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Solve the following quadratic equation:
$x^2-3\sqrt{5}x+10=0$

PQ and PR are tangents in the given figure. Find PQ (in cm).

Solve each of the following systems of eqautions by the method of cross-multiplication:

$\frac{x}{a}+\frac{y}{b}=a+b$

$\frac{x}{a^2}+\frac{y}{b^2}=2$

3 people A, B and C are sitting in a circular fashion. Find the probability that A and B do not sit together.

What do you understand by the quantity called ‘area’?

A sum of Rs $1000$ is invested at $12\%$ simple interest per year. Find the sum of all the interest generated every year for a period of $30$ years.

Fill in the blank with a modal auxiliary indicating duty.
You _______ to serve the country like your father did years ago.

Given a cube of mass 1 kg and its volume is 500 ml. Find the density of the cube.

If the height and radius of a cone of volume $V$ are doubled, then the volume of the cone will be _____.

A point P(-2, 3) is reflected in the line x = 2 to point P'. Find the coordinates of P'.

Find the roots of the following quadratic equation by the method of completing the square.
$4x^2+4\sqrt{3}x+3=0$

On which axis do the following points lie?

(i) P (5,0) (ii) Q(0 -2) (iii) R(-4,0) (iv) S (0,5)

If the probability of the occurrence of a certain event E is $\frac{5}{13}$ , then the odds against its occurrence is

Prove the following identities:

$\frac{tanA}{(1+ta{n}^{2}A{)}^2}+\frac{cotA}{(1+co{t}^{2}A{)}^2}=sinAcosA$

A (5, 4), B (-3, -2) and C (1, -8) are the vertices of a triangle ABC. Find :

(i) the slope of the altitude of AB,

(ii) the slope of the median AD and

(iii) the slope of the line parallel to AC.

Construct a triangle of sides 4 cm, 5cm and 6cm and then a triangle similar to it whose sides areof the corresponding sides of the first triangle.

Give the justification of the construction.

A wall of thickness $10cm$ and height $5m$ has to be built across a play ground of length $15m$. How many bricks are needed to construct the wall, if each brick has dimensions of $20cm\times 10cm\times 5cm.$?
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P (-3, 2) is the mid-point of line segment AB as shown in the given figure. Find the co-ordinates of points A and B.

In the figure below, ABCD is a square. Find the coordinates of B, C, D.

In a throw of a die, the probability of getting a prime number is

A (-1,0), B(1,3) and D(3,5) are the vertices of a parallelogram ABCD.Find the co-ordinates of vertex C.

Dividing the natural numbers by 2, we get Is this an arithmetic sequence?

(1) If ‘m’ and ‘n’ are the roots of the equation x² − 6x + 2 = 0 find the value of

(i) (m + n) mn

(ii)

(2) If ‘a’ and ‘b’ are the roots of the equation 3m² = 6m + 5, find the value of

(i)

(ii) (a + 2b) (2a + b)

(3) If ‘p’ and ‘q’ are the roots of the equations 2a² − 4a + 1 = 0, find the value of

(i) (p + q)² + 4pq

(ii) p³ + q³

(4) From the quadratic equation whose roots are

(5) Find the value of ‘k’ so that the equation x² + 4x + (k + 2) = 0 has one root equal to zero.

(6) Find the value of ‘q’ so that the equation 2x² − 3qx = 5q = 0 has one root which is twice the other.

(7) Find the value of ‘p’ so that the equation 4x² − 8px + 9 = 0 has roots whose difference is 4.

(8) If one root of the equations x² + px + q = 0 is 3 times the other prove that 3p² = 16q

If the sides of two similar triangles are in the ratio of 4 : 9, then the areas of these triangles are in the ratio ____.

The missing terms in the arithmetic sequence __,__, 5,8,11,__,__,20 are

I purchased a hair dryer for $Rs.5400$ including $8\%$ VAT. find the price before VAT was added.

Find out which of the following sequences are arithmetic progressions. For those which ar3e arithmetic progressions, find out the common difference.
(i) $3,6,12,24\dots$
(ii) $0,-4,-8,-12\dots$
(iii) $\frac{1}{2},\frac{1}{4},\frac{1}{6}.\frac{1}{8},\dots$
(iv) $12,2,-8,-18,\dots$
(v) $3,3,33,3,\dots$
(vi) $p,p+90,p+180,p+270,\dots$ where $p=(999{)}^{999}$
(vii) $1.0,1.7,2.4,3.1,\dots$
(viii) $-225,-425,-625,-825,\dots$
(ix) $10+10+2^5,10+2^6,10+2^7,\dots$
(x) $a+b,(a+1)+b,(a+1)+(b+1),(a+2)+(b+1),(a+2)+(b+2),\dots$
(xi) $1^2,3^2,5^2,7^2,\dots$
(xii) $1^2,5^2,7^2,7^3,\dots$

ABC is an isosceles triangle and AC, BC are the tangents at M and N respectively. DE is the diameter of the circle. $\angle$ADP = $\angle$BEQ = ${100}^{\circ }$. What is value of $\angle$PRD?