1. Choose the correct option in each case from the following questions:
(i) If a principal becomes twice of it in 10 years, then the rate of simple interest per annum is:
(ii) The product of two roots of the equation x² – 7x + 3 = 0 is:
(iii) The length of two chords AB and CD of a circle of center O are equal and ∠AOB = 60°, then ∠COD is :
(iv) If the ratio of the volume of two right circular cones is 1 : 4 and the ratio of radii of their bases is 4:5, then the ratio of their heights is :
(v) If sinθ – cosθ = 0, (0° < θ < 90°) and secθ + cosecθ = x, then the value of x is:
(vi) The mode of 1, 3, 2, 8, 10, 8, 3, 2 ,8 , 8, is:
2. Fill up the blanks (any five):
(i) Anisur invests Rs 500 for 9 months in a business and Devid invests Rs 600 for 5 months in the same business, the ratio of their profits will be ____.
(ii) The roots of the quadratic equation ax²+ 2bx + c = 0 (a ≠ 0) are real and unequal, then b²= ____.
(iii) If sum of two angles is ____, then they are called supplementary angles.
(iv) Maximum value of sin 3θ is ____.
(v) One solid sphere is melted and a solid right circular cylinder is made, then ____ of sphere and the cylinder will be equal.
(iv) Ages of some students are (in years) 10, 11, 9, 7, 13, 8, 14; the median of the ages of those students is ____ years.
3. Write true or false (any five).
(i) The amount of Rs 2p in t years at the rate of simple interest of r/2 % annum is Rs (2p+ptr/100).
(ii) If 2a = 3b = 4c then a : b : c = 2 : 3 : 4.
(iii) If the ratio of the lengths of three sides of a triangle is 5 : 12 : 13, then the triangle will always be a right angled triangle.
(iv) The angle formed by rotating a ray about it end point in anticlockwise direction is positive.
(v) If n is even number, then median is the mean of (n/2)th and (n/2-1)th observation.
(vi) If the length of the radius of the base of a right circular cone be halved and its height be doubled, then the volume remains same.
4. Answer the following questions (any ten):
(i) If the ratio of a principal and the amounts for 5 years is 5 : 6, then find the rate of simple interest per annum.
(ii) In a business, A and B get Rs 1,050 as profit. If the principal and profit of A be Rs 900 and Rs 630 respectively. Find the principal of B.
(iii) If x∝y, y∝z and z∝x, find the product of three variation constants.
(iv) If the roots of the quadratic equation 5×²– 2x + 3 = 0 be α and β, find the value of 1/α+1/β .
(v) The point O is situated within the rectangular region ABCD in such a way that OB = 6 cm, OD = 8 cm and OA = 5 cm. Determine the length of OC.
(vi) In a right-angled triangle ABC, ∠ABC= 90°, AB = 3 cm, BC = 4 cm and the perpendicular BD on the side AC from the point B which meets the side AC at the point D. Determine the length of BD.
(vii) The lengths of radii of two circles are 8 cm and 3 cm and the distance between two centres is 13 cm. What is the length of the direct common tangent of two circles?
(viii) What is the circular measure of an angle formed by the rotation of hour hand of a clock in one-hour duration?
(ix) If tan 4θ tan 6θ = 1 and 6θ is a positive acute angle, find the value of θ.
(x) The height of a right circular cone is 12 cm and its volume are 100π cm³. Find the lateral height of the cone.
(xi) Curved surface areas of two spheres are in a ratio 1: 4. Find the ratio of their volumes.
(xii) If u_i=(x_i-35)/10, ∑fiui=30 and ∑fi =60, then determine the value of x ̅.
5. Answer any one question: –
(i) The price of a machine in a factory of your uncle depreciates at the rate of 10% every year. If its present price is Rs 6,000 then what will be its price after 3 years?
(ii) Three friends invested Rs 1,20,000 Rs 1,50,000 and Rs, 1,10,000 respectively to purchase a bus. The first person is a driver and the other two are conductors. They decided to divide 2/5 th of the profit among themselves in the ratio of 3 : 2 : 2 according to their work and remaining in the ratio of their capitals. If they earn Rs 29,260 in one month, find the share of each of them.
6. Solve any one question:
(i) Solve: 1/(x-3)-1/(x+5)=1/6
(ii) The product of two consecutive positive odd number is 143. Construct the equation and determine the numbers by applying Sridhara Acharyaa’s formula.
7. Answer any one question:
(i) x = 2 +3 and x + y = 4, then find the simplest value of xy+1/xy.
(ii) If a∝b and b∝c, then prove that a³+ b³+ c³∝3abc.
8. Answer any one question:
(i) If x:a=y:b=z:c then show that: x³./a³+y³/b³+z³/c³=3xyz/abc.
(ii) If (ay-bx)/c=(cx-az)/b=(bz-cy)/a, then prove that x/a=y/b=z/c.
9. Answer any one question:
(i) Prove that angles in the same segment of a circle are equal.
(ii) Prove that if two tangents are drawn to a circle from a point outside it, then the line segments joining the point of contacts and the exterior point are point.
10. answer any one question:
(i) Two circles intersect each other at the points P and Q. If the diameters of the two circles are PA and PB respectively, then prove that A, Q, B are collinear.
(ii) ABC is a right angled triangle whose A= 90°, AD is perpendicular on BC.
11. Answer any one question:
(i) Draw the mean proportional of line segments of lengths 4 cm and 3cm.
(ii) Draw a circle of radius 3 cm. Construct a tangent to the circle at a point A on the circle.
12. Answer any two questions:
(i) If sin 17° =x/y, show that sec 17°- sin73° =x²/(y√(y²-x²))
(ii) If the sum of two angle is 135° and their difference is π/12, then determine the sexagesimal and circular value of two angles.
(iii) Find the value of: (5cos² π/3+4sec² π/6-tan² π/4)/(sin² π/6+cos^2 π/6)
13. Answer any one question:
(i) If the angle of elevation of a cloud from a point h meters above a lake is α and the angle of depression of its reflection in the lake is β. Prove that the distance of the cloud from the point of observation is (2h secα)/(tanβ-tanα).
(ii) The heights of two towers are 180 meters and 60 meters respectively. If the angle of elevation of the top of the first tower from the foot of the second tower is 60°, then find the angle of elevation of the top of the second tower from the foot of the first.
14. Answer any two questions:
(i) The length of outer and inner radii of a hollow right circular pipe are 5 cm 4 cm respectively. If the total surface area of the pipe is 1188 sq.cm. find the length of the pipe.
(ii) A hemisphere pot with internal radius of 9 cm is completely filled with water in cylindrical bottles with a diameter of 3 cm and height of 4 cm, then find the number of bottles to be required to make the pot empty.
(iii) The diameter of the base of a right circular cone is 21 meters and height is 14 meters. What will be the expenditure to colour the the curved surface at the
rate of Rs 1.50 per square meter?
- Answer any two questions:
(i) Find the mean of marks obtained by the girl students if their cumulative frequencies are as follows:
(ii) Find the median of data from the following frequency distribution table:
(iii) Find the mode of data from the following frequency distribution table: