Let A = {x \(\in \) N| 1 < x < 4}, B = {x \(\in \) W| 0 ≤ x < 2) and C = {x \(\in \) N| x < 3} Then verify that
(i) A x (B U C) = (A x B) U (A x C)
(ii) A x (B ∩ C) = (A x B) ∩ (A x C)

Given A = {1,2,3}, B = {2,3,5}, C = {3,4} and D = {1,3,5}, check if (A ∩ C) x (B ∩ D) = (A x B) ∩ (C x D) is true?

Forensic scientists can determine the height (in cms) of a person based on the length of their thigh bone. They usually do so using the function h(b) = 2.47b + 54.10 where b is the length of the thigh bone.
(i) Check if the function h is one – one or not
(ii) Also find the height of a person if the length of his thigh bone is 50 cm.
(iii) Find the length of the thigh bone if the height of a person is 147.96 cm.

If the function f: R⟶ R defined by 
\(f(x)=\left\{\begin{array}{l} 2 x+7, x<-2 \\ x^{2}-2,-2 \leq x<3 \\ 3 x-2, x \geq 3 \end{array}\right.\)
(i) f( 4)
(ii) f( -2)
(iii) f(4) + 2f(1)
(iv) \(\frac { f(1)-3f(4) }{ f(-3) } \)

Find the domain of the function f(x) = \(\sqrt { 1+\sqrt { 1-\sqrt { 1-x^{ 2 } } } } \).

If f(x) = x², g(x) = 3x and h(x) = x - 2, Prove that (f o g) o h = f o (g o h).

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?

Let A = The set of all natural numbers less than 8, B = The set of all prime numbers less than 8, C = The set of even prime number. Verify that
A x ( B - C) = (A x B) - (A x C)

Find the remainder when 2⁸¹ is divided by 17.

Find the sum of all natural numbers between 602 and 902 which are not divisible by 4.

A brick staircase has a total of 30 steps. The bottom step requires 100 bricks. Each successive step requires two bricks less than the previous step.
(i) How many bricks are required for the top most step?
(ii) How many bricks are required to build the stair case?

If 1 + 2 + 3 +...+ k = 325, then find 1³ + 2³ + 3³ +...K³.

If 1³ + 2³ + 3³+...k³ = 44100 then find 1 + 2 + 3 +...+ k

The sum of the cubes of the first n natural numbers is 2025. then Find the value of n.

Find the G.P. in which the 2^nd term is \(\sqrt { 6 } \) and the 6^th term is 9\(\sqrt { 6 } \)

If l^th , m^th and n^th terms of an A.P are x, y, z respectively, then show that  (x - y)n + (y - z)l + (z - x)m = 0

One hundred and fifty students are admitted to a school. They are distributed over three sections A, B and C. If 6 students are shifted from section A to section C, the sections will have equal number of students. If 4 times of students of section C exceeds the number of students of section A by the number of students in section B, find the number of students in the three sections.

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.

Arul, Mohan and Ram working together can clean a store in 6 hours. Working alone, Mohan takes twice as long to clean the store as Arul does. Ram needs three times as long as Arul does. How long would it take each if they are working alone?

The sum of thrice the first number, second number and twice the third number is 5. If thrice the second number is subtracted from the sum of first number and thrice the third we get 2. If the third number is subtracted from the sum of twice the first, thrice the second, we get 1. Find the numbers.

Discuss the nature of solutions of the following system of equations
x + 2y - z = 6; -3x - 2y + 5z = -12; x - 2z = 3

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?

Discuss the nature of solutions of the following system of equations
\(\frac { y+z }{ 4 } =\frac { z+x }{ 3 } =\frac { x+y }{ 2 } \) x + y + z = 27

Let A = \(\left[ \begin{matrix} 1 & 2 \\ 1 & 3 \end{matrix} \right] \), B = \(\left[ \begin{matrix} 4 & 0 \\ 1 & 5 \end{matrix} \right] \), C = \(\left[ \begin{matrix} 2 & 0 \\ 1 & 2 \end{matrix} \right] \) Show that  (A − B)^T = A^T − B^T

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 similar to a given triangle PQR with its sides equal to \(\frac { 7 }{ 3 } \) of the corresponding sides of the triangle PQR (scale factor\(\frac { 7 }{ 3 } >1\))

Construct a triangle \(\triangle\)PQR such that QR = 5 cm, \(\angle\)P = 30^o and the altitude from P to QR is of length 4.2 cm.

In trapezium ABCD, AB || DC, E and F are points on non-parallel sides AD and BC respectively, such that EF || AB. Show that \(\frac { AE }{ ED } =\frac { BF }{ FC } \)

In figure DE || BC and CD. Prove that AD² = AB x AF

What length of ladder is needed to reach a height of 7 ft along the wall when the base of the ladder is 4 ft from the wall? Round off your answer to the next tenth place.

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).

PQ is a chord of length 8 cm to a circle of radius 5 cm. The tangents at P and Q intersect at a point T. Find the length of the tangent TP.

If the points A(2, 2), B(–2, –3), C(1, –3) and D(x, y) form a parallelogram then find the value of x and y.

A quadrilateral has vertices A(- 4, - 2), B(5, - 1), C(6, 5) and D(- 7, 6). Show that the mid-points of its sides form a parallelogram.

Find the equation of the perpendicular bisector of the line joining the points A(-4, 2) and B(6, -4).

Without using distance formula, show that points (-2, -1) , (4, 0), (3, 3) and (-3, 2) are the vertices of a parallelogram

The graph relates temperatures y (in Fahrenheit degree) to temperatures x (in Celsius degree) Write an equation of the line

A circular garden is bounded by East Avenue and Cross Road. Cross Road intersects North Street at D and East Avenue at E. AD is tangential to the circular garden at A(3, 10). Using the figure.

Where does the Cross Road intersect the
(i) East Avenue ?
(ii) North Street ?

You are downloading a song. The percent y (in decimal form) of mega bytes remaining to get downloaded in x seconds is given by y = -0.1x + 1.
Find the total MB of the song.

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

if cos\(\theta \) + sin\(\theta \) =\(\sqrt { 2 } \) cos \(\theta \), then prove that cos\(\theta \) - sin\(\theta \) =\(\sqrt { 2 } \) sin\(\theta \) 

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

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 traveler approaches a mountain on highway. He measures the angle of elevation to the peak at each milestone. At two consecutive milestones the angles measured are 4° and 8°. What is the height of the peak if the distance between consecutive milestones is 1 mile. (tan4° =0.0699, tan8° =0.1405)

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.

A bird is flying from A towards B at an angle of 35°, a point 30 km away from A. At B it changes its course of flight and heads towards C on a bearing of 48° and distance 32 km away.
How far is B to the West of A? (sin 55° = 0.8192, cos 55° = 0.5736, sin 42° = 0.6691.cos 42° = 0.7431)

A bird is flying from A towards B at an angle of 35°, a point 30 km away from A. At B it changes its course of flight and heads towards C on a bearing of 48° and distance 32 km away.
How far is C to the North of B? (sin 55° = 0.8192, cos 55° = 0.5736, sin 42° = 0.6691. cos 42° = 0.7431)

From a window (h metres high above the ground) of a house in a street, the angles of elevation and depression of the top and the foot of another house on the opposite side of the street are θ₁ and θ₂ respectively. Show that the height of the opposite house is h \(\left( 1+\frac { cot{ \theta }_{ 2 } }{ { cot\theta }_{ 1 } } \right) \)

An industrial metallic bucket is in the shape of the frustum of a right circular cone whose top and bottom diameters are 10 m and 4 m and whose height is 4 m. Find the curved and total surface area of the bucket.

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.

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

If the ratio of radii of two spheres is 4 : 7, find the ratio of their volumes.

A hemispherical section is cut out from one face of a cubical block  such that the diameter l of the hemisphere is equal to side length of the cube. Determine the surface area of the remaining solid.

A hollow metallic cylinder whose external radius is 4.3 cm and internal radius is 1.1 cm and whole length is 4 cm is melted and recast into a solid cylinder of 12 cm long. Find the diameter of solid cylinder.

The slant height of a frustum of a cone is 4 m and the perimeter of circular ends are 18 m and 16 m. Find the cost of painting its curved surface area at Rs.100 per sq. m

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.

Marks of the students in a particular subject of a class are given below:

Marks	0-10	10-20	20-30	30-40	40-50	50-60	60-70
Number of students	8	12	17	14	9	7	4

Find its standard deviation.

The consumption of number of guava and orange on a particular week by a family are given below.

Number of Guavas	3	5	6	4	3	5	4
Number of Oranges	1	3	7	9	2	6	2

Which fruit is consistently consumed by the family?

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.

Three fair coins are tossed together. Find the probability of getting
(i) all heads
(ii) atleast one tail
(iii) at most one head
(iv) at most two tails

Two customers Priya and Amuthan are visiting a particular shop in the same week (Monday to Saturday). Each is equally likely to visit the shop on any one day as on another day. What is the probability that both will visit the shop on
(i) the same day
(ii) different days
(iii) consecutive days?

The probability of happening of an event A is 0.5 and that of B is 0.3. If A and B are mutually exclusive events, then find the probability that neither A nor B happen.

Two dice are rolled once. Find the probability of getting an even number on the first die or a total of face sum 8.

The frequency distribution is given below.

x	k	2k	3k	4k	5k	6k
f	2	1	1	1	1	1

In the table, k is a positive integer, has a variance of 160. Determine the value of k.