If f(x) = 2x² and g(x) = \(\frac{1}{3x}\), then f o g is

Let f(x) = \(\sqrt { 1+x^{ 2 } } \) then

If f : R⟶R is defined by (x) = x² + 2, then the preimage 27 are _________

If function f : N⟶N, f(x) = 2x then the function is, then the function is ___________

If f(x) = 2 - 3x, then f o f(1 - x) = ?

If f is constant function of value \(\frac { 1 }{ 10 } \), the value of f(1) + f(2) + ... + f(100) is _________

An A.P. consists of 31 terms. If its 16^th term is m, then the sum of all the terms of this A.P. is

If the sequence t₁, t₂, t₃... are in A.P. then the sequence t₆, t₁₂, t₁₈,.... is 

What is the HCF of the least prime and the least composite number?

In an A.P if the p^th term is q and the q^th term is p, then its n^th term is ____________

\(\frac {3y - 3}{y} \div \frac {7y - 7}{3y^{2}}\) is

If number of columns and rows are not equal in a matrix then it is said to be a

Which of the following is correct
(i) Every polynomial has finite number of multiples
(ii) LCM of two polynomials of degree 2 may be a constant
(iii) HCF of 2 polynomials may be constant
(iv) Degree of HCF of two polynomials is always less then degree of LCM

Consider the following statements:
(i) The HCF of x+y and x⁸-y⁸ is x+y
(ii) The HCF of x+y and x⁸+y⁸ is x+y
(iii) The HCF of x-y nd x⁸+y⁸ is x-y
(iv) The HCF of x-y and x⁸-y⁸ is x-y

The real roots of the quadratic equation x²-x-1 are ___________

The perimeters of two similar triangles ∆ABC and ∆PQR are 36 cm and 24 cm respectively. If PQ = 10 cm, then the length of AB is

A tangent is perpendicular to the radius at the

S and T are points on sides PQ and PR respectively of \(\Delta PQR\) If PS = 3cm, AQ = 6 cm, PT = 5 cm, and TR = 10 cm and then QR

The height of an equilateral triangle of side a is

A line which intersects a circle at two distinct points is called ____________

If (5, 7), (3, p) and (6, 6) are collinear, then the value of p is

If A is a point on the Y axis whose ordinate is 8 and B is a point on the X axis whose abscissae is 5 then the equation of the line AB is

If the mid-point of the line segment joining \(A\left( \frac { x }{ 2 } ,\frac { y+1 }{ 2 } \right) \) and B(x + 1, y-3) is C(5, -2) then find the values of x, y ____________

The lines y = 5x - 3, y = 2x + 9 intersect at A. The coordinates of A are ___________

If x = a tan\(\theta \) and y = b sec\(\theta \) then

(1 + tan \(\theta \) + sec\(\theta \)) (1 + cot\(\theta \) - cosec\(\theta \)) is equal to 

(cosec²θ - cot²θ) (1 - cos²θ) is equal to ___________

If x = r sin θ cos φ y = r sin θ. Then x² + y² + z²___________

If sin(α + β) = 1 then cos(α - β) can be reduced to ___________

A shuttle cock used for playing badminton has the shape of the combination of

The volume (in cm³) of the greatest sphere that can be cut off from a cylindrical log of wood of base radius 1 cm and height 5 cm is

How many balls, each of radius 1 cm, can be made from a solid sphere of lead of radius cm?

The radius of a wire is decreased to one-third of the original. If volume the same, then the length will be increased _______of the original.

A solid frustum is of height 8 cm. If the radii of its lower and upper ends are 3 cm and 9 cm respectively, then its slant height is ___________

A floating boat having a length 3m and breadth 2m is floating on a lake. The boat sinks by 1 cm when a man gets into it. The mass of the man is (density of water is 10000 kg/m³)

The sum of all deviations of the data from its mean is

The probability a red marble selected at random from a jar containing p red, q blue and r green marbles is

Th4e batsman A is more consistent than batsman B if ___________

A letter is selected at random from the the word 'PROBABILITY'. The probability that its is nota vowel is _______.

In one thousand lottery tickets, there are 50 prizes to be given. The probability of happenning of the event is ___________

Let A = {1,2,3}, B = {4, 5, 6,7}, and f = {(1, 4),(2, 5),(3, 6)}  be a function from A to B. Show that f is one – one but not onto function.

Using the functions f and g given below, find f o g and g o f. Check whether f o g = g o f
 f(x) = \(\frac { x+6 }{ 3 } \), g(x) = 3 - x

State whether the graph represent a function. Use vertical line test.

Let A = {0, 1, 2, 3} and B = {1, 3, 5, 7, 9} be two sets. Let f: A \(\rightarrow\)B be a function given by  f(x) = 2x + 1. Represent this function as  a table.

Let A = {0, 1, 2, 3} and B = {1, 3, 5, 7, 9} be two sets. Let f: A \(\rightarrow\)B be a function given by  f(x) = 2x + 1. Represent this function as a graph.

Find the number of terms in the following G.P.
4, 8,16,…,8192

Check whether the following sequences are in A.P. or not?
 \(3\sqrt { 2 } ,5\sqrt { 2 } ,7\sqrt { 2 } ,9\sqrt { 2 } \),.....

Solve x² + 2x - 2 = 0 by formula method

Determine the nature of the roots for the following quadratic equations
\(9{ y }^{ 2 }-6\sqrt { 2 } y+2\) = 0

The perimeters of two similar triangles ABC and PQR are respectively 36 cm and 24 cm. If PQ = 10 cm, find AB

In the Figure, AD is the bisector of \(\angle\)BAC, if A = 10 cm, AC = 14 cm and BC = 6 cm. Find BD and DC.

The line r passes through the points (–2, 2) and (5, 8) and the line s passes through the points (–8, 7) and (–2, 0). Is the line r perpendicular to s ?

Find the slope of the line which is parallel to 3x - 7y = 11

Find the equation of a straight line passing through the mid-point of a line segment joining the points (1, -5), (4, 2) and parallel to: Y axis

prove that sec\(\theta \) - cos\(\theta \) = tan \(\theta \) sin\(\theta \) 

prove the following identity.
\(\frac { cos\theta }{ 1+sin\theta } \) = sec \(\theta \) - tan \(\theta \)

A sphere, a cylinder and a cone  are of the same radius, where as cone and cylinder are of same height. Find the ratio of their curved surface areas.

An aluminium sphere of radius 12 cm is melted to make a cylinder of radius 8 cm. Find the height of the cylinder.

If the range and the smallest value of a set of data are 36.8 and 13.4 respectively, then find the largest value.

The standard deviation and coefficient of variation of a data are 1.2 and 25.6 respectively. Find the value of mean.

Let f = {(2, 7); (3, 4), (7, 9), (-1, 6), (0, 2), (5,3)} be a function from A = {-1,0, 2, 3, 5, 7} to B = {2, 3, 4, 6, 7, 9}. Is this
(i) an one-one function
(ii) an onto function,
(iii) both one and onto function?

Let A = {1, 2, 3, 4, 5}, B = N and f: A \(\rightarrow\)B be defined by f(x) = x². Find the range of f. Identify the type of function.

Which of the following list of numbers form an AP? If they form an AP, write the next two terms:
1,-1,-3, -5, ...

Given A = \(\left( \begin{matrix} p & 0 \\ 0 & 2 \end{matrix} \right) \), B = \(\left( \begin{matrix} 0 & -q \\ 1 & 0 \end{matrix} \right) \), C = \(\left( \begin{matrix} 2 & -2 \\ 2 & 2 \end{matrix} \right) \) and if BA = C², find p and q.

If 9x⁴ + 12x³ + 28x² + ax + b is a perfect square, find the values of a and b.

Find the square root of the following polynomials by division method
37x² - 28x³ + 4x⁴ + 42x + 9

A chess board contains 64 equal squares and the area of each square is 6.25 cm², A border round the board is 2 cm wide.

O is any point inside a triangle ABC. The bisector of \(\angle AOB\), \(\angle BOC\) and \(\angle COA\) meet the sides AB, BC and CA in point D, E and F respectively. Show that AD x BE x CF = DB x EC x FA

The area of a triangle is 5 sq. units. Two of its vertices are (2,1) and (3, –2). The third vertex is (x, y) where y = x + 3. Find the coordinates of the third vertex.

To a man standing outside his house, the angles of elevation of the top and bottom of a window are 60° and 45° respectively. If the height of the man is 180 cm and if he is 5 m away from the wall, what is the height of the window?(\( \sqrt { 3 } \) = 1.732)

The top of a 15 m high tower makes an angle of elevation of 60° with the bottom of an electronic pole and angle of elevation of 30° with the top of the pole. What is the height of the electric pole?

Two ships are sailing in the sea on either side of the lighthouse. The angles of depression of two ships as observed from the top of the lighthouse are 60° and 45° respectively. If the distance between the ships is 200\(\left( \frac { \sqrt { 3 } +1 }{ \sqrt { 3 } } \right) \) metres, find the height of the lighthouse.

If tanθ+sinθ=P; tanθ-sinθ=q P.T P²-q²=4\(\sqrt{pq}\)

The internal and external diameters of a hollow hemispherical vessel are 20 cm and 28 cm respectively. Find the cost to paint the vessel all over at Rs. 0.14 per cm².

A conical container is fully filled with petrol. The radius is 10 m and the height is 15 m. If the container can release the petrol through its bottom at the rate of 25 cu. metre per minute, in how many minutes the container will be emptied. Round off your answer to the nearest minute.

A vessel is in the form of a hemispherical bowl mounted by a hollow cylinder. The diameter is 14 cm and the height of the vessel is 13 cm. Find the capacity of the vessel.

Find the standard deviation of the data 2, 3, 5, 7, 8. Multiply each data by 4. Find the standard deviation of the new values.

48 students were asked to write the total number of hours per week they spent on watching television. With this information find the standard deviation of hours spent for watching television.

A and B are two candidates seeking admission to IIT. The probability that A getting selected is 0.5 and the probability that both A and B getting selected is 0.3. Prove that the probability of B being selected is allmost 0.8.

If f(x) = \(\frac { x-1 }{ x+1 } \), x ≠ 1 show that f(f(x)) = -\(\frac{1}{x}\), provided x ≠ 0.

How many terms of the AP: 24, 21, 18, ... must be taken so that their sum is 78?

Find two consecutive natural numbers whose product is 20.

Construct a triangle similar to a given triangle PQR with its sides equal to \(\frac { 2 }{ 3 } \) of the corresponding sides of the triangle PQR (scale factor \(\frac { 2 }{ 3 } <1\)).

Draw a circle of radius 4.5 cm. Take a point on the circle. Draw the tangent at that point using the alternate segment theorem.

Find the equation of a straight line Passing through (1, -4) and has intercepts which are in the ratio 2:5

Find the value of k if the points A(2, 3), B(4, k) and (6, -3) are collinear.

Express cot 85° + cos 75° in terms of trigonometric ratios of angles between 0° and 45°.

A spherical ball of iron has been melted and made into small balls. If the radius of each smaller ball is one-fourth of the radius of the original one, how many such balls can be made?

Final the probability of choosing a spade or a heart card from a deck of cards.