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Prove that (n³- n) is divisible by 6

Determine the ratio of the volume of a cube to that of a sphere which will exactly fit inside the cube.

Three years ago Preethi ‘s age was 3 times the square of Swati’s age. Six years hence, Swati’s age will be two – third of Preethi’s age. Find their Present ages.

In an A.P. the first term is 2 and the sum of the first five terms is one fourth of the next five terms. Show that 20^th terms is - 112.

The mid-point of the line segment joining $(a+b-c,2a)$ and $(2a-b+c,2b)$ is $(6,4)$. Then the values of $a$ and $b$ respectively are

The distance between the tops of two trees $20m$ and $28m$ high is $17m.$ The horizontal distance between the feet of the trees is

In the figure P , Q and R are the points where the incircle of $△ABC$ touches the sides :

Using the information given , $\angle QPR$ = ____

Mr. Gupta opened a recurring deposit account in a bank. He deposited Rs 2,500 per month for two years. At the time of maturity he got Rs 67,500. Find:

(i) the total interest earned by Mr. Gupta

(ii) the rate of interest per annum.

The radius of a cylindrical tank is 28m. If its capacity is equal to that of a rectangular tank of size 28m x 16m x 11m, then its depth is

The figure given below shows a relation from $P$ to $Q.$ Complete the Range = {-2, 2, -3, 3, -5, 5, -6, __}from the diagram given?

1. 6

The 17th term of an A.P. is 5 more than twice its 8th term. If the 11th term of the A. P. is 43, find the nth term.

Solve each of the following given systems of equations graphically and find the vertices and area of the triangle formed by these lines and the y-axis:

$2x-3y+6=0,2x+3y-18=0.$

In $△$ABC, B is a right angle, and BD is perpendicular to AC and DE is perpendicular to BC. Then $AD.DC=$

f(x) = $a^x$ (where a >1) defined on f: R $\to (0,\infty )$ is -

A rectangle whose area is $180m^2$has its sides respectively diminished by $7m$ and $4m$ so as to form a square. Find the side of the square.

A man has to clean a window at a height of 5 m on a building. He needs to reach a point 1.3m below the window to clean it. What should be the length of the ladder that he should use which, when inclined at an angle of 60^o to the horizontal, would enable him to reach the required position?

The sum of the areas of two squares is 640 $m^2$. If the difference in the perimeters be 64 m, find the sides of the two squares.

In the given triangle ABC which is right-angled at B, if sin A =$\frac{12}{13}$, then find the value of (sinA cosA + sinC cosC).

The shape of a glass (tumbler) is usually in the form of

(a) a cylinder

(b) frustum of a cone

(c) a cone

(d) a sphere

How will you make the following equation linear?

$\frac{5}{x-1}$ + $\frac{1}{y-2}$ = 2

$\frac{6}{x-1}$ - $\frac{3}{y-2}$ = 1

If a chord AB subtends an angle of ${60}^o$ at the centre of a circle, then the angle between the tangents to the circle drawn from A and B is

(a) ${30}^o$ (b) ${60}^o$ (c) ${90}^o$ (d) ${120}^o$

Write the zeros of the polynomial $x^2-x-6$.

The sum of three numbers in A.P. is 12 and the sum of their cubes is 288. Find the numbers.

A bucket, made of metal sheet, is in the form of a cone whose height is 35 cm and radii of circular ends are 30 cm and 12 cm. How many litres of milk it contains if it is full to the brim ? If the milk is sold at Rs. 40 per litre, find the amount received by the person.

$\frac{co{s}^3\theta +si{n}^3\theta }{cos\theta +sin\theta }+\frac{co{s}^3\theta -si{n}^3\theta }{cos\theta -sin\theta }=2$

In Fig, ABC is a quadrant of a circle of radius 14 cm and a semicircle is drawn with BC as diameter. Find the area of the shaded region.

John travels $250\text{m}$ to North but then back-tracks to South for $100\text{m}$ to pick up a friend. What is John’s total displacement?

Prove that if the diagonals of a quadrilateral bisect each other, then it is a parallelogram. [4 MARKS]

Find the volume of water in a solid bottle in the form of cylindrical bottom and conical top. The cylinder is of radius 2.1 cm and height 14 cm . The height of the cone is 9 cm.

If α and $\frac{1}{\alpha }$ are the zeroes of

P(x)=4x²-2x+(k-4),

Find k.

Three bells toll together at intervals of 30, 36 and 45 minutes respectively. At what time will they toll together again if they toll simultaneously at 8 A.M?

Question 15
A building is in the form of a cylinder surmounted by a hemispherical vaulted dome and contains $41\frac{19}{21}{m}^3$ of air. If the internal diameter of dome is equal to its total height above the floor, find the height of the building?

The points (-3, 2), (2, -1) and (a, 4) are collinear. Find a.

The following table shows the class mark and frequency. Find the highest class interval that the data can have

$\begin{array}{cc}\text{Class Mark}& \text{Frequency}\\ 55& 8\\ 65& 6\\ 75& 18\\ 85& 25\\ 95& 12\end{array}$

Factorise the given quadratic equation

$x^2+5x+6=0.$

We can draw one tangent to the given circle through point ____

By reducing the selling price of an article by $Rs.50,$ a gain of $5\%$ turns into a loss of $5\%.$ Find the original selling price of the article.

ABCD is a parallelogram where A(x, y), B(5, 8), C(4, 7) and D(2, -4). Find :

(i) co-ordinates of A

(ii) equation of diagonal BD.

Three circles touch each other externally. The distance between their centres is $5cm,6cm$ and $7cm.$ Find the radii of the circles.

If -1,1 & 3 are the zeroes of a cubic polynomial, which of the following is the correct expression of the polynomial?

Which of the following equations has the sum of its roots as 3?

If the probability of winning a game is 0.3, then the probability of losing it is:

The modal class in the following frequency distribution is: