This question paper contains 30 questions. Time allowed is 3 hours and Maximum Marks are 80.

All questions are compulsory. The question paper consists of 30 questions divided into four sections - A, B, C and D. Section A contains 6 questions of 1 mark each. Section B contains 6 questions of 2 marks each, Section C contains 10 questions of 3 marks each and Section D contains 8 questions of 4 marks each. There is no overall choice. However, an internal choice has been provided in two questions of 1 mark each, two questions of 2 marks each, four questions of 3 marks each and three questions of 4 marks each. You have to attempt only one of the alternatives in all such questions.

### Section A

Question numbers 1 to 6 carry 1 mark each.

**1.** If HCF (336, 54) = 6, find LCM (336, 54).

**2.** Find the nature of roots of the quadratic equation 2x^{2} – 4x + 3 = 0.

**3.** Find the common difference of the Arithmetic Progression (A.P.)

**4.** Evaluate :

sin^{2} 60° + 2 tan 45° – cos^{2} 30°

OR

If sin A = 3/4, calculate sec A.

**5.** Write the coordinates of a point P on x-axis which is equidistant from the points A(– 2, 0) and B(6, 0).

**6.** In Figure 1, ABC is an isosceles triangle right angled at C with AC = 4 cm. Find the length of AB.

OR

In Figure 2, DE ∥ BC. Find the length of side AD, given that AE = 1·8 cm, BD = 7·2 cm and CE = 5·4 cm.

### Section B

Question numbers 7 to 12 carry 2 marks each.

**7.** Write the smallest number which is divisible by both 306 and 657.

**8.** Find a relation between x and y if the points A(x, y), B(– 4, 6) and C(– 2, 3) are collinear.

OR

Find the area of a triangle whose vertices are given as (1, – 1) (– 4, 6) and (– 3, – 5).

**9.** The probability of selecting a blue marble at random from a jar that contains only blue, black and green marbles is 1/5. The probability of selecting a black marble at random from the same jar is 1/4. If the jar contains 11 green marbles, find the total number of marbles in the jar.

**10.** Find the value(s) of k so that the pair of equations x + 2y = 5 and 3x + ky + 15 = 0 has a unique solution.

**11.** The larger of two supplementary angles exceeds the smaller by 18°. Find the angles.

OR

Sumit is 3 times as old as his son. Five years later, he shall be two and a half times as old as his son. How old is Sumit at present ?

**12.** Find the mode of the following frequency distribution :

### Section C

Question numbers 13 to 22 carry 3 marks each.

**13.** Prove that 2 + 5√3 is an irrational number, given that 3 is an irrational number.

OR

Using Euclid’s Algorithm, find the HCF of 2048 and 960.

**14.** Two right triangles ABC and DBC are drawn on the same hypotenuse BC and on the same side of BC. If AC and BD intersect at P, prove that AP × PC = BP × DP.

OR

Diagonals of a trapezium PQRS intersect each other at the point O, PQ ∥ RS and PQ = 3RS. Find the ratio of the areas of triangles POQ and ROS.

**15.** In Figure 3, PQ and RS are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting PQ at A and RS at B. Prove that ∠AOB = 90°.

**16.** Find the ratio in which the line x – 3y = 0 divides the line segment joining the points (– 2, – 5) and (6, 3). Find the coordinates of the point of intersection.

**17.** Evaluate :

**18.** In Figure 4, a square OABC is inscribed in a quadrant OPBQ. If OA = 15 cm, find the area of the shaded region. (Use π = 3·14)

OR

In Figure 5, ABCD is a square with side 2√2 cm and inscribed in a circle. Find the area of the shaded region. (Use π = 3·14)

**19.** A solid is in the form of a cylinder with hemispherical ends. The total height of the solid is 20 cm and the diameter of the cylinder is 7 cm. Find the total volume of the solid. (Use π = 22/7)

**20.** The marks obtained by 100 students in an examination are given below :

Find the mean marks of the students.

**21.** For what value of k, is the polynomial

f(x) = 3x^{4} – 9x^{3} + x^{2} + 15x + k

completely divisible by 3x^{2} – 5 ?

OR

Find the zeroes of the quadratic polynomial 7y^{2} – 11/3 y – 2/3 and verify the relationship between the zeroes and the coefficients.

**22.** Write all the values of p for which the quadratic equation x^{2} + px + 16 = 0 has equal roots. Find the roots of the equation so obtained.

### Section D

Question numbers 23 to 30 carry 4 marks each.

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

**24.** Amit, standing on a horizontal plane, finds a bird flying at a distance of 200 m from him at an elevation of 30°. Deepak standing on the roof of a 50 m high building, finds the angle of elevation of the same bird to be 45°. Amit and Deepak are on opposite sides of the bird. Find the distance of the bird from Deepak.

**25.** A solid iron pole consists of a cylinder of height 220 cm and base diameter 24 cm, which is surmounted by another cylinder of height 60 cm and radius 8 cm. Find the mass of the pole, given that 1 cm^{3} of iron has approximately 8 gm mass. (Use π = 3·14)

**26.** Construct an equilateral ΔABC with each side 5 cm. Then construct another triangle whose sides are 2/3 times the corresponding sides of ΔABC.

OR

Draw two concentric circles of radii 2 cm and 5 cm. Take a point P on the outer circle and construct a pair of tangents PA and PB to the smaller circle. Measure PA.

**27.** Change the following data into ‘less than type’ distribution and draw its ogive :

**28.** Prove that :

OR

Prove that :

**29.** Which term of the Arithmetic Progression –7, –12, –17, –22, ... will be –82 ? Is –100 any term of the A.P. ? Give reason for your answer.

OR

How many terms of the Arithmetic Progression 45, 39, 33, ... must be taken so that their sum is 180 ? Explain the double answer.

**30.** In a class test, the sum of Arun’s marks in Hindi and English is 30. Had he got 2 marks more in Hindi and 3 marks less in English, the product of the marks would have been 210. Find his marks in the two subjects.