Math Question and Answers


To grasp the understanding of the basic concepts of mathematics it is essential for the students to solve questions, they give an idea of how math can help them in real-world situations and also enhancing their cognitive and inquisitive skills. The math questions allow creative minds to explore the different types of problems, recall the concepts, and apply them to reach a solution.

Solving maths questions instills a strong mathematical acumen in the young minds and also helps as a great tool for the teachers and the parents in assessing the student's understanding of any particular topic. Hence math questions and answers should be solved step by step to gain the most information out of it. Math worksheets are an important tool in continuing efforts to motivate students during class and to engage their minds. The math questions and answers stimulate imaginative thought as well.

Math Question and Answer

A bus stop is barricaded from the remaining part of the road, by using 50 hollow cones made of recycled cardboard. Each cone has a base diameter of 40 cm and height 1 m.

100 surnames were randomly picked up from a local telephone directory and a frequency distribution of the number of letters in the English alphabet in the surnames was found as follows:

1500 families with 2 children were selected randomly, and the following data were recorded: Compute the probability of a family, chosen at random, having i) 2 girls ii) 1 girl iii) No girl

A capsule of medicine is in the shape of a sphere of diameter 3.5 mm. How much Medicine (in mm3) is needed to fill this capsule?

Find the zero of the polynomials in each of the following case: (i) p(x) = x + 5 (ii) p(x) = x - 5 (iii) p(x) = 2x+ 5 (iv) p(x) = 3x - 2 (v) p(x) = 3x (vi) p(x) = ax,a ≠ 0 (vii) p(x) = cx + d,c ≠ 0

A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc

A circular park of radius 20m is situated in a colony. Three boys Ankur, Syed and David are sitting at equal distance on its boundary each having a toy telephone in his hands to talk each other

A city has two main roads which cross each other at the center of the city. These two roads are along the North-South direction and East-West direction. All other streets of the city run parallel

A company manufactures car batteries of a particular type. The lives (in years) of 40 such batteries were recorded as follows: Construct a grouped frequency distribution table for this data

A conical pit of top diameter 3.5 m is 12 m deep. What is its capacity in kilolitres

A conical tent is 10 m high and the radius of its base is 24 m. Find i) Slant height of the tent. ii) Cost of the canvas required to make the tent, if the cost of 1 m² canvas is ₹ 70

A cubical box has each edge 10 cm and another cuboidal box is 12.5 cm long, 10 cm wide and 8 cm high. i) Which box has the greater lateral surface area and by how much?

A cylindrical pillar is 50 cm in diameter and 3.5 m in height. Find the cost of painting the curved surface of the pillar at the rate of ₹12.50 per m

A cuboidal water tank is 6 m long, 5 m wide and 4.5 m deep. How many liters of water can it hold? (1 m^3 = 1000 L). The cuboidal water tank can hold 135000 liters of water.

A cuboidal vessel is 10 m long and 8 m wide. How high must it be made to hold 380 cubic metres of a liquid? The cuboidal vessel must be made 4.75 m high.

A dome of a building is in the form of a hemisphere. From inside, it was whitewashed at the cost of ₹4989.60. If the cost of white washing is ₹20 per square meter, find inside surface area of dome

A farmer was having a field in the form of a parallelogram PQRS. She took any point A on RS and joined it to points P and Q. In how many parts the fields is divided? What are shapes of these parts?

A floral design on a floor is made up of 16 tiles which are triangular, the sides of the triangle being 9 cm, 28 cm, and 35 cm. Find the cost of polishing the tiles at the rate of 50p per cm^2.

A field is in the shape of a trapezium whose parallel sides are 25 m and 10 m. The non-parallel sides are 14 m and 13 m. Find the area of the field

A godown measures 40m × 25m × 15m. Find the maximum number of wooden crates each measuring 1.5m × 1.25m × 0.5m that can be stored in the godown

A hemispherical bowl is made of steel, 0.25 cm thick. The inner radius of the bowl is 5 cm. Find the outer curved surface area of the bowl

A heap of wheat is in the form of a cone whose diameter is 10.5 m and height is 3 m. Find its volume. The heap is to be covered by canvas to protect it from rain. Find the area of the canvas required.

A hemispherical bowl made of brass has inner diameter 10.5 cm. Find the cost of tin-plating it on the inside at the rate of ₹16 per 100 cm. The cost of tin-plating is ₹27.72

A joker’s cap is in the form of a right circular cone of base radius 7cm and height 24cm. Find the area of the sheet required to make 10 such caps

A hemispherical tank is made up of an iron sheet 1 cm thick.If the inner radius is 1m,then find the volume of the iron used to make the tank

A kite in the shape of a square with a diagonal 32cm and an isosceles triangle of base 8cm and sides 6cm each is to be made of three different shades as shown. How much paper of each shade used

A lead pencil consists of a cylinder of wood with a solid cylinder of graphite filled in the interior. If length of the pencil is 14 cm, find volume of the wood and that of the graphite

A matchbox measures 4 cm × 2.5 cm × 1.5 cm. What will be the volume of a packet containing 12 such boxes?

A metal pipe is 77 cm long. The inner diameter of a cross section is 4 cm, the outer diameter being 4.4 cm (see Fig. 13.11). Find its i) Inner curved surface area ii) Outer curved surface area

A park, in the shape of a quadrilateral ABCD, has ∠C = 90°, AB = 9 m, BC = 12 m, CD = 5 m and AD = 8 m. How much area does it occupy?

A patient in a hospital is given soup daily in a cylindrical bowl of diameter 7cm.If bowl is filled with soup to height of 4cm,how much soup has to prepare dailytoserve 250 patients

A plastic box 1.5 m long, 1.25 m wide and 65 cm deep, is to be made. It is opened at the top. Ignoring the thickness of the plastic sheet, determine: i) The area of the sheet required

A random survey of the number of children of various age groups playing in a park was found as follows: Draw a histogram to represent the data above

A right circular cylinder just encloses a sphere of radius r. Find i) Surface area of the sphere, ii) Curved surface area of the cylinder, iii) Ratio of the areas obtained in (i) and (ii).

A rhombus shaped field has green grass for 18 cows to graze. If each side of the rhombus is 30 m and its longer diagonal is 48 m, how much area of grass field will each cow be getting?

A right triangle ABC with sides 5 cm, 12 cm and 13 cm is revolved about the side 12 cm. Find the volume of the solid so obtained

A river 3 m deep and 40 m wide is flowing at the rate of 2 km per hour. How much water will fall into the sea in a minute?

A small indoor greenhouse (herbarium) is made entirely of glass panes (including base) held together with tape. It is 30 cm long, 25 cm wide and 25 cm high. i) What is the area of the glass?

A soft drink is available in two packs - i) a tin can with a rectangular base of length 5 cm and width 4 cm, having a height of 15 cm, and ii) a plastic cylinder with a circular base of diameter 7 cm

A solid cube of side 12 cm is cut into 8 cubes of equal volume. What will be side of new cube? Also, find ratio between their surface areas

A survey conducted by an organization for the cause of illness and death among the women between the ages 15 - 44 (in years) worldwide, found the following figures (in%): (i) Represent it graphically

A study was conducted to find out the concentration of sulphur dioxide in the air in parts per million (ppm) of a certain city. The data obtained for 30 days is as follows

A triangle and a parallelogram have the same base and the same area. If the sides of the triangle are 26cm, 28cm, and 30cm, and the parallelogram stands on the base 28cm, find the height

A traffic signal board, indicating ‘SCHOOL AHEAD’, is an equilateral triangle with side ‘a’. Find the area of the signal board, using Heron’s formula. If its perimeter is 180 cm.

A villager Itwaari has a plot of land of the shape of a quadrilateral. The Gram Panchayat of the village decided to take over some portion of his plot from one of the corners to construct a Health

A village, having a population of 4000, requires 150 litres of water per head per day. It has a tank measuring 20 m × 15 m × 6 m. For how many days will the water of this tank last?

A wooden bookshelf has external dimensions as follows: Height = 110 cm, Depth = 25 cm, Breadth = 85 cm. The thickness of the plank is 5 cm everywhere. The external faces are to be polished

AB and CD are respectively the smallest and longest sides of a quadrilateral ABCD (see Fig. 7.50). Show that ∠A > ∠C and ∠B > ∠D

AB is a line segment and P is its mid-point. D and E are points on the same side of AB such that Show that ∠BAD = ∠ABE and ∠EPA = ∠DPB (See the given figure). i) ΔDAP ≅ ΔEBP ii) AD = BE

ABC and ADC are two right triangles with common hypotenuse AC. Prove that ∠CAD = ∠CBD

ΔABC and ΔDBC are two isosceles triangles on the same base BC and vertices A and D are on the same side of BC.If AD is extended to intersect BC at P, show that i) ΔABD ≅ ΔACD ii) ΔABP ≅ ΔACP

ABC is a right-angled triangle in which ∠A = 90° and AB = AC. Find ∠B and ∠C

ABC and DBC are two isosceles triangles on the same base BC. Show that ∠ABD = ∠ACD

ABC is a triangle in which altitudes BE and CF to sides AC and AB are equal (see the given figure). Show that (i) ΔABE ≅ ΔACF (ii) AB = AC, i.e., ABC is an isosceles triangle.

ΔABC is an isosceles triangle in which AB = AC. Side BA is produced to D such that AD = AB.Show that ∠BCD is a right angle

ABC is a triangle. Locate a point in the interior of ∆ABC which is equidistant from all the vertices of ∆ABC

ABC is a triangle right angled at C. A line through the mid-point M of hypotenuse AB and parallel to BC intersects AC at D. Show that i) D is the mid-point of AC ii) MD ⊥ AC iii) CM = MA = 1/2 AB

ABC is an isosceles triangle in which altitudes BE and CF are drawn to equal sides AC and AB respectively (see Fig. 7.31). Show that these altitudes are equal

ABC is an isosceles triangle with AB = AC. Draw AP ⊥ BC to show that ∠B = ∠C

ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If ∠DBC = 70°, ∠BAC is 30° find ∠BCD. Further if AB = BC, find ∠ECD

ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD (see Fig. 8.21). Show that (i) ΔAPB ≅ ΔCQD (ii) AP = CQ

ABCD is a quadrilateral in which AD = BC and ∠DAB = ∠CBA . Prove that i) △ ABD ≅ △ BAC ii) BD = AC iii) ∠ABD = ∠BAC

ABCD is a parallelogram. The circle through A, B and C intersect CD (produced if necessary) at E. Prove that AE = AD

ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA. AC is a diagonal. Show that: (i) SR || AC and SR = 1/2AC (ii) PQ = SR (iii) PQRS is a parallelogram.

ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA respectively. Show that quadrilateral PQRS is a rhombus

ABCD is a rectangle in which diagonal AC bisects ∠A as well as ∠C. Show that: (i) ABCD is a square (ii) diagonal BD bisect ∠B as well as ∠D

ABCD is rhombus and P, Q, R and S are midpoints of the sides AB, BC, CD, DA respectively. Show that quadrilateral PQRS is rectangle

ABCD is a trapezium with AB || DC. A line parallel to AC intersects AB at X and BC at Y. Prove that ar (ADX) = ar (ACY)

ABCD is a rhombus. Show that diagonal AC bisects ∠A as well as ∠C and diagonal BD bisects ∠B as well as ∠D

ABCD is trapezium in which AB || DC, BD is a diagonal, and E is the mid-point of AD. A line is drawn through E parallel to AB intersecting BC at F. Show that F is the mid-point of BC.

ABCD is trapezium where AB || CD, AD = BC. Show i) ∠A = ∠B ii) ∠C = ∠D iii) ∆ABC ≅ ∆BAD iv) diagonal AC = diagonal BD [Hint: Extend AB and draw line through C || to DA intersecting AB produced at E.]

AC and BD are chords of a circle which bisect each other. Prove that (i) AC and BD are diameters, (ii) ABCD is a rectangle

AD and BC are equal perpendiculars to a line segment AB. Show that CD bisects AB

AD is an altitude of an isosceles triangle ABC in which AB = AC.Show that i) AD bisects BC ii) AD bisects ∠A

An organisation selected 2400 families at random and surveyed them to determine a relationship between income level and the number of vehicles in a family. The information gathered is listed below:

An isosceles triangle has perimeter 30 cm and each of the equal sides is 12 cm. Find the area of the triangle

An umbrella is made by stitching 10 triangular pieces of cloth of two different colours, each piece measuring 20cm, 50cm, and 50cm. How much cloth of each colour is required for the umbrella?

Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number.

BE and CF are two equal altitudes of a triangle ABC. Using RHS congruence rule, prove that the triangle ABC is isosceles. Δ ABC is isosceles

Bisectors of angles A, B and C of a triangle ABC intersect its circumcircle at D, E and F respectively. Prove that the angles of the triangle DEF are 90° - 1/2 A, 90° - 1/2 B, 90° - 1/2 C

Check whether 7 + 3x is a factor of p(x) = 3x^3 + 7x

Check which of the following are solutions of the equation x - 2y = 4 and which are not: i) (0, 2) ii) (2, 0) iii) (4, 0) iv) (√2, 4√2) v) (1, 1)

Classify the data in Q.1 above as primary or secondary data

Classify the following as linear, quadratic and cubic polynomials: i) x² + x ii) x - x3 iii) y + y² + 4 iv) 1 + x v) 3t vi) r² vii) 7x3

Classify the following numbers as rational or irrational: i) 2 - √5 ii) (3 + √23) - √23 iii) 2√7 ÷ 2√7 iv) 1/√2 v) 2π

Classify the following numbers as rational or irrational: i) √23 ii) √225 iii) 0.3796 iv) 7.478478... v) 1.101001000100001...

Complete the hexagonal and star shaped rangolies by filling them with as many equilateral triangles of side 1 cm as you can.Count the number of triangles in each case.

Consider two 'postulates' given below: i) Given any two distinct points A and B there exists a third point C which is in between A and B. ii) There exist at least three points that are not on the same

Construct a right triangle whose base is 12 cm and sum of its hypotenuse and other side is 18 cm

Construct a triangle ABC in which BC = 7 cm, ∠B = 75° and AB + AC = 13 cm

Construct a triangle ABC in which BC = 8cm, ∠B = 45° and AB - AC = 3.5 cm

Construct a triangle PQR in which QR = 6cm, Q = ∠60° and PR - PQ = 2cm

Construct an angle of 45° at the initial point of a given ray and justify the construction

Construct an equilateral triangle, given its side and justify the construction. An equilateral triangle has three equal sides and three angles equal to 60º each.

Construct an angle of 90 degree at the initial point of a given ray and justify the construction

Construct a triangle XYZ in which ∠Y = 30°, ∠Z = 90° and XY + YZ + ZX = 11 cm

Construct the following angles and verify by measuring them by a protractor: (i) 75 (ii) 105 (iii) 135

Construct the angles of the following measurements: (i) 30 (ii) 22(1/2) (iii) 15

Curved surface area of a right circular cylinder is 4.4 m². If the radius of the base of the cylinder is 0.7 m, find its height.

Curved surface area of a cone is 308 cm and its slant height is 14 cm. Find i) Radius of the base ii) Total surface area of the cone

D and E are points on sides AB and AC respectively of ∆ABC such that ar (DBC) = ar (EBC). Prove that DE || BC

D, E and F are respectively the mid-points of the sides BC, CA and AB of a ΔABC. Show that: i) BDEF is a parallelogram. ii) ar (DEF) = 1/4 ar (ABC) iii) ar (BDEF) = 1/2 ar (ABC)

Determine which of the following polynomials has (x + 1) a factor: i) x^3 + x^2 + x + 1 ii) x^4 + x^3 + x^2 + x + 1 iii) x^4 + 3x^3 + 3x^2 + x +1 iv) x^3 - x^2 - (2 + √2)x + √2

Diagonals AC and BD of a quadrilateral ABCD intersect at O in such a way that ar (AOD) = ar (BOC). Prove that ABCD is a trapezium

Diagonal AC of a parallelogram ABCD bisects ∠A (see the given figure). Show that i) it bisects ∠C also, ii) ABCD is a rhombus

Diagonals AC and BD of a quadrilateral ABCD intersect each other at P. Show that ar (ΔAPB) × ar (ΔCPD)= ar (ΔAPD) × ar (ΔBPC)

Diagonals AC and BD of a trapezium ABCD with AB || DC intersect each other at O. Prove that ar (AOD) = ar (BOC). Hence it is proved that ar (AOD) = ar (BOC)

Diameter of the base of a cone is 10.5 cm and its slant height is 10 cm. Find its curved surface area

Draw different pairs of circles. How many points does each pair have in common? What is the maximum number of common points?

Does Euclid fifth postulate imply the existence of parallel lines? Explain

Draw the graph of each of the following linear equations in two variables: i) x + y = 4 ii) x - y = 2 iii) y = 3x iv) 3 = 2x + y

Eleven bags of wheat flour, each marked 5 kg, actually contained the following weights of flour (in kg): 4.97, 5.05, 5.08, 5.03, 5.00, 5.06, 5.08, 4.98, 5.04, 5.07, 5.00

Expand each of the following, using suitable identities: i) (x + 2y + 4z)² ii) (2x - y + z)² iii) (-2x + 3y + 2z)² iv) (3a - 7b - c)² v) (-2x + 5y - 3z)² vi) [(1/4)a - (1/2)b + 1]²

Evaluate the following products without multiplying directly: (i) 103 × 107 (ii) 95 × 96 (iii) 104 × 96 103×107, 95×96, and 104×96

Evaluate the following using suitable identities: (i) (99)^3 (ii) (102)^3 (iii) (998)^3

Express the following in the form of p/q, where p and q are integers and q ≠ 0. i) 0.6 ii) 0.47 iii) 0.001. 0.6, 0.47, 0.001 can be expressed in the form of p/q as 2/3, 43/90, and 1/999.

Express 0.999 ... in form of p/q. Are you surprised by your answer? With your teacher & classmates discuss why the answer makes sense?

Express the following linear equations in the form ax + by + c = 0 and indicate the values of a, b, c in each case: i) 2x + 3y = 9.35 ii) x - y/5 -10 = 0 iii) -2x + 3y = 6 iv) x = 3y v) 2x = - 5y

Factorise: 27x³ + y³ + z³ - 9xyz

Factorise each of the following: i) 27y³ + 125z³ ii) 64m³ - 343n³ = (4m)³ - (7n)³ [Hint: See Question 9]

Factorise each of the following: i) 8a³ + b³ + 12a²b + 6ab² ii) 8a³ - b³ - 12a²b + 6ab² iii) 27 - 125a³ - 135a + 225a² iv) 64a³ - 27b³ - 144a²b + 108ab² v) 27p³ - 1/216 - (9/2)p² + (1/4)p

Factorise: (i) 12x^2 - 7x + 1 (ii) 2x^2 + 7x + 3 (iii) 6x^2 + 5x - 6 (iv) 3x^2 - x - 4

Factorise: (i) 4x² + 9y² + 16z² + 12xy - 24yz - 16xz (ii) 2x² + y² + 8z² - 2√2xy + 4√2yz - 8xz

Factorise: (i) x^3 - 2x^2 - x + 2 (ii) x^3 - 3x^2 - 9x - 5 (iii) x^3 + 13x^2 + 32x + 20 (iv) 2y^3 + y^2 - 2y - 1

Factorise the following using appropriate identities: (i) 9x^2 + 6xy + y^2 (ii) 4y^2 - 4y + 1 (iii) x^2 - y^2/100

Fill in the blanks i) The centre of a circle lies in____of the circle. (exterior / interior) ii) A point, whose distance from the centre of a circle is greater than its radius lies in____of the circle

Find five rational numbers between 3/5 and 4/5

Find: i) 6^1/2 ii) 32^1/5 iii) 125^1/3

Find i) The lateral or curved surface area of a closed cylindrical petrol storage tank that is 4.2 m in diameter and 4.5 m high. ii) How much steel was actually used, if 1/12

Find i) 9^3/2 ii) 32^2/5 iii) 16^3/4 iv) 125^(-1/3)

Find six rational numbers between 3 and 4

Find p(0), p(1) and p 2) for each of the following polynomials: (i) p(y) = y2 - y + 1 (ii) p(t) = 2 + t + 2t2 - t2 iii) p(x) = x3 iv) p(x) = (x - 1)(x + 1)

Find the amount of water displaced by a solid spherical ball of diameter. i) 28 cm ii) 0.21 m

Find the area of a quadrilateral ABCD in which AB = 3 cm, BC = 4 cm, CD = 4 cm, DA = 5 cm and AC = 5 cm

Find the area of a triangle two sides of which are 18cm and 10cm and the perimeter is 42cm

Find the capacity in litres of a conical vessel with i) radius 7 cm, slant height 25 cm ii) height 12 cm, slant height 13 cm

Find the cost of digging a cuboidal pit 8 m long, 6 m broad and 3 m deep at the rate of inr 30 per m3

Find the mean salary of 60 workers of a factory from the following table

Find the mode of 14, 25, 14, 28, 18, 17, 18, 14, 23, 22, 14, 18

Find the radius of a sphere whose surface area is 154 cm2

Find the remainder when x^3 + 3x^2 + 3x + 1 is divided by i) x + 1 ii) x - (1/2) iii) x iv) x + π v) 5 + 2x

Find the remainder when x3 - ax2 + 6x - a is divided by x - a

Find the surface area of a sphere of diameter: i) 14 cm ii) 21 cm iii) 3.5 m

Find the total surface area of a cone, if its slant height is 21 m and diameter of its base is 24 m

Find the surface area of a sphere of radius: i) 10.5 cm ii) 5.6 cm iii) 14 cm

Find the value of k, if x - 1 is a factor of p(x) in each of the following cases: (i) p(x) = x2 + x + k (ii) p(x) = 2x2 + kx + √2 (iii) p(x) = kx2 - √2x + 1 (iv) (x) = kx2 - 3x + k

Find the total surface area of a hemisphere of radius 10 cm. (Use π = 3.14)

Find the value of k, if x = 2, y = 1 is a solution of the equation 2x + 3y = k

Find the value of the polynomial 5x - 4x² + 3 at i) x = 0 ii) x = -1 iii) x = 2

Find the volume of a sphere whose radius is i) 7 cm ii) 0.63 m

Find the volume of a sphere whose surface area is 154 cm2

Find the volume of the right circular cone with i) radius 6 cm, height 7 cm ii) radius 3.5 cm, height 12 cm

Find three irrational numbers between the rational numbers 5/7 and 9/11.

From the choices given below, choose the equation whose graphs are given in Fig. 4.6 and Fig. 4.7. For fig. 4.6 (i) y = x (ii) x + y = 0 (iii) y = 2x (iv) 2 + 3y = 7x

Give a definition for each of the following terms. Are there other terms that need to be defined first? What are they, and how might you define them?

Give five examples of data that you can collect from your day-to-day life

Give one example each of a binomial of degree 35, and of a monomial of degree 100.

Give one example of a situation in which: (i) The mean is an appropriate measure of central tendency. (ii) The mean is not an appropriate measure of central tendency, but the median is an appropriate

Give possible expressions for the length and breadth of each of the following rectangles, in which their areas are given: i) Area: 25a² - 35a + 12 ii) Area: 35y² + 13y - 12.

Give the equations of two lines passing through (2, 14). How many lines can pass through (2,14) and why?

Give the geometric representation of y = 3 as an equation i) in one variable ii) in two variables

Give the geometric representations of 2x + 9 = 0 as an equation (i) in one variable (ii) in two variables

Given below are the seats won by different political parties in the polling outcome of a state assembly elections: i) Draw a bar graph to represent the polling results.

How will you describe the position of a table lamp on your study table to another person?

How many litres of milk can a hemispherical bowl of diameter 10.5 cm hold? By using the concept of volume, we have found that 0.303 litres of milk can be held in the bowl

How would you rewrite Euclid’s fifth postulate so that it would be easier to understand?

If a line intersects two concentric circles (circles with the same center) with center O at A, B, C and D, prove that AB = CD

If a point C lies between two points A and B such that AC = BC, then prove that AC = 1/2 AB. Explain by drawing the figure

If the diagonals of a parallelogram are equal, then show that it is a rectangle

If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle

If E, F, G, and H are mid-points of the sides of parallelogram ABCD, show that ar (EFGH) = 1/2 ar (ABCD)

If the non-parallel sides of a trapezium are equal, prove that it is cyclic

If the lateral surface of a cylinder is 92.4 cm3 and its height is 5 cm, then find (i) radius of its base (ii) its volume Use π = 3.14

If the point (3, 4) lies on the graph of the equation 3y = ax + 7, find the value of a

If the triangle ABC in the Question 7 above is revolved about the side 5 cm, then find the volume of the solid so obtained. Find also the ratio of the volumes of the two solids

If the volume of a right circular cone of height 9 cm is 48π cm3, find the diameter of its base

If two circles intersect at two points, prove that their centres lie on the perpendicular bisector of the common chord

If the work done by a body on application of a constant force is directly proportional to the distance travelled by the body, express this in the form of an equation in two variables and draw graph.

If two equal chords of a circle intersect within the circle, prove that line joining the point of intersection to the center makes equal angles with the chords

If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to corresponding segments of the other chord. Draw perpendicular from center to chord

If x + y + z = 0, show that x³ + y³ + z³ = 3xyz.

In a cricket match, a batswoman hits a boundary 6 times out of 30 balls she plays. Find the probability that she did not hit a boundary

In a hot water heating system, there is a cylindrical pipe of length 28 m and diameter 5 cm. Find the total radiating surface in the system

In a huge park people are concentrated at three points (see Fig. 7.52): A: where there are different slides and swings for children, B: near which a man-made lake is situated.

In a mathematics test given to 15 students, the following marks (out of 100) are recorded: 41, 39, 48, 52, 46, 62, 54, 40, 96, 52, 98, 40, 42, 52, 60 Find the mean, median and mode of this data

In a triangle ABC, E is the mid-point of median AD. Show that ar (BED) = 1/4 ar(ABC)

In a parallelogram ABCD, E and F are the mid-points of sides AB and CD respectively (see the given figure). Show that the line segments AF and EC trisect the diagonal BD.

In a triangle locate a point in its interior which is equidistant from all the sides of the triangle

In ∆ABC ∆DEF, AB=DE, AB||DE, BC=EF , BC||EF. A,B & C are joined to D, E and F. Show i)ABED is parallelogram ii)BEFC is parallelogram iii)AD||CF, AD = CF iv)ACFD is parallelogram v)AC=DF vi)∆ABC≅∆DEF

In ΔABC,AD is the perpendicular bisector of BC.Show that ΔABC is an isosceles triangle in which AB=AC.If in ΔABC,AD bisects BC,then ΔABC is an isosceles triangle with AB=AC.

In any triangle ABC, if the angle bisector of ∠A and perpendicular bisector of BC intersect, prove that they intersect on the circumcircle of the triangle ABC

In an isosceles triangle ABC, with AB = AC, the bisectors of ∠B and ∠C intersect each other at O. Join A to O. Show that: i) OB = OC ii) AO bisects ∠A

In countries like USA and Canada, temperature is measured in Fahrenheit, whereas in countries like India, it is measured in Celsius. Here is a linear equation that converts Fahrenheit to Celsius

In Fig. 10.36, A, B and C are three points on a circle with center O such that ∠BOC = 30º and ∠AOB = 60º. If D is a point on the circle other than the arc ABC, find ∠ADC

In Fig. 10.37, ∠PQR = 100° where P, Q and R are points on a circle with center O. Find ∠OPR. The angle subtended by an arc at the center is double the angle subtended by it at any point

In Fig. 10.38, ∠ABC = 69° and ∠ACB= 31°, find ∠BDC

In Fig. 10.39, A, B, C and D are four points on a circle. AC and BD intersect at a point E such that ∠BEC = 130° and ∠ECD = 20°. Find ∠BAC

In Fig. 13.12, you see the frame of a lampshade. It is to be covered with a decorative cloth. The frame has a base diameter of 20 cm and height of 30 cm. A margin of 2.5 cm

In Fig. 6.13, lines AB and CD intersect at O. If ∠AOC + ∠BOE = 70° and ∠BOD = 40°, find ∠BOE and reflex ∠COE

In Fig. 5.10, if AC = BD, then prove that AB = CD

You know 1/7 = 0.14258. Can you predict what the decimal expansions of 2/7, 3/7, 4/7, 5/7, 6/7 are, without actually doing the long division? If so, how?

Yamini and Fatima, two students of Class IX of a school, together contributed ₹100 towards the Prime Minister’s Relief Fund to help the earthquake victims. Write a linear equation.

XY is a line parallel to side BC of a triangle ABC. If BE || AC and CF || AB meet XY at E and F respectively, show that ar (ABE) = ar (ACF)

Write True or False: Give reasons for your answers. (i) Line segment joining the centre to any point on the circle is a radius of the circle. (ii) A circle has only finite number of equal chords.

Write three numbers whose decimal expansions are non-terminating and non-recurring.

Write numbers in decimal and write decimal expansion type: i 36/100 ii 1/11 iii 4 1/8 iv 3/13 v 2/11 vi 329/400.

Write the following cubes in expanded form: i) (2x +1)³ ii) (2a - 3b)³ iii) (3x/2 + 1)³ iv) (x - 2y/3)³

Write the degree of each of the following polynomials: i) 5x3 + 4x2 + 7x ii) 4 - y2 iii) 5t - √7 iv) 3

Write the coefficients of x² in each of the following: i) 2 + x² + x ii) 2 - x² + x³ iii) (π/2)x² + x iv) √2x - 1

Four solutions for each of the following equations 2x + y = 7, πx + y = 9 and x = 4y

Write the answer of each of the following questions: (i) What is the name of horizontal and the vertical lines drawn to determine the position of any point in the Cartesian plane?

Without actually calculating the cubes, find the value of each of the following: i) (-12)³ + (7)³ + (5)³ ii) (28)³ + (-15)³ + (-13)³.

Why is Axiom 5, in the list of Euclid’s axioms, considered a ‘universal truth’? (Note that the question is not about the fifth postulate.)

Which one of the following options is true, and why? y = 3x + 5 has (i) A unique solution (ii) Only two solutions (iii) Infinitely many solutions

Which of the following statements are true and which are false? Give reasons for your answers. i) Only one line can pass through a single point. ii) There are an infinite number of lines

Which of the following figures lie on the same base and between the same parallels. In such a case, write the common base and the two parallels

Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer. i) 4x² - 3x + 7 ii) y² + √2 iii) 3√t + t√2 iv) y + (2/y) v) x10 + y3 + t50

What length of tarpaulin 3m wide will be required to make conical tent of height 8m and base radius 6m? Assume that the extra length of material that will be required for stitching margins and wastage

What can the maximum number of digits be in the repeating block of digits in the decimal expansion of 1/17? Perform the division to check your answer.

What are the possible expressions for the dimensions of the cuboids whose volume are given below? i) Volume: 3x² - 12x ii) Volume: 12ky² + 8ky - 20k

Visualize 3.765 on the number line, using successive magnification

Visualise 4.26 on the number line, up to 4 decimal places

Verify whether the following are zeroes of the polynomial. (i) p(x) = 3x + 1, x = -(1/3) (ii) p(x) = 5x - π , x = 4/5 (iii) p(x) = x2 - 1, x = 1, -1 (iv) p(x) = (x + 1)(x - 2), x = -1, 2

Verify that x³ + y³ + z³ - 3xy = 1/2 (x + y + z)[(x - y)² + (y - z)² + (z - x)²]

Verify: i) (x³ + y³) = (x + y)(x² - xy + y²) ii) (x³ - y³) = (x - y)(x² + xy + y²) i) (x³ + y³) = (x + y)(x² - xy + y² ) (x + y)(x² - xy + y²) = x(x² - xy + y² ) + y(x² - xy + y²)

Use the Factor Theorem to determine whether g(x) is a factor of p(x) in the following cases: (i) p(x)=2x^3 + x^2-2x -1,g(x)=x+1 (ii) p(x)=x^3 + 3x^2+3x+ 1,g(x)=x+2 (iii) p(x)=x^3 - 4x^2+x+ 6,g(x)=x- 3

Use suitable identities to find the following products: (i) (x + 4) (x + 10) (ii) (x + 8) (x - 10) (iii) (3x + 4) (3x - 5) (iv) (y^2 + 3/2)(y^2 - 3/2) (v) (3 - 2x) (3 + 2x)

Two sides AB and BC and median AM of one triangle ABC are respectively equal to sides PQ and QR and median PN of ΔPQR (see Fig. 7.40). Show that: (i) Δ ABM ≅ Δ PQN (ii) Δ ABC ≅ Δ PQR

Two congruent circles intersect each other at points A and B. Through A any line segment PAQ is drawn so that P, Q lie on the two circles. Prove that BP = BQ

Two circles intersect at two points B and C. Through B, two line segments ABD and PBQ are drawn to intersect the circles at A, D, P and Q respectively (see Fig. 10.40). Prove that ∠ACP = ∠QCD

Two circles of radii 5 cm and 3 cm intersect at two points and the distance between their centers is 4 cm. Find the length of the common chord

Twenty-seven solid iron spheres, each of radius r and surface area S are melted to form a sphere with surface area S′. Find the i) Radius r′ of the new sphere, ii) Ratio of S and S′

Two chords AB and CD of lengths 5 cm and 11 cm respectively of a circle are parallel to each other and are on opposite sides of its center. If the distance between AB and CD is 6 cm, find the radius.

To know the opinion of the students about the subject statistics, a survey of 200 students was conducted. The data is recorded in the following table. Find the probability.

Three coins were tossed 30 times simultaneously. Each time the number of Heads occurring was noted down as follows: Prepare a frequency distribution table for the data given above.

Three coins are tossed simultaneously 200 times with the following frequencies of different outcomes: If the three coins are simultaneously tossed again, compute the probability of 2 heads coming up

There is a slide in a park. One of its side walls has been painted in some colour with a message KEEP THE PARK GREEN AND CLEAN. If the sides of the wall are 15m, 11m and 6m, find the area

The volume of a right circular cone is 9856 cm3. If the diameter of the base is 28 cm, find i) height of the cone ii) slant height of the cone iii) curved surface area of the cone

The value of π up to 50 decimal places is given below: 3.14159265358979323846264338327950288419716939937510 i) Make a frequency distribution of the digits from 0 to 9 after the decimal point.

Three girls Reshma, Salma and Mandip are playing a game by standing on a circle of radius 5m drawn in a park. Reshma throws a ball to Salma, Salma to Mandip, Mandip to Reshma.

Thirty children were asked about the number of hours they watched TV Programmes in the previous week. The results were found as follows: i) Make a grouped frequency distribution table for this data,

The triangular side walls of a flyover have been used for advertisements. The sides of the walls are 122m, 22m, and 120m. The advertisements yield an earning of ₹ 5000 per m2 per year.

The taxi fare in a city is as follows: for the first kilometre ₹ 8 and for the subsequent distance ₹ 5 per km, x km distance covered, and total fare as ₹ y, write and draw linear equation.

The students of a Vidyalaya were asked to participate in a competition for making and decorating penholders in the shape of a cylinder with a base, using cardboard. Each penholder was to be of radius

The slant height and base diameter of a conical tomb are 25 m and 14 m respectively. Find the cost of white washing its curved surface at the rate of ₹210 per 100 m

The runs scored by two teams A and B on the first 60 balls in a cricket match are given below: Represent the data of both the teams on the same graph by frequency polygons.

The side AB of a parallelogram ABCD is produced to any point P. A line through A and parallel to CP meets CB produced at Q and then parallelogram PBQR is completed. Show that ar (ABCD) = ar (PBQR).

The relative humidity (in %) of a certain city for a month of 30 days was as follows: (i) Construct a grouped frequency distribution table with classes 84 - 86, 86 - 88, etc.

The radius of a spherical balloon increases from 7 cm to 14 cm as air is being pumped into it. Find the ratio of surface areas of the balloon in the two cases

The paint in a certain container is sufficient to paint an area equal to 9.375 m^2. How many bricks of dimensions 22.5 cm × 10 cm × 7.5 cm can be painted out of this container?

The lengths of two parallel chords of a circle are 6 cm and 8 cm. If the smaller chord is at distance 4 cm from the center, what is the distance of the other chord from the center?

The length, breadth and height of a room are 5 m, 4 m, and 3 m respectively. Find the cost of white washing the walls of the room and ceiling at the rate of ₹7.50 per m^2

The length of 40 leaves of a plant are measured correct to one millimetre, and the obtained data is represented in the following table: i) Draw a histogram to represent the given data.

The inner diameter of a cylindrical wooden pipe is 24 cm and its outer diameter is 28 cm. The length of the pipe is 35 cm. Find the mass of the pipe, if 1 cm3 of wood has a mass of 0.6 g

The inner diameter of a circular well is 3.5 m. It is 10 m deep. Find i) Its inner curved surface area ii) The cost of plastering this curved surface at the rate of ₹40 per m².

The heights of 50 students, measured to the nearest centimeters, have been found to be as follows: i) Represent the data given above by a grouped frequency distribution table

The height of a cone is 15 cm. If its volume is 1570 cm3, find the radius of the base Use π = 3.14

The front compound wall of a house is decorated by wooden spheres of diameter 21 cm, placed on small supports as shown in Fig 13.32. Eight such spheres are used for this purpose, and are to be painted

The following table gives the lifetimes of 400 neon lamps: (i) Represent the given information with the help of a histogram. (ii) How many lamps have a life time of more than 700 hours?

The following observations have been arranged in ascending order. If the median of the data is 63, find the value of x. 29, 32, 48, 50, x, x + 2, 72, 78, 84, 95

The following table gives the distribution of students of two sections according to the marks obtained by them: Represent the marks of the students of both the sections on the same graph

The following number of goals were scored by a team in a series of 10 matches: 2, 3, 4, 5, 0, 1, 3, 3, 4, 3 Find the mean, median and mode of these scores

The following data on the number of girls (to the nearest ten) per thousand boys in different sections of Indian society is given below. i) Represent the information above by a bar graph.

The floor of a rectangular hall has a perimeter 250 m. If the cost of painting the four walls at the rate of ₹ 10 per m^2 is ₹ 15000, find the height of the hall.

The diameter of the moon is approximately one-fourth of the diameter of the earth. What fraction of the volume of the earth is the volume of the moon

The distance (in km) of 40 engineers from their residence to their place of work was found as follows: Construct a grouped frequency distribution table with class size 5 for the data given

The diameter of the moon is approximately one fourth of the diameter of the earth. Find the ratio of their surface areas

The diameter of a sphere is decreased by 25%. By what percent does its curved surface area decrease

The diameter of a roller is 84 cm and its length is 120 cm. It takes 500 complete revolutions to move once over to level a playground. Find the area of the playground in m

The curved surface area of a right circular cylinder of height 14 cm is 88 cm2. Find the diameter of the base of the cylinder

The diameter of a metallic ball is 4.2 cm. What is the mass of the ball, if the density of the metal is 8.9 g per cm

The cost of a notebook is twice the cost of a pen. Write a linear equation in two variables to represent this statement. (Take the cost of a notebook to be ₹ x and that of a pen to be ₹ y).

The circumference of the base of a cylindrical vessel is 132 cm and its height is 25 cm. How many litres of water can it hold? 1000 cm3 1 l Capacity of cylindrical vessel is 34.65 l

The capacity of a cuboidal tank is 50000 litres of water. Find the breadth of the tank, if its length and depth are respectively 2.5 m and 10 m

The capacity of a closed cylindrical vessel of height 1 m is 15.4 litres. How many square metres of metal sheet would be needed to make it

The blood groups of 30 students of Class VIII are recorded as follows: A, B, O, O, AB, O, A, O, B, A, O, B, A, O, O, A, AB, O, A, A, O, O, AB, B, A, O, B, A, B, O. Represent this data

The angles of quadrilateral are in the ratio 3 : 5 : 9 : 13. Find all the angles of the quadrilateral

Suppose you are given a circle. Give a construction to find its centre

State whether the following statements are true or false. (i) Every natural number is a whole number.(ii) Every integer is a whole number. (iii) Every rational number is a whole number.

Simplify: i) 2^2/3. 2^1/5 ii) (1/3^3)7 iii) 11^1/2/11^1/4 iv) 7^1/2. 8^1/2

State whether following statements are true or false i. Every irrational number is real number ii. Every point on number line is of form √m, where m is natural number

Simplify each of the following expressions: (i) (3 + √3)(2 + √2) (ii) (3 + √3)(3 - √3) (iii) (√5 + √2)² (iv) (√5 - √2)(√5 + √2)

Sides of a triangle are in the ratio of 12:17:25 and its perimeter is 540cm. Find its area

Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other

Show that the diagonals of a square are equal and bisect each other at right angles

Show that the diagonals of parallelogram divide it into 4 triangles of equal area

Show that the angles of an equilateral triangle are 60° each

Show that in a right-angled triangle, the hypotenuse is the longest side

Show that of all line segments drawn from a given point not on it, the perpendicular line segment is the shortest

Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus

Show how √5 can be represented on the number line.

Show that if diagonals of quadrilateral are equal & bisect at right angles, then it is square

See Fig. 3.14, and write the following: (i) The coordinates of B. (ii) The coordinates of C (iii) The point identified by the coordinates (-3, -5) (iv) The point identified by the coordinates (2, -4)

Shanti Sweets Stall was placing an order for making cardboard boxes for packing their sweets. Two sizes of boxes were required. The bigger of dimensions 25cm×20cm×5 cm and smaller of dimensions 15cmx

Represent √9.3 on the number line

Refer to Table 14.7, Chapter 14. i) Find the probability that a student obtained less than 20% in the mathematics test. ii) Find the probability that a student obtained marks 60 or above

Refer to Q.8, Exercise 14.2. What is the empirical probability that an engineer lives: i) less than 7 km from her place of work? ii) more than or equal to 7 km from her place of work?

Refer to Example 5, Section 14.4, Chapter 14. Find the probability that a student of the class was born in August

Recall that two circles are congruent if they have the same radii. Prove that equal chords of congruent circles subtend equal angles at their centers

Recall, π is defined as the ratio of circumference (say c) of a circle to its diameter (say d). That is, π = c/d. This seems to contradict the fact that π is irrational.

Rationalize the denominators of the following: i) 1/√7 ii) 1/(√7 - √6) iii) 1/(√5 + √2) iv) 1/(√7 - 2)

Radha made a picture of an aeroplane with coloured paper as shown in Fig 12.15. Find the total area of the paper used

Prove that the line of centers of two intersecting circles subtends equal angles at the two points of intersection

Prove that the circle drawn with any side of a rhombus as diameter, passes through the point of intersection of its diagonals

Prove that if chords of congruent circles subtend equal angles at their centers, then the chords are equal

Prove that a cyclic parallelogram is a rectangle

Plot the points (x, y) given in the following table on the plane, choosing suitable units of distance on the axes

Parveen wanted to make a temporary shelter for her car, by making a box-like structure with tarpaulin that covers all the four sides and the top of the car (with the front face as a flap which

Parallelogram ABCD and rectangle ABEF are on the same base AB and have equal areas. Show that the perimeter of the parallelogram is greater than that of the rectangle.

P and Q are respectively the mid-points of sides AB and BC of a triangle ABC and R is the mid-point of AP, show that i) ar (PRQ) = 1/2 ar (ARC) ii) ar (RQC) = 3/8 ar (ABC) iii) ar (PBQ) = ar (ARC)

P and Q are any two points lying on the sides DC and AD of a parallelogram ABCD. Show that arAPB=arBQC

Line l is the bisector of an angle ∠A and B is any point on l.BP and BQ are perpendiculars from B to the arms of ∠A.Show that:i) ΔAPB≅ΔAQB ii) BP=BQ or B is equidistant from the arms of ∠A

Look at the several examples of rational numbers in the form p/q (q ≠ 0) where p and q are integers with no common factors other than 1 and having terminating decimal representation (expansions).

Let the vertex of an angle ABC be located outside a circle and let the sides of the angle intersect equal chords AD and CE with the circle. Prove that ∠ABC is equal to half the difference of

l and m are two parallel lines intersected by another pair of parallel lines p and q. Show that ΔABC≅ΔCDA

It is required to make a closed cylindrical tank of height 1 m and base diameter 140 cm from a metal sheet. How many square metres of the sheet are required for the same?

It is given that ∠XYZ = 64º and XY is produced to point P. Draw a figure from the given information. If ray YQ bisects ∠ZYP, find ∠XYQ and reflex ∠QYP. ∠XYQ= 122° and ∠QYP= 302°.

It costs ₹2200 to paint the inner curved surface of a cylindrical vessel 10 m deep. If the cost of painting is at the rate of ₹20 per m2, find (i) inner curved surface area of the vessel

Is zero a rational number? Can you write it in the form p/q, where p and q are integers and q ≠ 0?

In which quadrant or on which axis do each of the points (-2, 4), (3, -1), (-1, 0), (1, 2) and (-3, -5) lie? Verify your answer by locating them on the Cartesian plane.

In the given figure, AC = AE, AB = AD and ∠BAD = ∠EAC. Show that BC = DE

In right triangle ABC,right angled at C,M is the mid-point of hypotenuse AB.C is joined to M and produced to a point D such that DM=CM.Point D is joined to point B.Show that: i) ΔAMC≅ΔBMD

In Question 4, point C is called a mid-point of line segment AB. Prove that every line segment has one and only one mid-point

In quadrilateral ACBD, AC = AD and AB bisects ∠A. Show that ΔABC ≅ ΔABD. What can you say about BC and BD?

In Q.5, Exercise 14.2, you were asked to prepare a frequency distribution table, regarding the concentration of sulphur dioxide in the air in parts per million of a certain city for 30 days.

In Q.1, Exercise 14.2, you were asked to prepare a frequency distribution table regarding the blood groups of 30 students of a class. Use this table to determine the probability has blood group AB

In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see Fig. 8.20). Show that: (i) ΔAPD ≅ ΔCQB (ii) AP = CQ (iii) ΔAQB ≅ ΔCPD (iv) AQ = CP (v) APCQ is a parallelogram

In Fig.9.29, ar (DRC) = ar (DPC) and ar (BDP) = ar (ARC). Show that both the quadrilaterals ABCD and DCPR are trapeziums

In Fig.9.28, AP || BQ || CR. Prove that ar (AQC) = ar (PBR)

In Fig. 9.27, ABCDE is a pentagon. A line through B parallel to AC meets DC produced at F. Show that i) ar (ACB) = ar (ACF) ii) ar (AEDF) = ar (ABCDE)

In Fig.9.23, E is any point on median AD of ∆ ABC. Show that ar (ABE) = ar (ACE)

In Fig. 9.25, diagonals AC and BD of quadrilateral ABCD intersect at O such that OB = OD. If AB = CD, then show that: i) ar (DOC) = ar (AOB) ii) ar (DCB) = ar (ACB) iii) DA || CB

In Fig. 9.15, ABCD is a parallelogram, AE ⊥ DC and CF ⊥ AD. If AB = 16 cm, AE = 8 cm and CF = 10 cm, find AD

In Fig. 9.16, P is point in the interior of parallelogram ABCD. Show (i) ar (APB) + ar (PCD) = 1/2 ar (ABCD) (ii) ar (APD) + ar (PBC) = ar (APB) + ar (PCD) [Hint: Through P draw line parallel to AB.]

In Fig. 9.17, PQRS and ABRS are parallelograms and X is any point on side BR. Show (i) ar (PQRS) = ar (ABRS) (ii) ar (AXS) = 1/2 ar (PQRS)

In Fig. 9.24, ABC and ABD are two triangles on the same base AB. If line-segment CD is bisected by AB at O, show that ar(ABC) = ar (ABD).AO and BO are medians of triangles ADC and BDC

In Fig. 9.34, ABC is a right triangle right angled at A. BCED, ACFG and ABMN are squares on the sides BC, CA and AB respectively. Line segment AX ⊥ DE meets BC at Y. Show that: i) ΔMBC ≅ ΔABD

In Fig.9.33, ABC and BDE are two equilateral triangles such that D is the mid-point of BC. If AE intersects BC at F, show that i) ar (BDE) =1/4 ar (ABC) ii) ar (BDE) = 1/2 ar (BAE) iii) ar (ABC) =

In Fig. 9.31, ABCD, DCFE and ABFE are parallelograms. Show that ar (ADE) = ar (BCF)

In Fig. 9.32, ABCD is a parallelogram and BC is produced to a point Q such that AD = CQ. If AQ intersect DC at P, show that ar (BPC) = ar (DPQ). [Hint: Join AC.]

In Fig. 9.30, D and E are two points on BC such that BD = DE = EC. Show that ar (ABD) = ar (ADE) = ar (AEC).

In the given figure, PR > PQ and PS bisects ∠QPR. Prove that ∠PSR > ∠PSQ

In Fig. 6.44, the side QR of ∠PQR is produced to a point S. If the bisectors of ∠PQR and ∠PRS meet at point T, then prove that ∠QTR = 1/2 ∠QPR

In the given figure, ∠B < ∠A and ∠C < ∠D. Show that AD < BC

In the given figure sides AB and AC of ΔABC are extended to points P and Q respectively. Also, ∠PBC < ∠QCB. Show that AC > AB

In Fig. 6.43, if PQ ⊥ PS, PQ || SR, ∠SQR = 28° and ∠QRT = 65° then find the values of x and y

In Fig. 6.42, if lines PQ and RS intersect at point T, such that ∠PRT = 40°, ∠RPT = 95° and ∠TSQ = 75°, find ∠SQT

In Fig. 6.40, ∠X = 62°, ∠XYZ = 54°. If YO and ZO are the bisectors of ∠XYZ and ∠XZY respectively of ∠XYZ, find ∠OZY and ∠YOZ

In Fig. 6.41, if AB || DE, ∠BAC = 35° and ∠CDE = 53°, find ∠DCE

In Fig. 6.39, sides QP and RQ of ∆PQR are produced to points S and T respectively. If ∠SPR =135° and ∠PQT = 110°, find ∠PRQ

In Fig. 6.33, PQ and RS are two mirrors placed parallel to each other. An incident ray AB strikes the mirror PQ at B, the reflected ray moves along the path BC and strikes the mirror RS at C

In Fig. 6.32, if AB || CD, ∠APQ = 50° and ∠PRD = 127°, find x and y

In Fig. 6.31, if PQ || ST, ∠PQR = 110° and ∠RST = 130°, find ∠QRS. [Hint: Draw a line parallel to ST through point R.]

In Fig. 6.30, if AB||CD, EF ⊥ CD and ∠GED =126°, find ∠AGE, ∠GEF and ∠FGE

In Fig. 6.28, find the values of x and y and then show that AB||CD

In Fig. 6.29, if AB || CD, CD || EF and y : z = 3 : 7, find x

In Fig. 6.17, POQ is a line. Ray OR, is perpendicular to line PQ. OS is another ray lying between rays OP and OR. Prove that ∠ROS = 1/2 (∠QOS - ∠POS)

In Fig. 6.16, if x + y = w + z, then prove that AOB is a line

In Fig. 6.15, ∠PQR = ∠PRQ then prove that ∠PQS = ∠PRT

In Fig. 6.14, lines XY and MN intersect at O. If ∠POY = 90° and a : b = 2 : 3, find c