HIGH ORDER THINKING SKILL QUESTIONS (CHAPTERWISE)
SUBJECT - MATHEMATICS
CLASS - IX
CHAPTER 1- NUMBER SYSTEM
1. Solve the equation : 22x + 2 = 23x -1
2. If (x+y)-1 (x-1 + y-1) = xp . yq, prove that p + q + 2 = 0
4 3 5
3. Simplify: √225 + √81 - 8 √216 + 15 √32
3 6
4. Arrange in ascending order : √18 , √144 , √6
𝑝
5. Convert the following in 𝑞 form:(I) 0.2222… (ii) 6.656565….
3 + √2
6. Find the value of ‘a’ and ‘b’ in = a + b√2
3 − √2
CHAPTER 2- POLYNOMIALS
1. If (x+1) and (x-1) are the factors of x4 + px3 + 2x2 - 3x + q, find the values of p and q.
2. By using Factor Theorem, factorise :
(I) 2x3 - 3x2 - 17x + 30 (ii) x3 -6x2 + 11x - 6
1 1 1
3. If x + 𝑥 = 2 . Find : (I) x2 + 𝑥 2 (ii) x4 + 𝑥 4
4. If a2 + b2 = 13 and ab = 6, find a3 + b3
5. Factorise :(I) x6 - y6 (ii) x9 - y9
CHAPTER 3- COORDINATE GEOMETRY
1. The lengths of the perpendicular PM and PN drawn from a point P, on x-axis and y-axis are 3 and
2 units respectively. Find the coordinates of P,M and N.
2. What will be the coordinates if it lies on y-axis at a distance of -7 units from the x-axis?
3. Plot two points P(0,-4) and Q(0,4) on the graph paper. Now , plot R and S such that ∆PQR and
∆PQS are isosceles ∆.
CHAPTER 4- LINEAR EQUATIONS IN TWO VARIABLES
1. If x=2 and y=5 is a solution of a2x + ay + 3 = 0, then find the value of ‘a’
2. On her birthday, Anisha donates 2 toffee to each child of an orphanage and 15 chocolates to
adults working there. Taking the total items distribuited as x and the number of children as y , write
a linear equation in 2 variables for the above situtation.
(i) Write the equation in standard form
(ii) How many children are there if 61 items were distributed?
(iii) What value does Anisha posses?
3. If x=3k-2 and y=2k is a solution of the equation 4x - 7y + 12 = 0, find k
4. If (m-2 , 2m+1) lies on the graph of the equation 2x + 3y - 10 = 0, find the value of m
5. Find any four different values of the equation 2x - 5y =10
CHAPTER 5- INTRODUCTION TO EUCLID’S GEOMETRY
1. Prove that through a given point, there can be only one line parallel to a given line.
2. Can you think of a real-life scenario where Euclid's axioms are used?
3. 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.
4. Prove that every line segment has one and only one mid-point.
5. State the fifth postulate of Euclid.
CHAPTER- 6 LINES AND ANGLES
Q1- A transversal intersects two lines in such a way that the two interior angles on the same side of
the transversal are equal. Will the two lines always be parallel? Give reason for your answer.
Q2- A triangle ABC is right angled at A. L is a point on BC such that AL ⊥ BC. Prove that ∠BAL =
∠ACB.
Q3- Prove that the sum of the three angles of a triangle is 180°.
Q4- A transversal intersects two parallel lines. Prove that the bisectors of any pair of corresponding
angles so formed are parallel.
Q5- Prove that two lines that are respectively perpendicular to two intersecting lines intersect each
other. [Hint: Use proof by contradiction].
Q6- If a transversal intersects two lines such that the bisectors of a pair of corresponding angles are
parallel, then prove that the two lines are parallel.
Q7- How many triangles can be drawn having its angles as 45°, 64° and 72°? Give reason for your
answer.
Q8- If one of the angles formed by two intersecting lines is a right angle, what can you say about the
other three angles? Give reason for your answer.
Q9- Two lines l and m are perpendicular to the same line n. Are l and m perpendicular to each
other? Give reason for your answer
Q10- Prove that through a given point, we can draw only one perpendicular to a given line. [Hint:
Use proof by contradiction].
CHAPTER- 7 TRIANGLES
Q1- M is a point on side BC of a triangle ABC such that AM is the bisector of ∠BAC. Is it true to
say that perimeter of the triangle is greater than 2 AM? Give reason for your answer.
Q2- ABC is an isosceles triangle with AB = AC and BD and CE are its two medians. Show that BD
= CE.
Q3- Bisectors of the angles B and C of an isosceles triangle with AB = AC intersect each other at O.
BO is produced to a point M. Prove that ∠MOC = ∠ABC.
Q4- Bisectors of the angles B and C of an isosceles triangle ABC with AB = AC intersect each other
at O. Show that external angle adjacent to ∠ABC is equal to ∠BOC.
Q5- Prove that if in two triangles two angles and the included side of one triangle are equal to two
angles and the included side of the other triangle, then the two triangles are congruent.
Q6- Two lines l and m intersect at the point O and P is a point on a line n passing through the point
O such that P is equidistant from l and m. Prove that n is the bisector of the angle formed by l and m.
Q7- AB and CD are the smallest and largest sides of a quadrilateral ABCD. Out of ∠B and ∠D
decide which is greater.
Q8- . ABCD is quadrilateral such that AB = AD and CB = CD. Prove that AC is the perpendicular
bisector of BD.
Q9- In a right triangle, prove that the line-segment joining the mid-point of the hypotenuse to the
opposite vertex is half the hypotenuse.
Q10- BE and CF are two equal altitudes of a triangle ABC. Using RHS congruence rule, prove that
the triangle ABC is isosceles.
CHAPTER-8 QUADRILATERALS
Q1- ABCD is a rhombus and P, Q, R and S are the mid-points of the sides AB, BC, CD and DA
respectively. Show that the quadrilateral PQRS is a rectangle.
Q2- ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA
respectively. Show that the quadrilateral PQRS is a rhombus.
Q3- 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
Q4-ABCD is a rectangle in which diagonal AC bisects ∠ A as well as ∠ C. Show that: (i) ABCD is a
square (ii) diagonal BD bisects ∠ B as well as ∠ D.
Q5- If the diagonals of a parallelogram are equal, then show that it is a rectangle.
Q6- The line segment joining the mid-points of two sides of a triangle is parallel to the third side.
Q7- The line drawn through the mid-point of one side of a triangle, parallel to another side bisects
the third side.
Q8- . D, E and F are respectively the mid-points of the sides AB, BC and CA of a triangle ABC.
Prove that by joining these mid-points D, E and F, the triangles ABC is divided into four congruent
triangles.
Q9- : PQ and RS are two equal and parallel line-segments. Any point M not lying on PQ or RS is
joined to Q and S and lines through P parallel to QM and through R parallel to SM meet at N. Prove
that line segments MN and PQ are equal and parallel to each other
Q10- Can the angles 110º, 80º, 70º and 95º be the angles of a quadrilateral? Why or why not?
CHAPTER-9 CIRCLES
Q1- Equal chords of a circle subtend equal angles at the centre.
Q2- If the angles subtended by the chords of a circle at the centre are equal, then the chords are
equal.
Q3- Recall that two circles are congruent if they have the same radii. Prove that equal chords of
congruent circles subtend equal angles at their centres.
Q4- 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. If the
distance between Reshma and Salma and between Salma and Mandip is 6m each, what is the
distance between Reshma and Mandip?
Q5- 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.
Find the length of the string of each phone
Q6- The angle subtended by an arc at the centre is double the angle subtended by it at any point on
the remaining part of the circle.
Q7- f a line segment joining two points subtends equal angles at two other points lying on the same
side of the line containing the line segment, the four points lie on a circle (i.e. they are concyclic).
Q8- 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.
Q9- If circles are drawn taking two sides of a triangle as diameters, prove that the point of
intersection of these circles lie on the third side.
Q10- Prove that the quadrilateral formed (if possible) by the internal angle bisectors of any
quadrilateral is cyclic.
CHAPTER-10 HERON’S FORMULA
Q1- A triangular park ABC has sides 120m, 80m and 50m (see Fig. 10.4). A gardener Dhania has to
put a fence all around it and also plant grass inside. How much area does she need to plant? Find the
cost of fencing it with barbed wire at the rate of `20 per metre leaving a space 3m wide for a gate on
one side.
Q2- The sides of a triangular plot are in the ratio of 3 : 5 : 7 and its perimeter is 300 m. Find its area.
Q3- 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, what will be the
area of the signal board?
Q4- An isosceles triangle has perimeter 30 cm and each of the equal sides is 12 cm. Find the area of
the triangle
Q6- Find the area of a triangle, two sides of which are 8 cm and 11 cm and the perimeter is 32 cm
Q7- The triangular side walls of a flyover have been used for advertisements. The sides of the walls
are 122 m, 22 m and 120 m . The advertisements yield an earning of ` 5000 per m2 per year. A
company hired one of its walls for 3 months. How much rent did it pay?
Q8- A field in the form of a parallelogram has sides 60 m and 40 m and one of its diagonals is 80 m
long. Find the area of the parallelogram.
Q9- A rectangular plot is given for constructing a house, having a measurement of 40 m long and 15
m in the front. According to the laws, a minimum of 3 m, wide space should be left in the front and
back each and 2 m wide space on each of other sides. Find the largest area where house can be
constructed.
Q10- A field is in the shape of a trapezium having parallel sides 90 m and 30 m. These sides meet
the third side at right angles. The length of the fourth side is 100 m. If it costs Rs 4 to plough 1m2 of
the field, find the total cost of ploughing the field.
CHAPTER- 11 SURFACE AREA AND VOLUME
Q1A cylindrical water tank of diameter 3 meters and height 4 meters is placed inside a cubic room
with a side of 5 meters. Will the room have enough space for the tank? Justify your answer with
calculations. If 20% of the water in the cylindrical tank is used every day in a school, how many
days will it take to empty the tank completely?
Q2 A cone and a cylinder have the same base and the same height. If the cylinder is filled with water
and you pour this water into the cone, what fraction of the cone will be filled? Explain why he cost
of filling a conical and a cylindrical container with milk is different. Which container is more
cost‑effective if both have the same base and height?
Q3 An ice cream cone has a hemispherical top of radius 3centimeters and a conical part of height
4centimeters. Find the total surface area that needs to be wrapped with a wrapper, ignoring any
overlap. If wrapping the ice cream cone costs 0.5rs per square centimeter, what will be the total cost
of the wrapper required?
Q4 A solid metallic sphere is melted to form a cylindrical rod of diameter 10centimeters and length
14centimeters. Find the radius of the sphere and justify your method. From one solid iron sphere,
how many cylindrical rods can be made if the diameter of the rod is half the diameter of the sphere?
Q5 A hemispherical dome of a building has an internal diameter of 21 meters. Find the cost of
painting its interior surface area at the rate of 50 rupees per square meter. If the dome of a building
develops cracks covering 25% of its surface area, how much area needs to be repaired?
Q6 An artist is making a decorative lamp by joining a cylinder and a cone. The cylinder has a radius
of 5centimeters and a height of 10centimeters. The cone has the same base and a slant height of
13centimeters. Find the total surface area of the lamp that needs to be painted. If you reduce the
material used for making a lamp by 30%, which parameter (height or diameter) should be adjusted to
save more material?
Q7 A solid cone of base diameter 14centimeters and slant height 25centimeters is cut from a solid
cylinder of the same base diameter and height. Find the volume of the remaining solid and justify
why. Compare the cost of painting a decorative piece made from a cone and a cylinder if the cost per
square centimeter is different for both shapes. Which part is more expensive? Why?
Q8 A tent is in the form of a cone of slant height 13 meters and base diameter 24 meters. What area
of canvas is required to make the tent, assuming 10 percent waste due to stitching? A tent is made in
the shape of a cone. If 15% of the canvas is wasted due to stitching, how much total canvas is
required?
Q9A cubic box of side 10centimeters is to be packed inside a spherical shell. What must be the
internal diameter of the shell, and why? A cubic box is placed inside a spherical shell. Will the box
fit completely if the shell’s internal diameter is slightly smaller? Discuss what happens and justify
why this consideration is important in packaging.
Q10 A solid sphere is placed inside a cylindrical container such that the diameter of the sphere is
equal to the diameter of the cylinder, and the sphere just fits in. Compare the surface area and
volume of the sphere and cylinder, and justify the results. Compare the surface area and volume of a
sphere and a cylinder when both have the same diameter. In what practical situations would you
prefer one over the other? Justify with examples.
Hots of class 9 th chapter 12 Statistics
1.The mean of 100 observations is 50. If one of the observation which was 50 is replaced by 150
then what will be the resulting mean?
2. There are 50 numbers. Each number is subtracted from 53 and the mean of the numbers so
obtained is found to be –3.5. What is the mean of given numbers?
3. What is the mean of first 10 natural numbers?
4. The mean of 10, 15, x, 5, 15 is 15. What is the value of x?
5. In the frequency distribution.
Class intervals Frequencies
0 –10 5
10 – 20 15
20 – 30 10
30 – 40 2
40 – 50 3
What is the cumulative frequency corresponding to class 40 – 50?
6. The average mark of boys in an examination is 68 & that of girls in 89. If the
average mark of all candidates in that examination is 80 , find the ratio of the no. of
boys to the number of girls that appeared in the examinations.
7. In x standard, these are three sections A, B, C with 25, 40 and 30 students
respectively. The average mark of section A is 70%, of section B is 65% and of
section B is 50%. Find the average marks of the entire X standard
8. Find the median of the following data 19, 25, 59, 48, 35, 31, 30, 32, 51. If 25 is
replaced by 52, what will be the new median.
9. Prove that the sum of the deviations of an individual's observations from the mean is zero
10. The average score of girls in class examinations in a school is 67 and that of boys is 63. The
average score for the whole class is 64.5. Find the percentage of girls and boys in the class.
11. The mean of 200 items was 50. Later on, it was discovered that the two items
were misread as 92 and 8 instead of 192 and 88. Find the correct mean.
The mean of 200 items was 50. Later on, it was discovered that the two items were
misread as 92 and 8 instead of 192 and 88. Find the correct mean.
12. The width of each of five continuous classes in a frequency distribution is 5 and the lower
class limit of the lowest class is 10. What is the upper class limit of the highest class?
