#### 10th Public Exam March 2019

10th Standard

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Maths

Time : 02:30:00 Hrs
Total Marks : 100
I. Answer all the 15 questions
15 x 1 = 15
1. If the third term of a G.P is 2, then the product of first 5 terms is

(a)

${ 5 }^{ 2 }$

(b)

${ 2 }^{ 5 }$

(c)

10

(d)

15

2. The common difference of the A.P. $\frac { 1 }{ 6 } ,\frac { 1 }{ 3 } ,\frac { 5 }{ 6 }$ is ______________

(a)

$\frac { 1 }{ 3 }$

(b)

$\frac { 1 }{ 6 }$

(c)

$\frac { 1 }{ 2 }$

(d)

$\frac { 1 }{ 4 }$

3. Padma (x) is 3 years elder to Radha (y). The equation connecting them is______.

(a)

x+y=3

(b)

x=y+3

(c)

x+3=y

(d)

None of these

4. If A is of order 3 x 4 and B is of order 4 x 3, then the order of BA is

(a)

3 x 3

(b)

4 x 4

(c)

4 x 3

(d)

not defined

5. If the line segment joining the points A(3, 4) and B(14,-3) meets the x-axis at P,then the ratio in which P divides the segment AB is

(a)

4 :3

(b)

3 : 4

(c)

2 : 3

(d)

4 :1

6. If the slope of the line joining (-6, 13) and (3, a) is $\frac{-1}{3}$ then the value of a is ____________.

(a)

5

(b)

-10

(c)

-5

(d)

10

7. The sides of two similar triangles are in the ratio 2:3, then their areas are in the ratio

(a)

9:4

(b)

4:9

(c)

2:3

(d)

3:2

8. The perimeters of two similar triangles are 24 cm and 18 cm respectively. If one side of the first triangle is 8 cm, then the corresponding side of the other triangle is

(a)

4 cm

(b)

3 cm

(c)

9 cm

(d)

6 cm

9. In ΔABC, AD is median and also bisects ㄥA. If AB = 16 cm, BC = 8 cm, then AC=

(a)

$\left( \sqrt { 4 } \right) ^{ 2 }$

(b)

8

(c)

42

(d)

2

10. $1-\frac { { sin }^{ 2 }\theta }{ 1+cos\theta } =$

(a)

$cos\theta$

(b)

$tan\theta$

(c)

$cot\theta$

(d)

$cosec\theta$

11. $\frac { 1+{ tan }^{ 2 }\theta }{ 1+{ cot }^{ 2 }\theta } =$

(a)

${ cos }^{ 2 }\theta$

(b)

${ tan }^{ 2 }\theta$

(c)

${ sin }^{ 2 }\theta$

(d)

${ cot }^{ 2 }\theta$

12. ${ sin }^{ 2 }\theta +\frac { 1 }{ 1+{ tan }^{ 2 }\theta } =$

(a)

${ cosec }^{ 2 }\theta +{ cot }^{ 2 }\theta$

(b)

${ cosec }^{ 2 }\theta -{ cot }^{ 2 }\theta$

(c)

${ cot }^{ 2 }\theta -{ cosec }^{ 2 }\theta$

(d)

${ sin }^{ 2 }\theta -{ cos }^{ 2 }\theta$

13. If tanθ-cotθ=3, then tan2θ+cot2θ=______________

(a)

11

(b)

9

(c)

$\frac{1}{3}$

(d)

3

14. If the surface area of a sphere is 100$\pi$ cm2 , then its radius is equal to

(a)

25cm

(b)

100cm

(c)

5cm

(d)

10cm

15. The probability that a leap year will have 53 Fridays or 53 Saturdays is

(a)

$2 \over 7$

(b)

$1\over7$

(c)

$4\over13$

(d)

$3\over7$

16. II. Answer any 10 questions. Question number 30 is compulsory. select any 9 questions form the first 14 questions
10 x 2 = 20
17. Find the ${ 18 }^{ th }$ and ${ 25 }^{ th }$terms of the sequence defined by

${ a }_{ n }=\begin{cases} n\left( n+3 \right) ,\quad if\quad n\quad \in \quad N\quad and\quad n\quad is\quad even \\ \frac { 2n }{ { n }^{ 2\quad }+\quad 1 } ,\quad if\quad n\quad \in \quad N\quad and\quad n\quad is\quad odd. \end{cases}$

18. How many terms are there in the A.P. 10, 13, 16, ....., 43?

19. Solve the following quadratic equations by factorization method. $3x-\frac { 8 }{ x } =2$.

20. Simplify: $\frac { { x }^{ 2 }+6\sqrt { 3 } x+27 }{ { x }^{ 2 }+\sqrt { 3 } }$

21. Find the product of the matrices,if exists $\begin{pmatrix} 3 & -2 \\ 5 & 1 \end{pmatrix}\begin{pmatrix} 4 & 1 \\ 2 & 7 \end{pmatrix}$

22. AB and CD are two chords of a circle which intersect each other externally at P.If AB = 4 cm BP = 5 cm and PD = 3 cm, then find CD.

23. In a $\triangle ABC,AD$ is the internal bisector of $\angle A$ meeting BC at D. If AB = x, AC = x–2, BD = x+2 and DC = x–1 find the value of x.

24. The diagonals of a quadrilateral ABCD cut at K. If AK = 2.4 cm, KC = 1.6 cm, BK = 1.5 cm, KD = 1 cm, prove that AB II DC.

25. Prove the following identities

${sin \ \theta \over 1 - cos\ \theta} = cosec\ \theta + cot \ \theta$

26. A solid right circular cylinder has radius 7 cm and height 20 cm. Find its (i) curved surface area and (ii) total surface area. ( Take $\pi =\frac { 22 }{ 7 }$)

27. The volume of a cone with circular base is 216$\pi$ cu.cm. If the base radius is 9 cm, then find the height of the cone.

28. The volume of right circular cylinder is 448πcu. cm and height is 14 cm. Find its C.S.A.

29. Find the variance of the following data:
2, 4, 5, 6, 8, 17.

30. Find the probability of getting a multiple of 2 in the throw of a die.

1. Show that the roots of the equation $x^{ 2 }+2(a+b)x+2(a^{ 2 }+b^{ 2 })=0$ are unreal.

2. Find the product of the matrices, if exists $\left( \begin{matrix} 6 \\ -3 \end{matrix} \right) \left( 2\quad -7 \right)$

31. II. Answer any 9 questions. Question number 45 is compulsory. select any 8 questions form the first 14 questions.

9 x 5 = 45
32. Use Venn diagrams to verify De Morgan’s law for set difference $A\setminus (B\cap C)=(A\setminus B)\cup (A\setminus C).$

33. A function f : [-7,6) $\Rightarrow$ R is defined as follows

$f\left( x \right) =\begin{cases} { x }^{ 2 }+2x+1; &-7\le x<-5\\ x+5;&-5\le x\le 2 \\ x-1; &2< x <6\end{cases}$

Find $\frac { 4f\left( -3 \right) +2f\left( 4 \right) }{ f\left( -6 \right) -3f\left( 1 \right) }$

34. The ${ 4 }^{ th }$ term of a geometric sequence is$\frac { 2 }{ 3 }$ and the seventh term is $\frac { 16 }{ 81 } .$  Find the geometric sequence.

35. If S1, S2, S3 , be the sums of n terms of 3 arithmetic series, the first term of each being 1 and the respective common differences 1, 2, 3, prove, that S1 + S3 = 2S2

36. Simplify:

$\frac { { (a+b) }^{ 2 }-{ c }^{ 2 } }{ { a }^{ 2 }-{ (b+c) }^{ 2 } } +\frac { { b }^{ 2 }-({ c+a) }^{ 2 } }{ { c }^{ 2 }-{ (a+b) }^{ 2 } }$

37. A triangle has vertices at (6 , 7), (2 , -9) and (-4 , 1). Find the slopes of its medians.

38. Find the equations of the sides of the triangle whose vertices are (2, 1), (-2, 3) and (4, 5).

39. The points D and E are on the sides AB and AC of $\triangle ABC$ respectively, such that $DE\parallel BC.$ If AB = 3 AD and the area of $\triangle ABC$ is 72 cm2 then find the area of the quadrilateral DBCE.

40. In ΔPQR, G is the mid-point of PR and H is the mid-point of QR. What is the ratio of area ( ΔGHR) to area (ΔPQR)?

41. A boy is standing at some distance from a 30 m tall building and his eye level from the ground is 1.5 m. The angle of elevation from his eyes to the top of the building increases from ${ 30 }^{ \circ }$ to ${ 60 }^{ \circ }$as he walks towards the building. Find the distance he walked towards the building.

42. The diameter of a road roller of length 120 cm is 84 cm. If it takes 500 complete revolutions to level a playground, then find the cost of levelling it at the cost of 75 paise per square metre. (Take $\pi =\frac { 22 }{ 7 }$ )

43. A hollow cylindrical pipe is of length 40 cm. Its internal and external radii are 4 cm and 12 cm respectively. It is melted and cast into a solid cylinder of length 20 cm. Find the radius of the new solid.

44. A solid cylinder has total surface area of 462 sq.cm. Its C.S.A. is one third of its T.S.A. Find the volume of cylinder.

45. Two dice are rolled once. Find the probability of getting an even number on the second die or the total of face numbers 10

1. The GCD of $​​​​{ x }^{ 4 }+3{ x }^{ 3 }+5{ x }^{ 2 }+26x+56$ and ${ x }^{ 4 }+{ 2x }^{ 3 }-4{ x }^{ 2 }-{ x }+28$ is ${ x }^{ 2 }+5{ x }+7$.

2. Solve : $\begin{pmatrix} x & 1 \end{pmatrix}\begin{pmatrix} 1 & 0 \\ \quad -2 & -3 \end{pmatrix}\begin{pmatrix} x \\ 5 \end{pmatrix}=(0).$

46. IV. Answer both the questions choosing either of the alternatives.
2 x 10 = 20
1. Draw a circle of radius 4.8 cm. Take a point on the circle. Draw the tangent at that point using the tangent-chord theorem.

2. Construct a cyclic quadrilateral ABCD in which AB = 6 cm, AC = 7 cm, BC = 6 cm, and AD = 4.2 cm.

1. Draw the graph of $y=-3{ x }^{ 2 }.$

2. Draw the graph of $y={ x }^{ 2 }-x-8$ and hence find the roots of${ x }^{ 2 }-2x-15=0$