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The sum of all positive integers n for which $\frac{1^3+2^3....+(2n{)}^3}{1^2+2^2+...n^2}$ is also an integer is.

Simplify $\frac{9x^2+12x+4}{4x^2-27x-7}÷\frac{12x^2+5x-2}{16x^2-1}.$

Two dice are rolled simultaneously. Find the probability that number appearing on the first die is of the form of 5p or 2p and the number appearing on the second die is of the form 3q. (where 'p' and‘q’ can take values 1, 2 or 3).

In the figure given below, PA and PB are tangents to the circle from point P. Which of the following is the correct relation between PA and PB?

Rasheed got a playing top (lattu) as his birthday present, which surprisingly had no colour on it. He wanted to colour it with his crayons. The top is shaped like a cone surmounted by a hemisphere. The entire top is 5 cm in height and the diameter of the top is 3.5 cm. Find the area he has to colour $(take\pi =\frac{22}{7})$.

Which of the following cannot be a cyclic quadrilateral?

Pair the cards of obstacles and the solutions to cross it.

A train take a 2 hours less for a journey of 300 km . If the speed of the is increased by 5 km / hr from its usual speed , find the usual speed of the train.

In the given figure, O is the centre of the circumcircle ABC. Tangents at A and C intersect at P. Given angle $AOB={140}^{\circ }$ and angle $APC={80}^{\circ }$; find the angle BAC.

Which of the following is a zero of $(49x^2-1)+(1+7x{)}^2$ ?

Frame the quadrilateral whose solution set is {–1/2, 3/4}

if a : b = 5:3, find 5a + 8b : 6a – 7b.

If $I_n=\int \frac{{\sin }^{2}nx}{{\sin }^{2}x}dx,$ then the value of $I_{n+1}-2I_n+I_{n-1}$ is:

(where $n\in N$ and $C$ is integration constant)

The areas of three rectangular field are 165m²,195m²and 285m²​​. The three fields are divided into some smaller rectangular parts of equal areas. If the breath of each part is 3m, then find the length of the each part.

In given figure, DE||BC, if AB = 7.6 cm, AD = 1.9 cm, then AE : EC is :

As observed from the top of a lighthouse, the angles of depression of two ships are ${30}^{\circ }$ and ${45}^{\circ }$. If one ship is exactly behind the other on the same side of the lighthouse and is 100m apart, find the height of the lighthouse from the sea-level.

Which of the following equations are quadratic?

1. 2

If $\alpha$, $\beta$ are the zeros of the quadratic polynomial $f\left(x\right)=2x^2-5x+7$. find a polynomial whose zeros are $2\alpha +3\beta$ and $3\alpha +2\beta$

A vessel is filled with liquid, $3$ parts of which are water and $5$ parts syrup. How much of the mixture must be drawn off and replaced with water so that the mixture may be half water and half syrup?

1. 30