Let n(A) = m and n(B) = n then the total number of non-empty relations that can be defined from A to B is

If g = {(1,1), (2,3), (3,5), (4,7)} is a function given by g(x) = αx + β then the values of α and β are

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

If the order pairs (a, -1) and (5, b) blongs to {(x, y) | y = 2x + 3}, then a and b are __________

The function t which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined Fahrenheit degree is 95, then the value of  C \(t(C)=\frac { 9c }{ 5 } +32\) is ___________

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

The least number that is divisible by all the numbers from 1 to 10 (both inclusive) is

If 6 times of 6^th term of an A.P. is equal to 7 times the 7^th term, then the 13^th term of the A.P. is

The difference between the remainders when 6002 and 601 are divided by 6 is ____________

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

The solution of (2x - 1)² = 9 is equal to

For the given matrix A = \(\left( \begin{matrix} 1 \\ 2 \\ 9 \end{matrix}\begin{matrix} 3 \\ 4 \\ 11 \end{matrix}\begin{matrix} 5 \\ 6 \\ 13 \end{matrix}\begin{matrix} 7 \\ 8 \\ 15 \end{matrix} \right) \) the order of the matrix A^T is

Which of the following is correct
(i) Every polynomial has finite number of multiples
(ii) LCM of two polynimials 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

If \(\frac { p }{ q } =a\) then \(\frac { { p }^{ 2 }+{ q }^{ 2 } }{ { p }^{ 2 }-{ q }^{ 2 } } \) ___________

The product of the sum and product of roots of equation (a²-b²)x²-(a+b)²x+(a³-b³) = 0 is ___________

In a given figure ST || QR, PS = 2 cm and SQ = 3 cm. Then the ratio of the area of \(\triangle\)PQR to the area \(\triangle\)PST is 

A tangent is perpendicular to the radius at the

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

Two concentric circles if radii a and b where a>b are given. The length of the chord of the circle which touches the smaller circle is ____________

Three circles are drawn with the vertices of a triangle as centres such that each circle touches the other two if the sides of the triangle are 2cm,3cm and 4 cm. find the diameter of the smallest circle.

The slope of the line joining (12, 3), (4, a) is \(\frac 18\). The value of ‘a’ is

A straight line has equation 8y = 4x + 21. Which of the following is true

Find the value of 'a' if the lines 7y = ax + 4 and 2y = 3 - x are parallel

Find the equation of the line passing through the point (0, 4) and is parallel to 3x+5y+15 = 0 the line is ___________

a cot \(\theta \) + b cosec\(\theta \) = p and b cot \(\theta \) + a cosec\(\theta \) = q then p²- q² is equal to 

The angle of elevation of a cloud from a point h metres above a lake is \(\beta \). The angle of depression of its reflection in the lake is 45°. The height of location of the cloud from the lake is

The value of (tan1^o tan2^o tan3^o..... tan89^o) is ___________

The maximum value of sin θ is ___________

If a sin (90 - θ) of (90^o - θ) = cos(90^o - θ)tan equal to ___________

In a hollow cylinder, the sum of the external and internal radii is 14 cm and the width is 4 cm. If its height is 20 cm, the volume of the material in it is

A spherical ball of radius r₁ units is melted to make 8 new identical balls each of radius r₂ units. Then r₁:r₂ is

The radio of base of a one 5 cm and to height 12 cm. The slant height of the cone ___________

The volume of a frustum if a cone of height L and ends-radio and r₁ and r₂ is ___________

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 range of the data 8, 8, 8, 8, 8. . . 8 is

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

If the observations 1, 2, 3, ... 50 have the variance V₁ and the observations 51, 52, 53, ... 100 have the variance V₂ then \(\frac { { V }_{ 1 } }{ { V }_{ 2 } } \) is ___________

In a competition containing two events A and B, the probability of winning the events A and B are \(\frac { 1 }{ 3 } \) and \(\frac { 1 }{ 4 } \) respectively and the probability if winning both events is ___________

When three coins are tossed, the probability of getting the same face on all the three coins is ___________

Using horizontal line test (Fig.1.35(a), 1.35(b), 1.35(c)), determine which of the following functions are one – one.

Find k if f o f(k) = 5 where f(k) = 2k - 1.

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 set of ordered pairs.

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 an arrow .

Find the n^th term of the following sequences,
0,\(\frac { 1 }{ 2 } ,\frac { 2 }{ 3 } \),..

Which of the following sequences form a Geometric Progression?
\(\frac { 1 }{ 2 } \), 1, 2, 4,....

Solve the following quadratic equations by factorization method\(\sqrt { 2 } { x }^{ 2 }+7x+5\sqrt { 2 } =0\)

In the matrix A = \(\left[ \begin{matrix} 8 \\ -1 \\ \begin{matrix} 1 \\ 6 \end{matrix} \end{matrix}\begin{matrix} 9 \\ \sqrt { 7 } \\ \begin{matrix} 4 \\ 8 \end{matrix} \end{matrix}\begin{matrix} 4 \\ \frac { \sqrt { 3 } }{ 2 } \\ \begin{matrix} 3 \\ -11 \end{matrix} \end{matrix}\begin{matrix} 3 \\ 5 \\ \begin{matrix} 0 \\ 1 \end{matrix} \end{matrix} \right] \), write The order of the matrix

Observe Fig and find \(\angle\)P

D is the mid point of side BC and AE \(\bot \) BC. If BC = a, AC = b, AB = c, ED = x, AD = p and AE = h, prove that
\({ c }^{ 2 }={ p }^{ 2 }-ax+\frac { { a }^{ 2 } }{ 4 } \)

A line makes positive intercepts on coordinate axes whose sum is 7 and it passes through (-3,8). Find its equation

Find the equation of a straight line passing through the point P(-5, 2) and parallel to the line joining the points Q(3, -2) and R(-5, 4).

Find the intercepts made by the following lines on the coordinate axes. 4x + 3y + 12 = 0

Find the angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of a tower of height \(10\sqrt { 3 } m\)

A road is flanked on either side by continuous rows of houses of height \( 4\sqrt { 3 } \)m with no space in between them. A pedestrian is standing on the median of the road facing a row house. The angle of elevation from the pedestrian to the top of the house is 30°. Find the width of the road.

A cylindrical drum has a height of 20 cm and base radius of 14 cm. Find its curved surface area and the total surface area.

If the total surface area of a cone of radius 7cm is 704 cm², then find its slant height.

Find the standard deviation of first 21 natural numbers.

Two coins are tossed together. What is the probability of getting different faces on the coins?

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 oneone and onto function?

A functionf: [-7,6) \(\rightarrow\) R is defined as follows.

\(\cfrac { 4f(-3)+2f(4) }{ f(-6)-3f(1) } \)

In a flower bed, there are 23 rose plants in the first row, 21 in the second, 19 is the third, and so on. There are 5 rose plants in the last row. How many rows are there in the flower bed?

In a three-digit number, when the tens and the hundreds digit are interchanged the new number is 54 more than three times the original number. If 198 is added to the number, the digits are reversed. The tens digit exceeds the hundreds digit by twice as that of the tens digit exceeds the unit digit. Find the original number.

Solve \(\frac { x }{ 2 } -1=\frac { y }{ 6 } +1=\frac { z }{ 7 } +2\); \(\frac { y }{ 3 } +\frac { z }{ 2 } =13\)

If α and β are the roots of the polynomial f(x) = x² - 2x + 3, find the polynomial whose roots are
\(\frac { \alpha -1 }{ \alpha +1 } ,\frac { \beta -1 }{ \beta +1 } \)

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.

In figure \(\angle\)QPR = 90^o, PS is its bisector.If ST\(\bot \)PR, prove that STx (PQ + PR) = PQ x PR.

Two trains leave a railway station at the same time. The first train travels due west and the second train due north. The first train travels at a speed of 20 km/hr and the second train travels at 30 km/hr. After 2 hours, what is the distance between them?

The floor of a hall is covered with identical tiles which are in the shapes of triangles. One such triangle has the vertices at (-3, 2), (-1, -1) and (1, 2). If the floor of the hall is completely covered by 110 tiles, find the area of the floor.

If \(\frac { cos\alpha }{ cos\beta } \) = m and \(\frac { cos\alpha }{ sin\beta } \) = n, then prove that (m² + n²) cos²\(\beta\) = n²

If a cos\(\theta \) - bsin\(\theta \) = c, then prove that (a sin\(\theta \) + bcos\(\theta \)) = \(\pm \sqrt { { a }^{ 2 }+{ b }^{ 2 }-{ c }^{ 2 } } \)

if sin\(\theta \) (1 + sin²\(\theta \)) = cos²\(\theta \), then prove that cos⁶\(\theta \) - 4cos⁴\(\theta \) + 8cos²\(\theta \) = 4

If 15tan² θ+4 sec² θ=23 then find the value of (secθ+cosecθ)² -sin² θ

Find the volume of the iron used to make a hollow cylinder of height 9 cm and whose internal and external radii are 21 cm and 28 cm respectively

Calculate the mass of a hollow brass sphere if the inner diameter is 14 cm and thickness is 1mm, and whose density is 17.3 g/ cm³.

A container open at the top is in the form of a frustum of a cone of height 16 cm with radii of its lower and upper ends are 8 cm and 20 cm respectively. Find the cost of milk which can completely fill a container at the rate of Rs. 40 per litre.

The temperature of two cities A and B in a winter season are given below.

Find which city is more consistent in temperature changes?

If A, B, C are any three events such that probability of B is twice as that of probability of A and probability of C is thrice as that of probability of A and if P(A ∩ B) = \(\frac{1}{6}\) , P(B ∩ C) = \(\frac{1}{4}\), P(A ∩ C), \(\frac{1}{8}\), P(A P(A U B U C) = \(\frac{9}{10}\) , P(A ∩ B ∩ C) = \(\frac{1}{15}\), then find P(A), P(B) and P(C)?

If the range and coefficient of range of the data are 20 and 0.2 respectively, then find the largest and smallest values of the data.

Let A = {1, 2} and B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}, Verify whether A x C is a subset of B x D?

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

A two digit number is such that the product of its digits is 18, when 63 is subtracted from the number, the digits interchange their places. Find the number.

Draw the two tangents from a point which is 10 cm away from the centre of a circle of radius 5 cm. Also, measure the lengths of the tangents.

Take a point which is 11 cm away from the centre of a circle of radius 4 cm and draw the two tangents to the circle from that point.

Find the equation of a line whose intercepts on the x and y axes are given below. -5, \(\frac 34\)

Find a relation between x and y if the points (x,y) (1, 2) and (7, 0) are collinear.

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

A wooden article was made by scooping out a hemisphere from each end of a cylinder as shown in figure. If the height of the cylinder is 10 cm and its base is of radius 3.5 cm find the total surface area of the article.

Which team is more consistent?