9th class Maths NCERT Solution and previous Question

X Chapter For  Science Test Paper And Study Materials 1  Matter in Our Surroundings 2  Is Matter Around Us Pure 3  Atoms and Molecules 4  Structure of the Atom 5  The Fundamental Unit of Life 6  Tissues 7  Diversity in Living Organisms 8  Motion 9  Force and Laws of Motion 10  Gravitation 10. Thrust and Pressure 11  Work and Energy 12  Sound 13  Why Do We Fall Ill 14  Natural Resources 15  Improvement in Food Resources

CBSE Class IX Chapters for Mathematics Test Paper And Study Materials IX Number System IX Polynomial(Algebra) IX Introduction to Euclid Geometry IX Lines and Angles (Geometry) IX Triangle(Geometry) IX CO-Ordinate Geometry IX Area of Triangle(Heron Formula) IX Linear equations in two variables IX Quadrilaterals IX Area of Parallelogram IX Geometry: Circles IX Constructions IX Surface Areas and Volumes IX Statistics IX Probability

cbse maths guess 9th inequalities of triangle

1. If one angle of a triangle is equal to the sum of the other two angles, then the triangle is
(A) an isosceles triangle (B) an obtuse triangle (C) an equilateral triangle (D) a right triangle

2. An exterior angle of a triangle is 105° and its two interior opposite angles are equal. Each of these equal angles is

(A) 37+ 1/2° (B)52+ 1/2° (C) 72+ 1/2° (D) 75°

3. The angles of a triangle are in the ratio 5 : 3 : 7. The triangle is

(A) an acute angled triangle (B) an obtuse angled triangle (C) a right triangle (D) an isosceles triangle

4 . If one of the angles of a triangle is 130°, then the angle between the bisectors of the other two angles can be
(A) 50° (B) 65° (C) 145° (D) 155°

CBSE Test sample paper- IX Mathematics (Congruent triangle)

IX Mathematics (Congruent triangle)

1. If one angle of a triangle is equal to the sum of the other two angles, then the triangle is

(A) an isosceles triangle (B) an obtuse triangle (C) an equilateral triangle (D) a right triangle
2. An exterior angle of a triangle is 105° and its two interior opposite angles are  equal. Each of these equal angles is
(A) 37+ 1/2°  (B)52+ 1/2° (C) 72+ 1/2° (D) 75°
3. The angles of a triangle are in the ratio 5 : 3 : 7. The triangle is 
(A) an acute angled triangle (B) an obtuse angled triangle  (C) a right triangle (D) an isosceles triangle
4 . If one of the angles of a triangle is 130°, then the angle between the bisectors of  the other two angles can be
(A) 50° (B) 65° (C) 145° (D) 155°
5. The sum of two angles of a triangle is equal to its third angle. Find the third angles.
(a) 900  (b) 450 (c) 600 (d) 700                                                     

                                                            Section B

Problem related to factorization,ratiolization

Problem-1 If the values of a – b and ab are 6 and 40 respectively, find the values of  a² + b² and (a + b)².

Solution: a² + b² = (a – b)² + 2ab = 6² + 2(40) = 36 + 80 = 116

  (a + b)² = (a – b)² + 4ab  = 6² + 4(40)= 36 + 160 = 196

Problem-2. If (x + p)(x + q) = x² – 5x – 300, find the value of p² + q².

Solution:

By product formula, we have (x + p) (x + q) = x² + (p + q)x + pq.

So, by comparison, we get p + q = –5, pq = –300.

Now, we have p² + q² = (p + q)² – 2 pq = (–5)² –2(–300) = 25 + 600 = 625.

Problem-3. If (x + a)(x + b)(x + c) ≡ x³ – 6x² + 11x – 6, find the value of a² + b² + c².

Solution:  From the product formula, we have

(x+a)(x+b)(x+c) = x³ + (a + b + c)x² + (ab + bc + ca)x + abc.

Comparing, we get a + b + c = –6, ab + bc + ca = 11, abc = –6.

∴ a² + b² + c² = (a + b + c)² –2 (ab + bc + ca) = (– 6)² – 2(11) = 36 – 22 = 14.

Problem-4 If a+b=2 and a2+b2=8,find a3+b3 and a4+b4.

CBSE FORMATIVE TEST PAPER CLASS 9th MATHS

Section – A (1 marks)
1. Abscissa of all the points on the x-axis is

(A) 0 (B) 1(C) 2 (D) any number

2. Ordinate of all points on the x-axis is

(A) 0 (B) 1 (C) – 1 (D) any number

3. Any point on the y-axis is of the form

(A) (x, 0) (B) (x, y) (C) (0, y) (D) ( y, y)

4. Any point on the y-axis is of the form

(A) (x, 0) (B) (x, y) (C) (0, y) (D) ( y, y)

5. The things which are double of the same thing are

(A) equal (B) unequal (C) halves of the same thing (D) double of the same thing


Section-B ( 4 marks)

9th Test Paper : HERON’S FORMULA & SURFACE AREAS AND VOLUMES

Section A 1 mark each
1. Heron was born in ____________ in ___________.
2. Heron formula is used to find area of triangle when _______________.
3. Find the area of a triangle, two sides of which are 8 cm and 11 cm and the perimeter is 32 cm.
Section B 2 marks each

CBSE 9th Class Model Question Papers chapterTriangle

CLASS 9TH CBSE MATHS
CHAPTER - TRIANGLE
CBSE TEST PAPER-1

1. PQ = PR of < QPR and S and T are point on PR and PQ such that ∠PQS = ∠PRT . Prove that Δ PQS ≅ Δ PRT.

2. Two lines AB and CD intersect each other at the point O such that BC || DA and BC = DA. Show that O is the midpoint of both the line-segments AB and CD( join B-C and A-D)

3. In triangle P Q R , PQ > PR and QS and RS are the bisectors of ∠Q and ∠R, respectively.

Show that SQ > SR

4. ABC is an isosceles triangle with AB = AC and BD and CE are its two medians. Show that BD = CE.

5. D and E are points on side BC of a Δ ABC such that BD = CE and AD = AE. Show that Δ ABD ≅ Δ ACE.

6. CDE is an equilateral triangle formed on a side CD of a square ABCD (join AE and BE). Show that Δ ADE ≅ Δ BCE.

7. BA ⊥ AC, DE ⊥ DF such that BA = DE and BF = EC. Show that Δ ABC ≅ Δ DEF.

8. Q is a point on the side SR of a Δ PSR such that PQ = PR. Prove that PS > PQ.

9. S is any point on side QR of a Δ PQR. Show that: PQ + QR + RP > 2 PS.

10. D is any point on side AC of a Δ ABC with AB = AC. Show that CD < BD.

11. l || m and M is the mid-point of a line segment AB. Show that M is also the mid-point of any line segment CD, having its end points on l and m, respectively.

12. 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.

13. 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.

14. AD is the bisector of ∠BAC. of DABC . Prove that AB > BD.

15. ABC is a right triangle and right angled at B such that ∠BCA = 2 ∠BAC. AD perpendicular to BC. Show that hypotenuse AC = 2 BC.

16. 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.

17. If the bisector of an angle of a triangle also bisects the opposite side, prove that the triangle is isosceles.

18. S is any point in the interior of Δ PQR. Show that SQ + SR < PQ + PR. { Produce QS to intersect PR at T}     

CBSE TEST PAPER-2

1 marks questions

1. Which of the following is not a criterion for congruence of triangles?

(A) SAS (B) ASA (C) SSA (D) SSS

2 . If AB = QR, BC = PR and CA = PQ, then

(A) Δ ABC ≅ Δ PQR (B) Δ CBA ≅ Δ PRQ (C) Δ BAC ≅ Δ RPQ (D) Δ PQR ≅ Δ BCA

3 . In Δ ABC, AB = AC and ∠B = 50°. Then ∠C is equal to

(A) 40° (B) 50° (C) 80° (D) 130°

2 marks questions

1. In triangles ABC and PQR, ∠A = ∠Q and ∠B = ∠R. Which side of Δ PQR should be equal to side AB of Δ ABC so that the two triangles are congruent? Give reason for your answer.

2 . In triangles ABC and PQR, ∠A = ∠Q and ∠B = ∠R. Which side of Δ PQR should be equal to side BC of Δ ABC so that the two triangles are congruent? Give reason for your answer.

3 . AB is a line segment and line l is its perpendicular bisector. If a point P lies on l, show that P is equidistant from A and B.

3 marks questions

1. S is any point in the interior of Δ PQR. Show that SQ + SR < PQ + PR.

2. If the bisector of an angle of a triangle also bisects the opposite side, prove that the triangle is isosceles.

3. P is a point on the bisector of ∠ABC. If the line through P, parallel to BA meets BC at Q, prove that BPQ is an isosceles triangle.

4 marks questions

1. Prove that sum of any two sides of a triangle is greater than twice the median with respect to the third side

2. Show that in a quadrilateral AB + BC + CD + DA < 2 (BD + AC)

3. In a right triangle, prove that the line-segment joining the mid-point of the hypotenuse to the opposite vertex is half the hypotenuse.

9th Maths COORDINATE GEOMETRY(Assignments)

Assignments 
1. Solve the equation 2x + 1 = x – 2 & represent the solution on :
The number line Cartesian plane
Q. 2. The linear equation that converts the Fahrenheit to Celsius is as follows :-
F=(9/5)c + 32
Draw the graph of the linear equation .
If the temperature is 950F , what is the temperature in Celsius .
Is there a temperature which is numerically the same in both Fahrenheit and Celsius ? If yes, find it.

Q. 3. Express the following information in the form of linear equation :
In a one day International cricket match between India & Srilanka played in Nagpur, two Indian batsman together scored 176 runs .
The cost of a notebook is twice the cost of a pen.


Q. 4. Find two solutions for each of the following equations :
4x + 3y = 12
2x + 5y = 0
px + y = 3

Q. 5. Express each of the following linear equation in the standard form & hence find the values of a, b & c in each case :
x-[y/5]-10=0
2x = -5y

Q. 6. Check which of the following are solutions of the equation x – 2y = 4.

1. √2 ,4√2 
2. (1,1) 
3. (4,0) 

Q. 7. (a). How many solutions does the following linear equation & why ?
3x – y + 5 = 0

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

Q. 8. (a). Given the point (1,2), find the equation of a line on which it lies. How many such equations are there ?

(b)Give an equation of two lines passing through (2,14). How many more Such lines are there & why ?

Q. 9. Draw the graph of each of the following linear equation in two variables :
3 = 2x + y
y = 3x

Q. 10. The taxi fare in the city is as follows : For the first kilometer, the are is Rs. 8 & for the subsequent distance it is Rs. 5 per km. Write a linear equation for this information & draw its graph .


Q.11. Write the quadrant in which each of the following points lie :

(i) (–3, –5) (ii) (2, –5) (iii) (–3, 5) Also, verify by locating them on

the Cartesian plane.

Q. 12 Solve the equation 3x + 2 = 2x – 2 and represent the solution on the Cartesian plane

Q. 13. The taxi fair in a city is as follows:

For the first kilometer, the fare is Rs 10 and for the subsequent distance it is Rs 6 per km. Taking the distance covered as x km and total fare as Rs y, write a linear equation for this information and draw its graph. From the graph, find the fare for travelling a distance of 4 km

Q.14. Is (1, 8) the only solution of y = 3x + 5? Give reasons.

Q.15. Write the coordinates of a point on x-axis at a distance of 4 units from origin in the positive direction of x-axis and then justify your answer.

CBSE NCERT COORDINATE GEOMETRY for class 9

Plotting of points in the cartesian plane:

• In the Cartesian plane, the horizontal line is called the x-axis and the vertical line is called the y-axis,

• The coordinate axes divide the plane into four parts called quadrants,

• The point of intersection of the axes is called the origin,

• Abscissa or the x-coordinate of a point is its distance from the y-axis and the ordinate or the y-coordinate is its distance from the x-axis,

• (x, y) are called the coordinates of the point whose abscissa is x and the ordinate is y,

• Coordinates of a point on the x-axis are of the form (x, 0) and that of the point on the y-axis is of the form (0, y),

• The coordinates of the origin are (0, 0),

• Sign of the coordinates of a point in the first quadrant are (+, +), in the second

quadrant (–, +), in the third quadrant (–, –) and in the fourth quadrant (+, –).

Write the correct answer in each of the following :

1. Point (–3, 5) lies in the

(A) first quadrant (B) second quadrant (C) third quadrant (D) fourth quadrant

2. Signs of the abscissa and ordinate of a point in the second quadrant are respectively

(A) +, + (B) –, – (C) –, + (D) +, –

3. Point (0, –7) lies

(A) on the x –axis (B) in the second quadrant (C) on the y-axis (D) in the fourth quadrant

4. Point (– 10, 0) lies

(A) on the negative direction of the x-axis  (B) on the negative direction of the y-axis

(C) in the third quadrant   (D) in the fourth quadrant

5. Abscissa of all the points on the x-axis is

(A) 0 (B) 1(C) 2 (D) any number

6. Ordinate of all points on the x-axis is

(A) 0 (B) 1 (C) – 1 (D) any number

7. The point at which the two coordinate axes meet is called the

(A) abscissa (B) ordinate (C) origin (D) quadrant

8. A point both of whose coordinates are negative will lie in

(A) I quadrant (B) II quadrant (C) III quadrant (D) IV quadrant

9. Points (1, – 1), (2, – 2), (4, – 5), (– 3, – 4)

(A) lie in II quadrant (B) lie in III quadrant (C) lie in IV quadrant (D) do not lie in the same quadrant

10. If y coordinate of a point is zero, then this point always lies

(A) in I quadrant (B) in II quadrant (C) on x - axis (D) on y - axis

11. The points (–5, 2) and (2, – 5) lie in the

(A) same quadrant (B) II and III quadrants, respectively

(C) II and IV quadrants, respectively (D) IV and II quadrants, respectively

12. If the perpendicular distance of a point P from the x-axis is 5 units and the foot of the perpendicular lies on the negative direction of x-axis, then the point P has

(A) x coordinate = – 5 (B) y coordinate = 5 only

(C) y coordinate = – 5 only (D) y coordinate = 5 or –5

13. On plotting the points O (0, 0), A (3, 0), B (3, 4), C (0, 4) and joining OA, AB, BC and CO which of the following figure is obtained?

(A) Square (B) Rectangle (C) Trapezium (D) Rhombus

14. If P (– 1, 1), Q (3, – 4), R(1, –1), S(–2, –3) and T (– 4, 4) are plotted on the graph paper, then the point(s) in the fourth quadrant are (A) P and T (B) Q and R (C) Only S (D) P and R

15. If the coordinates of the two points are P (–2, 3) and Q(–3, 5), then (abscissa of P) – (abscissa of Q) is  (A) – 5 (B) 1 (C) – 1 (D) – 2

16. If P (5, 1), Q (8, 0), R (0, 4), S (0, 5) and O (0, 0) are plotted on the graph paper, then the point(s) on the x-axis are  (A) P and R (B) R and S (C) Only Q (D) Q and O

17. Abscissa of a point is positive in

(A) I and II quadrants (B) I and IV quadrants (C) I quadrant only (D) II quadrant only

Answers 1. (B) 2. (C) 3. (C) 4. (A) 5. (D) 6. (A) 7. (C) 8. (C) 9. (D) 10. (C)11. (C) 12. (D) 13. (B) 14. (B) 15. (B)  16. (D) 17. (B)

2 MARKS QUESTIONS

1. Points A (5, 3), B (– 2, 3) and D (5, – 4) are three vertices of a square ABCD. Plot these points on a graph paper and hence find the coordinates of the vertex C.

2. Write the coordinates of the vertices of a rectangle whose length and breadth are 5 and 3 units respectively, one vertex at the origin, the longer side lies on the x-axis and one of the vertices lies in the third quadrant.

3. Plot the points P (1, 0), Q (4, 0) and S (1, 3). Find the coordinates of the point R such that PQRS is a square.

4. Plot the points A (1, – 1) and B (4, 5)

(i) Draw a line segment joining these points. Write the coordinates of a point on this line segment between the points A and B.

(ii) Extend this line segment and write the coordinates of a point on this line which lies outside the line segment AB.

ANSWERS

1. C(–2, – 4)         2. (0, 0), (–5, 0), (0, –3)            3. (4, 3)          4. (i) (2, 1), (ii) (5, 7)

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