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If the length of the tangents from P(1, -1) and Q(3, 3) to a circle are $\sqrt{2}\mathrm{and}\sqrt{6}$ then the length of a tangent from R(-1, -5) to the same circle is .

For the three events A, B and C, P(exactly one of the events A or B occurs) = P(exactly one of the events B or C occurs) = P(exactly one of the events C or A occurs) = p and P (all the three events occur simultaneously) $=p^2$, where 0 < p < $\frac{1}{2}$. Then, the probability of at least one of the three events A, B and C occurring is

Question 8
A grouped frequency table with class intervals of equal sizes using 250 – 270 (270 not included in this interval) as one of the class intervals is constructed for the following data.

268, 220, 368, 258, 242, 310, 272, 342

310, 290, 300, 320, 319, 304, 402, 318

406, 292, 354, 278, 210, 240, 330, 316,
406, 215, 258, 236.

The frequency of class 310 - 330 is:

A) 4
B) 5
C) 6
D) 7

In an examination, the ratio of passes to failures was 4 : 1. Had 30 less appeared and 20 less passed, the ratio of passes to failures would have been 5 : 1 Find the number of students who appeared for the examination.

Suppose x candidates passed and y failed;
therefore, $\frac{x}{4}=\frac{4}{1}\dots \left(I\right)$

In the second case: no. of students appeared = x + y - 30 and no. of those who passed = x - 20:

$\therefore$ No. of failed = $x+y-30-(x-20)=y-10$

Given: $\frac{x-20}{y-10}=\frac{5}{1}\dots \left(II\right)$

Plot the points A(2,9), B(-1,3) and C(6,3) on a graph paer.On the same graph paper,draw the locus of point A so that the area of $△ABC$ remains the same as A moves.

One of the solutions of the pair of equations $\frac{a}{x}-\frac{b}{y}=0\text{and}\frac{a{b}^2}{x}+\frac{a^{2}b}{y}=a^2+b^2\text{is:}$

Equation of the circle passing through $(4,3)$ and centre at the origin is _______ .

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?

If $p\ne q$ and the sequences p, a, b, q and p, m, n, q each are in AP, then $\frac{b-a}{n-m}$ is

If x+y=3^1/2 x-y= 2^1/2

Then the expression 8xy(x² + y²) has the value ?

Find the value of a if 3 is one of the solution of the equation $2x^2+ax+6=0$.

The sum of the first 20 terms of an AP is 1280, and the common difference is 20. Then the fourth term of the AP will be __.

The ratio whose terms differ by 40 is 2 : 7. The smallest number is ___.

How many terms of the AP 27,24, 21,..... are required to get the sum as 0?

$△ABC$ is an equilateral triangle with side AC = 6cm. Find the area of the incircle.

Prove that SinA+sinB=sin(A+B)cos(A+B)/2

Question 31

Fill in the blanks to make the statement true.

(-a) + b = b + additive inverse of ___ .

Rohit walks 80 m towards left and then turns right. How much distance should he walk now so that the minimum distance between the initial and final point is 100 m? ___

Question 4
Construct a tangent to a circle of radius 4 cm from a point which is at a distance of 6 cm from its centre of circle.

In the above figure, BC is a tangent at B. A is the centre of the circle. If AB = BC, then find the value of $\angle BAC$

$ta{n}^2\theta si{n}^2\theta$ is equal to

Shobo s mother present age is six times shobo s age . Shobo s age five years from now will be one third of his mother's present age . What are their present age?

The list price of a pair of shoes is Rs. 450. If Amit paid Rs. 45 as dales tax for it, find the rate of sales tax.

Is the roots of the equation x² + 2cx + ab = 0 are real and unequal prove that the equation x² - 2(a + b) + a² + b² + 2c² = 0 has no real roots.

Four numbers in an AP sum up to 32. The ratio of product of first and last term to that of the two middle terms is 7:15. Find the numbers

For any positive integer n , prove that n^3−n divisible by 6

Solve the following inequation and find the cardinality of the solution set.

$\frac{2x+3}{3}\ge \frac{3x-1}{4}$ where x is positive even integer.

The reflection of A (2, 3) in the line y = x is A' (3, 2). Write down the co-ordinates of the image of each of the points reflected in the line y = x
(i) (4, 1)
(ii) (- 2, - 3).

Without using trigonometric tables, prove that:

$\left(i\right)tan{48}^{\circ }tan{23}^{\circ }tan{42}^{\circ }tan{67}^{\circ }=1$

$\left(ii\right)tan{7}^{\circ }tan{23}^{\circ }tan{60}^{\circ }tan{67}^{\circ }tan{83}^{\circ }=\sqrt{3}$

$\left(iii\right)cot{12}^{\circ }cot{38}^{\circ }cot{52}^{\circ }cot{60}^{\circ }cot{78}^{\circ }=\frac{1}{\sqrt{3}}$ [3 MARKS]

Explain how square root of 2 is irrational.

√3 and-√3 are the zeroes of polynomial x^3-4x^2-3x+12. Find the third zero

If a- b, 0 and a + b are consecutive terms of an AP then

Water in stored up to its brim in the right circular cylinder of volume $100c{m}^3$. How many bottles of $10c{m}^3$ will be required to empty the cylinder?

A sphere of diameter $12cm$, is dropped in a right circular cylindrical vessel, partly filled with water. If the sphere is completely submerged in water, the water level in the cylindrical vessel rises by $\frac{32}{9}cm$. The diameter of the cylindrical vessel is _____.

The area of triangle whose vertices are (2,3),(-1,0) and (2,-4), is

If X and Y are two finite sets, then what is n(X) + n(Y)?

Find the numbers such that their mean proportion is 14 and third proportion is 112.

If $f:R\to R$ satisfies $f(x+y)=f\left(x\right)+f\left(y\right)$, for all $x,y\in R$ and $f\left(1\right)=7$ ,then ${\sum }_{r=1}^{n}f\left(r\right)$ is

If the difference of 25th and 20th term of an increasing AP is 45, then what is the value of d?

In the given figure, R is an external point and RP and RQ are tangents drawn to the circle from R. $\angle POQ={90}^{\circ }$. The quadrilateral OPRQ is a ___ and the $\angle$POR is___.