STANDARD 10 , GUJARAT BOARD, STANDARD MATHEMATIC QUESTION PAPER, MARCH 2023

 Time : 3 hrs. Total Marks : 80

1. Write in clear and legible handwriting.

2. This question paper has four sections A,B,C and D and question number from 1 to 39

3. All questions are compulsory. Internal options are given

4. Numbers to the right represents the mark to for the question

5. Draw the neat diagram , wherever necessary

6. New section should be written on new page and write the answer in the numerical order

SECTION A(16 marks)

Answer the following as per the instruction given. This section has 16 questions and each carries 1 mark each. 

Write true or false for the following questions ( 1 to 4)

1. 8sec2𝜃 -  8tan2𝜃 = 8

2. 7 x 11 x 13 + 13  is a prime number

3. ½, ⅓, ¼ are arithmetic progression

4. If one of the root of the quadratic equation X2 - 4X + m  = 0 is 3 then  m=3

Fill in the blanks to make each statement true. ( 5 to 8)

5. If H.C.F ( 12, k) = 6 and LCM(12,k) = 36 then k= —---

6. If 𝜶 and 𝜷 are zeros of polynomial 3x - X2 + 8  then 𝜶𝜷 = —-

7. If 27x + 63y = 45 and 63x + 27y = 135 then x + y = —- 

8. Coordinates of midpoint of line segment AB  joining A(2a - b, b) and B(b, 2a-b) is ____

Answer in one sentence, one word or one number

9. Which is the median class for the following frequency distribution.

Class	0-10	10-20	20-30	30-40	40-50
Frequency	7	15	13	17	10

10.  If P(A) -  P(A bar ) = 0.8 than find the value of P(A)

11. Find the diameter of the circle whose circumference and area are equal in number.

12. For what value of acute angle 𝜃, cot2𝜃.cot7𝜃 =1

Select the proper alternative to make each statement true.

13.  If a sphere of radius r divided into 4 equal parts, than total surface area of each part is

1. 𝛑r2

2. 2𝛑r2

3. 3𝛑r2

4. (½)𝛑r2

14. If the pair of linear equation 2x + 2y +2 =0 and 4x + ky + 8= 0 has unique solution, then k ≠

1. 4

2. 2

3. -4

4. 8

15. In the given figure PA and PB are tangent to the circle with centre O. ∠AOP = 550 than ∠APB= 

1. 350

2. 700

3. 1250

4. 1100

16. An unbiased coin is tossed thrice,what is the probability of getting at least two heads.

1. ⅜

2. ⅛

3. ½

4. ⅔

SECTION B (20 marks)

Solve the following questions showing calculations. ( 17 to 26) , Each carries two(2) marks.

17. Prove that 5 + 2√7 is irrational

18. Find the zeros of quadratic equations 6X2 - 13X + 6 = 0

19. Verify whether the linear pair of equations (4/3)x + 2y = 8 and 2x + 3y = 12 are consistent or not.

OR

Solve the pair of equations by substitution method.

x + y = 4 and 2x = 8 - 3y

20. If P, Q, R are interior angle of a triangle , prove that sec[(P+Q)/2] = cosec(R/2)

21. If 2 sin𝜃 + cos𝜃 = 2, find the value of tan𝜃

OR

2tan245 + x - sin260 = 2, then find the value of x 

22. As shown in the figure , quadrilateral PQRS is drawn to  circumscribe a circle, prove that 

Prove that PQ + RS = QR + SP

OR

Two concentric circles are of radii 29 cm and 21 cm. Find the length of chord of the larger circle which touches the smaller circle.

23. For the  following grouped frequency distribution find the mode.

class	10-25	25-40	40-55	55-70	70-85	85-100
frequency	2	3	7	6	6	6

24. Salma and Mona are friends. What it the probability that both have

1. Different birthday 

2. Same birthday in the year 2019.

25. Find the root of the quadratic equation  5x = 6 + 2/x by completing the square method.

OR

Find the root of quadratic equation √2x2 + 7x + 5√2 = 0

26. In the given figure if PQ ∥ BC , find the value of AB

SECTION C (24 marks)

Answer the following questions showing  calculation (27 to 34) each carries 3 marks.

27.  On dividing 3x3 + x2+ 2x + 5 by a polynomial g(x), the quotient and remainder were 3x - 5 and 9x + 10 respectively, find the g(x)

28.  Sum of areas of two squares is 468 sqm. If  the difference of their perimeter is 24 m. find the sides of two squares

29. For what value of n , nth term of two AP 65, 67,69 ,.... AND 10,17, 24 equal.

OR

Find the sum of all terms of AP -2, -5,-8…-227..

30.  If P(2,3), Q(3,-2), R(-3,-5) and S(-4,-2) are vertices of a quadrilateral. Find the area of quadrilateral PQRS.

31. The length of the minute hand of a clock is 14 cm. Find the area swept by minute hand in 15 minutes. Find the distance to be swept to complete one revolution.

OR

In the given figure OACB is quadrant of a circle with centre O and having diameter of 7 cm. If OD is 2 cm, Find the area of the 

1. Quadrant OACB

2. Shaded region.

32. A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is 28 cm and total height of the vessel is 26 cm.Find the inner surface area of the vessel.

33. Following frequency distribution shows the age of 100 persons. Find the median of the data.

Age in years	20-30	30-40	40-50	50-60	60-70	70-80
No of persons.	15	16	38	15	9	7

OR

The mean of following frequency distribution is 18. Find the missing frequency.

class	11-13	13-15	15-17	17-19	19-21	21-23	23-25
frequency	7	6	f	13	20	5	4

34. Prove that the length of the tangents drawn from an external point to a circle are equal.

SECTION D (20 marks)

Solve the following,( 35 to 39) Each carries 4 marks each.

35. A boat goes 40 km upstream and 49 km downstream in 15 hours. In the same river it can go 25 km upstream and 35 km downstream in 10 hours. Determine the speed of the stream and that of the boat in still water.

36. As observed from the top of 100 m high hill, the angle of depression to the top of a tower is 300 and angle of depressions to the bottom of the tower is 450.Find the height of the tower and distance between base of tower and base of the hill.

37. A container shaped like a right circular cylinder having radius 6 cm and height 15 cm  is full of ice cream. The ice cream is to be filled in cone of height 12 cm and radius 3 cm, with a hemispherical shape on the top. Find the number of such cones which can be filed with the ice cream.

38. In ΔXYZ if XY2 + XZ2 = YZ2 , then prove that ∠X = 900 

OR

BL and CM are medians of ΔABC right angled at A. 

Prove that 4(BL2 + CM2) = 5BC2

39. Draw a right triangle with sides (other than hypotenuse) are of length 4 cm and 3 cm. Then construct another triangle whose  sides are  5/3 times corresponding sides of the given triangle.

OR

Draw a circle radius of 4.5 cm. From  a point 7.5 cm away from its centre, construct the pair of tangents to the circle and measure their lengths. Write steps of construction.