Time Allowed: 3 hours Maximum Marks: 80
General Instructions:
1. This Question Paper has 5 Sections A-E.
2. Section A has 20 MCQs carrying 1 mark each.
3. Section B has 5 questions carrying 02 marks each.
4. Section C has 6 questions carrying 03 marks each.
5. Section D has 4 questions carrying 05 marks each.
6. Section E has 3 case based integrated units of assessment carrying 04 marks each. Maximum Marks: 80
7. All Questions are compulsory. However, an internal choice in 2 Qs of 5 marks, 2 Qs of 3 marks and 2 Questions of 2 marks has been provided. An internal choice has been provided in the 2 marks questions of Section E.
8. Draw neat figures wherever required. Take π =22/7 wherever required if not stated.
Section A
1.
a) −1/5
b)-5
c) 1/5
d) 5
View AnswerAns. c) 1/5
2. If the line represented by the equation 3x + ky = 9 passes through the points (2, 3), then the value of k is
a) 2
b) 1
c) 3
d) 4
View AnswerAns. b) 1
3. In which quadrant does the point (-7, -4) lie?
a) IV
b) 1
c) II
d) III
View AnswerAns. d) III
4. A histogram is a pictorial representation of the grouped data in which class intervals and frequency are respectively taken along
a) horizontal axis only
b) horizontal axis and vertical axis
c) vertical axis and horizontal axis
d) vertical axis only
View AnswerAns. b) horizontal axis and vertical axis
5. The equation x – 2 = 0 on number line is represented by
a) infinitely many lines
b) two lines
c) a point
d) a line
View AnswerAns. c) a point
6. The basic facts which are taken for granted, without proof, are called
a) theorems
b) axioms
c) propositions
d) lemmas
View AnswerAns. b) axioms
7. In the adjoining figure, AOB is a straight line. If x : y : z = 4 : 5 : 6, then y = ?
a) 72°
b) 48°
c) 60°
d) 80°
View AnswerAns. c) 60°
8. In Triangle ABC which is right angled at B. Given that AB = 9cm, AC = 15cm and D, E are the mid-points of the sides AB and AC respectively. Find the length of BC?
a) 13cm
b) 13.5cm
c) 12cm
d) 15cm
View AnswerAns. c) 12cm
9. (x2 – 4x – 21) = ?
a) (x – 7) (x + 3)
b) ( x – 7) ( x – 3)
c) (x + 7) (x + 3)
d) (x + 7) (x – 3)
View AnswerAns. a) (x – 7) (x + 3)
10. Any solution of the linear equation 2x + 0y + 9 = 0 in two variables is of the form
Ans. Option (a)
11. ABCD is a parallelogram in which diagonal AC bisects ∠BAD. If ∠BAC = 35°, then ∠ABC =
a) 70°
b) 120°
c) 110°
d) 90°
View AnswerAns. c) 110°
12. If bisector of at ∠A and ∠B of a quadrilateral ABCD intersect each other at p, ∠B and ∠C at Q, ∠C and ∠D at R and, ∠D and ∠A at S then PQRS is a
a) Rectangle
b) Parallelogram
c) Rhombus
d) Quadrilateral whose opposite angles are supplementary
View AnswerAns. a) Rectangle
13. In the given figure, AB and CD are two intersecting chords of a circle. If ∠CAB = 40° and ∠BCD = 80°, then ∠CBD = ?
a) 60°
b) 50°
c) 70°
d) 80°
View AnswerAns. a) 60°
14. The simplest rationalising factor of
Ans. Option (b)
15. If a linear equation has solutions (1, 2), (-1, -16) and (0, -7), then it is of the form
a) y = 9x – 7
b) 9x – y + 7 = 0
c) x – 9y = 7
d) x = 9y – 7
View AnswerAns. a) y = 9x – 7
16. Line sgements AB and CD intersect at O such that AC||DB. If ∠CAB = 45° and ∠CDB = 55°, then ∠BOD =
a) 135°
b) 80°
c) 100°
d) 90°
View AnswerAns. b) 80°
17. To draw a histogram to represent the following frequency distribution :
The adjusted frequency for the class 25-45 is
a) 6
b) 5
c) 2
d) 3
View AnswerAns. c) 2
18. The number of spherical bullets each 5 dm in diameter which can be cast from a rectangular block of lead 11 m long, 10 m broad and 5 high is
a) 8400.
b) 5600.
c) 6300.
d) 4200.
View AnswerAns. a) 8400.
19. Assertion (A): The side of an equilateral triangle is 6 cm then the height of the triangle is 9 cm.
Reason (R): The height of an equilateral triangle is √3/2 a.
a) Both A and R are true and R is the correct explanation of A.
b) Both A and R are true but R is not the correct explanation of A.
c) A is true but R is false.
d) A is false but R is true.
View AnswerAns. d) A is false but R is true.
20. Assertion (A): The point (1, 1) is the solution of x + y = 2.
Reason (R): Every point which satisfy the linear equation is a solution of the equation.
a) Both A and R are true and R is the correct explanation of A.
b) Both A and R are true but R is not the correct explanation of A.
c) A is true but R is false.
d) A is false but R is true.
View AnswerAns. a) Both A and R are true and R is the correct explanation of A.
Section B
21. The perimeter of an isosceles triangle is 42 cm and its base 1.5 times each of the equal sides. Find the height of the triangle. (Given, √7 = 2.64.)
View AnswerAns.
22. In given figure, if OA = 10, AB = 16 and OD ⊥ to AB. Find the value of the CD.
Ans.
23. The surface areas of two spheres are in the ratio of 4 : 25. Find the ratio of their volumes.
View AnswerAns.
24. In a given figure, O is the centre of a circle and PQ is a diameter. If ∠ROS = 40°, find ∠RTS
OR
In given figure, ∠ABC = 69°, ∠ACB = 31°, find ∠BDC.
Ans.
25. A three-wheeler scooter charges ₹ 15 for first kilometer and ₹ 8 each for every subsequent kilometer. For a distance of x km, an amount of y is paid. Write the linear equation representing the above information.
View AnswerAns. Given, a three-wheeler scooter charges Rs. 15 for first kilometre and Rs. 8 each for every subsequent kilometre. For a distance of x km, an amount of Rs. y is paid.
= 15 × 1 + (x – 1) × 8 = y
= 8x – y + 7 = 0
OR
The cost of a notebook is twice the cost of a pen. Write a linear equation in two variables to represent this statement. (Take the cost of a notebook to be ₹ x and that of a pen to be ₹ y).
View AnswerAns.
Let the cost of a notebook be ₹ x.
Let the cost of a pen be ₹ y.
We need to write a linear equation in two variables to represent the statement, “Cost of a notebook is twice the cost of a pen”. Therefore, we can conclude that the required statement will be x = 2y.
Section C
26. Find the values of a and b in each of
Ans.
27. If x + y + z = 0, show that x3 + y3 + z3 = 3xyz.
View AnswerAns.
28. One side of an equilateral triangle is 8 cm. Find its area by using Heron’s Formula. Find its altitude also.
View AnswerAns.
OR
The triangular side walls of a flyover have been used for advertisements. The sides of the walls are 13 m, 14 m and 15 m. The advertisements yield an earning of Rs2000 per m2 a year. A company hired one of its walls for 6 months. How much rent did it pay?
View AnswerAns.
29. Write linear equation 3x + 2y =18 in the form of ax + by + c = 0. Also write the values of a, b and c. Are (4, 3) and (1, 2) solution of this equation?
View AnswerAns. We have the equation as 3x + 2y = 18
In standard form
3x + 2y – 18 = 0
Or 3x + 2y + (-18) =0
But standard linear equation is
ax + by + c = 0
On comparison we get, a = 3, b = 2, c = -18
If (4, 3) lie on the line, i.e., solution of the equation LHS = RHS
∴ 3(4) + 2(3) = 18
12 + 6 = 18
18 = 18
As LHS = RHS, Hence (4, 3) is the solution of given equation.
Again for (1,2)
3x + 2y = 18
∴ 3(1)+2(2)=18
3 + 4 = 18
7 = 18
LHS ≠ RHS
Hence (1, 2) is not the solution of given equation.
Therefore (4,3) is the point where the equation of the line 3x + 2y = 18 passes through where as the line for the equation 3x + 2y =18 does not pass through the point (1,2).
30. Through A, B and C, lines RQ, PR and QP have been drawn, respectively parallel to sides BC, CA and AB of a △ABC as shown in Fig., Show that BC = 1/2QR
Ans. Given,
PQ||AB, PR||AC and RQ||BC.
In quadrilateral BCAR,
BR||CA and BC||RA
∴ BCAR is a parallelogram
∴ BC = AR …(i)
Now, in quadrilateral BCQA,
BC||AQ and AB||QC
∴ BCQA is a parallelogram
∴ BC = AQ …(ii)
Adding Eqn. (i) and (ii), we get
2BC = AR + AQ
2BC = RQ
BC= QR/2
Hence proved.
OR
Prove that the line segment joining the mid-points of two sides of a triangle is parallel to the third side.
View AnswerAns. Given ABC in which E and F are mid points of side AB and AC respectively.
To prove: EF||BC
Construction: Produce EF to D such that EF = FD. Join CD
Proof: In ΔAEF and ΔCDF
AF=FC [∵ F is the mid-point]
∠1=∠2 [∵ vertically opposite angles]
EF=FD [By construction]
∴ ΔAEF ≅ ΔCDF [By SAS]
And AE= BE [By CPCT]
AE= BE [∵ E is the mid-point]
And ∴BE = CD
AB || CD [∴∠BAC =∠CAD]
∴ BCDE is a parallelogram
EF || BC Hence proved
31. In fig find the vertices’ co-ordinates of ABC
Ans. (A) (0, 0) (B) (3, 4) (c) (-4, 4)
Section D
32. If x is a positive real number and exponents are rational numbers, simplify
Ans.
OR
Ans.
33. In Fig, name the following:
i. Five line segments
ii. Five rays
iii. Four collinear points
iv. Two pairs of non-intersecting line segments
View AnswerAns.
34. In the given figure, AB || CD || EF, OR ∠DBG=x, ∠EDH=y ∠AEB=z, ∠EAB=90° ∠BEF =65°. Find the values of x, y and z.
Ans.
OR
In each of the figures given below, AB || CD. Find the value of x° in each other case.
Ans.
35. Using factor theorem, factorize the polynomial: x3 – 6x2 + 3x + 10
View AnswerAns.
36. Read the following text carefully and answer the questions that follow:
Child labour refers to any work or activity that deprives children of their childhood. It is a violation of children’s rights. This can them mentally or physically. It also exposes them to hazardous situations or stops them from going to school. Naman got data on the number of child labors (in million) in different country that is given below.
i. What is the difference between highest no child labor and the minimum no of child labor?
View AnswerAns. The highest no child labor are in India and the lowest no child labor are in United states
No of child labor in India = 20,000,000
No of child labor in United states = 8,00,000
The difference = 20,000,000 – 8,00,000
= 19,200,000
ii. What is the percentage of no. of child labor in Peru over the no. of child labor in India?
View AnswerAns. No. of child labor in Peru = 4,000,000
No. of child labor in India = 20,00,000
iii. What is the total no. of child labor in the countries having child labor more than 2 million?
View AnswerAns. The countries having child labor more than 2 million are
Egypt = 3 Million
Brazil = 3.5 million
Peru = 4 million
Bangladesh = 4.4 million
India = 20 million
Total no of these labor child = 3 + 3.5 + 4 + 4.4 + 20 = 34.9 Million.
OR
How many countries are having child labor more than Mexico?
View AnswerAns. The countries having child labor more than Mexico are:
Philippines = 1.8 Million
Egypt = 3 Million
Brazil = 3.5 million
Peru = 4 million
Bangladesh = 4.4 million
India = 20 million
Thus 6 countries are having child labor more than Mexico.
37. A construction company purchased a big cylindrical vessel to keep some liquid on it. Before using this vessel, the company decided to paint it properly. It costed ₹3300 to paint the inner curved surface of this 10 m deep cylindrical vessel at the rate of ₹ 30 per m2
i. Find the inner curved surface area of the vessel,
View AnswerAns. cost of painting inner curved surface area of vessel
= cost of painting per m2 x Inner curved surface of vessel
⇒ Rs. 3300 = Rs. 30 × Inner curved surface of vessel
⇒ Inner curved surface of vessel = 110 m2
ii. Find the inner radius of the base and capacity of the vessel.
View AnswerAns.
38. Read the following text carefully and answer the questions that follow:
As shown in the village of Surya there was a big pole PC. This pole was tied with a strong wire of 10 m length. Once there was a big spark on this pole, thus wires got damaged very badly. Any small fault was usually repaired with the help of a rope which normal board electricians were carrying on bicycles. This time electricians need a staircase of 10 m so that it can reach at point P on the pole and this should make 60° with line AC.
i. Show that Δ APC and Δ BPC are congruent.
View AnswerAns. In △ APC and △ BPC
AP = BP (Given)
CP = CP (common side)
∠ACP = ∠BCP = 90°
By RHS criteria △APC ≅ △BPC
ii. Find the value of ∠x.
Ans. In △ACP
∠APC + ∠PAC + ∠ACP = 180°
∠x + 60° + 90° = 180° (angle sum property of △)
∠x = 180° – 150° = 30°
∠x = 30°
[/expand]iii. What is the value of ∠PBC?
View AnswerAns. In △ APC and △ BPC
Corresponding part of congruent triangle
∠PAC = ∠PBC
∠PBC = 60° (given ∠PAC = 60°)
OR
Find the value of ∠y.
View AnswerAns. In △ APC and △ BPC
Corresponding part of congruent triangle
∠X = ∠Y
∠Y = 30 (given ∠X = 30)