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Find that sum of first $30$ integer divisible by $6$.

On a graph paper plot the points A(-4,-4), B(4,-4), C(2,2)and D(-2,2). Then the point of intersection of the locus of a point equidistant from A and B and the locus of a point equidistant from A and D lies in the

Find the volume of a hemisphere whose radius is 3 inches.​

Solve the following quadratic equation by factorization.
$3x^2-2\sqrt{6x}+2=0$

The perimeter of a rectangle is 240 cm. If its length is decreased by 10% and its breadth is increased by 20% ,we get the AZ same perimetre. Find the length,breadth of the rectangle.

If A = {2, 3, 5, 6}, B = {4, 8, 15, 17} a R b $\Rightarrow$ a divides b. Find Relation R.

The locus of the position of a person at the same distance from a tower is a _____.

Who is standing second to the right of K?

In the given figure AB and DC are two chords of a circle with centre O, which when produced meet at P. If PB = 7 cm, BA = 10 cm and PO =12cm, find the radius of the circle.

Find the value of x which will satisfy the equation:

$xcos\left(a\right)cos({90}^{\circ }-a)tan\left(a\right)tan({90}^{\circ }-a)sec\left(a\right)cosec\left(a\right)=1$

In a college, $30$ students fail in Physics, $25\%$ fail in Mathematics and $10$ fail in both. One student is chosen at random. The probability that she fails in Physics, if she failed in Mathematics is

(a) $\frac{1}{10}$
(b) $\frac{2}{5}$
(c) $\frac{9}{20}$
(d) $\frac{1}{3}$

Determine $k$, so that $k^2+4k+8,2k^2+3k+6$ and $3k^2+4k+4$ are three consecutive terms of an AP.

In the given figure, QR is parallel to AB and DR is parallel to QB.
Prove that : $P{Q}^2=PD\times PA$.

In the figure below, is any side of ΔABC parallel to X axis? If so, the side is ______

Haneef is processing sugarcane juice to make molasses, which is poured into moulds in the shape of a frustum of a cone having the diameters of the two circular faces as 30 cm and 35 cm and the vertical height of the mould is 14 cm. If each cm³ of molasses has a mass of about 1.2 g then the mass of the molasses that can be poured into each mould will be about

If (2x + iy) + (y - 3ix) = 12 - 3i, then X = _____ y = _____

The string of a kite is 100m long and it makes an angle of ${60}^{\circ }$ with the horizontal. Find the height of the kite from the ground, assuming that there is no slack in the string.

Look at the following figure. Start by finding the value for x₁, then for x₂, then x₃, and so on..Then the value for x₆ is

When 0^o< θ< 90^o, and 2cosθ=1 the value of θ is

Chords AB & CD (not the diameters) of a circle intersect at point P inside the circle.

Find the sum of the angles subtended at the centre of the circle by the arcs AC & BD in terms of ∠APC.

Simplify
$\frac{1}{x+p}+\frac{1}{x+q}+\frac{1}{x+r}+\frac{px}{x^3+p{x}^2}+\frac{qx}{x^3+p{x}^2}+\frac{rx}{x^3+rx}$

Kim has a long pole of length 2m which makes an angle of ${20}^{\circ }$ with vertical. Kim takes a rope of length 5m and ties it to the pole. Now Kim moves along a circle so that the angle between initial and final rope positions is ${22}^{\circ }$. What is the radius and angle subtended by the arc formed by Kim’s movement?

From a solid cylinder whose height is 8 cm and radius 6 cm, a conical cavity of height 8 cm and of base radius 6 cm is hollowed out. Find the volume of the remaining solid. Also, find the total surface area of the remaining solid. $[Take\pi =\mathrm{3.14.}]$

Two circular pieces of equal radii and maximum area, touching each other are cut out from a rectangular card board of dimensions $14cm\times 7cm$. Find the area of the remaining card board. (Use $\pi =\frac{22}{7}$)

sin (A + B) = 1, cos (A – B) = 1, $0^{\circ }<A+B\le {90}^{\circ }$, Find A and B.

Prove the following trigonometric identities:

$\frac{1-cos\theta }{sin\theta }=\frac{sin\theta }{1+cos\theta }$

If 2 cos 3$\theta$ = 1 then $\theta$ = ?

(a) ${10}^o$ (b) ${15}^o$ (c) ${20}^o$ (d) ${30}^o$

The wheel of a motor cycle is of radius 35 cm. How many revolutions per minute must the wheel make so as to keep a speed of 66 km/hr?

The sum of $4th$ and $8th$ terms of an A.P. is $24$ and the sum of the $6th$ and $10th$ terms is $34.$ Find the first term and the common difference of the A.P.

Write the first five terms of each of the following sequences whose nth terms are:

(i) $a_n=3n+2$
(ii) $a_n=\frac{n-2}{3}$
(iii) $a_n=3^n$
(iv) $a_n=\frac{3n-2}{5}$
(v) $a_n=(-1{)}^n,2^n$
(vi) $a_n=\frac{n(n-2)}{2}$
(vii) $a_n=n^2-n+1$
(viii) $a_n=-2n^2-3n+2$
(ix) $a_n=\frac{2n-3}{6}$

The sum of first n terms of an A.P. whose first term is 8 and the common difference is 20 is equal to the sum of first 2n terms of another A.P. Whose first term is -30 and common difference is 8. Fnd n.

The cost of fencing a square lawn at Rs.14 per metre is Rs.28000. Find the cost of moving the lawn at Rs.54 per 100 $m^2$.

From the top of a tower h metre high, the angles of depression of two objects, which are in the line with the foot of the tower are $\alpha$ and $\beta$ $(\beta >\alpha )$. Find the distance between the two objects.

Question 2
The following data gives the information on the observed lifetimes (in hours) of 225 electrical components:

$\begin{array}{ccccccc}\text{Lifetimes(in hours)}& 0-20& 20-40& 40-60& 60-80& 80-100& 100-120\\ \text{Frequency}& 10& 35& 52& 61& 38& 29\end{array}$

Determine the modal lifetimes of the components.

Using the method of completing the square, find the roots of $4m^2+4\surd 3m+3=0$.

A shopkeeper sells an article at the listed price of Rs 1500 & the rate of VAT is 12% at each stage of the sale.

If the shopkeeper pays a VAT of Rs 36 to the government, what was the price inclusive of tax at which the shopkeeper purchased the article from the wholesaler?

$\Delta ABC$ is right-angled at A and $AD\perp BC$. If BC = 13 cm and AC = 5 cm, find the ratio of the areas of $\Delta ABCand\Delta ADC$.

If Market Value (MV) ___ Face Value (FV) then the share is at discount.

Find the ${31}^{st}$ term of an A. P. whose ${10}^{th}$ term is $38$ and ${16}^{th}$ term is $74$.

In the given figure, PA and PB are two tangents drawn from an external point P to a circle with centre C and radius 4 cm. If PA $\perp$ PB, then the length of each tangent is

(a) 3 cm (b) 4 cm (c) 5 cm (d) 6 cm

A solid consisting of a right circular cone of height $120cm$ and radius $60cm$ standing on a hemisphere of radius $60cm$ is placed upright in a right circular cylinder full of water such that it touches the bottoms. Find the volume of water left in the cylinder, if the radius of the cylinder is $60cm$ and its height is $180cm$.