English Medium कक्षा १० - CBSE Question Bank Solutions

A ladder rests against a vertical wall at an inclination α to the horizontal. Its foot is pulled away from the wall through a distance p so that its upper end slides a distance q down the wall and then the ladder makes an angle β to the horizontal. Show that `p/q = (cos β - cos α)/(sin α - sin β)`

In which ratio the y-axis divides the line segment joining the points (5, – 6) and (–1, – 4)?

`sqrt(3)` cos²A + `sqrt(3)` sin²A is equal to ______.

The total surface area of a solid hemisphere of radius 7 cm is ______.

Assertion (A): The ratio in which the line segment joining (2, -3) and (5, 6) internally divided by x-axis is 1:2.

Reason (R): as formula for the internal division is `((mx_2 + nx_1)/(m + n) , (my_2 + ny_1)/(m + n))`

Prove that `tan θ/(1 - cot θ) + cot θ/(1 - tanθ)` = 1 + sec θ cosec θ

Ramesh made a bird-bath for his garden in the shape of a cylinder with a hemispherical depression at one end. The height of the cylinder is 1.45 m and its radius is 30 cm. Find the total surface area of the bird-bath.

If the vertices of a parallelogram PQRS taken in order are P(3, 4), Q(–2, 3) and R(–3, –2), then the coordinates of its fourth vertex S are ______.

If sin θ + cos θ = `sqrt(2)` then tan θ + cot θ = ______.

Statement A (Assertion): If the coordinates of the mid-points of the sides AB and AC of ∆ABC are D(3, 5) and E(–3, –3) respectively, then BC = 20 units.

Statement R (Reason): The line joining the mid-points of two sides of a triangle is parallel to the third side and equal to half of it.

Find an acute angle θ when `(cos θ - sin θ)/(cos θ + sin θ) = (1 - sqrt(3))/(1 + sqrt(3))`

There are two identical solid cubical boxes of side 7 cm. From the top face of the first cube a hemisphere of diameter equal to the side of the cube is scooped out. This hemisphere is inverted and placed on the top of the second cube’s surface to form a dome. Find

1. the ratio of the total surface area of the two new solids formed
2. volume of each new solid formed.

A tiling or tessellation of a flat surface is the covering of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. Historically, tessellations were used in ancient Rome and in Islamic art. You may find tessellation patterns on floors, walls, paintings etc. Shown below is a tiled floor in the archaeological Museum of Seville, made using squares, triangles and hexagons.

A craftsman thought of making a floor pattern after being inspired by the above design. To ensure accuracy in his work, he made the pattern on the Cartesian plane. He used regular octagons, squares and triangles for his floor tessellation pattern

Use the above figure to answer the questions that follow:

1. What is the length of the line segment joining points B and F?
2. The centre ‘Z’ of the figure will be the point of intersection of the diagonals of quadrilateral WXOP. Then what are the coordinates of Z?
3. What are the coordinates of the point on y-axis equidistant from A and G?
   OR
   What is the area of Trapezium AFGH?

Solve the quadratic equation: x² – 2ax + (a² – b²) = 0 for x.

If the sum of the roots of the quadratic equation ky² – 11y + (k – 23) = 0 is `13/21` more than the product of the roots, then find the value of k.

If x = –2 is the common solution of quadratic equations ax² + x – 3a = 0 and x² + bx + b = 0, then find the value of a²b.

Khurja is a city in the Indian state of Uttar Pradesh famous for the pottery. Khurja pottery is traditional Indian pottery work which has attracted Indians as well as foreigners with a variety of tea sets, crockery and ceramic tile works. A huge portion of the ceramics used in the country is supplied by Khurja and is also referred as "The Ceramic Town". One of the private schools of Bulandshahr organised an Educational Tour for class 10 students to Khurja. Students were very excited about the trip. Following are the few pottery objects of Khurja. Students found the shapes of the objects very interesting and they could easily relate them with mathematical shapes viz sphere, hemisphere, cylinder etc.

Maths teacher who was accompanying the students asked the following questions:

1. The internal radius of hemispherical bowl (filled completely with water) in I is 9 cm and the radius and height of the cylindrical jar in II are 1.5 cm and 4 cm respectively. If the hemispherical bowl is to be emptied in cylindrical jars, then how many cylindrical jars are required?
2. If in the cylindrical jar full of water, a conical funnel of the same height and same diameter is immersed, then how much water will flow out of the jar?

3 cubes each of 8 cm edge are joined end to end. Find the total surface area of the cuboid.

If x = 3 is one root of the quadratic equation 2x² + px + 30 = 0, find the value of p and the other root of the quadratic equation.

The distance of the point (–6, 8) from x-axis is ______.