Date: March 2019

Duration: 3h

(i) All questions are compulsory.

(ii) The question paper consists of 30 questions divided into four sections – A, B, C and D.

(iii) **Section** **A** comprises 6 questions of **1** **mark **each. **Section** **B** contains 6 questions of **2 marks** each. **Section** **C** contains 10 questions of **3** **marks** each. **Section** **D** contains 8questions of** 4 marks** each.

(iv) There is no overall choice. However, an internal choice has been provided in two questions of 1mark,** two** questions of 2 marks, four questions of 3 marks each and three questions of 4 marks each. You have to attempt only** one **of the alternative in all such questions.

(v) Use of calculators is not permitted.

Two concentric circles of radii a and b (a > b) are given. Find the length of the chord of the larger circle which touches the smaller circle.

Chapter: [3.01] Circles

In Figure 1, PS = 3 cm, QS = 4 cm, ∠PRQ = θ, ∠PSQ = 90°, PQ ⊥ RQ and RQ = 9 cm. Evaluate tan θ.

Chapter: [6.01] Lines (In Two-dimensions)

If tan α =`5/12` find the value of sec α.

Chapter: [2.02] Arithmetic Progressions

Write the discriminant of the quadratic equation (x + 5)^{2} = 2 (5x − 3).

Chapter: [2.03] Quadratic Equations

Find after how many places of decimal the decimal form of the number `27/(2^3. 5^4. 3^2)` will terminate.

Chapter: [1.01] Real Numbers

Express 429 as a product of its prime factors.

Chapter: [1.01] Real Numbers

Find the sum of the first 10 multiples of 6.

Chapter: [2.02] Arithmetic Progressions

Find the positive value of m for which the distance between the points A(5, −3) and B(13, m) is 10 units.

Chapter: [4.01] Heights and Distances

A die is thrown once. Find the probability of getting a composite number

Chapter: [5.01] Probability [5.01] Probability

A die is thrown once. Find the probability of getting a prime number.

Chapter: [5.01] Probability

Cards numbered 7 to 40 were put in a box. Poonam selects a card at random. What is the probability that Poonam selects a card which is a multiple of 7?

Chapter: [5.01] Probability [5.01] Probability

Points A(3, 1), B(5, 1), C(a, b) and D(4, 3) are vertices of a parallelogram ABCD. Find the values of a and b.

Chapter: [3.02] Triangles

Points P and Q trisect the line segment joining the points A(−2, 0) and B(0, 8) such that P is near to A. Find the coordinates of points P and Q.

Chapter: [3.03] Constructions

**Solve the following pair of linear equations:**

3x − 5y = 4

2y + 7 = 9x

Chapter: [2.01] Pair of Linear Equations in Two Variables

If HCF of 65 and 117 is expressible in the form 65n − 117, then find the value of n.

Chapter: [1.01] Real Numbers

On a morning walk, three persons step out together and their steps measure 30 cm, 36 cm, and 40 cm respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps?

Chapter: [4.01] Heights and Distances

In the quadratic equation kx^{2} − 6x − 1 = 0, determine the values of k for which the equation does not have any real root.

Chapter: [2.03] Quadratic Equations

A, B and C are interior angles of a triangle ABC. Show that

sin `(("B"+"C")/2) = cos "A"/2`

Chapter: [4.02] Trigonometric Identities [4.03] Introduction to Trigonometry

A, B and C are interior angles of a triangle ABC. Show that

If ∠A = 90°, then find the value of tan`(("B+C")/2)`

Chapter: [4.02] Trigonometric Identities [4.03] Introduction to Trigonometry

If tan (A + B) = 1 and tan(A-B)`=1/sqrt3` , 0° < A + B < 90°, A > B, then find the values of A and B.

Chapter: [4.03] Introduction to Trigonometry

In Figure 2, PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q intersect at a point T. Find the length TP.

Chapter: [3.03] Constructions

Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

Chapter: [3.01] Circles

A class teacher has the following absentee record of 40 students of a class for the whole term. Find the mean number of days a student was absent.

Number of days: | 0-6 | 6-12 | 12-18 | 18-24 | 24-30 | 30-36 | 36-42 |

Number of students: | 10 | 11 | 7 | 4 | 4 | 3 | 1 |

Chapter: [5.02] Statistics

A car has two wipers that do not overlap. Each wiper has a blade of length 21 cm sweeping through an angle of 120°. Find the total area cleaned at each sweep of the blades. `("Take" π =22/7)`

Chapter: [7.01] Areas Related to Circles

The perpendicular from A on side BC of a Δ ABC meets BC at D such that DB = 3CD. Prove that 2AB^{2} = 2AC^{2 }+ BC^{2}.

Chapter: [3.02] Triangles

AD and PM are medians of triangles ABC and PQR, respectively where Δ ABC ~ Δ PQR, prove that `("AB")/("PQ") = ("AD")/("PM")`

Chapter: [3.02] Triangles

Check whether g(x) is a factor of p(x) by dividing polynomial p(x) by polynomial g(x),

where p(x) = x^{5} − 4x^{3} + x^{2} + 3x +1, g(x) = x^{3} − 3x + 1

Chapter: [2.04] Polynomials

Prove that is `sqrt3` irrational number.

Chapter: [1.01] Real Numbers

Find the largest number which on dividing 1251, 9377 and 15628 leave remainders 1, 2 and 3 respectively.

Chapter: [1.01] Real Numbers

Find the area of the triangle ABC with the coordinates of A as (1, −4) and the coordinates of the mid-points of sides AB and AC respectively are (2, −1) and (0, −1).

Chapter: [3.02] Triangles

Two numbers are in the ratio of 5: 6. If 7 is subtracted from each of the numbers, the ratio becomes 4: 5. Find the numbers.

Chapter: [1.01] Real Numbers

Water is flowing at the rate of 2.52 km/h through a cylindrical pipe into a cylindrical tank, the radius of whose base is 40 cm. If the increase in the level of water in the tank, in half an hour is 3.15 m, find the internal diameter of the pipe.

Chapter: [7.02] Surface Areas and Volumes [7.02] Surface Areas and Volumes

In Figure 3, a decorative block is shown which is made of two solids, a cube, and a hemisphere. The base of the block is a cube with an edge 6 cm and the hemisphere fixed on the top has a diameter of 4⋅2 cm. Find

**(a) **the total surface area of the block.**(b)** the volume of the block formed. `("Take" pi = 22/7)`

Chapter: [7.02] Surface Areas and Volumes [7.02] Surface Areas and Volumes

A bucket open at the top is in the form of a frustum of a cone with a capacity of 12308.8 cm^{3}. The radii of the top and bottom circular ends are 20 cm and 12 cm, respectively. Find the height of the bucket and the area of metal sheet used in making the bucket. (use *π* = 3.14)

Chapter: [4.01] Heights and Distances

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, prove that the other two sides are divided in the same ratio

Chapter: [3.02] Triangles

Prove that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Chapter: [3.02] Triangles

Change the following distribution to a 'more than type' distribution. Hence draw the 'more than type' ogive for this distribution.

Class interval: | 20−30 | 30−40 | 40−50 | 50−60 | 60−70 | 70−80 | 80−90 |

Frequency: | 10 | 8 | 12 | 24 | 6 | 25 | 15 |

Chapter: [5.02] Statistics

The shadow of a tower standing on level ground is found to be 40 m longer when the Sun's altitude is 30° than when it was 60°. Find the height of the tower. (Given `sqrt3` = 1.732)

Chapter: [4.01] Heights and Distances

If m times the m^{th} term of an Arithmetic Progression is equal to n times its n^{th} term and m ≠ n, show that the (m + n)^{th} term of the A.P. is zero.

Chapter: [2.02] Arithmetic Progressions

The sum of the first three numbers in an Arithmetic Progression is 18. If the product of the first and the third term is 5 times the common difference, find the three numbers.

Chapter: [2.02] Arithmetic Progressions

A shopkeeper buys some books for Rs 80. If he had bought 4 more books for the same amount, each book would have cost Rs 1 less. Find the number of books he bought.

Chapter: [1.01] Real Numbers

Construct a pair of tangents to a circle of radius 4 cm from a point which is at a distance of 6 cm from its centre.

Chapter: [3.01] Circles

Prove the following:

`1/(1+sin^2theta) + 1/(1+cos^2theta) + 1/(1+sec^2theta) + 1/(1+cosec^2theta) = 2`

Chapter: [4.03] Introduction to Trigonometry [4.03] Introduction to Trigonometry

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