10th trigonometrical Identities

10th trigonometrical Identities

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CBSE CLASS 10 Trigonometric Ratios-Key Concept

10th trigonometrical Identities

Key Concept of Trigonometric Ratios

Trigonometric Ratios of 30 degree and more

CBSE Class X Self Evaluation Tests For Maths : Real Numbers

1. The values of the remainder r, when a positive integer a is divided by 3 are 0 and 1 only. Justify your answer

No. According to Euclid’s division lemma,

a = 3q + r, where 0 ≤ r < 3 and r is an integer. Therefore, the values of r can be 0, 1 or 2.

2. Can the number 6n, n being a natural number, end with the digit 5? Give reasons.

 No, because 6ⁿ = (2 × 3)ⁿ = 2ⁿ × 3ⁿ, so the only primes in the factorization of 6ⁿ are 2 and 3, and not 5. Hence, it cannot end with the digit 5.

3. Write whether every positive integer can be of the form 4q + 2, where q is an integer. Justify your answer.

No, because an integer can be written in the form 4q, 4q+1, 4q+2, 4q+3.

4. “The product of two consecutive positive integers is divisible by 2”. Is this statement true or false? Give reasons.

True, because n (n+1) will always be even, as one out of n or (n+1) must be even

5. “The product of three consecutive positive integers is divisible by 6”. Is this statement true or false”? Justify your answer.

True, because n (n+1) (n+2) will always be divisible by 6, as at least one of the factors will be divisible by 2 and at least one of the factors will be divisible by 3.

6. Write whether the square of any positive integer can be of the form 3m + 2, where m is a natural number. Justify your answer.

No. Since any positive integer can be written as 3q, 3q+1, 3q+2,

therefore, square will be 9q2 = 3m, 9q2 + 6q + 1 = 3 (3q2 + 2q) + 1 = 3m + 1, 9q2 + 12q + 3 + 1 = 3m + 1.

7. A positive integer is of the form 3q + 1, q being a natural number. Can you write its square in any form other than 3m + 1, i.e., 3m or 3m + 2 for some integer m? Justify your answer.

No. (3q + 1)2 = 9q2 + 6q + 1 = 3 (3q2 + 2q) = 3m + 1.

8. The numbers 525 and 3000 are both divisible only by 3, 5, 15, 25 and 75. What is HCF (525, 3000)? Justify your answer.

HCF = 75, as HCF is the highest common factor

9. Explain why 3 × 5 × 7 + 7 is a composite number.

3×5×7+7 = 7 (3×5 + 1) = 7 (16), which has more than two factors

10. Can two numbers have 18 as their HCF and 380 as their LCM? Give reasons.

No, because HCF (18) does not divide LCM (380).

11. Without actually performing the long division, find if 987/10500 will have terminating or non-terminating (repeating) decimal expansion. Give reasons for your answer.

Terminating decimal expansion, because 987/ 10500 = 47/ 500  and 500 =5³ 2²

12. A rational number in its decimal expansion is 327/7081. What can you say about the prime factors of q, when this number is expressed in the form p/q ? Give reasons.

Since 327.7081 is a terminating decimal number, so q must be of the form 2^m.5ⁿ;m, n are natural numbers

Self Evaluation Tests Paper For Maths : Real Numbers-2

Practice paper for class 10 chapter-Polynomials

1. Every linear equation in two variables has ___ solution(s).

(a) No (b) one (c) two (d) infinitely many

2. For a pair to be consistent and dependent the pair must have

(a) no solution (b) unique solution (c) infinitely many solutions (d) none of these

3. Graph of every linear equation in two variables represents a ___

(a) point (b) straight line(c) curve (d) triangle

4. Each point on the graph of pair of two lines is a common solution of he lines in case of

(a) Infinitely many solutions (b) only one solution (c) no solution (d) none of these

5.The pair of linear equations is said to be inconsistent if they have

(a) only one solution (b) no solution (c) infinitely many solutions. (d) both a and c

6. Find the value of k so that the equations x + 2y = – 7, 2x + ky + 14 = 0 will represent coincident
lines.

7. Give linear equations which is coincident with 2 x + 3y - 4 = 0 Find the value of K so that the pair of linear equations :

(3 K + 1) x + 3y – 2 = 0

(K² + 1) x + (k–2)y – 5 = 0 is inconsistent.

8. Solve for x and y :

2^x + 3^y = 17

2^{x + 2} – 3 ^y+1 = 5.

9. The area of a rectangle remain the same if its length is increased by 7 cm and the breadth is decreased  by 3 cm. The area remains unaffected if length is decreased by 7 cm and the breadth is increased by 5 cm. Find length and breadth.

10. A no. consists of three digits whose sum is 17. The middle one exceeds the sum of other two by 1. If the digits are reversed, the no. is diminished by 396. Find the no.

CBSE TEST PAPER MATHEMATICS (Class-10) SIMILAR TRIANGLE

1. In an equilateral Δ ABC, the side BC is trisected at D. Prove that 9AD2 = 7AB2

2.  P and Q are points on sides AB and AC respectively, of ΔABC. If AP = 3 cm,PB = 6 cm, AQ = 5 cm and QC = 10 cm, show that BC = 3 PQ.

3. The image of a tree on the film of a camera is of length 35 mm, the distance from the lens to the film is 42 mm and the distance from the lens to the tree is 6 m. How tall is the portion of the tree being photographed?

4. . Prove that in any triangle the sum of the squares of any two sides is equal to twice the square of half of the third side together with twice the square of the median, which bisects the third side.

5.  If a straight line is drawn parallel to one side of a triangle intersecting the othertwo sides, then it divides the two sides in the same ratio.

6. If ABC is an obtuse angled triangle, obtuse angled at B and if AD ^ CB Prove that  AC²=AB²+ BC²+2 BC x BD

7. If a straight line divides any two sides of a triangle in the same ratio, then the line must be parallel to the third side.

8. ABCD is a quadrilateral with AB =AD. If AE and AF are internal bisectors of ΔABC, D and E are points on AB and AC respectively such that AD/ DB = AEC/EC and ΔABC is isosceles.

9. In a ΔABC, points D, E and F are taken on the sides AB, BC and CA respectively such that DE IIAC and FE II AB.

10. . Prove that three times the sum of the squares of the sides of a triangle is equal to four times the sum of the squares of the medians of the triangle.

11. If a perpendicular is drawn from the vertex of a right angled triangle to its hypotenuse, then the triangles on each side of the perpendicular are similar to the whole triangle.

12. A man sees the top of a tower in a mirror which is at a distance of 87.6 m from the tower. The mirror is on the ground, facing upward. The man is 0.4 m away from the mirror, and the distance of his eye level from the ground is 1.5 m. How tall is the tower? (The foot of man, the mirror and the foot of the tower lie along a straight line).

13. In a right Δ ABC, right angled at C, P and Q are points of the sides CA and CB respectively, which divide these sides in the ratio 2: 1. Prove that

(I) 9AQ²= 9AC²+4BC² (II) 9 BP²= 9 BC²+ 4AC² (III) 9 (AQ²+BP²) = 13AB²

14. ABC is a triangle. PQ is the line segment intersecting AB in P and AC in Q such that PQ parallel to BC and divides Δ ABC into two parts equal in area. Find BP: AB.

15. P and Q are the mid points on the sides CA and CB respectively of triangle ABC right angled at C. Prove that4(AQ²+BP²) = 5 AB²

CBSE CLASS 10 MCQ TRIGONOMETRY

1. If cos A = 4/5 , then the value of tan A is

(A) 3/5             (B)3/4              (C)4/3              (D)5/3

2. If sin A = 1/2 , then the value of cot A is

(A) 3                (B) 1/3             (C) 3/2             (D) 1

3. The value of the expression [cosec (75° + q) – sec (15° – q – tan (55° + q+ cot (35° – q)] is

(A) – 1             (B) 0                (C) 1                (D) 3/2

4. Given that sinq= a/b , then cosq is equal to



5. If cos (a + b) = 0, then sin (a - b) can be reduced to

(A) cos b                     (B) cos 2b                   (C) sin α                      (D) sin 2a

6. The value of (tan1° tan 2° tan3° ... tan 89°) is

(A) 0                (B) 1                (C) 2                            (D)1 / 2

7. If cos 9a= sinα and 9a < 90°, then the value of tan5a is

(A) 1/√3                       (B) √ 3                         (C) 1                (D) 0

8. If DABC is right angled at C, then the value of cos (A+B) is

(A) 0                (B) 1                            (C) 1/2                         (D)√3/2

9. If sinA + sin²A = 1, then the value of the expression (cos²A + cos⁴A) is

(A) 1                (B) 1/2                         (C) 2                            (D) 3

10. Given that sina= 1/2 and cosb =1/2 , then the value of (a + b) is

(A) 0°                           (B) 30°                         (C) 60°            (D) 90°

11. The value of the expression [sin² 22⁰ sin² 68⁰ / cos² 22⁰ cos² 68⁰       +  sin ² 63⁰ cos  63⁰ sin 27⁰ ]   is         

(A) 3                (B) 2                (C) 1                            (D) 0

12. If 4 tanq = 3, then [4sinq - cosq ] / [4sinq + cos q ] is equal to

(A) 2/3                                     (B) 1/3                         (C) 1/2             (D) 3/4

13. If sinq – cosq = 0, then the value of (sin⁴q + cos⁴qθ) is

(A) 1                (B) 3/4                         (C) 1/2                         (D) 1/4

14. sin (45° + q) – cos (45° – q) is equal to

(A) 2cosq                    (B) 0                (C) 2 sin q                   (D) 1

15. A pole 6 m high casts a shadow 2 √3m long on the ground, then the Sun’s elevation is

(A\) 60°                        (B) 45°                         (C) 30°                        (D) 90°

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)

1. A point lies on the x-axis at a distance of 7 units from the y-axis. What are its coordinates? What will be the coordinates if it lies on y-axis at a distance of –7 units from x-axis?


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


3. A triangle ABC is right angled at A. L is a point on BC such that AL ^ BC. Prove that ∠< BAL = ∠ <ACB.


4. Two lines are respectively perpendicular to two parallel lines. Show that they are parallel to each other.

FORMATIVE TEST 2 GUESS PAPER CLASS 10 MATHS

Section -A

CBSE GUESS

1.The pair of equations x + 2y + 5 = 0 and –3x – 6y + 1 = 0 have

(A) a unique solution (B) exactly two solutions        (C) infinitely many solutions (D) no solution

2. If a pair of linear equations is consistent, then the lines will be

(A) parallel      (B) always coincident             (C) intersecting or coincident             (D) always intersecting

3. The pair of equations y = 0 and y = –7 has

(A) one solution          (B) two solutions        (C) infinitely many solutions              (D) no solution

4.If sinA + sin2A = 1, then the value of the expression (cos2A + cos4A) is

(A) 1                                        (B)1/2                          (C) 2                                                  (D) 3

5. Given that sinα =1/2 and cosβ =1/2 , then the value of (α + β) is

(A) 0°              (B) 30°                                    (C) 60°                                                           (D) 90°

Section - B

1.For which values of p and q, will the following pair of linear equations have infinitely many solutions?

4x + 5y = 2    and            (2p + 7q) x + (p + 8q) y = 2q – p + 1.

2. In a competitive examination, one mark is awarded for each correct answer while1/2 mark is deducted for every wrong answer. Jayanti answered 120 questions and got 90 marks. How many questions did she answer correctly

3. Given that α + β = 90°, show that  = sinα

4. A two-digit number is obtained by either multiplying the sum of the digits by 8 and then subtracting 5 or by multiplying the difference of the digits by 16 and then adding 3. Find the number

5.An aeroplane leaves an Airport and flies due North at 300 km/h. At the same time, another aeroplane leaves the same Airport and flies due West at 400 km/h. How far apart the two aeroplanes would be after 1and 1/2  hours?       Or,

In an equilateral triangle ABC, D is a point on side BC such that BD =1/3 BC. Prove that 9 AD²= 7 AB².

6. Prove that the area of the semicircle drawn on the hypotenuse of a right angled triangle is equal to the sum of the areas of the semicircles drawn on the other two sides of the triangle.

7. Show that tan⁴θ + tan²θ = sec⁴θ – sec²θ.

8. If sin θ + cos θ = √3 , then prove that tan θ + cot θ = 1

9. Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians

10. In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

MATH 10th Quadratic Equations Important quesrtions

1. Represent the following situation mathematically:

John and Jivanti together have 45 marbles. Both of them lost 5 marbles each and the product of the number of marbles they now have is 124. We would like to find out how many marbles the had to start with.

2. A cottage industry produces certain number of toys in a day. The cost of production of each toy (inrupees) was found to be 55 minus the number of the toys produced in a day.On aparticular day, thetotal cost of production was Rs. 750. We would like to find out the number of toys produced on that day.

3. Represent the following situations in the form of quadratic

equations: Rohan’s mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. We would like to find Rohan’s present age.

4. A train travels a distance of 480 km at a uniform speed. If the speed had been 8km/h less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train.

5. Find two consecutive positive integers, sum of whose squares is 365.

6. Find the root of the equation; x -1/x = 3

7. Two water taps together can fill a tank in 9 hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill thetank.

8. An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore (without taking into consideration the time they stop at intermediate stations). If the average speed of the express train is 11 km/h more than that of the passenger train, find the average speed of the two trains.

9. Find the value of k of the following quadratic equations, so that they have two equal root of kx (x – 2) + 6 = 0.

10.A motor boat whose speed is 18 km/h in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.

CBSE NCERT MATH 10th Chapter 6 Triangles Test Paper

Triangle test paper-1
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Triangle test paper-2
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Triangle test paper-3
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Introduction_to_trigonometry_test_paper-4
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10th_similar_triangle_solved
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CBSE NCERT MATH 10th Sample Question Papers for Term I

(September 2011)

Science (Class IX & X)

Mathematics (Class IX & X)

Social Science (Class IX)

CBSE Maths Sample Papers X 2011- 2012 Sem. 1

CBSE NCERT MATH 10th Chapter 8 Introduction to Trigonometry Test Paper

Trigonometry_identities_test_paper_1
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Trigonometry_identities_test_paper_2
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Trigonometry_identities_test_paper_3
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CBSE NCERT MATH 10th Chapter 3 Pair of Linear Equations in Two Variables Test Paper

Linear equations 2-variables_MCQ-test paper-1
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CBSE NCERT MATH 10th Chapter 2 Polynomials Test paper

Polynomial_cbse_test_paper-1

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CBSE NCERT MATH 10th Chapter Real Numbers Test Papers

Real_numbers_test_paper.1

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10th_real_number_objectives_questions.pdf

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cbse-math-triangles NCERT optional question solved

Prove that the sum of the squares of the diagonals of parallelogram is equal to the sum of the squares of its sides.

Given that ABCD be a parallelogram.

Construction : We have to Draw AF and DE perpendicular on side DC and on extended side AB to use Pythagoras theorem

In ΔDEA, DE2 + EA2 = DA2 … (i)

Similarly, In ΔDEB,

DE2 + EB2 = DB2

DE2 + (EA + AB)2 = DB2 ( EB =EA + AB)

DE2 + EA2 + AB2 + 2EA × AB == DB2

(DE2 + EA2) + AB2 + 2EA × AB = DB2

DA2 + AB2 + 2EA × AB = DB2 … (ii)( Using -----1)

In ΔAFC, AC2 = AF2 + FC2

= AF2 + (DC − FD)2

= AF2 + DC2 + FD2 − 2DC × FD

= (AF2 + FD2) + DC2 − 2DC × FD

=AC2 = AD2 + DC2 − 2DC × FD ( using AD2 = AF2 + FD2)… (iii)

Now,given that  ABCD is a parallelogram, AB = CD … (iv) And, BC = AD … (v)

Look In ΔDEA and ΔADF,

∠DEA = ∠AFD (Both 90°)

∠EAD = ∠ADF (EA || DF -Alternate angles )

AD = AD (Common)

∴ ΔEAD cong. ΔFDA (AAS congruence criterion)

⇒ EA = DF … (vi)
Adding equations (i) and (iii), we obtain

DA2 + AB2 + 2EA × AB + AD2 + DC2 − 2DC × FD = DB2 + AC2

DA2 + AB2 + AD2 + DC2 + 2EA × AB − 2DC × FD = DB2 + AC2

BC2 + AB2 + AD2 + DC2 + 2EA × AB − 2AB × EA = DB2 + AC2

[Using equations (iv) and (vi)]

AB2 + BC2 + CD2 + DA2 = AC2 + BD2

Q. In Fig. is the perpendicular bisector of the line segment DE, FA perpendiculat OB and F E intersects OB at the point C. Prove that 1/ OA + 1/OB = 1/OC

Mathematicians born in India

Mathematicians