If n(A x B) = 6 and A = {1,3} then n(B) is

If {(a, 8 ),(6, b)}represents an identity function, then the value of a and b are respectively

If f : R⟶R is defined by (x) = x² + 2, then the preimage 27 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) = ax - 2, g(x) = 2x - 1 and fog = gof, the value of a is ___________

If f is identify function, then the value of f(1) - 2f(2) + f(3) is: 

If the HCF of 65 and 117 is expressible in the form of 65m - 117 , then the value of m 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

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

A square is drawn by joinintg the mid points of the sides of a given square in the same way and this process continues indefinitely. If the side of the first square is 4 cm, then the sum of the area of all the squares is ____________

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

The values of a and b if 4x⁴ - 24x³ + 76x² + ax + b is a perfect square are

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

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

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 I for ___________

If in triangles ABC and EDF,\(\cfrac { AB }{ DE } =\cfrac { BC }{ FD } \) then they will be similar, when

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

If ABC is a triangle and AD bisects A, AB = 4cm, BD = 6cm, DC = 8cm then the value of AC is ____________

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

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.

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

When proving that a quadrilateral is a trapezium, it is necessary to show

Find the value of P, given that the line  \(\frac { y }{ 2 } =x-p\) passes through the point (-4, 4) is ____________

In a right angle triangle, right angled at B, if the side BC is parallel to x axis, then the slope of AB is ___________

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

If (sin α + cosec α)² + (cos α + sec α)² = k + tan²α + cot²α, then the value of k is equal to

If sin A + sin²A = 1, then the value of the expression (cos²A + cos⁴A) is ___________

If 4 tan θ = 3, then \(\left( \frac { 4sin\theta -cos\theta }{ 4sin\theta +cos\theta } \right) \) is equal to ___________

(cosec²θ - cot²θ) (1 - cos²θ) is 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 frustum of a right circular cone is of height 16 cm with radii of its ends as 8 cm and 20 cm. Then, the volume of the frustum is

The ratio of the volumes of two spheres is 8 : 27. If r and R are the radii of sphere respectively, Then (R - r) : r is ___________

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.

When Karuna divided surface area of a sphere by the sphere's volume, he got the answer as \(\frac { 1 }{ 3 } \). What is the radius of the sphere?

Which of the following is not a measure of dispersion?

The probability of getting a job for a person is \(\frac{x}{3}\). If the probability of not getting the job is \(\frac{2}{3}\)  then the value of x is

The range of first 10 prime number 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 ___________

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

In each of the following cases state whether the function is bijective or not. Justify your answer.
i. f : R ⟶ R defined by f(x) = 2x + 1
ii. f : R ⟶ R defined by f(x) = 3 - 4x²

Represent the function f(x) =\(\sqrt { 2x^{ 2 }-5x+3 } \) as a composition of two functions.

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 bea relation. Show that R is a function and find its domain, co-domain and the range of R.

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. Letf: A \(\rightarrow\)B be a function given by  f(x) = 2x + 1. Represent this function as a graph.

A man has 532 flower pots. He wants to arrange them in rows such that each row contains 21 flower pots. Find the number of completed rows and how many flower pots are left over.

Find the sum of
1² + 2² +...+ 19²

Find the excluded values, if any of the following expressions.
\(\frac { { x }^{ 2 }+6x+8 }{ { x }^{ 2 }+x-2 } \)

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

Show that \(\triangle\) PST~\(\triangle\) PQR 

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.

Find the area of the triangle formed by the points (1, –1), (–4, 6) and (–3, –5)

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 line passing through the point A(1,4) and perpendicular to the line joining points (2, 5) and (4, 7).

Prove that tan²\(\theta \)-sin² \(\theta \) = tan² \(\theta \) sin² \(\theta \)

calculate \(\angle \)BAC in the given triangles (tan 38.7° = 0.8011 )

Find the diameter of a sphere whose surface area is 154 m².

The external radius and the length of a hollow wooden log are 16 cm and 13 cm respectively. If its thickness is 4 cm then find its T.S.A.

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

Write the sample space for selecting two balls from a bag containing 6 balls numbered 1 to 6 (using tree diagram).

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

find 2f(-4) + 3f(2)

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.

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

If α and β are the roots of the polynomial f(x) = x² - 2x + 3, find the polynomial whose roots are
α + 2, β + 2

Find the square root of the following expressions
\(\left[ \sqrt { 15 } { x }^{ 2 }+\left( \sqrt { 3 } +\sqrt { 10 } \right) x+\sqrt { 2 } \right] \left[ \sqrt { 5 } { x }^{ 2 }+\left( 2\sqrt { 5 } +1 \right) x+2 \right] \left[ \sqrt { 3 } { x }^{ 2 }+\left( \sqrt { 2 } +2\sqrt { 3 } \right) x+2\sqrt { 2 } \right] \)

Reduce the given Rational expressions to its lowest form
\(\frac { 10{ x }^{ 3 }-25{ x }^{ 2 }+4x-10 }{ -4-10{ x }^{ 2 } } \)

A two digit number is such that the product of its digits is 12. When 36 is added to the number the digits interchange their places. Find the number.

There are two paths that one can choose to go from Sarah’s house to James house. One way is to take C street, and the other way requires to take B street and then A street.How much shorter is the direct path along C street? (Using figure).

In Fig, ABC is a triangle with \(\angle\)B=90^o, BC=3cm and AB=4 cm. D is point on AC such that AD=1 cm and E is the midpoint of AB. Join D and E and extend DE to meet CB at F. Find BF.

Find the value of k, if the area of a quadrilateral is 28 sq.units, whose vertices are (–4, –2), (–3, k), (3, –2) and (2, 3)

A kite is flying at a height of 75m above the ground,the string attached to the kite is temporarily tied to a point on the ground.The inclination of the string with the ground is \(60°\).find the length of the string ,assuming that there is no slack in the string.

A bird is sitting on the top of a 80 m high tree. From a point on the ground, the angle of elevation of the bird is 45°. The bird flies away horizontally in such away that it remained at a constant height from the ground. After 2 seconds, the angle of elevation of the bird from the same point is 30°. Determine the speed at which the bird flies.(\(\sqrt { 3 } \) = 1.732)

From the top of a tower 50 m high, the angles of depression of the top and bottom of a tree are observed to be 30° and 45° respectively. Find the height of the tree.(\(\sqrt { 3 } \) = 1.732)

Express the ratios cos A, tan A and see A in terms of sin A.

A right angled triangle PQR where ∠Q = 90^o is rotated about QR and PQ. If QR = 16 cm and PR = 20 cm, compare the curved surface areas of the right circular cones so formed by the triangle.

Nathan, an engineering student was asked to make a model shaped like a cylinder with two cones attached at its two ends. The diameter of the model is 3 cm and its length is 12 cm. If each cone has a height of 2 cm, find the volume of the model that Nathan made.

A hemispherical bowl is filled to the brim with juice. The juice is poured into a cylindrical vessel whose radius is 50% more than its height. If the diameter is same for both the bowl and the cylinder then find the percentage of juice that can be transferred from the bowl into the cylindrical vessel.

A teacher asked the students to complete 60 pages of a record note book. Eight students have completed only 32, 35, 37, 30, 33, 36, 35 and 37 pages. Find the standard deviation of the pages yet to be completed by them.

A game of chance consists of spinning an arrow which is equally likely to come to rest pointing to one of the numbers 1, 2, 3, …12. What is the probability that it will point to (i) 7 (ii) a prime number (iii) a composite number?

Three unbiased coins are tossed once. Find the probability of getting atmost 2 tails or atleast 2 heads.

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 \(\frac { 7 }{ 4 } \) of the corresponding sides of the triangle PQR (scale factor \(\frac { 7 }{ 4 } \)>1)

Construct a \(\triangle\)PQR in which QR = 5 cm, \(\angle\)P = 40^o and the median PG from P to QR is 4.4 cm. Find the length of the altitude from P to QR.

Find the equation of a line whose intercepts on the x and y axes are given below. 4, -6

Find the area of the triangle formed by the points P(-1, 5, 3), Q(6, -2) and R(-3, 4).

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 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.

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