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### Light Reﬂection and Refraction

1.The bending of a beam of light when it passes obliquely from one medium to another is known as       .

A.  reflection

B.  refraction

C.  dispersion

D.  deviation

2.The part of the lens through which the ray of light passes without suffering deviation is called                .

A.  optical centre

B.  focus

C.  centre of curvature

D.  pole

3. Convex lens always gives a real image if the object is situated beyond               .

A.  optical centre

B.  centre of curvature

C.  Focus

4. Parallel rays of light entering a convex lens always converge at              .

A.  centre of curvature

B.  the principal focus

C.  optical centre

D.  the focal plane

5. Where should an object be placed so that a real and inverted image of the same size is obtained, using a convex lens?

A.  Between O and F

B.  At F

C.  At 2 F

D.  At infinity

6. SI unit of the power of a lens is

A.      dioptre

B.      cm

C.      metre

D.    watt

7.  1 D is the power of the lens of focal length of

A.  10 cm

B.  100 cm

C.  1/ 10 cm

D.  1/100 cm

8. In a simple microscope lens used is              .

A.  biconvex

B.  biconcave

C.  plano convex

D.  cylindrical

9. Reciprocal of focal length in metres is known as the     of a lens.

A. focus

B. power

C. power of accommodation

D. far point

10. A convex lens is called             .

A. converging lens

B. diverging lens

C. both converging and diverging lens

D. refracting lens

11. A positive magnification greater than unity indicates                .

A. real image

B. virtual image

C. neither real not virtual image

D. distorted image

12. The power of a convex lens of focal length 50 cm is    .

A. + 2D

B. - 2D

C.  50 D

D.  - 5D

13            The focal length of a lens whose power is -1.5 D is

A.             -66.66 cm

B.      + 1.5 m

C.   + 66.66 cm

D.    -1.5 m

14. Real images formed by single convex lenses are always          .

A.  on the same side of the lens as the object

B.  Inverted

C.  Erect

D.  smaller than the object

15. An object is placed 12 cm from a convex lens whose focal length is 10 cm. The image must be.

A.  virtual and enlarged

B.  virtual and reduced in size

C.  real and reduced in size

D.  real and enlarged

16. When a person uses a convex lens as a simple magnifying glass, the object must be placed at a distance.

A.  less than one focal length

B.  more than one focal length

C.  less than twice the focal length

D.  more than twice the focal length

17. The image produced by a concave lens is       .

A.  always virtual and enlarged

B.  always virtual and reduced in size

C.  always real

D.  sometimes real, sometimes virtual

18.  A virtual image is formed by                .

A.  a slide projector in a cinema hall

B.  the ordinary camera

C.  a simple microscope

D.  Telescope

19.  An object is placed 25 cm from a convex lens whose focal length is 10 cm. The image distance is         cm.

A.  50 cm

B.  16.66 cm

C.  6.66 cm

4.  10 cm

20. The least distance of distinct vision is               .

A.  25 cm

C.  25 m

C.  0.25 cm

D.  2.5 m

## REAL NUMBERS

EUCLID'S DIVISION LEMMA

Given positive integers a and b. there exist unique integers q and r satisfying a = bq + r where
0<  r <b,
Here we call 'a' as a dividend, 'b' as divisor 'q' as quotient and 'r' as the remainder,

Dividend = (Divisor x Quotient) + Remainder

If in Euclid's lemma r= 0 then b would be HCF of 'a' and 'b'.

### IMPORTANT QUESTIONS

Show that any positive even integer is of the form 6q, or 6q+ 2, or 6q + 4. where q is some integer.

Solution: Let x be any positive integer such that x >6, Then, by Euclid's algorithm, x = 6q +r for some integer q > 0 and 0 <  r < 6.
Therefore, x= 6q or 6q+1 or 6q + 2 or 6q + 3 or 6q + 4 or 6q + 5 .
Now, 6q is an even integer being a multiple of 2.
We know that the sum of  two even integers are always even integers.
Therefore, 6q+2 and 6q + 4 are even integers .
Hence any positive even integer is of the form 6q. or 6q + 2, or 6q + 4. where q is some integer,

#### Questions for practice

1. Show that any positive even integer is of the form 4q or 4q + 2, where q is some integer.

2. Show that any positive odd integer is of the form 4q + 1, or 4q+3, where is some integer.

3. Show that any positive odd integer is of the form 6q + 1. or 6q + 3, or 6q + 5, where is some integer.

4. Use Euclid's division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.

5. Use Euclid's division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m +8.

EUCLID'S DIVISION ALGORITHM

Euclid's division algorithm is a technique to compute the Highest Common Factor (HCF) of two given positive integers. Recall that the HCF of two positive integers a and b is the largest positive integer d that divides both a and b.

To obtain the HCF of two positive integers, say c and d, with c > d, follow the steps below:

Step 1: Apply Euclid's division lemma, to c and d. So, we find whole numbers, q and r such that c =dq+r. 0 < r < d

Step 2: If r=0. d is HCF of c and d. If r is not equal 0 apply the division lemma to d and r .

Step 3: Continue the process till the remainder is zero. The divisor at this stage will be the required HCF.

This algorithm works because of HCF (c,d) = HCF (d,r) where the symbol HCF (c,d) denotes the HCF of c and d, etc.

IMPORTANT QUESTIONS

Use Euclid's division algorithm to find the HCF of 867 and 255

Solution: Since 867 > 255, we apply the division lemma to 867 and 255 to obtain 867 = 255  X  3 + 102.
Since remainder    $102&space;\neq&space;0$ , we apply the division lemma to 255 and 102 to obtain.
255 = 102 X 2 + 51
We consider the new divisor 102 and new remainder 51, and apply the division lemma to obtain 102 = 51 X 2 + 0
Since the remainder is zero, the process stops.
Since the divisor at this stage is 51
Therefore, HCF of 867 and 255 is 51.

Questions for practice

1. Use Euclid's algorithm to find the HCF of 4052 and 12576.

2.Use Euclid's division algorithm to find the HCF of 135 and 225.

3. Use Euclid's division algorithm to find the HCF of 196 and 38220.

4. Use Euclid's division algorithm to find the HCF of 455 and 42.

5. Using Euclid's algorithm, find the HCF of (i) 405 and 2520 (ii) 504 and 1188  (iii) 960 and 1575

## CLASS: X

### General Instruction:

(i) All the questions are compulsory.
(ii)The question paper costs of 40 questions divided into 4 sections ABC, and D
(iii) Section A comprises of 20 questions of 1 marks each. Section B comprises of 6 questions of 2 marks Section B comprises of 8 questions of 3 marks each. Section D comprises of 6  questions of 4 marks each.

#### SECTION -A

Questions 1 to 20 carry 1 mark each.

1. LCM of two co-prime numbers is always

(a) product of number
(b) sum of numbers
(c) difference of numbers.
(d)none

2 .For some integer q. every odd integer is of the form
(a)q
(b)q+1
(c)2q
(d)2q+1

3.$\sqrt{3}$ is

(a) a natural number
(b) a rational number
(c) not o a real number
(d) an irrational number

4. The product and sum of zeroes of the quadratic polynomial x2 + bx + c respectively are:

(a) b  ,   c
a      a

(a) c  ,   b
a      a

(a) c  ,   1
b

(a) c ,   -b
a      a

5. The ratio in which x-axis divides the line segment joining the points (5, 4) and (2-3) is
(a) 5:2
(b) 3:4
(C) 2:5
(d) 4: 3

6. Point on y-axis has coordinates

(a)(a,b)
(b)(a,0)
(c) (0,6)
(d) (-2, -b)

7 .From a point P,10 cm away from the centre of a circle, a tangent PT of length 8 cm is drawn Find the radius of the circle

(a)4 cm
(b) 5 cm
(c) 7 cm
(d)6 cm

8.

9. A box contains 3 blue, 2 white, and 5 red marbles. If a marble is drawn at random from the box then what is the probability that the marble will be blue ?

(a)  3,
10

(b) 1
2

(c) 1

(a)  0

11. The end points of diameter of circle are (2,4) and (-3,1). The radius of circle is _____

12. Value of tan 30 - cos 45 is ______

13. The value of san60° cos 30° - cos 60°sin30° is_____

14. 1f ∆ABC and ∆DEF are two triangles such that AB. = BC  = CA  =2 , Then ar(∆ABC) =
DE.     EF.     FD.   5            ar(∆DEF)

15. If (6, k) sa solution of the equation 3x + y-22=0, then the value of k is

16. Find the area of a sector of a circle with radii 6 cm if angle of the sectors is 90°.

17. IF P(E) = 0 47, what is the probability of 'not E'?

18. Find the 9th term from the end (towards the first term) of the AP 5,9.,13,..... ,185.

19. 1f  cot2© =tan1© , where 2θ nd 4θ are acute angles, find the value of sin 3θ.

20.ABC and BDE are two equilateral triangles such that D is the  midpoint of BC .Find the ratio of the areas of triangles ABC and BDE

### SECTION B Questions 21 to 26 carry 2 marks each

21. Three cards of spades are lost from a pack of 52 playing cards The remaining cards were well stuffed and then a card was drawn al random from them .Find the probability that the drawn cards of black colour

OR

Find the probability that a leap year should have exactly 52 Tuesday.

22. Two different dice are tossed together. Find the probability (i) that the Number on each dice is  Even (ii) that the sum of numbers appear on two dice is 5.

23. A wie s loped in the form of a circle of radan 28 cm .It reverted into a square form.Determine the side of the square.

24. Show that tan48° tan 23° tan 42°tan 67° =1

25. Find a quadratic polynomial whose zeroes are -5 and 7.

26. A quadrilateral ABCD is drawn to circumscribe circle Prove that AB + CD = AD + BC.

### SECTION -C Questions 27 to 34 carry marks each.

27.Find the HCF and LCM of 180 and 288 by prime factorisation method.

28 To conduct Sports Day activities, in your rectangular aided school ground ABCD, lines have been drawn with chalk powder at a distance of 1 m each 100 flower pots have been placed at a distance of 1m from each other along AD, as shown in the  figure Aditi name 1/5th the distance AD on the 2nd lne and posts a green fag Priyanka runs 1/4th the distance AD on the eighth line and posts a red flag
(i) What is the distance between both the flags?

(ii) If Manta has to post a blue flag exactly halfway between the line segment joining the two Nags, where should the post her flag?

29. Find the zeroes of the quadratic polynomial 9t - 6t + 1, and verify the relationship between the zeroes and the coefficients.

30. Solve the following system of equations graphically for x and y 3x + 2y= 12, 5x-2y=4

31. Prove that The lengths of the two tangents from an external point to a circle are equal.

32. Draw a line segment of length 9 cm and divide in the rate 3:4. Measure the two parts.

33. Prove that. A conec Ayrcos A CAY - 7+ A+ cor A

OR

Prove that

setan

14 sin A 1-m 4

34. In the below for sure OABC is inscribed quadrant OPBQ. If OA =20 cm, find the area of the shaded region.(Use Π= 3.14)

### SECTION -D Questions 35 to 40 carry marks rach

35. A 1.5 m tall boy is standing at some distance from a 30 m tall building. The angle of elevation from his eyes to the top of the building increases from 30° to 60° as he walks towards the building. Find the distance he walked towards the building.

36. The diagonal of a rectangular field is 60 metres more than the shorter side. If the longer side is 30 metres more than the shorter side, find the sides of the field.

37. Find the sum of first 24 terms of the list of numbers whose nth term is given by An= 3 + 2n

38.Prove that If Image is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio

OR

State and prove converse of Pythagoras theorem

39. A cubical block of side 7 cm is surmounted by a hemisphere. What is the greatest diameter the hemisphere can have? Find the surface area of the solid.

40.

### Sample Paper 1 | Based on full NCERT | MATHS | Class 10

SUBJECT: MATHEMATICS
SAMPLE PAPER 1
CLASS : X
MAX. MARKS: 80
DURATION : 3 HRS

General Instruction:

(i) All the questions are compulsory.

(ii) The question paper consists of 40 questions divided into 4 sections A, B, C, and D.

(iii) Section A comprises of 20 questions of 1 mark each. Section B comprises of 6 questions of 2 marks each. Section C comprises of 8 questions of 3 marks each. Section D comprises of 6 questions of 4 marks each.

(iv) Use of calculators is not permitted.

SECTION - A

1.The decimal expansion of pie
(a) is terminating
(b) is non terminating and recurring
(c) is non-terminating and non-recurring
(d) does not exist.

2.The product of L.C.M and H.C.F. of two numbers is equal to
(a) Sum of numbers
(b) Difference of numbers
(c) Product of numbers
(d) Quotients of numbers

3. What is the H.C.F. of two consecutive even numbers
(a) 1
(b)2
(c) 4
(d) 8

4. A quadratic polynomial can have at most... zeros
(a) 0
(b) 1
(c)2
(d) 3

5. Which are the zeroes of p(x) = (x - 1)(x-2):

(a) 1,-2
(b) - 1,2
(c) 1,2
(d)-1,-2

6. X-axis divides the join of A(2.-3) and B(5,6) in the ratio
(a) 3:5
(b) 2:3
(c) 2:1
(d) 1:2

7. If the distance between the points (8. p) and (4.3) is 5 then value of p is
(a) 6
(b) 0
(c) both (a) and (b)
(d) none of these

8. TP and TQ are the two tangents to a circle with center O so that angle POQ = 130°. Find angle PTQ.

(a) 50°
(b) 70
(c) 80"
(d) none of these

9. Cards are marked with numbers 1 to 25 are placed in the box and mixed thoroughly. What is the probability of getting a number 5?

(a) 1
(b) 0
(c) 1
25

(d) 1
5

10. The value of y for which the points A(1.4), B(3. y) and C-3, 16) collinear is

11. If  ∆ABC is right-angled at B, then the value of cos(A + C) is

12. If tanA =  , then the value of cosA is
3
13. In ABC, DE || BC and AD = 4 cm, AB =9 cm. AC = 13.5 cm then the value of EC is

14. The value of k for which the quadratic equation 4x 3kx +1=0 has real and equal roots

15. In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. Find the area of the sector formed by the arc

16. A bag contains lemon flavoured candies only. Malini takes out one candy without looking into the bag. What is the probability that she takes out a lemon flavoured candy?

17. In ∆ ABC, right-angled at B, AB = 5 cm and ㄥACB = 30° then find the length of the side BC

18. If sin 30 = cos(0-6) here, 30 and (0-6°) are acute angles, find the value of 0.

19. For what value of p, are 2p+1, 13,5p - 3 three consecutive terms of an AP?

20. The areas of two similar triangles ABC and ADEF are 144 cm and 81 cm, respectively. If the longest side of larger ABC be 36 cm, then find the longest side of the similar triangle ADEF.

SECTION B

21.15 cards, numbered 1,,. 3,. . ,15 are put in a box and mixed thoroughly. A card is drawn at random from the box. Find the probability that the card is drawn bears (i) an even number (ii) a number divisible by 3.

22. A card is drawn at random from a pack of 52 playing cards. Find the probability that the card is drawn is neither an ace nor a king.

23. The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in three minutes. [Use pie = 22/7]

24.If tan A = cot B, prove that A+B= 90°

25. If the product of the zeroes of the polynomial  $ax^2&space;-&space;6x&space;-&space;6$  is 4, then find the value of a. Also, find the sum of zeroes of the polynomial.

26. The two tangents from an external point P to a circle with centre O are PA and PB. If APB = 70°, what is the value of AOB?

SECTION C

27. Prove that   $2+5\sqrt{3}$  is an irrational number.

28. Find the zeroes of the quadratic polynomial   $6x^{2}&space;-&space;7x-&space;3$   and verify the relationship between the zeroes and the coefficients of the polynomial.

29. Given   linear equation 3x - 5y = 1 form another linear equation in these variables such that the geometric representation of pair so formed is: (i) intersecting lines (ii) coincident lines (iii) parallel lines.

30. To conduct Sports Day activities, in your rectangular shaped school ground ABCD, lines have been drawn with chalk powder at a distance of 1 m each. 100 flower pots have been placed at a distance of 1 m from each other along AD, as shown in the below figure. Niharika runs 1/4th the distance AD on the 2nd line and posts a green flag. Preet runs 1/5 th the distance AD on the eighth line and posts a red flag.

(i) What is the distance between both the flags?

(ii) If Rashmi has to post a blue flag exactly halfway between the line segment joining the two flags, where should she post her flag?

31. Prove that:

32. Construct a triangle ABC with BC = 7 cm. ㄥB = 60° and AB = 6 cm. Construct another triangle whose sides are times the corresponding sides of AABC.

OR

Draw a line segment of length 10 cm and divide it in the ratio 3:5. Measure the two parts.

33. In the below figure, AB and CD are two diameters of a circle (with centre O) perpendicular to each other and OD is the diameter of the smaller circle. If OA = 7 cm, find the area of the shaded region.

34. Prove that the intercept of a tangent between two parallel tangents to a circle subtends a right angle at the centre.

SECTION D

35. A 1.2 m tall girl spots a balloon moving with the wind in a horizontal line at a height of 88.2 m from the ground. The angle of elevation of the balloon from the eyes of the girl at any instant is 60°. After some time, the angle of elevation reduces to 30° (see the below figure). Find the distance travelled by the balloon during the interval.

36. In a class test, the sum of Shefali's marks in Mathematics and English is 30. Had she got 2 marks more in Mathematics and 3 marks less in English, the product of their marks would have been 210. Find her marks in the two subjects.

37. Show that a1 ......... form an AP where an is defined as the first 15 terms in each case.
=9-5n. Also find the sum of

38. Prove that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
OR
State and prove Basic proportionality theorem,

39. A solid toy is in the form of a hemisphere surmounted by a right circular cone. The height of the cone is 2 cm and the diameter of the base is 4 cm. Determine the volume of the toy. (Take pie =3.14)

40.

### CLASS 10 MCQ QUESTIONS | MATHS

1. If a and b are positive integers, then HCF (a, b) x LCM (a, b) =

(a)   a x b
(b) a + b
(c) a b
(d) a/b

2.If the HCF of two numbers is 1, then the two numbers are called

(a) composite
(b) relatively prime or co-prime
(c) perfect
(d) irrational numbers

The decimal expansion of      93      will be

1500
(a) terminating
(b) non-terminating
(c) non-terminating repeating
(d) non-terminating non-repeating.

4.  A quadratic polynomial whose sum and product of zeroes are –3 and 2 is
(a) x2 3x +2
(b) x2 + 3x + 2
(c) x2 + 2x 3.
(d) x2 + 2x + 3.

5.A point P divides the join of A(5, –2) and B(9, 6) are in the ratio 3 : 1. The coordinates of P are

(a) (4, 7)
(b) (8, 4)

(c) ( 11 , 5)
2
(d) (12, 8)

6. The distance of the point P(4, –3) from the origin is

(a)  1 unit
(b) 7 units
(c) 5 units
(d) 3 units

7. A point P is 26 cm away from the centre of a circle and the length of the tangent drawn from P to the circle is 24 cm. Find the radius of the circle.
(a) 11 cm         (b) 10 cm         (c) 16 cm         (d) 15 cm

8.Which measure of central tendency is given by the x – coordinate of the point of intersection of the more than ogive and less than ogive?

(a) mode
(b) median
(c) mean
(d) all the above three measures

9.   If the points (1, x), (5, 2) and (9, 5) are collinear then the value of x is

10.   Product tan10.tan20.tan30……tan890 is

11.   If ABC and DEF are similar triangles such that ÐA = 470 and ÐE = 830, then ÐC =

12.   The values of k for which the quadratic equation 2x2 + kx + 3 = 0 has real equal roots is

13.The value of k for which the system of equations x + 2y = 3 and 5x + ky + 7 = 0 has no solution is

14.How many three-digit numbers are divisible by 7?

15.The radii of two circles are 8 cm and 6 cm respectively. Find the radius of the circle having area equal to the sum of the areas of the two circles.

16.It is given that in a group of 3 students, the probability of 2 students not having the same birthday is 0.992. What is the probability that the 2 students have the same birthday?

 17.LCM of 6 and 20 is (a) 30                           (b) 60                 (c)120                        (d)none of these 18.  Given 15 cot A = 8, then sin A =