Posted: 01 Jan 2017 11:29 PM PST
Posted: 01 Jan 2017 10:55 PM PST
NCERT Mathematics Question Paper (Class - 9)
(Mathematics) Chapter 3 Coordinate Geometry
Question 1. How will you describe the position of a table lamp on your study table to another person?
(i) The perpendicular distance of the point P from the y - axis measured along the positive direction of the x - axis is PN = OM = 4 units.
(i) The x - coordinate of a point is its perpendicular distance from the y - axis measured along the x -axis (positive along the positive direction of the x - axis and negative along the negative direction of the x - axis). For the point P, it is + 4 and for Q, it is – 6. The x - coordinate is also called the abscissa.
Question 1.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?
(i) The coordinates of B.
(Mathematics) Chapter 4 Linear Equations in Two Variables
Question 1. 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 Rs x and that of a pen to be Rs y).
(i) 2x + 3y = 9.35
Question 1.Which one of the following options is true, and why? y = 3x + 5 has
(i) a unique solution
(i) 2x + y = 7
(i) (0, 2)
Question 1. Draw the graph of eachof the following linear equations in two variables:
(i) x + y = 4
(i) y = x
(i) Draw the graph of the linear equation above using Celsius for x-axis and Fahrenheit for y-axis.
Question 1. Give the geometric representations of y = 3 as an equation
(i) in one variable
(i) in one variable
(Mathematics) Chapter 7 Triangles
Question 1. In quadrilateral ACBD, AC = AD and AB bisects ∠ A (see Fig. 7.16). Show that Δ ABC Δ ABD. What can you say about BC and BD?
Question 2 . ABCD is a quadrilateral in which AD = BC and ∠ DAB = ∠ CBA (see Fig. 7.17). Prove that
(i) Δ ABD Δ BAC
Question 3. AD and BC are equal perpendiculars to a line segment AB (see Fig. 7.18). Show that CD bisects AB.
Question 5. 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 (see Fig. 7.20). Show that:
(i) Δ APB Δ AQB
Question 7 . AB is a line segment and P is its mid-point. D and E are points on the same side of AB such that ∠ BAD = ∠ ABE and ∠ EPA = ∠ DPB (see Fig. 7.22). Show that
(i) Δ DAP Δ EBP
Question 8. 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 (see Fig. 7.23). Show that:
(i) Δ AMC Δ BMD
Question 1. 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
Question 2. In Δ ABC, AD is the perpendicular bisector of BC (see Fig. 7.30). Show that Δ ABC is an isosceles triangle in which AB = AC.
Question 3. 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.
Question 4. ABC is a triangle in which altitudes BE and CF to sides AC and AB are equal (see Fig. 7.32). Show that
(i) Δ ABE Δ ACF
Question 1. Δ ABC and Δ DBC are two isosceles triangles on the same base BC and vertices A and D are on the same side of BC (see Fig. 7.39). If AD is extended to intersect BC at P, show that
(i) Δ ABD Δ ACD
Question 2. AD is an altitude of an isosceles triangle ABC in which AB = AC. Show that
(i) AD bisects BC
(i) Δ ABM Δ PQN
Question 5. ABC is an isosceles triangle with AB = AC. Draw AP ⊥ BC to show that ∠ B = ∠ C.
Question 1. Show that in a right angled triangle, the hypotenuse is the longest side.
Question 2. In Fig. 7.48, sides AB and AC of Δ ABC are extended to points P and Q respectively. Also, ∠ PBC < ∠ QCB. Show that AC > AB. 3. In Fig. 7.49, ∠ B < ∠ A and ∠ C < ∠ D. Show that AD < BC.
Question 3. 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.
Question 4. In Fig 7.51, PR > PQ and PS bisects ∠ QPR. Prove that ∠ PSR > ∠ PSQ.
Question 5. Show that of all line segments drawn from a given point not on it, the perpendicular line segment is the shortest.
Question 1 . ABC is a triangle. Locate a point in the interior of Δ ABC which is equidistant from all the vertices of Δ ABC.
Question 2. In a triangle locate a point in its interior which is equidistant from all the sides of the triangle.
Question 3. In a huge park, people are concentrated at three points (see Fig. 7.52):
A : where there are different slides and swings for children,
Question 4. Complete the hexagonal and star shaped Rangolies [see Fig. 7.53(i) and (ii)] by filling them with as many equilateral triangles of side 1 cm as you can. Count the number of triangles in each case. Which has more triangles?
(Mathematics) Chapter 8 Quadrilaterals
Question 1. The angles of quadrilateral are in the ratio 3 : 5 : 9 : 13. Find all the angles of the quadrilateral.
Question 6. Diagonal AC of a parallelogram ABCD bisects ∠ A (see Fig. 8.19). Show that
(i) it bisects ∠ C also
Question 7. ABCD is a rhombus. Show that diagonal AC bisects ∠ A as well as ∠ C and diagonal BD bisects ∠ B as well as ∠ D.
Question 8. ABCD is a rectangle in which diagonal AC bisects ∠ A as well as ∠ C. Show that:
(i) ABCD is a square
Question 9. In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see Fig. 8.20). Show that:
(i) Δ APD Δ CQB
Question 10. 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) Δ APB Δ CQD
Question 11. In Δ ABC and Δ DEF, AB = DE, AB || DE, BC = EF and BC || EF. Vertices A, B and C are joined to vertices D, E and F respectively (see Fig. 8.22). Show that :
(i) quadrilateral ABED is a parallelogram
Question 12. ABCD is a trapezium in which AB || CD and AD = BC (see Fig. 8.23). Show that:
(i) ∠ A = ∠ B
Question 1. ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA (see Fig 8.29). AC is a diagonal. Show that :
(i) SR || AC and SR = 1 2 AC
(i) D is the mid-point of AC
(Mathematics) Chapter 9 Areas of Parallelograms and Triangles
Question1. 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.
Question 1. 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.
(i) ar (APB) + ar (PCD) = 1 ar (ABCD) 2
(i) ar (PQRS) = ar (ABRS)
Question 1. In Fig.9.23, E is any point on median AD of a Δ ABC. Show that ar (ABE) = ar (ACE).
(i) BDEF is a parallelogram.
(i) ar (DOC) = ar (AOB)
(i) ar (ACB) = ar (ACF)
Question 1. 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.
(i) ar (BDE) = 1 4 ar (ABC)
(i) ar (PRQ) = 1 2 ar (ARC)
Question 8. 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) Δ MBC Δ ABD
(Mathematics) Chapter 10 Circles
Question 1. Fill in the blanks:
(i) The centre of a circle lies in of the circle. (exterior/ interior)
(i) Line segment joining the centre to any point on the circle is a radius of the circle.
Question 1. 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.
Question 1. Draw different pairs of circles. How many points does each pair have in common? What is the maximum number of common points?
Question 1. Two circles of radii 5 cm and 3 cm intersect at two points and the distance between their centres is 4 cm. Find the length of the common chord.
Question 1. In Fig. 10.36, A,B and C are three points on a circle with centre O such that ∠ BOC = 30° and ∠ AOB = 60°. If D is a point on the circle other than the arc ABC, find ∠ADC
Question 2. 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.
Question 3. In Fig. 10.37, ∠ PQR = 100°, where P, Q and R are points on a circle with centre O. Find ∠ OPR.
Question 4. In Fig. 10.38, ∠ ABC = 69°, ∠ ACB = 31°, find ∠ BDC.
Question 5. 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.
Question 6. 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.
Question 7. If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle.
Question 8. If the non-parallel sides of a trapezium are equal, prove that it is cyclic.
Question 9. 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 and P, Q respectively (see Fig. 10.40). Prove that ∠ ACP = ∠ QCD.
Question 10. 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.
Question 11. ABC and ADC are two right triangles with common hypotenuse AC. Prove that ∠ CAD = ∠ CBD.
Question 12. Prove that a cyclic parallelogram is a rectangle.
Question 1. Prove that the line of centres of two intersecting circles subtends equal angles at the two points of intersection.
Question 2. 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 centre. If the distance between AB and CD is 6 cm, find the radius of the circle.
Question 3. 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 centre, what is the distance of the other chord from the centre?
Question 4. 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 the angles subtended by the chords AC and DE at the centre.
Question 5. Prove that the circle drawn with any side of a rhombus as diameter, passes through the point of intersection of its diagonals.
Question 6. ABCD is a parallelogram. The circle through A, B and C intersect CD (produced if necessary) at E. Prove that AE = AD.
Question 7. AC and BD are chords of a circle which bisect each other. Prove that :
(i) AC and BD are diameters
(Mathematics) Chapter 11 Constructions
Question 1. Construct an angle of 900 at the initial point of a given ray and justify the construction.
Question 1. Construct a triangle ABC in which BC = 7cm, ∠B = 75° and AB + AC = 13 cm.
Question 2. Construct a triangle ABC in which BC = 8cm, ∠B = 45° and AB – AC = 3.5 cm.
Question 3. Construct a triangle PQR in which QR = 6cm, ∠Q = 60° and PR – PQ = 2cm.
Question 4. Construct a triangle XYZ in which ∠Y = 30°, ∠Z = 90° and XY + YZ + ZX = 11 cm.
Question 5. Construct a right triangle whose base is 12cm and sum of its hypotenuse and other side is 18 cm.
(Mathematics) Chapter 12 Heron's Formula
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