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

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

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

\((x-\frac { 1 }{ x } )={ x }^{ 2 }+\frac { 1 }{ { x }^{ 2 } } \) then f(x) =

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

If f(x) = ax - 2, g(x) = 2x - 1 and fog = gof, the value of a is ___________

The sum of the exponents of the prime factors in the prime factorization of 1729 is

Given F₁ = 1, F₂ = 3 and F_n = F_n-1 + F_n-2 then F₅ is

44 ≡ 8 (mod12), 113 ≡ 85 (mod 12), thus 44 x 113 ≡______(mod 12):

The sum of first n terms of the series a, 3a, 5a...is ____________

If (x - 6) is the HCF of x² - 2x - 24 and x² - kx - 6 then the value of k is

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

\(\frac { { x }^{ 2 }+7x12 }{ { x }^{ 2 }+8x+15 } \times \frac { { x }^{ 2 }+5x }{ { x }^{ 2 }+6x+8 } =\_ \_ \_ \_ \_ \_ \_ \_ \_ \)

Choose the correct answer
(i) Every scalar matrix is an identity matrix
(ii) Every identity matrix is a scalar matrix
(iii) Every diagonal matrix is an identity matrix
(iv) Every null matrix is a scalar matrix

If \(A=\left[ \begin{matrix} y & 0 \\ 3 & 4 \end{matrix} \right] \) and \(I=\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right] \) then A² = 16 for ___________

In ∆LMN, \(\angle\)L = 60^o, \(\angle\)M = 50^o. If ∆LMN ~ ∆PQR then the value of \(\angle\)R is

In figure CP and CQ are tangents to a circle with centre at O. ARB is another tangent touching the circle at R. If CP = 11 cm and BC = 7 cm, then the length of BR is

The height of an equilateral triangle of side a is

The perimeter of a right triangle is 36 cm. Its hypotenuse is 15 cm, then the area of the triangle is ____________

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 ____________

A man walks near a wall, such that the distance between him and the wall is 10 units. Consider the wall to be the Y axis. The path travelled by the man is

The straight line given by the equation x = 11 is

If the points (0, 0), (a, 0) and (0, b) are collinear, then ____________

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

The value of \(si{ n }^{ 2 }\theta +\frac { 1 }{ 1+ta{ n }^{ 2 }\theta } \) is equal to

The angle of depression of the top and bottom of 20 m tall building from the top of a multistoried building are 30° and 60° respectively. The height of the multistoried building and the distance between two buildings (in metres) is

Sin(45^o+ θ ) - cos(45^o - θ) is equal to ___________

If x = a sec θ and = b tan θ, then b²x² - a²y² is equal to ___________

The angle of depression of a boat from a \(50\sqrt { 3 } \) m high bridge is 30^o. The horizontal distance of the boat from the bridge is ___________

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 solid sphere of radius x cm is melted and cast into a shape of a solid cone of same radius. The height of the cone is

If S₁ denotes the total surface area at a sphere of radius ૪ and S₂ denotes the total surface area of a cylinder of base radius ૪ and height 2r, then ___________

The curved surface area of a cylinder is 264 cm² and its volume is 924 cm². The ratio of diameter to its height is ___________

A spherical steel ball is melted to make 8 new identical balls. Then the radius each new ball is how much times the radius of the original ball?

A purse contains 10 notes of Rs. 2000, 15 notes of Rs. 500, and 25 notes of Rs. 200.One note is drawn at random. What is the probability that the note is either a Rs. 500, note or Rs. 200 note?

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

The mean of 100 observations is 40 and their standard deviation is 3. The sum of squares of all observations is

If the data is multiplied by 4, then the corresponding variances is get multiplied by ___________

If the probability of non-happening of an event is, then probability of happening of the event is ___________

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

Let f = {(-1, 3), (0, -1), (2, -9)}. be a linear function from Z into Z. Find f(x).

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) = \(\frac{2}{x}\), g(x) = 2x²- 1

Let A =  {1,2, 3, 4} and B = {-1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} Let R = {(1, 3), (2, 6), (3, 10), (4, 9)} \(\subseteq \) A x B be a relation. Show that R is a function and find its domain, co-domain and the range of R.

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.

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 next three terms of the sequences.
5, 2, -1, -4,...,

Find the next three terms of the following sequence.
 \(\frac { 1 }{ 4 } ,\frac { 2 }{ 9 } ,\frac { 3 }{ 16 } \)......

Find the LCM of the given expressions.
4x²y, 8x³y²

Reduce the rational expressions to its lowest form
\(\frac { { x }^{ 2 }-16 }{ { x }^{ 2 }+8x+16 } \)

If \(\triangle\)ABC is similar to \(\triangle\)DEF such that BC = 3 cm, EF = 4 cm and area of \(\triangle\)ABC = 54 cm². Find the area of \(\triangle\)DEF.

Two circles with centres O and O' of radii 3 cm and 4 cm, respectively intersect at two points P and Q, such that OP and O'P are tangents to the two circles. Find the length of the common chord PQ.

Vertices of given triangles are taken in order and their areas are provided aside. In each case, find the value of ‘p’?

What is the slope of a line perpendicular to the line joining A(5, 1) and P where P is the mid-point of the segment joining (4, 2) and (-6, 4).

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 tan⁴\(\theta \) + tan²\(\theta \) = sec⁴\(\theta \) - sec²\(\theta \) .

The slant height of a frustum of a cone is 5 cm and the radii of its ends are 4 cm and 1 cm. Find its curved surface area.

Find the volume of a cylinder whose height is 2 m and whose base area is 250 m².

If P(A) = 0.37, P(B).= 0.42, P(A∩B) = 0.09 then find P(AUB).

Find the range and coefficient of range of the following data.
43.5, 13.6, 18.9, 38.4, 61.4, 29.8

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?

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

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

If the sum of the first 14 terms of an AP is 1050 and its first term is 10, find the 20th term.

The sum of the digits of a three-digit number is 11. If the digits are reversed, the new number is 46 more than five times the former number. If the hundreds digit plus twice the tens digit is equal to the units digit, then find the original three digit number?

Find the square root of the following expressions
16x² + 9y² - 24xy + 24x - 18y + 9

Find the values of ‘k’, for which the quadratic equation kx² - (8k + 4)x + 81 = 0 has real and equal roots?

Seven years ago, Varun's age was five times the square of Swati's age. Three years hence Swati's age will be two fifth of Varun's age. Find their present ages.

Two vertical poles of heights 6 m and 3 m are erected above a horizontal ground AC. Find the value of y.

In the given figure AB || CD || EF. If AB = 6cm, CD = x cm, EF = 4 cm, BD = 5 cm and DE = y can. Final x and y

PQRS is a rhombus. Its diagonals PR and QS intersect at the point M and satisfy QS = 2PR. If the coordinates of S and M are (1, 1) and (2, - 1) respectively, find the coordinates of P.

if cosec\(\theta \) + cot\(\theta \) = p, then prove that cos\(\theta \) = \(\frac { { p }^{ 2 }-1 }{ { p }^{ 2 }+1 } \)

Two ships are sailing in the sea on either sides of a lighthouse as observed from the ships are \(30°\) and \(45°\) respectively. if the lighthouse is 200 m high, find the distance between the two ships. \(\left( \sqrt { 3 } =1.732 \right) \)

An Aeroplane sets of from G on bearing of 24° towards H, a point 250 km away, at H it changes  course and heads towards J deviates further by 55° and a distance of 180 km away.
How far is H to the north of G?,
\(\left( \begin{matrix} sin24°=0.4067\quad sin11°=0.1908 \\ cos24°=0.9135\quad cos11°=0.9816 \end{matrix} \right) \)

The shadow of a tower, when the angle of elevation of the sum is 45^o is found to be 10 metres, longer than when it is 60^o. find the height of the tower

The frustum shaped outer portion of the table lamp has to be painted including the top part. Find the total cost of painting the lamp if the cost of painting 1 sq.cm is Rs. 2.

The volume of a cylindrical water tank is 1.078 x 10⁶ litres. If the diameter of the tank is 7m, find its height.

The volume of a cone is 1005\(\frac{5}{7}\)cu. cm. The area of its base is 201\(\frac{1}{7}\)sq. cm. Find the slant height of the cone.

Find the standard deviation of the following data 7, 4, 8, 10, 11. Add 3 to all the values then find the standard deviation for the new values.

Two dice are rolled. Find the probability that the sum of outcomes is (i) equal to 4 (ii) greater than 10 (iii) less than 13.

The King, Queen and Jack of the suit spade are removed from a deck of 52 cards. One card is selected from the remaining cards. Find the probability of getting
(i) a diamond
(ii) a queen
(iii) a spade
(iv) a heart card bearing the number 5.

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?

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.

Construct a triangle similar to a given triangle LMN with its sides equal to \(\frac { 4 }{ 5 } \) of the corresponding sides of the triangle LMN (scale factor \(\frac { 4 }{ 5 }<1\)).

Construct a △PQR which the base PQ = 4.5 cm, ∠R = 35^oand the median RG  from R to PG is 6 cm

Find the equation of a straight line Passing through (-8, 4) and making equal intercepts on the coordinate axes

Find the area of the quadrilateral whose vertices, taken in order, are (-4, -2), (-3, -5), (3, -2) and (2, 3).

From a point on a bridge across a river, the angles of depression of the banks on opposite sides at the river are 30° and 45°, respectively. If the bridge is at a height at 3 m from the banks, find the width at the river.

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.