question 39 :
A right triangle, whose sides are 3 cm and 4 cm (other than hypotenuse) is made to revolve about its hypotenuse. Find the volume of the double cone so formed.
Question 40 :
A copper rod of diameter 1 cm and length 8 cm is drawn into a wire of length 18 m of uniform thickness. Find the thickness of the wire.
Question 41 :
How many shots each having diameter 3 cm can be made from a cuboidal lead solid of dimensions 9cm × 11cm × 12cm?
Question 42 :
A drinking glass is in the shape of a frustum of a cone of height 14 cm. The diameters of its two circular ends are 4 cm and 2 cm. Find the capacity of the glass.
Question 43 :
A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the inner surface area of the vessel.
Question 44 :
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.
Question 45 : <img style='object-fit:contain' src='https://teachmint.storage.googleapis.com/question_assets/cbse_ncert/61b19c34273b230584979a93.JPG' />
In the above image, an open metal bucket is in the shape of a frustum of a cone, mounted on a hollow cylindrical base made of the same metallic sheet. The diameters of the two circular ends of the bucket are 45 cm and 25 cm, the total vertical height of the bucket is 40 cm and that of the cylindrical base is 6 cm. Find the area of the metallic sheet used to make the bucket, where we do not take into account the handle of the bucket.Take $\pi$ = $\frac{22}{7}$ .
Question 46 : <img style='object-fit:contain' src='https://teachmint.storage.googleapis.com/question_assets/cbse_ncert/61b19c32273b230584979a91.JPG' />
The radii of the ends of a frustum of a cone 45 cm high are 28 cm and 7 cm (see the above image). Find its curved surface area(Take $\pi$ = $\frac{22}{7}$ ).
Question 47 :
A right triangle, whose sides are 3 cm and 4 cm (other than hypotenuse) is made to revolve about its hypotenuse. Find the surface area of the double cone so formed.
Question 48 :
What is the formulae for curved surface area of solid hemisphere?
Question 49 :
16 glass spheres each of radius 2 cm are packed into a cuboidal box of internal dimensions 16 cm × 8 cm × 8 cm and then the box is filled with water. Find the volume of water filled in the box.
Question 50 : <img style='object-fit:contain' src='https://teachmint.storage.googleapis.com/question_assets/cbse_ncert/61b19c38273b230584979a98.JPG' />
In the above image, a wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown. If the height of the cylinder is 10 cm, and its base is of radius 3.5 cm, find the total surface area of the article.
