If A = {x : x is a multiple of 3} and B = {x : x is a multiple of 5} then A - B is _____.

If f(x) =\({x+1\over x-1}\) is a real function,\(\neq\) 1, then \(f[f\{f(2)\}]\) is ______.

If A, B, C are in A.P. then \({sinA-sinC\over cosC-cosA}\) is equal to ______.

Find the number of chord that can be drawn through 16 points on a circle

If in the expansion of (a + b)ⁿ and (a + b)^{n + 3}, the ratio of the coefficients of second and third terms and third and fourth terms respectively are equal then n is _____.

The sum of infinity of the G.P. a, ar, ar², ar³, ...... \(\infty\) is ______.

The vertices of a triangle are (6, 0), (0, 6) and (6, 6). The distance between its circumcentre and centriod is ______.

The circle x²+y²+2gx+2fy+c = 0 does not intersect x − axis if ______.

The ratio in which the line joining (4, -3, 2) and (6, -5, -1) is divided by YZ-plane is _______.

\(\overset{lim}{x\rightarrow 0} \frac{sin2x}{x}\) is _______.

Six boys and six girls sit in a row. The probability that all girls sit together is _______.

If A = {x : x = n² , n = 1,2,3}, then find the number of proper subsets.

Let f(x)=x² and g(x)=2x+1 be two real functions.Find  \(\left(\frac{f}{g}\right)(x)\)

Find the radian measures corresponding to the following degree measures:-47^o30'

Prove that \(\sum _{ t=1 }^{ n-1 }{ t(t+1) } =\frac { n(n-1)(n+1) }{ 3 } \) , for all natural numbers \(n\ge 2\)

Express the complex number in the form a+ib: (1-i)⁴

Find all pairs of consecutive even positive integers both of which are larger than 10 such that their sum is less than 28.

In how many ways can 5 girls and 3 boys be seated in a row so that no two boys are together?

Using binomial theorem, expand the following expansions.
\((1+x+{ x }^{ 2 })^{ 3 }\)

If x,y,z are positive integers, then prove that (x + y)(y + z)(z + x) > 8xyz.

Determine \(\angle B\) of the triangle with vertices A(-2,1), B(2,3) and C(-2,-4).

Find the equation of the ellipse whose major axis on x-axis, passing through points (4, 3) and (6,2).

Find the equation of the set of points which are equidistant from the points (1, 2, 3) and (3, 2, –1).

Evaluate the following limits \(\lim _{z \rightarrow 1} \frac{z^{\frac{1}{3}}-1}{z^{\frac{1}{6}}-1}\)

Write down the contrapositive of the following statements.
If PT Usha stands first in 200 m race, then she will be awarded.

Find the mean deviation about the mean of the following data:

Two dice are rolled. Let E_1, E_2, and E₃ be the events of getting a sum of 4, 5 and respectively.
Which events are elementary events?

Write the following intervals in set-builder from
[-23,5)

Let \(\lambda\) be a non-zero number and f : R \(\rightarrow\) R be a function defined by \(f(x)=\frac { x }{ \lambda } \forall x\epsilon R\). Find\(\left( \frac { 1 }{ \lambda } \right) f\)

In a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length of minor arc of the chord.

Prove that \(1+2+3......n<\frac { 1 }{ 8 } { (2n+1) }^{ 2 }\) , for all natural numbers n.

If z₁ = 3+2i and z₂ = 2-4i, then verify that \({ \left| { z }_{ 1 }+{ z }_{ 2 } \right| }^{ 2 }+ { \left| { z }_{ 1 }-{ z }_{ 2 } \right| }^{ 2 }=2({ \left| { z }_{ 1 } \right| }^{ 2 }+{ \left| { z }_{ 2 } \right| }^{ 2 })\)

Solve 5x - 3 < 7 when
(i) x is an integer,
(ii) x is a real number

In a monthly test, the teacher decides that there will be three questions, one from each of the Exercise 7, 8 and 9 of textbook. If there are 11 questions in Exercise 7, 15 in Exercise 8 and 10 in Exercise 9, in how many ways can three questions be selected ?

Draw the shape of the hyperbola \(\frac { { y }^{ 2 } }{ 9 } -\frac { { x }^{ 2 } }{ 27 } =1\) and find their centre, transverse axis, conjugate axis, value of c, vertices, directrices, foci,eccentricity and latusrectum. 

Write first five terms of sequence
(ii)   a_n = 2n²- n + 1

Find the distance between following parallel lines.
l (x + y) + h = 0 and l (x + y) - r = 0

Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
16x² +y² = 16

Find the values of x, if the distance between two points (x,-8,4) and (3,-5,4) is 5

Evaluate the following limits  \(\lim _{x \rightarrow 0} \frac{\log \left(1+x^{3}\right)}{\sin ^{3} x}\)

If p and q are two statements given by 
p : 25 is a multiple of 5.  
q : 25 is a multiple of 8.
Write the compound statement by using the connective 'and' and check its validity.

Calculate the mean deviation about median for the following data.

A die is thrown 500 times with frequencies for the outcomes 1,2,3,4,5 and 6 as given in the following table

Find the probability of getting outcome 3?

If X = {a,b,c,d} and Y = {f,b,d,g}, then find 
Y - X

Determine the domain and range of the relation R, where R = {(2x+3, x³)} : x is a prime number less than 10}.

In any ΔABC,prove that \({ a }^{ 3 }sin(B-C)+{ b }^{ 3 }sin(C-A)+{ C }^{ 3 }sin(A-B)=0\)

Prove by the principle of mathematical induction that, for all \(n\in N\),4ⁿwhen divided by 3, the remainder is always 1.

If z = x + iy, w = \(\frac { 1-iz }{ z-i } \) and \(|w|=1\) then show that z is purely real.

Solve the following system of inequalities.
\(\frac { 2x+1 }{ 7x-1 } \)>5, \(\frac { x+7 }{ x-8 } \)>2

If ²²P_r+1: ²⁰P_r+2 = 11:52, find r.

If the coefficients of 2nd, 3rd and 4th terms in the expansion of (1+x)ⁿ are in AP, then find the value of n.

If x, 2y and 3z are in AP, where the distinct numbers x, y, z are in GP, then find the common ratio of the GP.

Which of the lines 2x - y + 3 = 0 and ax - 4y - 7 = 0, is farther from the origin?

The cable of a uniformly loaded suspension bridge hangs in the form of a parabola. The roadway which is horizontal and 100 m long is supported by vertical wires attached to the cable, the longest wire being 30 m and the shortest being 6 m. Find the length of a supporting wire attached to the roadway 18 m from the middle.

Find the coordinates of centroid of \(\Delta ABC\) , where vertices are A(x₁, y₁, z₁), B(x₂, y₂, z₂) and C(x₃, y₃, z₃).

Evaluate sin ( x + 1)

By giving a counter example, shows that the following statement is not true.
p: If all the angles of a triangle are equal, than the triangle is an obtuse angled triangle.

Calculate mean and variance of the given data.

One card is drawn from a well-shuffled deck of 52 cards.Calculate the probability that the card will be
a black card

Find the domain of the function f given by f(x)=\(1\over \sqrt{[x]^2-[x]-6}\)

Prove the following: tan x = \({4\tan x(1-\tan^2 \ x)\over 1-6\ \tan^2 \ x+\tan^4 \ x}\)

Prove the following by using the principle of mathematical induction for all n ∊ N:a+ar+ar2+.......+ar^n-1 =\(\frac { a({ r }^{ n }-1) }{ r-1 } \)

Show that a real value of x will satisfy the equation \(\frac{1-ix}{1+ix}=a-ib\) if a + b² = 1where a and b are real.

Solve the inequalities graphically x+y \(\ge\) 4, 2x - y > 0

Find if  \(^{ 5 }P_{ r }=^{ 6 }P_{ r-1 }\)

Find the value of \(\left( { a }^{ 2 }+\sqrt { { a }^{ 2 }-1 } \right) ^{ 4 }+({ a }^{ 2 }-\sqrt { { a }^{ 2 }-1)^{ 4 } } \)

Show that \(\frac { 1\times { 2 }^{ 2 }+2\times { 3 }^{ 2 }+...+n\times (n+1)^{ 2 } }{ { 1 }^{ 2 }\times 2+{ 2 }^{ 2 }3+...+{ n }^{ 2 }\times (n+1) } =\frac { 3n+5 }{ 3n+1 } \).

Find the equation of the line passing through the point of intersection of the lines x - 3y + 1 = 0 and 2x + 5y - 9 = 0 and whose distance from the origin is √5.

A rod of length 12cm moves with its ends always touching the coordinate axes. Determine the equation of the locus of a point P on the rod, which is 3 cm from the end in contact with the x-axis.

Find the lengths of the medians of the triangle with vertices A (0, 0, 6), B (0,4, 0) and (6, 0, 0).

Find the derivative of \(\frac { { (px }^{ 2 }+qx+r) }{ ax+b } \) (it is to be understood that a, b, c, d, p, q, rand s are fixed non-zero constants and m and n are integers)

Give three examples of sentences which are not statements. Give reasons for the answers.

Calculate the mean deviation about median age for the age distribution of 100 persons gives below:

[Hint Convert the given data into continuous frequency distribution by subtracting 0.5 from the lower limit and adding 0.5 to the upper limit of each class interval]

Refer to question 6 above, state true or false (give reason for your answer)
(i) A and B are mutually exclusive
(ii) A and B are mutually exclusive and exhaustive
(iii) A = B'
(iv) A and C are mutually exclusive
(v) A and B' are mutually exclusive
(vi) A', B', C are mutually exclusive and exhaustive.