1. Choose the correct option in each case from the following questions:
(i) In a partnership business, the ratio of share of profit of two friends is 1/2:1/3 then the ratio of their principal is –
(ii) If p + q =√13 and p – q =√5 then the value of pq is –
(iii) 0 is the center of a circle and AB a diameter. ABCD is a cyclic quad-rilateral. ∠ABC = 65°, ∠DAC = 40°, then the measure of ∠BCD is-
(iv) If tanα+cotα=2 then the value of tan13α+cot13α is –
(v) If two cubes of length of each side 2√6 cm are placed side by side, then the length of the diagonal of the cuboid so produced is –
(a) 10 cm
(b) 6 cm
(c) 2 cm
(d) 12 cm
(vi) The mean of data x1, x2, x3,……., x10 is 20 then the mean of x1 + 4, x2 + 4, x3 + 4,……., x10 + 4 will be-
2. Fill up the blanks (any five):
(i) A person deposited Rs. 100 in a bank and gets the amount Rs.121 after two years. The rate of compound interest is ____%.
(ii) If the product and sum of two quadratic surds is a rational number then surds are ____ surds.
(iii) If the bases of two triangles are situated on same line and the other vertex of the two triangles are common, then the ratio of the areas of two triangles are __ to the ratio of their bases.
(iv) The simplest value of cos53°/sin37° is ___.
(v) Number of surfaces of a solid right circular cylinder is ___.
(iv) The variables x1, x2, …, x100 are in ascending order of their magnitude, then the median of the variables is __.
3. Write true or false (any five).
(i) The difference between the simple interest and the compound interest of Rs. 100 in 1 year at the rate of 10% p.a. is Re. 1.
(ii) The compound ratio of ab:c²,bc:a and ca:b² is 1:1.
(iii) Only one circle can be drawn through three non-colinear points.
(iv) sin 30° + sin 60° > sin 90°.
(v) The ratio of the volume of a right circular cone and a right circular cylinder with same base and height is 1 : 3.
(vi) Value of median of data 2, 3, 9, 10, 9, 3, 9 is 10.
4. Answer the following questions (any ten):
(i) Find the capital which gives Re. 1 as interest per month at 5% rate of interest per annum.
(ii) In a partnership business the ratio of capitals of three men is 3: 5: 8. The share of profit of the first member is Rs. 60 less than that of the third member, then what is the total profit in the business?
(iii) If a/2=b/3=c/4=(2a-3b+4c)/p, then find p.
(iv) x∝y² and y=2a when x=a, then show that y²=4ax.
(v) In a trapezium ABCD, BC||AD and AD = 4 cm. The two diagonals AC and BD intersect at the point O in such a way that AO/OC=DO/OB=1/2. Calculate the length of BC.
(vi) Two chords AB and AC of a circle are mutually perpendicular to each other. If AB = 4 cm and AC = 3 cm, find the length of the radius of the circle.
(vii) In △ABC, ∠ABC = 90 and BD⊥AC, if AB = 5cm, BC = 12 cm, then find the length of BD.
(viii) Find the value/values of θ (0° ≤θ≤90°)for which 2 sinθ cosθ=cosθ.
(ix) If sin10θ=cos8θ〗 and 10θ is a positive acute angle, then find the value of tan 9θ.
(x) The length, breadth and height of a cuboidal room be a unit, b unit, and c unit respectively and a + b + c=25, ab + bc + ca = 240.5 then find the length of the longest rod to be kept inside the room.
(xi) The area of curved surface of a right circular cone is5times of that of the base of the cone, find the ratio of the height and the radius of the base of the cone.
(xii) The mid value of first (2n + 1) consecutive natural number is (n+103)/3 Find the value of n.
5. Answer any one question: –
(i) If interest is compounded half yearly what will be the compound interest and amount on Rs. 8,000 at the rate of 10% compound interest per annum for 1.5 years?
(ii) Two friends start a partnership business investing Rs. 40,000 and Rs. 50,000 respectively. There is an agreement between them that 50% of the profit will be divided equally and rest amount of profit will be distributed between them in the ratio of their principal. If the share of profit of 1st friend is Rs. 800 less than that of the 2ndfriend, find the share of profit of the 1st friend.
6. Solve any one question:
(i) Determine the equation whose roots are the square of the roots of the equation x² + x + 1 = 0.
(ii) If the price of 1 dozen pen is reduced by Rs 6, then 3 more pens will be got in Rs 30. calculate the price of one dozen pen before the reduction of price.
7. Answer any one question:
(i) Simplify: (4√3)/(2-√2)-30/(4√3-√18)-√18/(3-√12).
(ii) If (1/x-1/y)∝1/(x-y) , then show that (x² + y²) ∝ xy.
8. Answer any one question:
(i) If (3x – 2y) : (x + 3y) = 5 : 6, then find the value of (2x + 5y) : (3x + 4y).
(ii) If (b+c-a)/(y+z-x)=(c+a-b)/(z+x-y)=(a+b-c)/(x+y-z) , then prove that a/x=b/y=c/z
9. Answer any one question:
(i) Prove that semicircular angle is a right angle.
(ii) If two circles touch each other externally then the point of contact will lie on the line segment joining the two centers – Prove it.
10. answer any one question:
(i) If a quadrilateral ABCD is circumscribed about a circle with center O, prove that AB + CD = BC + DA.
(ii) If in △ABC, ∠A is right angle and BP and CQ are two medians, then prove that 5 BC² = 4(BP² + CQ²).
11. Answer any one question:
(i) Draw a triangle ABC of which BC = 7 cm, AB = 5 cm and AC = 6 cm. Then draw the circumcircle of △ABC. (Only traces of construction are required)
(ii) Construct a circle of radius 4 cm and draw two tangents to the circle from an external point at a distance of 6.5 cm from the center of the circle.
12. Answer any two questions:
(i) In △ABC, C= 90°, if BC = m and AC = n then prove that, m sinA+n sinB=√(m^2+n²)
(ii) Find the value of 4/3 cot²30°+3 sin²60°-2cosec^2 60°-3/4 tan² 30°
(iii) If ∠P+∠Q=90° then show that sinP/cosQ-sinP cosQ=cos²P
13. Answer any one question:
(i) From a quay of a river, 600 meters wide, two boats start in two different directions to reach the opposite side of the river. The first boat moves making an angle of 30° with this bank and the second boat moves making an angle 90° with direction of the first boat. What will be the distance between the two boats when both of them reach the other side?
(ii) The length of the flag post at the roof of a three storied building is 3.6 meter. The angles of elevation of the top and foot of the post are 50° and 45° respectively from a point on the road. Find the height of the building. [Take tan 50° = 1.2]
14. Answer any two questions:
(i) If 64 buckets of water are withdrawn from a cubical water tank, full of water, then 1/3 of water in the tank still remains. If the length of the side of the water tank is 1.2 meter then what is the capacity (in liter) of each bucket? (1 cubic decimeter = 1 liter)
(ii) The diameter of cross-section of a wire is reduced by 50%. If the volume remains constant, what percent of length of the wire should be increased?
(iii) 77 sq. m tripal is required to make a right circular conical tent. If the slant height of the tent is 7 m, then what is the area of the base of the tent?
15. Answer any two questions:
(i) If the arithmetic mean of the following frequency distribution is 54, then find the value of K:
(ii) Making frequency distribution table from the given cumulative frequency distribution table, find the mode of the data:
(iii) Find the mean of 52 students of class X in a school by using direct method and assumed mean method from the table given below :