K-12 Math Questions and Answers

Question 19: What term is used to describe a line segment that is smaller than the diameter of the circle with endpoints on the circumference of a circle ?

Question 20: What is common term that is used to describe the distance from the center of a circle to a point on its circumference?

Question 21: What is the longest chord in a circle known as?

Question 22: How many minutes are there in 4 days?

Question 23: How many seconds are there in a day?

Question 24: A bodybuilder eats 3 dozen eggs in a day. How many eggs will he eat in a year?

Question 25: Division is a repeated form of which other mathematical operation?

Question 26: What do you get when you divide the number by itself (except 0) ?

Question 27: A number gives a quotient of 5 when divided by 5. What is the number.

Question 28: Find the largest four digit number which is divisible by 2, 3, 4, 5 and 6.

Question 29: How many right angle can we have in a triangle?

Question 30: What is the measure of each interior angle of a rectangle?

Question 31: What is the sum of all the internal angles of a rectangle?

Question 32: In a right-angled triangle, what statement can be made about the other two angles ?

Question 33: What is the sum of the angles that form a linear pair?

Question 34: What do we call a triangle whose two out of three sides are equal ?

Question 35: If two angles of a triangle are 60° each. Name the triangle.

Question 36: What is the term used to describe a triangle that has all its three sides unequal?

Question 37: In a triangle, one angle is a complement of the other angle. What is the value of the third angle?

Question 38: What term can be used to describe the angles of a regular pentagon?

Question 39: If a pair of vertically opposite angles is also supplementary. Find the value of either of the angles.

Question 40: If two angles are supplementary and one of them is an acute angle, what must the other angle be?

Question 41: What has only one endpoint and can be extended endlessly in the other direction?

Question 42: What do we call the part of a line that has two distinct end points?

Question 43: How many straight lines can be drawn through two distinct points?

Question 44: What term can be used to describe a location in space that has no length and no width?

Question 45: What is the minimum number of line segments that can be used to form a closed figure?

Question 46: What is common term that is used to describe the distance from the center of a circle to a point on its circumference?

Question 47: What term is used to describe a line segment that is smaller than the diameter of the circle with endpoints on the circumference of a circle ?

Question 48: What is the size of the diameter in terms of the radius of a circle ?(Write your answer in words.)

Question 49: What is the common term for the perimeter of a circle?

Question 50: Give an example of a closed curve.

Question 51: How many diameters can be drawn in a circle?

Question 52: How many diameters can be drawn in a circle?

Question 53: A Square is an example of what kind of basic geometrical figure?

Question 54: What is a closed plane figure made up of line segments called?

Question 55: What is a polygon with 4 sides called?

Question 56: The side of an equilateral triangle is given as a. Find the perimeter of the equilateral triangle using a.

Question 57: How many 8-digit whole numbers are there in the base-10 system?

Question 58: What is a polygon with 5 sides called?

Question 59: What is a closed plane figure made up of line segments called?

Question 60: How many seconds are there in a day?

Question 61: How many minutes are there in 6 days?

Question 62: What is the term used to indicate the price or amount that we pay in order to buy a thing?

Question 63: What is the term used to indicate the price or amount at which something is sold?

Question 64: What term is used to describe numbers with the property that the sum of all factors of the number including 1 and the number itself is less than twice the number?

Question 65: What term is used to describe a number with the property that the sum of all factors of the number including 1 and the number itself is equal to twice the number?

Question 66: What term is used to describe a number with the property that the sum of all factors of the number including 1 and the number itself is greater than twice the number?

Question 67: A ratio consists of two terms. If, the second one is called its consequent, what is the first term called?

Question 68: To write a ratio in its simplest form, what number should be used to divide both the antecedent and the consequent?

Question 69: What mathematical operation needs to be applied on the lengths of the sides of a rectilinear figure to get the perimeter of the figure?

Question 70: What is the term used to describe fractions with different denominators?

Question 71: Brian spends $x daily and saves $y per week. What is his income for 9 weeks?

Question 72: Elizabeth earns $ x daily and spends $ y per week. What are her savings of 2 weeks?

Question 73: Paul got 24 holidays in 2020. What is the ratio of holidays to the number of working days?

Question 74: What is the ratio of the lengths of any two sides of an equilateral triangle?

Question 75: What is the ratio of the length of a side of a square to its perimeter?

Question 76: A cycle wheel has 40 spokes. Find the angle between a pair of the adjacent spokes.

Question 77: The diagonal of a rectangle is thrice the length of its smaller side. What is the ratio of its length and breadth?

Question 78: 2222 = 2000 + ____ + 2. What is the missing number?

Question 79: If a and b are two integers such that a is the predecessor of b, then find the value of a - b.

Question 80: How many triangles are there in the figure below ?

Question 81: What fraction of the following image is shaded?

Question 82: Sarah is arranging the tiles as shown below, if she continues it the same way, how many tiles will be there in the next column of yellow tiles?

Question 83: How many basic numerals are there in the Roman number system?

Question 84: In the Roman number system, which symbols can never be repeated in a number representation?

Question 85: Betty introduces Paul as the son of the brother of her mother. How is Paul related to Betty?

Question 86: Kenneth says, 'Sarah is the wife of the grandson of my mother.' What is the relation of Kenneth with Sarah?

Question 87: Pointing towards a man playing cricket, Helen says, "He is the brother of my uncle's daughter". How is the man related to Helen?

Question 88: Pointing towards a girl playing cricket, Anthony says, "Her mother is the only daughter of my mother-in-law". Who is Anthony to the girl?

Question 89: Pointing towards a man playing cricket, Paul says, "His son is my son's uncle". How is the man related to Paul?

Question 90: Robert is the brother of the son of Anthony's son. How is Robert related to Anthony?

Question 91: Find the missing term.

Question 92: Ronald, George, William, and Robert are playing cards. They are sitting such that all of them sit in four different directions (North, South, East, and West) and face the center. The partners will always face each other. Ronald and George are partners. If George is facing east, and William is sitting towards the left-hand side of George, which direction is Robert facing?

Question 93: Find the value of the following:
^@1 + \dfrac { 1 } { 1 + \dfrac {1} { 2 } }^@

Question 94: Find the predecessor of the smallest 3 digit number.

Question 95: Find the sum of the largest 2 digit number and the smallest 5 digit number.

Question 96: What is the minimum number of line segments required to make letter W ?

Question 97: How many months in a year have exactly 31 days?

Question 98: When 21 is added to 30 % of a number, it gives the number itself. Find the number.

Question 99: Argentina has a total land area of 2781498 square kilometers. What is this value if you round it off to the nearest one thousand?

Question 100: What are the sides of the given polygon called?

Question 101: Betty calculates that she has 63072000 seconds left before she turns 18 and can vote in the elections. How old is she now?

Question 102: How many one fourths will make one whole?

Question 103: Name the next figure in the pattern given below:

Question 104: Which shape has 3 vertices and 3 sides?

Question 105: The number of hours left in a day is double the number of hours already passed. How many hours are left in the day?

Question 106: The number of hours left in a day is half the number of hours already passed. How many hours are left in the day?

Question 107: Among five friends, Patricia is taller than William, but not as tall as Jeff. Barbara is taller than Kevin but shorter than William. Who is the shortest in the group ?

Question 108: Nancy introduces Kenneth as the son of the brother of her mother. How is Nancy related to Kenneth?

Question 109: Richard says, "Elizabeth is the wife of the grandson of my mother". Who is Elizabeth to Richard?

Question 110: Pointing towards a man on the stage, Barbara says, "He is the brother of my uncle's daughter". How is Barbara related to the man?

Question 111: Pointing towards a girl swimming in pool, Daniel says, "Her mother is the only daughter of my mother-in-law". How is the girl related to Daniel?

Question 112: What is the smallest possible decimal number upto 6 decimal places?

Question 113: Find the number of corners in the given shape.

Question 114: Find the number of corners in the given shape.

Question 115: How many rays can be drawn on a plane?

Question 116: Let ^@x^@ be a real number. What is the minimum value of ^@x^2 - 4x + 3^@?

Question 117: The number of hours left in a day is double the number of hours already passed. How many hours have already passed in the day?

Question 118: How many lines of symmetry are there in the given figure ?

Question 119: How many lines of symmetry are there in the given figure ?

Question 120: How many lines of symmetry are there in the given figure ?

Question 121: How many lines of symmetry are there in the given figure ?

Question 122: In this picture, consider the dotted line as a mirror and complete the figure using the mirror image on the right side. Find the shape of the complete figure after adding the mirror image.

Question 123: What is the value of ax² + bx + c, if x =

-b

Question 124: Find the greatest n digit number.

Question 125: Find the difference between the smallest n digit number and the greatest m digit number. (Assume n > m).

Question 126: From a deck of cards, Richard randomly withdraws a card. What is the probability that the card is an Ace?

Question 127: From a deck of cards, George randomly withdraws a card. What is the probability that the card is a red King?

Question 128: From a deck of cards, George randomly withdraws a card. What is the probability that the card is a face card (one of king, queen, or jack)?

Question 129: Joseph throws a dice. What is the probability that he will get an odd number?

Question 130: David throws a dice. What is the probability that he will roll a number less than 4?

Question 131: Brian throws a dice. What is the probability that he will roll a number more than 2?

Question 132: Mark tosses a dice twice. What is the probability that the sum of the values obtained in the throws is 9?

Question 133: If a die is thrown, what is the probability of getting an even number?

Question 134: From a deck of cards, Ruth withdrew a card randomly. What is the probability that the number on the card is a composite number?

Question 135: In a right-angled triangle, what statement can be made about the other two angles ?

Question 136: How many triangles are there in the figure below ?

Question 137: In a triangle, one angle is more than the sum of the other two angles. What type of triangle is this?

Question 138: Find the order of rotational symmetry in the given image.

Question 139: Four friends Edward, Karen, David, and Kevin are standing in a row. Edward is not standing at the first position. Karen is not standing at the last position. David is standing behind Karen. Kevin is standing between David and Edward.
Who is standing at last position?

Question 140: Paul is Sandra's brother, Sandra is Joseph's sister, and Joseph is X's father. Who is X to Paul?

Question 141: ABCD is a parallelogram. Find the ratio of the area of the triangle ABC to the area of the parallelogram ABCD.

Question 142: Find the mean of all the internal angles of a quadrilateral.

Question 143: If

p x m

q x n

then what is the relation between m and n?

Question 144: What property is described in the following statement:

Question 145: What property is described in the statement:

+ (

) = (

) +

Question 146: How many rational numbers are there between any two non equal rational numbers ?

Question 147: If the area and perimeter of a square are numerically equal, find the numerical value of its area.

Question 148: There are 1926 soldiers in a platoon. The commander of the platoon wants to impress the visiting general by arranging his soldiers in such a manner that they are arranged in n rows, and each row has n soldiers. He finds that he cannot do this with the number of soldiers he has. What is the minimum number of soldiers he needs additionally to do this kind of a formation?

Question 149: Write the rational number that does not have a reciprocal.

Question 150: Write the rational number that are equal to their reciprocals.

Question 151: Write the rational number that are equal to its negative.

Question 152: If

-3

, then X is _____ a rational number.

Question 153: What is the expansion of (a + b)²?

Question 154: What is the expansion of (a - b)²?

Question 155: Solve the following : ^@\left(\sqrt{ 8^2 + 6^2 }\right)^3^@

Question 156: Simplify (a^-4 + b^-4)^-1.

Question 157: Simplify ^@ \dfrac { \sqrt{ 5 } + 1 } { \sqrt{ 5 } - 1 } ^@.

Question 158: Simplify ^@ \dfrac { \sqrt{ 3 } + \sqrt{ 2 } } { \sqrt{ 3 } - \sqrt{ 2 } } ^@.

Question 159: Take a number x, and multiply it with 5. Take the cube of the resulting number. What is the ratio of this number to the cube of the original number?

Question 160: If one face of a cube has an area of 16 m², then what is the volume of the cube?

Question 161: Find the value of ^@\sqrt[3]{ -1728}^@ .

Question 162: What is the value of ^@\sqrt[3]{ 4.096 }^@ ?

Question 163: If a rhombus is re-shaped such that one of its diagonal increases by 4%, while other diagonal decreases by 4%. Find the percentage change in the area of rhombus.

Question 164: How many diagonals will be there in a 12 sides regular polygon ?

Question 165: If the diagonals of a rhombus are 32 and 60 cm, find its perimeter.

Question 166: If area of a rhombus is 384 cm² and one of its diagonal is 24 cm, find its perimeter.

Question 167: If each side of a cube is tripled, find the factor by which the volume of the cube will change.

Question 168: If each side of a cube is doubled, find the factor by which volume of the cube will change.

Question 169: Assume you roll a die 2 times. What is the probability that you will get exactly one 6?

Question 170: William works thrice as fast as Carol. If William can complete the given task in 36 minutes, how long will it take to complete the task when both William and Carol works together?

Question 171: Two taps A and B can fill a water tank in 5 and 4 hours respectively. If they are turned up alternatively for one hour each (starting with tap A), find the time taken to fill the tank.

Question 172: Barbara and David can finish a piece of work in ^@10^@ and ^@20^@ days respectively. If this work is done by both of them on alternate days (starting with Barbara), how many days will it take for them to finish the work^@?^@

Question 173: Nancy's roll number is a two digit number. Her friend Donna's roll number has the same digits as Nancy's roll number, but with digits interchanged. If they add their roll number, and divide the sum by 11. Find the remainder of this division.

Question 174: The cost of a bat is a two digit number (for example $xy). Its selling prices has same digits as cost with digits interchanged ($yx). If its profit margin is divided by 9, find the remainder of division.

Question 175: William picks a three digit number and subtract it from the same number after reversing order of its digits. If this difference is divided by 11, find the remainder.

Question 176: If a number is divided by 5, remainder is 2. What should be the last digit of the number?

Question 177: If a number is divided by 5, remainder is 4. If same number is divided by 2, remainder is 0. What should be the last digit of the number?

Question 178: Find the difference between the smallest 6 digit number and the largest 3 digit number.

Question 179: Find the equation of a straight line which is parallel to y-axis, and is at a distance of b from y-axis.

Question 180: The positive solutions of the equation ex + fy = g always lie in which quadrant?

Question 181: While doing operations on fractions, in which order should the operations be done among these four operations?
@^ + \space \times \space - \space \div@^

Question 182: The radius of a cylinder is halved and the height is tripled. What is the area of the curved surface when compared to the same area previously?

Question 183: Karen travels along a circular path, when she reaches the spot she started from, she finds that she has covered 132 metres. If she were to walk in a straight line through the center of the circle and reach the opposite side, how much distance did she travel?
(Assume π is

)

Question 184: Find the volume of the biggest cone that can fit inside a cube of side 3 cm.

Question 185: If a hemisphere and a cylinder stand on equal base and have the same height, find the ratio of their volumes.

Question 186: A sphere and a right circular cylinder have the same radius. If the volume of the sphere is double of the volume of the cylinder, what is the ratio of the height of the cylinder to its radius?

Question 187: A sphere and a cone have the same radius. If the volume of the sphere is half of the volume of the cone, find the ratio of the height and radius of the cone.

Question 188: If x is 10% less than y, then find the percentage by which y is more than x.

Question 189: If ^@a^x = \sqrt{b} ^@, ^@b^y = \sqrt{c} ^@ and ^@c^z = \sqrt{a} ^@, find the value of ^@xyz^@.

Question 190: If ^@ 3(u^2 + v^2 + w^2) = (u+v+w)^2, ^@ find the value of ^@u + v - 2w.^@

Question 191: If ^@x(2 - \sqrt{ 2 } ) = y(2 + \sqrt{ 2 } ) = 1^@, find the value of ^@x^2 - y^2^@.

Question 192: From a deck of cards, Jeff took out one card at random. What is the probability that he got a prime number?

Question 193: One morning after sunrise, Edward was standing facing a pole. The shadow of the pole fell exactly to his right. Which direction was he facing?

Question 194: Michelle is facing North. She turns 90° in clockwise direction and then turns 45° in anticlockwise direction. Which direction is her back facing now?

Question 195: What is the probability that a leap year will contain 53 Sundays?

Question 196: The numbers 1 to 13 are written on 13 pieces of paper and dropped into a box. Three of them are drawn at random. What is the probability that the three pieces of paper picked have numbers that are in arithmetic progression?

Question 197: Express the following numbers in the form of

and reduce it to the lowest terms.
0.6

Question 198: If a cube has surface area ^@S^@ and volume ^@V^@, then find the volume of the cube of surface area ^@3S^@.

Question 199: The letters of the word STATISTICAL are rearranged in a random order. What is the probability that the letters C and L have exactly 3 letters between them?

Question 200: 5 coins are tossed in parallel. What is the probability of getting at least one head?

Question 201: If there are ^@ n ^@ numbers of which one is ^@ \left( 1 - \dfrac{ 1 } { n^5 } \right) ^@ and all the others are ^@ 1's, ^@ then by how much is the arithmetic mean of these numbers less than ^@1.^@

Question 202: Jeff says, If Kevin gives me $30, he will have half as much as Daniel. But, if Daniel gives me $30 then three of us will have the same amount. What is the total amount of money that Jeff, Kevin, and Daniel have among themselves?

Question 203: In 2008, the meteorological office predicted the weathers completely right for the months of february and april, and completely wrong for all the other months. What is the probability that the forecast was wrong for a given day that year?

Question 204: Two dice are rolled. What is the probability that the two numbers add up to a prime number?

Question 205: You are given the following equation
DOG + DOG + DOG + DOG + DOG + DOG = BRO, where each letter stands for a digit.
What is the value of DOG?

Question 206: Which is largest among 2^1/2, 3^1/3, 8^1/8 and 9^1/9.

Question 207: In a quadrilateral ^@PQRS,^@ if ^@PQ \parallel RS, \angle S = 2 \angle Q, PS = q^@ and ^@RS = p.^@ Find the length of the side ^@PQ.^@

Question 208: AD is a diameter of a circle and AB is a chord. If AD = 26 cm, AB = 10 cm, find the perpendicular distance of AB from the centre of the circle.

Question 209: How many times during a 12 hour period does the hour hand and the minute hand of the clock cross each other?

Question 210: The area of an equilateral triangle with a side of 12 cm is:

Question 211: What is the area of an equilateral triangle with a side of ^@R \space cm^@?

Question 212: AC is a diameter of the circle and arc AXB = 2 arc BYC. Find ∠BOC.

Question 213: Two circles with centres O and O′ intersect at two points A and B. A line PQ is drawn parallel to OO′ through A(or B) intersecting the circles at P and Q. Find the ratio PQ:OO′.

Question 214: A chord of a circle is equal to its radius. Find the angle subtended by this chord at a point in the major segment.

Question 215: The base of an isosceles triangle is 24 cm and the perimeter is 50 cm. Find the area of the triangle, if the base is not equal to the other two sides.

Question 216: The two adjacent sides of a parallelogram are 3 cm and 4 cm. If one of its diagonals is 5 cm, find the area of the parallelogram.

Question 217: AB and AC are two chords of a circle such that AB = 2AC. If distances of AB and AC from the centre are 3 cm and 6 cm respectively, find the area of circle. (Assume π =3)

Question 218: Find the percentage increase in the area of a triangle if each side is increased by ^@n^@ times.

Question 219: Find the percentage increase in the area of a triangle if each side is increased to ^@5^@ times.

Question 220: The perimeter of a rhombus is 74 cm and one of its diagonals is 35 cm. What is the length of other diagonal?

Question 221: Two circles of radii 13 cm and 5 cm intersect at two points and the distance between their centres is 12 cm. Find the length of the common chord.

Question 222: ABCD is a parallelogram. The circle through A, B and C intersect CD (produced if necessary) at E. Find value of

Question 223: Find the area of a quadrilateral whose sides are 8 cm, 15 cm, 16 cm and 17 cm and the angle between first two sides is a right angle.

Question 224: If x + y - 4t = 0 then find the value of ^@ \dfrac{ x } { x - 2t } + \dfrac{ 2t } { y - 2t} ^@.

Question 225: Find the area of a quadrilateral ABCD where AB = 9 cm, BC = 10 cm, CD = 8 cm, AD = 15 cm, and AC = 17 cm.

Question 226: The sides of a triangle are 13 cm, 37 cm and 40 cm. The altitude to the longest side is:

Question 227: From a point in the interior of an equilateral triangle, perpendiculars are drawn on the three sides. The lengths of the perpendiculars are 4 cm, 6 cm, and 2 cm. Find the area of the triangle.

Question 228: From a point in the interior of an equilateral triangle, perpendiculars are drawn on the three sides. If the lengths of the perpendiculars are ^@a, b,^@ and ^@c^@, find the altitude of the triangle.

Question 229: From a point in the interior of an equilateral triangle, perpendiculars are drawn on the three sides. If the lengths of the perpendiculars are ^@a, b,^@ and ^@c,^@ find the area of the triangle.

Question 230: The perimeter of a triangle is 32 cm. One side of a triangle is 9 cm longer than the smallest side and the third side is 1 cm less than 4 times the smallest side. Find the area of the triangle.

Question 231: If we use (x+) to indicate the following sum:
1 + 2 + 3 + ... + x
then find the value of k in the following equation:
(15+) - (14+) = (k+).

Question 232: Find the area of the figure below. Also, find the altitude to the base of the triangle.

(All measurements are in centimeters)

Question 233: The sides of a quadrilateral, taken in order are ^@ 13 \space cm, 10 \space cm, 12 \space cm,^@ and ^@5 \space cm^@ respectively. The angle contained by the last two sides is a right angle. Find the area of the quadrilateral.

Question 234: The perimeter of a quadrilateral is ^@56 \space cm^@. If the first three sides of a quadrilateral, taken in order are ^@17 \space cm, 16 \space cm^@, and ^@15 \space cm^@ respectively, and the angle between fourth side and the third side is a right angle, find the area of the quadrilateral.

Question 235: The perimeter of an isosceles triangle is 64 cm. The ratio of the equal side to its base is 5 : 6. Find the area of the triangle.

Question 236: A garden is 30 meters wide and 40 meters long. There is a pathway that runs along the perimeter inside the garden. The area of the pathway is half of the area of the garden. What is the width of the pathway?

Question 237: If

= 1 (x,y ≠ 0), find the value of x³ + y³.

Question 238: ∠ABC is a right angle and a:b = 2:3. Find b.

Question 239: If a:b = 2:3, find b.

Question 240: If lines XY and MN intersect as shown below and a:b = 1:2, find c.

Question 241: In a quadrilateral ABCD, the angles A, B, C and D are in ratio 1:2:3:4. Find the measure of each angle of the quadrilateral.

Question 242: During a day, the hour hand and the minute hand of a clock form a right angle, at multiple times. For example, the two hands form a right angle at 9 am. How many times during a day (24 hours) will the two hands form a right angle?

Question 243: In a quadrilateral ABCD, O is a point inside the quadrilateral such that AO and BO are the bisectors of ∠A and ∠B respectively. Prove that ∠AOB =

(∠C + ∠D).

Question 244: In the parallellogram ABCD, the sum of angle bisectors of two adjacent angles is _______.

Question 245: If OD is perpendicular to AB, and ∠DOC = 30°, find ∠BOC - ∠AOC.

Question 246: In a parallelogram, prove that the ratio of any side to an altitude of adjacent side is same for all its sides.

Question 247: ABCD is a parallelogram. The angle bisectors of ∠A and ∠D meet at O. What is the measure of ∠AOD?

Question 248: The diameter of the circumcircle of a rectangle is ^@13 \space cm^@ and rectangle's width is ^@5 \space cm^@. Find the length of the rectangle.

Question 249: The length and breadth of a rectangle is ^@8 \space cm^@ and ^@15 \space cm^@. Find the radius of the circumcircle of this rectangle.

Question 250: In a parallelogram ABCD, ∠A - ∠C is

Question 251: In a rectangle ABCD, the diagonals bisect at O. What kind of a triangle AOB is?

Question 252: If two interior angles on the same side of a transversal intersecting two parallel lines are in the ratio 4:5, then find the greater of the two angles.

Question 253: Write down the expression that gives the products of 9 numbers which have the following properties:
- The GCD/HCF of each pair of numbers in that group is a
- The LCM of the 9 numbers is b

Question 254: Given the following equations:
5^(a+b) = 3125, and
3125^(a-b) = 5,
what is the value of b?

Question 255: Given a circle, a circumscribed square is one that completely encloses the circle, and an inscribed square is one that is completely enclosed by the circle, as shown below:

If the the difference in the areas of the two squares is 35 sq. m., then what is the area of the circle? (Assume π =

)

Question 256: In a rhombus, one of the diagonals is equal to a side of the rhombus. Find the angles of the rhombus.

Question 257: In a parallelogram ^@ABCD^@, the bisectors of ^@\angle A^@ and ^@\angle B^@ intersect at ^@S^@, ^@ \angle B ^@ and ^@ \angle C ^@ at ^@R^@, ^@ \angle C^@ and ^@ \angle D ^@ at ^@Q^@ and ^@ \angle D^@ and ^@\angle A^@ at ^@P^@. What kind of a quadrilateral is ^@PQRS^@?

Question 258: In a rectangle ABCD, E, F, G, and H are the midpoints of the four sides, what kind of shape is represented by EFGH.

Question 259: The corners of a square with side 10 is cut away to form an octagon with all sides equal. What is the length of a side of the octagon so formed?

Question 260: A wireframe is bent into a circle of diameter 56 is reshaped as a rhombus. What is the length of the side of the resulting rhombus? (assume π =

)

Question 261: Brian covers a certain distance in a certain time. If

of the distance is covered in

of the time with speed V₁ and the rest of the

distance in

of the time with speed V₂, then find the ratio V₁:V₂.

Question 262: If the diagonal of a square is decreased by 15%, then by what percent does the area of the square decrease?

Question 263: A square of area 64 cm² is inscribed into a semi-circle. What is the area of the semi-circle?

Question 264: AC is the diameter of the semicircle and BD is perpendicular to AC. If AB = 8 cm and BC = 2 cm, find the length of BD.

Question 265: The diagonals of a rhombus ABCD intersect at O. If AB = 5 cm, diagonal BD = 8 cm, find the length of other diagonal AC.

Question 266: In a parallelogram ABCD, the bisector of ∠A also bisects the side BC. If AB = 6 cm, find the length of side AD.

Question 267: If N = 111111², find the sum of digits in the expansion of N.

Question 268: If Jeff drives at three fourth of his usual speed, he covers a certain distance in 3 hours more than the time he takes while driving at his usual speed. Find the time taken by him to cover this distance with his usual speed.

Question 269: A triangle ^@ABC^@ with ^@\angle ABC = 90^\circ^@, has the length of the side ^@AB = 128 \space cm^@ and of ^@BC = 96 \space cm^@. What is the length of the perpendicular line from side ^@AC^@ to point ^@B^@?

Question 270: ^@ \dfrac{ 1 } { 1 + \sqrt{2} } + \dfrac{ 1 } { \sqrt{2}+\sqrt{3} } + \dfrac{ 1 } { \sqrt{3}+\sqrt{4 } } + \ldots + \dfrac{ 1 } { \sqrt{ 15 } + \sqrt{ 16 } } = ? \space\space\space\space\space^@

Question 271: ^@ABCD^@ is a parallelogram where ^@P^@ and ^@R^@ are the midpoints of sides ^@BC^@ and ^@DC^@ respectively. If the line ^@PR^@ intersects the diagonal ^@AC^@ at ^@Q^@, prove that ^@AC = 4CQ^@.

Question 272: Kenneth is brother of Robert. Lisa is sister of Kenneth. Mark is brother of Laura. Laura is daughter of Robert. Richard is father of Lisa. Who is the uncle of Mark?

Question 273: In the image given below, if the radius of all the large circles is r, then what is the radius of the inner smaller circle?

Question 274: The two sides of a rectangle are x and x + 1. If the length of the diagonal of the rectangle is 5 cm, then what is the area of the rectangle?

Question 275: We draw a square inside a rectangle. The ratio of the rectangle's width to the square's side is ^@2:1^@, and the ratio of the rectangle's length to its width is ^@4:1^@. What percentage of the rectangle's area is inside the square?

Question 276: n is a positive integer. Let's say that {n} denotes the sum of all the positive divisors of n except for n itself. For example, {4} = 1 + 2 = 3 and {8} = 1 + 2 + 4 = 7. What is {{{496}}}?

Question 277: Consider a square ABCD of area 25 cm². L is the midpoint of AB, M the midpoint of BC, N the midpoint of CD, and O the midpoint of DA. These points are used to construct a new square LMNO. The same process is repeated on LMNO to construct a smaller square QRST (where Q is the midpoint of LM and so on). What is the perimeter of square QRST?

Question 278: In a triangle ABC, AD is the median. F is the point on AC such that line BF bisect AD at E. Prove that AC = 3AF.

Question 279: In a triangle ^@ABC^@, ^@AD^@ is a median. ^@F^@ is a point on ^@AC^@ such that the line ^@BF^@ bisects ^@AD^@ at ^@E^@. If ^@AD = 9 \space cm^@ and ^@AF = 3 \space cm^@, find the measure of ^@AC^@.

Question 280: ABCD is a parallelogram with E and F as the mid-points of side AB and CD. Prove that the line segment AF and CE trisect the diagonal BD.

Question 281: Prove that the line segment joining the midpoints of the diagonals of a trapezium is parallel to each of the parallel sides and is equal to half of the difference of these sides.

Question 282: In a triangle, ABC, D is a point on AB such that AB = 4AD and E is a point on AC such that AC = 4AE. Prove that BC = 4ED.

Question 283: A Trapezoid ^@ PQRS ^@ has two parallel bases ^@PQ^@ and ^@RS.^@ The diagonals of the Trapezoid intersect at a point ^@ O.^@ Suppose that ^@ PQ = 10, SR = 15, ^@ and the area of the triangle ^@ POS ^@ is ^@30.^@ What is the area of the trapezoid ^@PQRS?^@

Question 284: Leap day, 29th February, in 2004 fell on a Sunday. On what day of the week will the leap day in 2012 fall?

Question 285: There is a sequence of numbers a₁, a₂, ... where a₁ = 2, a₂ = 3, and a_n =

a_n-1

a_n-2

for n ≥ 3. What is the value of a₇₅₆ ?

Question 286: How many 3-digit numbers satisfy the property that the digit in the middle is the average of the first and the third digit?

Question 287: The sum of the digits of a 2-digit number is subtracted from the number. The resulting number has the units digit of 6. Which numbers have this property?

Question 288: Coin A is flipped 4 times and coin B is flipped 5 times. What is the probability that the number of heads obtained from flipping the two coins is the same?

Question 289: Area of parallelogram ^@ABCD^@ is ^@x \space cm^2.^@ If ^@E, F, G,^@ and ^@H^@ are mid-points of the sides, find the area of ^@EFGH.^@

Question 290: ABCD is a square. Two arcs are drawn with A and B as centers, and radius equal to the side of square. If arcs intersects at point E, find the angle ∠ACE.

Question 291: ^@ABCD^@ is a rectangle and point ^@P^@ is such that ^@PA = 5 \text { cm }, PB = 4 \sqrt { 2 } \text { cm, } \text { and } PD = 3 \text { cm. }^@ Find the length of ^@PC^@.

Question 292: If there are 9 diagonals in a convex polygon, find the number of sides of the polygon.

Question 293: How many times do a clock's hour, minute and second hands overlap in a day?

Question 294: Assume that the symbol ^@\ddagger x \ddagger^@ denotes the largest integer not exceeding ^@x^@. For example, ^@\ddagger 3 \ddagger = 3^@, and ^@\ddagger 4.9 \ddagger = 4^@. What is the value of ^@\ddagger \sqrt{1}\ddagger + \ddagger\sqrt{2}\ddagger + \ddagger\sqrt{3}\ddagger \space + ...... + \ddagger\sqrt { 16 }\ddagger.^@

Question 295: You are given a set of tiles which are numbered from 1 to 121. You do the following operations repeatedly: you remove all those tiles that are numbered with a perfect square, and renumber the remaining tiles consecutively starting with 1. How many times must you perform the operation before you are left with 1 tile?

Question 296: {a_m} is a sequence of integers such that a₁ = 1, and a_m+n = a_m + a_n + mn, for all positive integers m and n. What is the value of a₁₁?

Question 297: If ^@ cot \theta = \dfrac{ a } { b } ^@ and ^@ 90^\circ > \theta > 0^\circ, ^@ find value of ^@ cosec \theta. ^@

Question 298: If ^@ sec \theta = \dfrac{ a } { b } ^@ and ^@ 90^\circ > \theta > 0^\circ, ^@ find value of ^@ tan \theta.^@

Question 299: If ^@\triangle ABC^@ is right angled at ^@C^@, find the value of ^@cos(A+B)^@.

Question 300: If cos θ + cos²θ = 1, then find the value of expression (sin²θ + sin⁴θ).

Question 301: What is the sum of all two-digit odd positive numbers?

Question 302: What is the sum of all three-digit numbers that are divisible by 13?

Question 303: Write the condition for the given system of equations to be inconsistent.
a₁x + b₁y + c₁ = 0, a₂x + b₂y + c₂ = 0

Question 304: Simplify sin⁶β + cos⁶β + 3sin²β cos²β.

Question 305: If ^@\alpha + \beta = 90^ \circ^@, simplify ^@ \sqrt{\cos \alpha \mathrm{cosec} \beta - \cos \alpha \sin \beta}. ^@

Question 306: If ^@ sin \theta + cos \theta = \sqrt{ 3 },^@ simplify ^@tan \theta + cot \theta.^@

Question 307: Simplify ^@ \dfrac{\cos \theta } { 1 + \sin \theta } + \dfrac{ 1 + \sin \theta } { \cos \theta } ^@.

Question 308: If sinθ + 2cosθ = 1, find the value of 2sinθ - cosθ.

Question 309: A spherical balloon of radius ^@ 15 ^@ feet subtends an angle ^@ 60^\circ ^@ at the eye of an observer. If the angle of elevation of its center is ^@ 45^\circ,^@ find the height of the center of the balloon.

Question 310: sin(45° + θ) - cos(45° - θ) = ?

Question 311: Simplify: 1 +

tan²α

1 + secα

Question 312: Prove that n²-n is even for every positive integer n.

Question 313: Prove that √2 is an irrational number.

Question 314: Prove that

√11

is an irrational number.

Question 315: Jason, Patricia, John, Daniel, Ronald, and Steven are playing game by forming two teams of three players each. In how many different ways can they be put into two teams of three players?

Question 316: Simplify ^@\dfrac{ 1 }{ 1 + sec \theta } + \dfrac {1} { 1 - sec \theta }^@.

Question 317: Simplify ^@\sqrt { \dfrac { 1 + cos \theta } { 1 - cos \theta } }^@.

Question 318: Simplify

cosθ - 2 cos³θ

2 sin³θ - sinθ

Question 319: Simplify (1 + tanθ + secθ) (1 + cotθ - cosecθ)

Question 320: Simplify

Question 321: Simplify

Question 322: Simplify: ^@ \dfrac{ \cos \theta } { 1 - \tan \theta } + \dfrac{ \sin^2 \theta } { \sin \theta - \cos \theta } ^@

Question 323: Simplify: ^@1 - \dfrac { sin^3 \theta + \cos^3 \theta } { sin \theta + cos \theta }^@

Question 324: Simplify: ^@ \dfrac{ \cos^2 \theta } { 1 - \tan \theta } + \dfrac{ \sin^3 \theta } { \sin \theta - \cos \theta } ^@

Question 325: Simplify: ^@ \dfrac{ \sin x - \sin y } { \cos x + \cos y } + \dfrac{ \cos x - \cos y } { \sin x + \ sin y } ^@

Question 326: Simplify cos⁶θ + sin⁶θ + 3 cos²θ sin²θ

Question 327: Simplify 3(sin⁴θ + cos⁴θ) - 2(sin⁶θ + cos⁶θ)

Question 328: Simplify ^@ \dfrac{ ( 1 + \cot \theta + \tan \theta) (\sin \theta - \cos \theta) } { \sec^3 \theta - \mathrm{cosec}^3 \theta } ^@

Question 329: If ^@ sec\theta + tan\theta = y, ^@ simplify ^@ \dfrac{ y^2 -1 } { y^2 + 1 } ^@ in terms of ^@ \theta. ^@

Question 330: If ^@ sec\theta + tan\theta = p, ^@ find value of ^@ sec\theta ^@ in terms of ^@ p. ^@

Question 331: Simplify cos³θ sin³(90°-θ) + sin³θ cos³(90°-θ) + 3 cos²θ sin²θ

Question 332: Simplify ^@ \dfrac{ \cos^2 10 ^\circ + \cos^2 80 ^\circ } { \sin^2 10 ^\circ + \sin^2 80 ^\circ } + \dfrac{ \sin (90 ^\circ - \theta ) \sin \theta } { \tan \theta } + \dfrac{ \cos (90 ^\circ - \theta ) \cos \theta } { \cot \theta } ^@

Question 333: Simplify: ^@ \dfrac{ \cos^2 20 ^\circ + \cos^2 70 ^\circ } { \sin^2 20 ^\circ + \sin^2 70 ^\circ } + \cos (20 ^\circ + \theta ) - \sin (70 ^\circ - \theta ) ^@

Question 334: Simplify: ^@ \cot^2 \theta \left( \dfrac{ \sec \theta - 1 } { 1 + \sin \theta } \right) + \sec^2 \theta \left( \dfrac{ \sin\theta - 1 } { 1 + \sec \theta } \right) ^@

Question 335: Simplify: ^@ \mathrm{cosec}^2 \theta \left( \dfrac{ \cos\theta - 1 } { 1 + \mathrm{cosec} \theta } \right) - \tan^2 \theta \left( \dfrac{ \mathrm{cosec} \theta - 1 } { 1 + \cos \theta } \right) ^@

Question 336: Simplify ^@ \sqrt { \dfrac{ 1 + \cos \theta } { 1 - \cos \theta } } + \sqrt { \dfrac{ 1 - \cos \theta } { 1 + \cos \theta } } ^@

Question 337: Simplify ^@ \sqrt { \dfrac{ \mathrm{cosec} \theta - 1 } { \mathrm{cosec} \theta + 1 } } + \sqrt { \dfrac{ \mathrm{cosec} \theta + 1 } { \mathrm{cosec} \theta - 1 } } ^@

Question 338: Solve quadratic equation a²b²x² + b²x - a²x - 1 = 0 using factorization.

Question 339: Solve quadratic equation ^@x^2 + (\dfrac { p } { p + q } + \dfrac { p + q } { p })x + 1^@ ^@= 0 ^@ using factorization.

Question 340: Solve quadratic equation ^@\dfrac {x} { x + 1 } + \dfrac { x + 1 } { x } = \dfrac { 25 } { 12 } ^@ (^@x \neq 0 ^@ and ^@ x \neq -1^@).

Question 341: The sum of a number and its reciprocal is ^@\dfrac { 34 } { 15 }^@. Find the number.

Question 342: The sum of the squares of two consecutive natural numbers is 313. Find the numbers.

Question 343: The sum of square of two positive numbers is 208. If square of the larger number is 18 times the smaller number, find the numbers.

Question 344: The speed of a boat in still water is ^@16 \text { km/hour }^@. It goes ^@60 \text { km }^@ upstream and return downstream to starting point in ^@8 \text { hours }^@. Find the speed of the stream.

Question 345: An equilateral triangle is inscribed in a circle of radius 18 cm. Find the length of the triangle's side.

Question 346: An equilateral triangle with a side of length 6√3 cm is inscribed in a circle. Find the radius of the circle.

Question 347: If tan θ + cot θ = 5, find the value of tan² θ + cot² θ.

Question 348: If ^@ \alpha ^@ and ^@ \beta ^@ are the zeros of polynomial ^@ x^2-x-6,^@ find a polynomial whose zeros are ^@ \dfrac{ \alpha^2 }{ \beta^2 } ^@ and ^@ \dfrac{ \beta^2 }{ \alpha^2 }. ^@

Question 349: In a two digit number, the units digit is twice the tens digit. If 18 is added to the number, the digits interchange their places. Find the number.

Question 350: ABC is an isosceles triangle with AB = AC. We extend the segment BA to D such that BA = AD. What is the value of ∠BCD?

Question 351: What is the probability that a leap year will contain 53 Tuesdays and 52 Wednesdays?

Question 352: Michael is on the left of the person sitting in center but is on the right of Ronald. Jennifer is on the right of Joseph and Mark is on the right of Jennifer. Mark is the second person from the person sitting in the center. Who is sitting in center.

Question 353: If f(x) = x², find the value of

f(x + r) - f(x)

Question 354: Look at the given figure. In how many ways can we divide the figure into two similar halves?

Question 355: We have 4 pants and 5 shirts of different colors. How many different outfits can we make?

Question 356: How many triangles are there in the given figure?

Question 357: How many more parts should be shaded so that

of the given figure is shaded?

Question 358: You are given a number N with the property that every next digit of N is greater than the previous digit (so second digit from the left is greater than the first digit, and third digit is greater than the second and so on). Now multiply N by 9. What is the sum of the digits of the resulting number?

Question 359: You are given a number ^@N^@ with the property that every next digit of ^@N^@ is greater than the previous digit (so second digit from the left is greater than the first digit, and third digit is greater than the second and so on). Now multiply ^@N^@ by ^@ 9. ^@ What is the sum of the digits of the resulting number?

Question 360: Anthony and David sit on the 6-member board of directors for company L. If the board is to be split up into two 3-person subcommittees, what is the percentage that David and Anthony are included in the same sub-committee?

Question 361: How many sides does a quadrilateral have?

Question 362: Write the adjacent sides of the following quadrilateral.

Question 363: Write the name of the diagonals of the following quadrilateral.

Question 364: Name a parallelogram in which all sides are equal.

Question 365: What is the multiplicative inverse of ^@ 7 ^@?

Question 366: How many lines of symmetry are there in a circle?

Question 367: How many paise are there in one rupee?

Question 368: Find the successor of the largest 6 digit number.

Question 369: Find the predecessor of the smallest 6 digit number.

Question 370: What is the term used to describe fractions where the denominator is greater than the numerator?

Question 371: What is the term used to describe fractions with the same denominator?

Question 372: What is the term used to describe fractions where the numerator is greater than the denominator?

Question 373: Write the numeric value for the Roman numeral ^@ L ^@.

Question 374: Write the roman numeral for ^@ 50 ^@.

Question 375: In Roman Number system, what is the symbol for zero?

Question 376: The age of a man is same as his wife’s age with the digits reversed. The sum of their ages is 77 years and the man is 9 years older than his wife. How old is the man?

Question 377: Two angles in a triangle are in the ratio 2:3. If the sum of these angles is equal to the third angle, then what is the value of third angle?

Question 378: The denominator of a rational number is greater than its numerator by ^@8.^@ If ^@8^@ is subtracted from the numerator and ^@7^@ is added to its denominator^@,^@ the new number becomes ^@\dfrac { 1 } { 24 }.^@ What is the original number^@?^@

Question 379: A child walked 5 m to cross a rectangular field diagonally. If breadth of field is 3 m, what is the length of rectangular filed?

Question 380: Two cubes have their volumes in the ratio 1:64. Find the ratio of their surface areas.

Question 381: How many surfaces are there in a right circular cylinder?

Question 382: Write the Roman numeral for the greatest one-digit number.

Question 383: If

, then what is the value of x?

Question 384: What is the difference between the greatest 4-digit number and the smallest 2-digit number?

Question 385: Find the greatest 4-digit number with all different digits.

Question 386: Form the smallest 4-digit number with 4 in the tens place by using different digits.

Question 387: I am a 4-digit number. If my digits are reversed, I become a 3-digit number. What is the digit in my ones place?

Question 388: If the digit 0 is to be repeated 3 times, then how many 4-digit numbers can be formed by using the digits 1 and 0?

Question 389: Find the number that has its successor as the smallest 3-digit number.

Question 390: Find the predecessor of the greatest 3-digit number.

Question 391: Is there any number that has no successor?

Question 392: What do we get on adding any two odd numbers?

Question 393: Which number will come next in the following series?
100, 200, 300, 400, 500, ......

Question 394: In multiplication, the number with which we multiply is called the _____.

Question 395: Observe the pattern and complete the series.
2, 4, 6, 8, 10, 12, _____, _____, _____, _____.

Question 396: Find the value of A.
A A
× A
---------
2 7 A
---------

Question 397: What do we get if a number is multiplied by 1 and the product is multiplied by 0?

Question 398: Fill in the blank.
0 ÷ 7 = _____

Question 399: Fill in the blank.
____ ÷ 4 = 0

Question 400: Mark has zero marbles. He wants to put them in 6 boxes. How many marbles are there in each box?

Question 401: Look at the figure given below. Find the fraction of the shaded part in the figure.

Question 402: A water bottle is one-third empty. What fraction of water remains in the bottle?

Question 403: Write the given fraction in words.

Question 404: The entrance ticket of a trade fair is shown below:

Cost of 1 ticket:	Child	Adult
	₹10	₹20

How much does it cost for a couple to visit the trade fair with their 1 child?

Question 405: Fill in the blank.
There are ____ quarters in a whole.

Question 406: Fractions having different denominators are called ________.

Question 407: How many grams make a kilogram?

Question 408: What is the standard unit of mass?

Question 409: How many 200 ml glasses will fill up a 1 l bottle?

Question 410: Which is the longest hand of the clock?

Question 411: The _____ hand is the slowest on the clock.

Question 412: How many weeks equal 14 days?

Question 413: How many days are there in 4 weeks?

Question 414: How many days of the week begin with the letter M?

Question 415: There are 365 days in a year and 200 years in a double century. How many days are there in a double century?

Question 416: If today is Friday. What day will it be tomorrow?

Question 417: What day came before yesterday, if the day after tomorrow is Saturday?

Question 418: How many months have only 30 days?

Question 419: The shortest month of the year is _____.

Question 420: Every year divisible by 4 is a _____.

Question 421: The year that is not a leap year is called an _____.

Question 422: How many days does a leap year have?

Question 423: How many weeks are there in an ordinary year?

Question 424: How many full weeks are there in 2 years?

Question 425: The part of a solid shape where its three faces meet is called a _____.

Question 426: The geometrical shapes that have only length and breadth are called _____.

Question 427: 2-dimensional shapes are also called _____.

Question 428: Name a solid shape with no vertex.

Question 429: How many vertices does a cuboid have?

Question 430: How many vertices does a cube have?

Question 431: In a cuboid, each vertex is formed by the meeting of ____ faces.

Question 432: Guess the given alphabet by looking at its mirror halves.

Question 433: How many edges does a cylinder have?

Question 434: How many shapes are there in a tangram?

Question 435: The given figure shows a number line. What is the value of A?

Question 436: The given figure has _______ line segments.

Question 437: How many line segments can be drawn by joining the given points?

Question 438: Which two shapes can be combined to make the given geometrical figure?

Question 439: Measure the length of the pencil with the help of the ruler given below:

Question 440: ________ is a type of line that continuously bends without making any sharp angles.

Question 441: What is the number just after the largest 4-digit number ?

Question 442: Find the successor of the largest 2 digit number.

Question 443: Find the predecessor of the smallest 2 digit number.

Question 444: How many straight lines can be found in the given figure?

Question 445: A closed planar shape with six sides is called a ______________.

Question 446: A number that is completely divisible by numbers other than 1 and itself is called a ________.

Question 447: Every multiple of a non-zero number is either _____ or _____ the given number.

Question 448: How many edges does a triangular prism have?

Question 449: The ______ of two or more numbers is the greatest number that exactly divides each of the numbers.

Question 450: Name a solid with four triangular faces.

Question 451: Which hand of the clock moves with the slowest speed?

Question 452: A _______ refers to any pattern that covers a flat surface like a painting on a canvas, using non-overlapping repetitions.

Question 453: Identify the shape that is used for tessellation in the given figure.

Question 454: Form the smallest 7-digit number without repeating the digits.

Question 455: Find the sum of the greatest 4-digit and the smallest 4-digit number.

Question 456: What will be the result, if a number is subtracted from itself?

Question 457: In multiplication, multiplicand × multiplier = ________________.

Question 458: If a non-zero number is divided by 1 then the quotient is ___________.

Question 459: ____ is a factor of every number.

Question 460: Which figure shows three-fourth fraction?

Question 461: If it is spring now, which season will come after 12 months?

Question 462: What is the difference between two consecutive odd numbers?

Question 463: How many months are there in a year?

Question 464: If it was Tuesday yesterday, what day is it today?

Question 465: If today is Thursday, what day will it be tomorrow?

Question 466: Which day comes between Friday and Sunday?

Question 467: Nancy did not go to school from Monday to Wednesday. For how many days was Nancy absent from the school?

Question 468: What is the greatest 3-digit number?

Question 469: What is the smallest 3-digit number?

Question 470: What is the 6^th day of the week?

Question 471: Add:

	1	0
+	8	0

Question 472: How many seconds are there in 1 minute?

Question 473: How many minutes does the minute hand takes to complete one round on the dial of the clock?

Question 474: How many rupees are there in 400 paise?

Question 475: What number is shown on the face of the die?

Question 476: What number is shown by the fingers of the hand?

Question 477: What comes between 3 and 5 ?

Question 478: What is the short form of 'first'?

Question 479: Add:

	0
+	2

Question 480: Add:

	4
+	3

Question 481: Add:

	2
+	4

Question 482: Add:

	5
+	5

Question 483: Add:

	0
+	6

Question 484: Add:

	3
+	7

Question 485: Add:

	0
+	8

Question 486: Add:

	1
+	9

Question 487: Add:

	7
+	0

Question 488: What do we get on adding a number to zero?

Question 489: What is the sum of 0 and 9 ?

Question 490: Use the doubles fact for addition.
4 + 5 = _____

Question 491: What do we get when we add the following?
7 + 6

Question 492: On the basis of given number line, fill in the blank.

16 ÷ 4 = ____

Question 493: What should we subtract from a number so that we get the number itself?

Question 494: Subtract:

	5
−	1

Question 495: Subtract:

	5
−	2

Question 496: What do we get when we subtract 8 from 16 ?

Question 497: Circle the combination of beakers that can be filled with the water in the bucket.

Question 498: Subtract.

	2	0
−	1	2

Question 499: IV workers are working on painting a fence. If each worker can paint IV meters of fence in an hour, how many meters can the workers paint, in total, over a period of 2 hours (answer in Roman Numerals)?

Question 500: _____ sq.m = 10000 sq.cm

Question 501: What do we call a rectangle with all sides of equal length?

Question 502: What do we call a quadrilateral bounded by two horizontal line segments and two vertical line segments?

Question 503: State true or false:
A plane has no boundary.

Question 504: How many rays can be drawn from a given point?

Question 505: An angle whose measure is more than 90° and less than 180° is called a/an ______.

Question 506: An angle whose measure is more than 0° and less than 90° is called a/an ______.

Question 507: An angle whose measure is 90° is called a/an ______.

Question 508: An angle whose measure is 180° is called a/an ______.

Question 509: An angle whose measure is 360° is called a/an ______.

Question 510: An angle whose measure is 0° is called a/an ______.

Question 511: Name the lines which cross paths at one point?

Question 512: Are AG and BE intersecting?

Question 513: What is the sum of two complementary angles?

Question 514: What is the sum of two supplementary angles?

Question 515: Name a closed figure bounded by three line segments.

Question 516: What do we call a triangle whose three sides are equal ?

Question 517: The numbers, one, two, three, and so on, that tell us about the count of objects or items are called ___________________

Question 518: Which mathematical operation is performed first in simplifying a numerical expression?

Question 519: Sharon has 8 fish. She keeps 4 fish in a single fish tank. How many fish tanks does Sharon need?

Question 520: The angles of a set square are 45°, 90° and _______.

Question 521: The set square is in the shape of ______.

Question 522: Name the angle.

Question 523: The fox is standing _____________ (inside/outside) the cage.

Question 524: The tiger is standing _____________ (inside/outside) the cage.

Question 525: Show that every positive even integer is of the form 2m and every positive odd integer is of the form (2m+1), where m is some integer.

Question 526: How many different angles can we count in the figure?

Question 527: How many different angles can we count in the figure?

Question 528: How many different angles can we count in the figure?

Question 529: The boy is standing on the _________ (top/bottom) of the building.

Question 530: The boy is standing at the _________ (top/bottom) of the building.

Question 531: Show that any number of the form 2ⁿ, where n belongs to natural numbers can never end with the digit 0.

Question 532: If a is a nonzero rational and √b is irrational number then show that a√b is irrational number.

Question 533: A hemisphere depression is cut out from one face of a cubical wooden block of length 'l' such that the diameter of the hemisphere is equal to the edge of the cube. Determine the surface area of the remaining solid.

Question 534: Identify the points that lie in the exterior of the circle.

Question 535: Count the number of chords shown in the circle.

Question 536: The 3^rd digit from the right of a number is k. Find its place value.

Question 537: How many vertices does a rectangle have?

Question 538: When a number is divided by its factor, the remainder will be ______.

Question 539: Determine the ratio of the volume of a cube to that of a sphere which will exactly fit inside the cube.

Question 540: Find the maximum volume of a cone that can be carved out of a solid hemisphere of radius r.

Question 541:

Identify the shape.

Question 542: What is the area of the largest triangle that can be inscribed in a semi-circle of radius ^@r^@ units?

Question 543: What do we call a definite sequence of figures, shapes, letters, or numbers?

Question 544: A chord of a circle of radius ^@ 42 \space cm ^@ subtends a right angle at the center. Find the area of the corresponding major segment. ^@ \left[ \pi = \dfrac { 22 } { 7 } \right] ^@

Question 545: Find the area of the square that can be inscribed in a circle of radius ^@6 \space cm^@.

Question 546: What is two-third of half?

Question 547: Prove that the lengths of tangents drawn from an external point ^@ A ^@ to the points ^@ P ^@ and ^@ Q ^@ on the circle are equal.

Question 548: Write the multiplication sentence that describes the model.

Question 549: In the given figure, two circles touch each other at a point ^@C^@. Prove that the common tangent to the circles at ^@C^@ bisects the common tangent at the points ^@P^@ and ^@Q^@.

Question 550: Two concentric circles have radii ^@a^@ and ^@b^@ ^@(a>b)^@. Find the length of the chord of the larger circle which touches the smaller circle.

Question 551: In the given figure, the radii of two concentric circles are ^@ 12 \space cm ^@ and ^@ 7 \space cm ^@. ^@ AB ^@ is the diameter of the bigger circle and ^@ BD ^@ is a tangent to the smaller circle touching it at ^@ D ^@. Find the length ^@ AD ^@.

Question 552: How many months in a year have only 31 days?

Question 553: From a point ^@ P ^@ outside a circle with center ^@ O ^@, tangents ^@PA^@ and ^@PB^@ are drawn to the circle. The line segment formed by joining the points ^@A^@ and ^@B^@ intersect the line segment ^@OP^@ at ^@M^@. Find the measure of ^@\angle AMP^@.

Question 554: ^@ AP ^@ and ^@ BP ^@ are the two tangents at the extremities of chord ^@ AB ^@ of a circle. Prove that ^@ \angle MAP ^@ is equal to ^@ \angle MBP ^@.

Question 555: In the given figure, ^@ AB ^@ is a chord of length ^@ 13 \space cm ^@ of a circle with center ^@ O ^@ and radius ^@ 9 \space cm ^@. The tangents at ^@ A ^@ and ^@ B ^@ intersect at ^@ P ^@. Find the length of ^@ PA ^@.

Question 556: What is one sixth of one sixth?

Question 557: Which is the hottest season?

Question 558: During which season should we use sunscreen and try to stay cool?

Question 559: Which day comes after Saturday?

Question 560: What comes before Thursday?

Question 561: If tomorrow is Saturday, which day was yesterday?

Question 562: Two tangents ^@ PA ^@ and ^@ PB ^@ are drawn to a circle with center ^@ O ^@ from an external point ^@ P ^@. Prove that ^@ \angle APB = 2 \angle OAB^@.

Question 563: How many days are there in a week?

Question 564: In the given figure, the incircle of ^@ \triangle ABC ^@ touches the sides ^@ BC, CA, ^@ and ^@ AB ^@ at ^@ P, Q, ^@ and ^@ R ^@ respectively.
Prove that ^@ (AR + BP + CQ) = (AQ + BR + CP) = \dfrac { 1 } { 2 } \text{(Perimeter of } \triangle ABC). ^@

Question 565: A circle is inscribed in a ^@ \triangle ABC ^@, touching ^@ BC, CA, ^@ and ^@ AB ^@ at points ^@ P, Q, ^@ and ^@ R ^@ respectively. If ^@ AB = 14 \space cm, AQ = 11 \space cm ^@ and ^@ CQ = 9 \space cm ^@ then find the length of ^@ BC ^@.

Question 566: The interval between sunrise and sunset is called _________.

Question 567: A circle is touching the side ^@ BC ^@ of ^@ \triangle ABC ^@ at ^@ P ^@ and touching ^@ AB ^@ and ^@ AC ^@ produced at ^@ Q ^@ and ^@ R ^@ respectively. Prove that ^@ AQ = \dfrac { 1 } { 2 } \text{(Perimeter of } \triangle ABC) ^@.

Question 568: ^@ PA ^@ and ^@ PB ^@ are tangents to a circle with center ^@ O ^@ from an external point ^@ P ^@, touching the circle at ^@ A ^@ and ^@ B ^@ respectively. Show that the quadrilateral ^@ AOBP ^@ is cyclic.

Question 569: In the given figure, tangents ^@ PQ ^@ and ^@ PR ^@ are drawn from an external point ^@ P ^@ to a circle with center ^@ O ^@, such that ^@ \angle RPQ = 60^\circ ^@. A chord ^@ RS ^@ is drawn parallel to the tangent ^@ PQ ^@. Find the measure of ^@ \angle RQS ^@.

Question 570: In the given figure, ^@ ABC ^@ is a right-angled triangle with ^@ AB = 7 \space cm ^@ and ^@ AC = 9 \space cm ^@. A circle with center ^@ O ^@ has been inscribed inside the triangle. Calculate the value of ^@r^@, the radius of the inscribed circle.

Question 571: A quadrilateral ^@ ABCD ^@ is drawn to circumscribe a circle, as shown in the figure. Prove that ^@ AB + CD = AD + BC ^@.

Question 572: From a point ^@ P ^@, two tangents ^@ PA ^@ and ^@ PB ^@ are drawn to a circle ^@ C(O,r) ^@. If ^@ OP = 2r ^@, show that ^@ \triangle APB ^@ is an equilateral triangle.

Question 573: In the given figure, ^@ XY ^@ and ^@ X'Y' ^@ are two parallel tangents to a circle with center ^@ O ^@ and another tangent ^@ AB ^@ with the point of contact ^@ C ^@ intersects ^@ XY ^@ at ^@ A ^@ and ^@ X'Y' ^@ at ^@ B ^@. Prove that ^@ \angle AOB = 90^\circ^@.

Question 574: In the given figure, ^@ XP ^@ and ^@ XQ ^@ are the two tangents to a circle with center ^@ O ^@ drawn from an external point ^@ X ^@. ^@ AB ^@ is another tangent touching the circle at ^@ R ^@. Prove that ^@ XA + AR = XB + BR ^@.

Question 575: What is the value of

rounded off to second decimal place?

Question 576: When a shape becomes exactly like another when we flip, slide, or turn it, it is called _______________.

Question 577: The angles of a quadrilateral are in AP whose common difference is ^@ 10^\circ ^@. Find the smallest angle of the quadrilateral.

Question 578: 2 liter of a medicine is to be packed in small bottles having a capacity of 250 ml each. How many such bottles are needed?

Question 579: Which season comes after summer?

Question 580: The sum of first ^@ n, \space 2n ^@, and ^@ 3n ^@ terms of an AP is ^@ S_1, \space S_2 ^@, and ^@ S_3 ^@ respectively. Prove that ^@ S_3 = 3(S_2 - S_1).^@

Question 581: If the sum of the first ^@ p ^@ terms of an AP is the same as the sum of its first ^@ q ^@ terms (where ^@ p \ne q ^@) then show that the sum of its first ^@(p + q)^@ terms is zero.

Question 582: If the sum of the first ^@ p ^@ terms of an AP be ^@ q ^@ and the sum of its first ^@ q ^@ terms be ^@ p ^@ then show that the sum of its first ^@ (p + q) ^@ terms is ^@ - (p + q) ^@.

Question 583: A 4-sided flat shape with straight sides where the opposite sides are parallel is called a ________.

Question 584: In the Roman Number system, X can be subtracted from ____ and ____ only.

Question 585: Linda built a cube out of 1-centimeter blocks, as shown below.

What is the volume of the cube?

Question 586: Write a fraction which is equivalent to one-fifth of one-fifth ?

Question 587: The angle of elevation of the top of a tower from the two points ^@ P ^@ and ^@ Q ^@ at distances of ^@ a ^@ and ^@ b ^@ respectively from the base and in the same straight line with it are complementary. Prove that the height of the tower is ^@ \sqrt{ab} ^@ where ^@ a > b ^@.

Question 588: From the top of a tower ^@ h \space meter ^@ high, the angle of depression of two objects, which are in line to the foot of the tower is ^@ \alpha ^@ and ^@ \beta ^@ ^@( \beta > \alpha)^@. Find the distance between the two objects.

Question 589: In the Hindu-Arabic system, the ten symbols, namely, 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are called _____.

Question 590: How many zeros are there in one lakh?

Question 591: The angle of elevation of the top of a tower as observed from a point on the ground is ^@ \alpha ^@ and on moving ^@ a \space meters ^@ towards the tower, the angle of elevation is ^@ \beta ^@. Prove that the height of the tower is ^@ \dfrac { a \tan \alpha \tan \beta } { \tan \beta - \tan \alpha }. ^@

Question 592: From a window ^@h \space meters ^@ high above the ground in a street, the angles of elevation and depression of the top and foot of the other house on the opposite side of the street are ^@\alpha^@ and ^@\beta^@ respectively. Show that the height of the opposite house is ^@ h (1 + \tan \alpha \space cot \beta) \space meters^@.

Question 593: Write a whole number whose predecessor is not a whole number?

Question 594: Which term of the ^@G.P. \space 4, 64, 1024, ....^@ upto ^@ n ^@ terms is ^@ 262144 ?^@

Question 595: Complete the number pattern.
20, 30, 40, 50, 60, .........

Question 596: A die is thrown once. What is the probability of getting a number lying between ^@ 1 ^@ and ^@ 5 ^@ ?

Question 597: Find the sign of the following multiplication sentence.

(−a) × (−b) × (−c)

Question 598: The product of two integers is always an ______.

Question 599: If a, b, and c are integers, then according to the associative law of multiplication, (a × b) × c must be equal to:

Question 600: According to the distributive law of multiplication over addition, a × (b + c) must be equal to:

Question 601: Write the missing fraction.

Question 602: If triangle ^@ABC^@ is congruent to triangle ^@DEF,^@ which side is equal to side ^@BC?^@

Question 603: If triangle ^@ABC^@ is congruent to triangle ^@DEF,^@ find the angle equal to angle ^@A .^@

Question 604: If the altitude from one vertex of a triangle bisects the opposite side, prove that the triangle is isosceles.

Question 605: If ^@D^@ is the midpoint of the hypotenuse ^@AC^@ of a right ^@ \triangle ABC^@, prove that ^@ BD = \dfrac { 1 } { 2 } AC ^@.

Question 606: If one acute angle is double than the other in a right-angled triangle, prove that the hypotenuse is double the smallest side.

Question 607: If two angles and the included side of one triangle are equal to the corresponding two angles and the included side of the other triangle, then prove that the two triangles are congruent.

Question 608: If ^@AB ^@ and ^@ CD ^@ are perpendicular to ^@ BC ^@ and ^@ AB = CD^@, show that ^@ EB = EC.^@

Question 609: In the figure, it is given that ^@AC = BC, \angle 4 = 2 \space \angle 1^@ and ^@ \angle 3 = 2 \space \angle 2 ^@. Prove that ^@\triangle ADC \cong \triangle BEC.^@

Question 610: If two angles and any side of a triangle are equal to the corresponding angles and side of another triangle, then prove that the two triangles are congruent.

Question 611: Prove that the perpendiculars drawn from the vertices of equal angles of an isosceles triangle to the opposite sides are equal.

Question 612: Prove that the sides opposite to equal angles of a triangle are equal.

Question 613: If the altitudes from the two vertices of a triangle to the opposite sides are equal, prove that the triangle is isosceles.

Question 614: If two sides ^@AB^@ and ^@BC,^@ and the median ^@AD^@ of ^@ \triangle ABC^@ are correspondingly equal to the two sides ^@PQ^@ and ^@QR,^@ and the median ^@PM^@ of ^@\triangle PQR.^@ Prove that ^@\triangle ABC \cong \triangle PQR.^@

Question 615: In ^@\triangle ABC^@, with ^@AB = AC^@, prove that the altitude from the vertex ^@A^@ bisects the side ^@BC^@.

Question 616: Prove that the sum of the lengths of the three altitudes of a triangle is less than the sum of the lengths of the three sides of the triangle.

Question 617: Prove that the sum of any two sides of a triangle is greater than twice the median drawn to the third side.

Question 618: Show that the perimeter of a triangle is greater than the sum of its three medians.

Question 619: In a right-angled triangle show that the hypotenuse is the longest side.

Question 620: ^@P^@ is any point in the interior of triangle ^@ABC^@. Prove that ^@AB + AC > BP + CP^@.

Question 621: If ^@ A ^@ and ^@B^@ are acute angles such that ^@cot{\space} A ^@ = ^@tan{\space} B^@, find the value of ^@(A + B )^@.

Question 622: If ^@ A^@, ^@B^@, and ^@C^@ are the angles of ^@\Delta ABC^@, show that ^@cot \bigg( \dfrac { B + C } { 2 } \bigg)^@ = ^@tan \bigg( \dfrac { A } { 2 } \bigg).^@

Question 623: If ^@ x^@ = ^@a{\space} cosec{\space} \theta^@ and ^@y^@ = ^@b{\space} sec{\space} \theta^@, prove that ^@\bigg ( \dfrac { a^2 } { x^2 } + \dfrac { b^2 } { y^2 } \bigg)^@ = 1.

Question 624: If ^@x^@ = ^@a{\space} cosec {\space} \theta {\space} sin{\space}\phi, ^@ ^@y^@ = ^@b{\space}cosec{\space}\theta{\space} cos{\space}\phi^@ and ^@z ^@ = ^@c{\space} cot{\space}\theta^@ then prove that ^@ \bigg( \dfrac { x^2 } { a^2 } + \dfrac { y^2 } { b^2 } \bigg) = \bigg( 1 + \dfrac { z^2 } { c^2 } \bigg).^@

Question 625: If ^@ cosec {\space} \theta - sin {\space} \theta = m^@ and ^@sec {\space} \theta - cos {\space} \theta = n^@, prove that ^@(m^2n)^\frac { 2 } { 3 } + (mn^2)^\frac { 2 } { 3 } = 1. ^@

Question 626: If ^@cosec \space \theta + cot \space \theta = m, ^@ show that ^@\dfrac { m^2 - 1 } { m^2 + 1 } = cos \space \theta.^@

Question 627: If ^@sin {\space} \theta + cos {\space} \theta = m ^@ and ^@sec {\space} \theta + cosec {\space} \theta = n^@, prove that ^@n(m^2 - 1) = 2m.^@

Question 628: If ^@ cot \space \theta + cosec \space \theta = \sqrt { 2 } \space cot \space \theta, ^@ show that ^@ cot \space \theta - cosec \space \theta = \sqrt { 2 } \space cosec \space \theta ^@

Question 629: If ^@ x = r{\space} sec{\space} \alpha {\space}cos{\space}\beta^@, ^@y = r{\space} sec {\space}\alpha{\space} sin{\space}\beta^@ and ^@z = r{\space} tan {\space}\alpha^@, prove that ^@{\space}x^2 + y^2 - z^2 = r^2^@ .

Question 630: Each one of ^@A^@ and ^@B^@ has some money. If ^@A^@ gives ^@$ 90^@ to ^@B^@ then ^@B^@ will have twice the money left with ^@A^@. But, if ^@B^@ gives ^@$ 30^@ to ^@A^@ then ^@A^@ will have thrice as much as is left with ^@B^@. How much money does each have?

Question 631: Find the ^@ n^{th} ^@ term of the ^@G.P. 7, 49, 343, .......^@

Question 632: If the length of a rectangle is increased by ^@14^@ units and its breadth is decreased by ^@7^@ units then the area of the rectangle is increased by ^@35^@ square units. However, if we decrease its length by ^@7^@ units and increase breadth by ^@3^@ units. Its area is decreased by ^@105^@ square units. Find the length and breadth of the rectangle.

Question 633: What is the sum of the first ^@ 5 ^@ terms of the geometric series ^@ 1, \dfrac{ 2 }{ 3 }, \dfrac{ 4 }{ 9 } \space ...? ^@

Question 634: The students of a school are made to stand in rows. If the number of students in each row is increased by ^@6^@, there would be ^@4 ^@ row less. If the number of students in each rows is reduced by ^@6^@, there would be ^@5^@ rows more. Find the number of students in the school.

Question 635: Solve the following pair of linear equations by the method of cross-multiplication. @^ \begin{aligned} a x - b y = a + b \\ b x + a y = b - a \end{aligned} @^

Question 636: Solve the following pair of linear equations by using cross multiplication. @^ \begin{aligned} &u x - v y = w \\ &v x - u y = 1 - w \end{aligned} @^

Question 637: What do we call a statement that requires a proof?

Question 638: Factorize: @^ \begin{aligned} p^3 + q^3 + p + q \end{aligned} @^

Question 639: Factorize: @^ (d - e)^3 + (e - f)^3 + (f - d)^3 @^

Question 640: Factorize: @^ (y^3 - y) @^

Question 641: Represent the complex number ^@ z = \sqrt{ 3 } + i ^@ in the polar form.

Question 642: Factorise: @^ \begin{aligned} 1 - 2cd - c^2 - d^2 \end{aligned} @^

Question 643: If the mean of ^@ 6 ^@ observations ^@ x, x + 2, x + 4, x + 6, x + 8, x + 10^@ is ^@ 10 ^@, find the value of ^@ x. ^@

Question 644: Simplify:
^@\Bigg ( \dfrac { \sqrt{ 6 } - \sqrt{ 5 } } { \sqrt{ 6 } + \sqrt{ 5 } } + \dfrac { \sqrt{ 6 } + \sqrt{ 5 } } { \sqrt{ 6 } - \sqrt{ 5 } } \Bigg )^@

Question 645: In ^@\Delta ABC^@, ^@AD \perp BC^@ and ^@AD^2 = BD \times CD^@. Prove that ^@\angle BAC = 90^\circ^@.

Question 646: ^@ ABCD ^@ is a trapezium in which ^@AB || DC^@ and ^@AB = 2DC^@. If the diagonals of trapezium intersect each other at a point ^@O^@, find the ratio of the areas of ^@\Delta AOB^@ and ^@\Delta COD^@.

Question 647: ^@ D^@, ^@E^@ and ^@F^@ are respectively the midpoints of sides ^@AB^@, ^@BC^@ and ^@CA^@ of ^@\Delta ABC^@. Find the ratio of the areas of ^@\Delta DEF^@ and ^@\Delta ABC^@.

Question 648: A ladder ^@ 40 \space m^@ long reaches a window which is ^@24 \space m^@ above the ground on one side of a street. Keeping its foot at the same point, the ladder is turned to the other side of the street to reach a window ^@32 \space m^@ high. Find the width of the street.

Question 649: A ladder ^@ 41 \space m^@ long just reaches the top of a building ^@40 \space m^@ high from the ground. Find the distance of the foot of the ladder from the building.

Question 650: In a rhombus of side ^@ 52 \space cm^@, one of the diagonals is ^@40 \space cm^@ long. Find the length of the second diagonal.

Question 651: Given a ^@ \Delta ABC^@ in which ^@\angle B = 90^\circ^@ and ^@AB = \sqrt{3} BC^@. Prove that ^@\angle C = 60^\circ^@.

Question 652: In a quadrilateral ^@ ABCD ^@, ^@\angle B = 90^\circ^@. If ^@AD^2 = AB^2 + BC^2 + CD^2^@, prove that ^@\angle ACD = 90^\circ^@.

Question 653: Draw an angle of ^@ 60^\circ ^@ .

Question 654: A man goes ^@ 16 \space m^@ due west and then ^@16 \space m^@ due north. How far is he from the starting point?

Question 655: Given a right-angled ^@\Delta ABC^@. The lengths of the sides containing the right angle are ^@7 \space cm^@ and ^@24 \space cm^@. A circle is inscribed in ^@\Delta ABC^@. Find the radius of the circle.

Question 656: In the given figure, points ^@ M ^@ and ^@ N ^@ divide the side ^@ AB ^@ of ^@\Delta ABC^@ into three equal parts. Line segments ^@MP^@ and ^@NQ^@ are both parallel to ^@BC^@ and meet ^@AC^@ at ^@P^@ and ^@Q^@ respectively. Prove that ^@P^@ and ^@Q^@ divide ^@AC^@ into three equal parts i.e ^@AC = 3AP = 3PQ = 3QC^@.

Question 657: Using ruler and compass only, draw a right angle.

Question 658: Three unbiased coins are tossed simultaneously. Find the probability of getting exactly ^@2^@ heads.

Question 659: A coin is tossed once. What is the probability of getting a head.

Question 660: A die is thrown once. Find the probability of getting a multiple of ^@3^@.

Question 661: Two coins are tossed simultaneously. What is the probability of getting at least one tail?

Question 662: Two coins are tossed simultaneously. What is the probability of getting both tails?

Question 663: Find the area of the hexagon ^@ABCDEF^@ in which each side measures ^@17 \space cm,^@ width ^@BD = 30 \space cm,^@ and height ^@CF = 33 \space cm.^@

Question 664: Find the area of the rhombus in which each side is ^@ 25 \space cm^@ long and one of whose diagonals is ^@14 \space cm^@.

Question 665: Find the area of the octagon ^@ ABCDEFGH ^@ having each side equal to ^@17 \space cm, CH = DG = 33 \space cm^@ and ^@EL \perp DG^@ such that ^@EL = 15 \space cm.^@

Question 666: If the area of the rhombus is ^@336 \space cm^2^@ and one of its diagonals is ^@14\space cm,^@ find its perimeter.

Question 667: What do we get when we subtract the following?
9 - 4

Question 668: Subtract the following:
11 - 6

Question 669: Find the Pythagorean triplet whose smallet member is ^@12.^@

Question 670: Express ^@25^@ as a sum of odd natural numbers.

Question 671: Evaluate:
^@ 29 \times 31 ^@

Question 672: How many lines of symmetry are there in the given figure ?

Question 673: A number is increased by ^@40\%^@ and then the increased number is decreased by ^@40\%^@. Find the net increase or decrease percent.

Question 674: A's income is ^@35\%^@ more than that of B. How many percent is B's income less than that of A?

Question 675: Find the percentage of pure gold in ^@ 10 ^@-carat gold, if 24-carat gold is ^@100\%^@ pure gold.

Question 676: How many faces does a triangular prism have?

Question 677: How many vertices does a square pyramid have?

Question 678: Find the length of the longest pole that can be put in a hall of dimensions ^@13 \space m^@ by ^@ 12 \space m ^@ by ^@ 5 \space m.^@

Question 679: What is the number of diagonals in a hexagon?

Question 680: Name the parallelogram in which the diagonals and the adjacent sides are equal.

Question 681: Find the measure of each exterior angle of a regular pentagon.

Question 682: Find the measure of each interior angle of a regular polygon of ^@10^@ sides.

Question 683: Find the number of sides of a regular polygon whose each exterior angle is ^@ 40 ^ \circ. ^@

Question 684: What is the sum of all interior angles of a polygon of ^@10^@ sides^@?^@

Question 685: In the given figure, ^@ABCD^@ is a parallelogram whose diagonals ^@AC^@ and ^@BD^@ intersect at ^@O^@. A line segment through ^@O^@ meets ^@AB^@ at ^@P^@ and ^@DC^@ at ^@Q^@.
Prove that ^@ar(\square APQD)^@ = ^@\dfrac { 1 } { 2 } ar(||gm\space ABCD)^@

Question 686: If the medians of a ^@\Delta ABC^@ intersect at ^@G^@. Prove that ^@ar(\Delta BGC) ^@ is equal to ^@ \dfrac { 1 } { 3 } ar(\Delta ABC) ^@.

Question 687: In the given figure, ^@D^@ is the midpoint of side ^@AB^@ of ^@\Delta ABC^@ and ^@P^@ is any point on ^@BC^@. If ^@CQ||PD^@ meets ^@AB^@ in ^@Q^@, prove that ^@ar(\Delta BPQ)^@ is equal to ^@\dfrac { 1 } { 2 } ar(\Delta ABC)^@.

Question 688: Represent ^@ \dfrac { 1 } { 2 }^@ on the number line.

Question 689: How many years are there in a decade?

Question 690: How many lines of symmetry does a rectangle have?

Question 691: Draw a circle with its two chords ^@PQ^@ and ^@RS^@ such that ^@PQ^@ is not parallel to ^@RS^@. Draw the perpendicular bisector of ^@PQ^@ and ^@RS^@. At what point do they intersect each other?
Justify the steps of construction.

Question 692: Draw a line segment ^@AB^@ and bisect it. Also, bisect one of the equal parts.

Question 693: In the examination hall, Barbara tried hard to simplify an expression in cotangents. But she got confused and was not able to complete it properly. After the examination, she seeks the help of Michelle for simplifying it. Then what would be Barbara's findings, if the expression was:
@^ \dfrac{ 3cot^3x - cotx }{ cot^4x - 3cot^2x } = ? @^

Question 694: Construct a right triangle whose base is ^@17 \space cm^@ and sum of its hypotenuse and another side is ^@23 \space cm^@.

Question 695: Robert is always curious in creating formulas, for solving mathematical problems, of his own by repeated trial and error method. One day he found a formula for finding the surface area of the hexagonal pyramid. What should be the correct formula for the surface area of the hexagonal pyramid?

Question 696: There are a cylinder and a cone with height, ^@h^@ and radius ^@r^@. There is also a sphere of the same radius. The height, ^@h^@ of the cylinder and the cone is ^@ 3 ^@ times the radius, ^@r^@. If ^@V_1^@ is the volume of the cylinder, ^@V_2^@ is the volume of the cone and ^@V_3^@ is the volume of the sphere, write the given volumes in ascending order.

Question 697: How many dollars are there in ^@700^@ cents?

Question 698: If ^@ x = 1 + \sqrt{ 2 } + \sqrt{ 3 }, ^@ then ^@ x^4 - 4x^3 -4x^2 + 16x = ? ^@

Question 699: The diameter of a solid metallic right circular cylinder is equal to its height. After cutting out the largest possible solid sphere ^@S^@ from the cylinder, the remaining material is recast to form a solid sphere ^@S_1^@. What is the ratio of the radius of the sphere ^@S^@ to that of sphere ^@S_1 ?^@

Question 700: How many parallelograms are there in the given figure?

Question 701: Express the given complex number in the form ^@ a + ib: ^@
^@ (1 - i )^{ 4 } ^@

Question 702: Express ^@(log _a \space x)(log _b \space a)^@ as a single logarithm.

Question 703: Find an equation of the circle with center at origin and radius ^@4^@.

Question 704: What is the sum of the first ^@ n ^@ terms of the geometric series ^@ 1, \dfrac{ 3 }{ 5 }, \dfrac{ 9 }{ 25 } \space ...? ^@

Question 705: What is the value of ^@ n ^@ if the sum of ^@ n ^@ terms of the ^@ G.P. \space 2, \dfrac{ 2 }{ 3 }, \dfrac{ 2 }{ 9 } \space ... is \dfrac{ 80 }{ 27 } ^@?

Question 706: Find the sum of the sequence ^@5, 55, 555, 5555, ... ^@ to ^@ n ^@ terms.

Question 707: Find the number of different signals that can be generated by arranging at least ^@2^@ flags in order(one below the other) on a vertical staff, if ^@5^@ different flags are available.

Question 708: Find the value of ^@log _{ 81 } 243^@.

Question 709: Find the slope of the line making the inclination of ^@45^\circ^@ with the positive direction of the ^@x^@-axis.

Question 710: Given ^@7^@ flags of different colors, how many different signals can be generated, if a signal requires the use of ^@3^@ flags one below the other?

Question 711: Find the center and the radius of the circle ^@5 x^2 + 5 y^2 - 2 x = 0^@

Question 712: Find the center and the radius of the circle ^@9 x^2 + 9 y^2 - 2 y = 0^@

Question 713: Solve ^@ x^2 + 5 = 0 ^@

Question 714: Convert the complex number ^@ \dfrac{ 4 }{ 1 - \sqrt{ 3 } i } ^@ into polar form.

Question 715: Suppose ^@ x - y = 1.^@ Find the value of ^@ x^4 - xy^3 - x^{ 3 }y - 3x^{ 2 }y + 3xy^{ 2 } + y^4 . ^@

Question 716: The figure ^@ ABCDEF ^@ is a regular hexagon. Evaluate the quotient ^@ \dfrac { \text{Area of hexagon } ABCDEF } { \text{Area of } \triangle BCE } ^@.

Question 717: If ^@ x > 0 ^@ and ^@ \left( x + \dfrac{ 1 } { x } \right)^2 = 36,^@ find the value of ^@ x^3 + \dfrac{ 1 } { x^3 }. ^@

Question 718: A palindrome number is the same either read from left to right or right to left, for example, ^@ 121. ^@ How many ^@ 5-^@digit palindrome numbers are there altogether^@?^@

Question 719: Let ^@ p ^@ be a prime number such that the next larger number is a perfect square. Find the sum of all such prime numbers. ^@(^@For example, if you think ^@ 11 ^@ and ^@ 13 ^@ are two such prime numbers, then the sum is ^@ 24.) ^@

Question 720: In the figure below, if ^@ \angle A + \angle B + \angle C + \angle D + \angle E + \angle F + \angle G = x ^@ degrees, then what is ^@ x ? ^@

Question 721: Given that ^@w > 0^@ and that ^@ w - \dfrac{1}{w} = 7^@, find the value of ^@\left(w + \dfrac{1}{w} \right)^2^@.

Question 722: Suppose ^@ a \ne 0, b \ne 0, c \ne 0 ^@ and ^@ \dfrac{ a } { b } = \dfrac{ b } { c } = \dfrac{ c } { a } .^@ Find the value of ^@ \dfrac{ a - b + c } {a + b - c }. ^@

Question 723: Suppose that ^@ a, b, c ^@ are distinct numbers such that ^@ ( c - b )^2 - 4( c - a )( a - b ) = 0, ^@ find the value of ^@ \dfrac{ c - a }{ a - b }. ^@

Question 724: From a group of ^@8^@ men and ^@7^@ women, 5 persons are to be selected to form a committee so that at least ^@4^@ men are there on the committee. In how many ways can it be done?

Question 725: Find the integer ^@ x ^@ that satisfies the equation ^@ x^5 - 101 x^3 - 999 x^{ 2 } + 100900 = 0 ^@.

Question 726: ^@ \triangle ABC ^@ is a triangle such that ^@ \angle C = 90^\circ. ^@ Suppose ^@ AC = 12 \space cm, AB = 13 \space cm ^@ and the perpendicular distance from ^@ C ^@ to ^@ AB ^@ is ^@ x \space cm. ^@ Find the value of ^@ x. ^@

Question 727: Consider the function ^@ f(x) = \dfrac{ 1 }{ 2^x + \sqrt{ 2 } }. ^@ Find the value of ^@ \sqrt{ 2 } \left[ f(-5) + f(-4) + f(-3) + f(-2) + f(-1) + f(0) + f(1) + f(2) + f(3) + f(4) + f(5) + f(6) \right]. ^@

Question 728: Let ^@\alpha^@ and ^@\beta^@ be the roots of ^@x^2 - 3 x + c = 0^@, where ^@c^@ is a real number. If ^@-\alpha^@ is a root of ^@x^2 + 3 x - c = 0^@, find the value of ^@\alpha\beta^@.

Question 729: If ^@ABC^@ is a triangle and ^@D^@ is a point on side ^@AB^@ with ^@AD = BD = CD^@, find the value of ^@\angle ACB^@.

Question 730: The ‘^@4^@’ button on my calculator is spoilt, so I cannot enter numbers which contain the digit ^@4^@. Moreover, my calculator does not display the digit ^@4^@ if ^@4^@ is part of an answer either. Thus I cannot enter the calculation ^@3 \times 24^@ and do not attempt to do so. Also, the result of multiplying ^@3^@ by ^@18^@ is displayed as ^@5^@ instead of ^@54^@ and the result of multiplying ^@6^@ by ^@8^@ is displayed as ^@8^@ instead of ^@48^@. If I multiply a positive one-digit number by a positive two-digit number on my calculator and it displays ^@26^@, how many possibilities could I have multiplied?

Question 731: Consider the ^@800^@-digit integer

^@234523452345 ..... 2345.^@

The first ^@m^@ digits and the last ^@n^@ digits of the above integer are crossed out so that the sum of the remaining digits is ^@2345^@. FInd the value of ^@ m + n ^@.

Question 732: Consider the ^@600^@-digit integer

^@234234234 ..... 234.^@

The first ^@m^@ and the last ^@n^@ digits of the above integer are crossed out so that the sum of the remaining digits is ^@234^@. Find the value of ^@ m + n ^@.

Question 733: Let ^@a^@ and ^@b^@ be positive real numbers such that

^@\dfrac{1}{a} - \dfrac{1}{b} - \dfrac{1}{ a + b } = 0^@

Find the value of ^@\left(\dfrac{b}{a} + \dfrac{a}{b} \right)^2^@.

Question 734: At a party, each person shakes hand with every other person. If there was a total of ^@28^@ handshakes, then how many persons were present there?

Question 735: How many digits will be there in the largest integer for which each pair of consecutive digits is a square^@?^@

Question 736: A two-digit number ^@ ab ^@ is multiplied by its reverse ^@ ba ^@. The ones (units) and tens digits of the four-digit resultant number are both ^@ 0. ^@ What is the value of the smallest such two-digit number ^@ ab ?^@

Question 737: How many three-digit integers less than ^@ 601 ^@ have exactly two different digits in their representation (for example, ^@ 232, ^@ or ^@ 466)? ^@

Question 738: A horse traveled at a constant speed of ^@30 \space km/h ^@ for the initial ^@2^@ hours and then at a constant speed of ^@20 \space km/h^@ for ^@3^@ hours. What was its average speed?

Question 739: In the diagram shown, ^@ ABCD ^@ is a square and point ^@ F ^@ lies on ^@ BC. ^@ ^@ \triangle DEC ^@ is equilateral and ^@ EB = EF. ^@ What is the measure of ^@ \angle EBC ? ^@

Question 740: The letters ^@ a, b ^@ and ^@ c ^@ stand for non-zero digits. The integer ^@ abc ^@ is a multiple of ^@ 3 ^@ the integer ^@ cbabc ^@ is a multiple of ^@ 15, ^@ and the integer ^@ abcba ^@ is a multiple of ^@ 8. ^@ What is the value of the integer ^@ cba? ^@

Question 741: A ^@3^@-digit number is of the form ‘high-low-high’ ^@ - ^@ that is the tens digit is smaller than both the hundreds digit and the units (or ‘ones’) digit. How many such ^@3^@-digit numbers are there?

Question 742: ^@ABCD^@ is a square with each side measuring ^@144 \space cm^@. ^@M^@ is a point on ^@CB^@ such that ^@CM = 36 \space cm^@. If ^@N^@ is a variable point on the diagonal ^@DB^@, find the least value of ^@CN + MN^@.

Question 743: Let ^@S^@ be the smallest positive multiple of ^@15^@ that comprises exactly ^@3k^@ digits with k ‘0’s, k ‘3’s and k ‘8’s. Find the remainder when ^@S^@ is divided by ^@8^@.

Question 744: In the figure below, ^@BAD, BCE, ACF^@ and ^@DEF^@ are straight lines. It is given that ^@BA = BC^@, ^@AD = AF^@, ^@EB = ED^@. If ^@\angle BED = x^\circ^@, find the value ^@x^@.

Question 745: Find the value of ^@\sqrt{ 70 + \sqrt{ 4896 } } + \sqrt{ 70 - \sqrt{ 4896 } }^@.

Question 746: Let ^@S = \{1, 2, 3, ....., 46, 47\}^@. What is the maximum value of ^@n^@ such that it is possible to select ^@n^@ numbers from ^@S^@ and arrange them in a circle in such a way that the product of any two adjacent numbers in the circle is less than ^@100?^@

Question 747: Show that there are infinitely many positive prime numbers.

Question 748: Prove that a positive integer ^@n^@ is a prime number if no prime number less than or equal to ^@\sqrt{n}^@ divides ^@n^@.

Question 749: Prove that the internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle.

Question 750: Draw a line parallel to the line ^@AB^@ and passing through ^@C^@ such that ^@C^@ is a point outside ^@AB.^@

Question 751: Draw any line segment ^@ \overline{ AB }. ^@ Mark any point ^@ M ^@ on it. Through ^@ M, ^@ draw a perpendicular to ^@ \overline{ AB }. ^@

Question 752: Draw a line segment ^@ \overline{ PQ }. ^@ Take a point ^@ R ^@ outside the line segment ^@ \overline{ PQ }. ^@ Through ^@ R, ^@ draw a line perpendicular to ^@ \overline{ PQ }. ^@

Question 753: Using a compass draw an angle of measure ^@ 135^\circ. ^@

Question 754: In an objective type question, ^@4^@ options are available out of which one is the correct answer. How likely is that the correct answer is chosen?

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K-12 Math Questions and Answers