CBSE MATH STUDY: Probability

CBSE MATH STUDY: Probability: "Probability1. A coin is tossed 1000 times with the following frequencies: Head: 455, Tail: 545 compute the probabi..."

Quadrilaterals and Parallelogram Test paper for class 9 CBSE/NCERT

    Prove that followings:

 1. A diagonal of a parallelogram divides it into two congruent triangles.

2. In a parallelogram, opposite sides and angle are equal.

3. If each pair of opposite sides of quadrilateral is equal, then it is a parallelogram.

4. If in a quadrilateral, each pair of opposite angles is equal, then it is a parallelogram.

5. The diagonals of a parallelogram bisect each other.

6. If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

7. A quadrilateral is a parallelogram if a pair of opposite sides is equal and parallel.

8. The line drawn through the mid-point of one side of a triangle, parallel to another side bisects the third side.

9. The line segment joining the mid- points of the two sides of a triangle is parallel to the third side.

10. Show that each angle of a rectangle is a right angle.

11. Show that the diagonal of a rhombus are perpendicular to each other.

12. ABC is an isosceles triangle in which AB=AC. AD bisects exterior angle PAC and CD||AB. Show that    (i) angle DAC=angle BCA and(ii) ABCD is a parallelogram (||gm).

13. Show that the bisectors of the angles of a parallelogram form a rectangle.

14. ABCD is a parallelogram (||gm) in which P and Q are mid-points of opposite side AB and CD. If AQ intersects DP at S and BQ intersects CO at R, show that
(i) APCQ is ||gm
(ii)DPBQ is ||gm
(iii) PSQR is ||gm]

15 If the diagonal of a parallelogram are equal, then show that it is a rectangle.

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

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

18. Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.


19. In a parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP=BQ. Show that
(I) tri APB cong Tri CQB
(ii) AP=CQ
(iv) tri AQB cong Tri CPD
(iv) AQ=CP
(v) APCQ is a parallelogram.

20. 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 =1/2 AC
(ii) PQ=SR (iii) PQRS is a parallelogram.

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

22 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.

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

24 ABC is a triangle right angle at C. A line through the mid-points M of hypotenuse AM 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.

25. Parallelograms on the same base and between the same parallels are equal in area.

10th Probability Test sample paper

Probability

1.     A coin is tossed 1000 times with the following frequencies:
Head: 455, Tail: 545 compute the probability for each event.

2.     Two coins are tossed simultaneously 500 times, and we get- Two heads: 105 times,
One head: 275 times, No head: 120 times, find the probability of occurrence of each of these events.

3.     A die is thrown 1000 times with the frequencies for the outcome 1, 2, 3, 4, 5 and 6 as given in the following table:

Outcome	1	2	3	4	5	6
Frequency	179	150	157	149	1175	190

Find the probability of getting each outcome.

4.     The record of a weather station shows that out of the past 250 consecutive days, its weather forecasts were correct 175 times.
(i) What is the probability that on a given day it was correct?
(ii) What is the probability that it was not correct on a given day?

5.     The percentage (%) of the marks obtained by a student in the monthly unit test are given below:

Unit test	I	II	III	IV	V
Percentage (%) of the marks obtained	69	71	73	68	74

Based on this data, find the probability that the student gets more than 70% marks in a unit test.

6.     An insurance company selected 2000 drivers at random (i.e. without any preference of one driver over another) in a particular city to find a relationship between age and accidents. The data obtained are given in the following table:

Age of  drivers (in years	Accidents in one year
	0	1	2	3	Over 3
18-29	440	160	110	61	35
30-50	505	125	60	22	18
Above 50	360	45	35	15	9

Find the probabilities of the following events for a driver chosen at random from the city:
(i) Being 18-29 years of age and having exactly 3 accidents in one year.
(ii) Being 30-50 years of age and having one or more accidents in a year.
(iii) Having no accidents in one year.

7.     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.

8.     1500 families with 2 children were selected randomly, and the following data were recorded:

Numbers of girls in a family	2	1	0
Number of families	475	814	211

Compute the probability of a family, chosen at random, having
(i) 2 girls
(ii) 1 girl
(iii) No girl.

9.     Three coins are tossed simultaneously 200 times with the following frequencies of different outcomes:

Outcome	3 heads	2 heads	1 head	No head
Frequency	23	72	77	28

If the three coins are simultaneously tossed again, compute the probability of 2 heads coming up.

10.   An organization selected 2400families 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 in the table below:

Monthly in come  (in Rs)	Vehicles per family
	0	1	2	Above 2
Less than 7000	10	160	25	0
7000-10000	0	305	27	2
10000-13000	1	535	29	1
13000-16000	2	469	59	25
16000 or more	1	579	82	88

Suppose a family is chosen. Find the probability that the family chosen is
(i) Earning Rs 10000-13000 per month and owing exactly 2 vehicles.
(ii) Earning Rs 16000 or more per month and owning exactly 1 vehicle.
(iii) Earning less than Rs 7000 per month and does not own any vehicle.
(iv) Earning Rs 13000-16000 per month and owning more than 2 vehicles.
(v) Owning not more than 1 vehicle.
11.   Eleven bags of wheat flour, each marked 5 kg, actually contained 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 find the probability that any of these Bags chosen at random contains more than 5 kg of flour.

exponants

Powers

x ^a x ^b = x ^{(a + b)}x ^a y ^a = (xy) ^a
(x ^a) ^b = x ^(ab)
x ^(a/b) = b^th root of (x ^a) = ( b^th (x) ) ^a
x ^(-a) = 1 / x ^a
x ^{(a - b)} = x ^a / x ^b

Logarithms

y = log_b(x) if and only if x=b ^y

log_b(1) = 0
log_b(b) = 1
log_b(x*y) = log_b(x) + log_b(y)
log_b(x/y) = log_b(x) - log_b(y)
log_b(x ⁿ) = n log_b(x)

log_b(x) = log_b(c) * log_c(x) = log_c(x) / log_c(b)

Table of Trigonometric Identities

Table of Trigonometric Identities

Reciprocal identities



Pythagorean Identities



Quotient Identities



Co-Function Identities



Even-Odd Identities



Sum-Difference Formulas



Double Angle Formulas



Power-Reducing/Half Angle Formulas



Sum-to-Product Formulas



Product-to-Sum Formulas

CBSE MATH STUDY: TEST PAPER Real Numbers

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CBSE MATH STUDY: VOLUMES Test paper for 9th

CBSE MATH STUDY: VOLUMES:
"JSUNIL TUTORIAL PUNJABI COLONY GALI 01 VOLUMES ..."

PH: 9835859669

JSUNIL TUTORIA

PUNJABI COLONY GALI 01

CLASS 9^th       SURFACE AREAS AND VOLUMES

1.    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.

2.    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.

3.    The hollow sphere, in which the circus motorcyclist performs his stunts, has a diameter of 7 m. Find the area available to the motorcyclist for riding.

4.    A wall of length 10 m was to be built across an open ground. The height of the wall is 4 m and thickness of the wall is 24 cm. If this wall is to be built up with bricks whose dimensions are 24 cm × 12 cm × 8 cm, how many bricks would be required?

5.    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?

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

7.    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.

8.    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.

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

10.  The diameter of a sphere is decreased by 25%. By what per cent does its curved surface area decrease?

11.  What length of tarpaulin 3 m wide will be required to make conical tent of height 8 m and base radius 6 m? Assume that the extra length of material that will be required for stitching margins and wastage in cutting is approximately 20 cm (Use p = 3.14).

12.   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 of the steel actually used was wasted in making the tank.

CBSE MATH STUDY: Triangles Oral Assessment:

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CBSE MATH STUDY: Quadrilaterals

CBSE MATH STUDY: Quadrilaterals: " 1. The angles of quadrilateral are in the ratio 3 : 5 : 9 : 13. Find all the angles of the quadrilateral. 2. If the diagonals of a par..."

Quadrilaterals

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

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

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

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

5. Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.

6. Diagonal AC of a parallelogram ABCD bisects angle A . Show that
(i) it bisects angle C also,
(ii) ABCD is a rhombus.

7. In parallelogram ABCD, two points P and Q are taken on
 diagonal BD such that DP = BQ. Show that:

8. ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD. Show that

9. In ∆ ABC and ∆ DEF, AB = DE, AB || DE, BC = EF and BC || EF. Vertices A, B and C are joined to vertices D, E and F respectively. Show that
(i) quadrilateral ABED is a parallelogram
(ii) quadrilateral BEFC is a parallelogram
(iii) AD || CF and AD = CF
(iv) quadrilateral ACFD is a parallelogram
(v) AC = DF

10. ABCD is a trapezium in which AB || CD and AD = BC. Show that

Key Points About Quadrilaterals

1. Sum of the angles of a quadrilateral is 360°.

2. A diagonal of a parallelogram divides it into two congruent triangles.

3. In a parallelogram,

(i) opposite sides are equal

(ii) opposite angles are equal

(iii) diagonals bisect each other

4. A quadrilateral is a parallelogram, if

(i) opposite sides are equal or

(ii) opposite angles are equal or

(iii) diagonals bisect each other or

(iv) a pair of opposite sides is equal and parallel

5. Diagonals of a rectangle bisect each other and are equal and vice-versa.

6. Diagonals of a rhombus bisect each other at right angles and vice-versa.

7. Diagonals of a square bisect each other at right angles and are equal, and vice-versa.

8. The line-segment joining the mid-points of any two sides of a triangle is parallel to the third side and is half of it.

9. A line through the mid-point of a side of a triangle parallel to another side bisects the third side.

10. The quadrilateral formed by joining the mid-points of the sides of a quadrilateral, in order, is a parallelogram.

9th Polynomials Test paper

Polynomials Test paper

1. Factorise using factor theorem

x3- 12x2 + 47x + 60 

6x3 + 11x2 – 4x – 4 

2. Factorise 

 p3(q – r)3 + q3(r – p)3 + r3(p – q)3 

3. Find value of  x3 - 8y3 – 36xy – 216 when x = 2y + 6. 

4. Solved  703 – 503 - 203 

5. Factorise  (i) a6 + 4a3 – 1  (ii) 125a3 – 343b3

6. Verify x3 – y3 = (x – y)(x2 + xy + y2) 

 
7. Evaluate using suitable identity 9993 99.83 

8. Verify a3 + b3 + c3 – 3abc = (x + y + z)[ (x – y)2 + (y – z)2 + (z – x)2 ]

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surface area and volume test paper for class 8,9

URFACE AREA AND VOLUME TEST PAPER 

1. The volume of a right circular cylinder is 1200 cm3. If the diameter of the base of a right circular cylinderis 8 cm, what will be its height?

2. A metallic cylindrical tube has thickness 1 cm and outer radius is 10 cm. Find the mass of such a 2 metre long tube, if the density of the metal is 3.5 g/cm3.

3. If the lateral surface area of a cylinder of height 7 cm is 90.5 cm2. Find the volume of the 
cylinder.

4. The height and slant height of a cone are 15 cm and 20 cm respectively. Find the volume of the cone.

5. A right triangle PQR with its sides 6 cm, 8 cm and 10 cm is shown in the figure. If the given triangle is revolved about the side 6 cm, find the volume of the solid so obtained.

6. A metallic solid cone is melted and cast into the form of a circular cylinder of the same base as that of the cone. If the height of the cylinder is 5 cm, what was the height of the cone?

7. The capacity of a hemispherical tank is 10,000 cm3. Find its radius.

8. A metallic wire 25 m long with diameter 1 mm is melted to form a sphere. Find the volume of the sphere.

9. A hemispherical bowl of steel is of thickness 0.5 cm. If the inner radius of bowl is 5 cm, find the volume of the steel used in making the bowl.

10. The diameter of a metallic ball is 3.2 cm. What is the mass of the ball, if the density of the metal is 6.7gper cm3?

Triangles Oral Assessment:

Triangles

A  Oral Assessment:

1. What do you understand by congruence ?

2. Give examples of congruent figures from your surroundings.

3. What is triangle ?

4. What are the various parts of a triangle ?

5. What do you understand by side of a triangle ?

6. What is angle ?

7. How many types of triangle are there ?

8. Classify triangles on the basis of their sides.

9. Classify triangles on the basis of their angles.

10. What is exterior angle of a triangle ? How is it different from the interior angle of the triangle?

What is the relation between them ?

B Name it

1. Two circles of same radii

2. Of all the line segments drawn to a line from a point not lying on it..................distance

is the shortest

3. Sides opposite the equal angles

4. Angle opposite to shorter side

5. Triangles having all sides equal

6. Triangles having two equal sides

7. Longest side in a right triangle

8. Hypotenuse and one side of a right triangle are respectively equal to hypotenuse and one

side of another right triangle (congruency criteria)

9. Side opposite to greater angle

10. Three sides of one triangle are equal to corresponding sides of another triangle (congruency

criteria)

11. Two angles and included side of one triangle are respectively equal to two angles and

included side of another (congruency criteria)

12. Sum of two sides in a triangle is ................ than third side