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Assignment3

Attempt all questions.

Part one is common for both ICSE and CBSE but choose your board for part two

In the given figure O is the center of the bigger circle an AC is its diameter. An other circle with AB as diameter is drawn. If AC = 54 cm and BC = 10 cm find the area of the shaded region.

Evaluate :-

Cosec (65⁰ + q) - sec (25⁰ - q) - tan (55⁰ - q) + cot(35⁰ + q)

In the fig. DE = 8 cm, EA = 10 cm and BE = 4cm, find CE.

Area of a sector of a circle measures 540 sq cm. What is the length of the arc of the sector, if radius of the circle is 27 cm?

In a D ABC, AD is the bisector of ÐA whereas D lies on BC, If AB = 4 cm, BC = 5.5 cm and CA = 6 cm. Find BD and DC.

When 0⁰ < q < 90⁰ solve the Equation

cosq + cosq = 2
cosecq+1 cosecq-1

In the adjacent figure AD II BC, find the value of x where AO = 3 cm, BO = (3x - 19) cm and CD = (x - 3) cm and DO = (x - 5) cm.

Two circle intersect at A and B & AC, AD are respectively diameters of the circles. Prove that C,B,D are collinear.

The sides of a quadrilateral ABCD touch a circle, prove that AB + CD = AD + BC

10.

In the given fig. AP is a tangent to a circle at P, ABC is secant and PD is the bisector of ÐBPC, prove that ÐBPD = ½ (ÐABP - ÐAPB). Also show that AP = AD

11.

Determine the value of k for which the mode of the following data is 7 :

3, 5, 5, 7, 3, 6, 7,
9, 6, 7, 3, 5, 3, 7, k.

also find the median of this data.

12.

Solve the following system of equation graphically -

2x - y = 1
x + 2y = 8

Also find the co-ordinates of the points where the lines meet the axis of y.

13.

Using the properties of proportion, solve for x

14.

The length of the hypotenuse of a right angled triangle exceeds the length of the base by 2cm and exceeds the altitude by 1 cm. Find the length of each side of the triangle

15.

If numerator and denominator of a fraction are increased by 3, the fraction becomes 3/4. If the numerator is increased by 2 and the denominator is multiplied by 2, the fraction becomes 1/2. Find the fraction.

16.

Prove that :-

17.

A page from the Pass Book of Mrs. Raj Rani is given below-

Date	Particulars	Amount Withdrawn Rs.	Amount deposited Rs.	Balance Rs.
06 Jan.96	Balance B/F	-	-	3,500.00
15 Feb.96	To cheque	1,000.00	-	2,500.00
12 March 96	By cheque	-	3,200.00	5,700.00
28 March 96	By cash	-	555.00	6,225.00
11th Nov.96	To cheque	4000.00	-	2,225.00
10th Dec.96	By cash	-	5,000.00	7,255.00

Find the interest Mrs. Raj Rani gets of the period Jan. 96 to Dec. 96 end 5% p.a. Simple interest, if she closes her account on 31st Dec. 96.

18.

The total salary of Roma is Rs 2,64,000 excluding H.R.A. during the year. She pays premium of Rs 10,000 annually towards L.I.C and contributes Rs 4,000 p.m. towards G.P.F. She also contributes Rs 8,000 p.a. to Unit Insurance Plan. She donates Rs 10,000 towards P.M. Relief Fund (100% tax exemption) and Rs 8,000 to a charitable Trust (50% tax exemption). Rs 2,700 are deducted each month from her salary as income tax for 11 months. Calculate the income tax payable, including surcharge, by her in the last month of year. Assume the following for calculating income tax.

(a) Standard deduction: 1/3 of total annual income subject to a maximum of Rs 20,000(Rs 25,000 if income is less than Rs 1 lakh.)

(b) Rate of Income tax:

Slab	Income tax
(i) Upto Rs 50,000	No tax
(ii) Rs 50,001 to Rs 60,000	10% of the amount exceeding Rs 50,000.
(iii) From Rs 60,001 to Rs 1,50,000	Rs 1,000+20% of the amount exceeding Rs 60,000
(iv) From Rs 1,50,001 onwards	Rs 19,000+30% of the amount exceeding Rs 1,50,000

(d) Surcharge 10% of the next tax payable

19.

The mean of the following frequency table is 50. But the frequencies F₁ and F₂ in classes 20-40 and 60-80 are missing. Find the missing frequencies:-

Class	Frequency
0-20	17
20-40	F1
40-60	32
60-80	F2
80-100	19
Total	120

20.

A cylindrical container is filled with ice-cream, whose radius is 6 cm and height is 15 cm. The whole ice-cream is distributed to 10 children in equal cones having hemispherical tops. If the height of the conical portion is four times the radius of its base, find the radius of the base of the ice-cream cone.

21.

The degree measure of an arc of a O is twice the angle subtended by it at any point of alternate segment of a circle with respect to arc. Prove using the above theorem, find the value of x in the given fig. if O is the center of the circle:

22.

Two pillars of equal height stand either side of a roadway which is 50 m wide. At a point P on the roadway between the pillars, elevation of the tops of pillars are 60° and 30°, find their height and the position of the point P.

23.

Construct a D ABC in which BC =7 cm, Ð A = 70° and foot of [ AD on BC is 4.5 cm away from B. Determine the length of the perpendicular AD. Write the steps of construction

24.

Solve the following quadratic Equation

Questions for ICSE's students

25.
26.	A sum of money is borrowed at 20% p.a. for 2 years, C.I. being reckoned half yearly. If the same sum were borrowed at the same rate for the same period, C.I. being reckoned yearly, the borrower would pay Rs. 241 less. What was the sum borrowe
27.	A point P(x, y) when reflected in the x-axis becomes P_x(3, -a), and when reflected in the y - axis becomes P_y(+b, 2). Find x,y, a,b
28.	A function is defined by :
29	A man sells Rs. 6400 worth of Rs. 100 shares for Rs. 108 each, and another lot worth Rs. 8600 of Rs. 100 shares for Rs. 125. What was the total amount realised by him
30.	In a right-angled D XYZ, Ð Y = 90⁰ and XY = YZ. Prove that the perpendicular drawn from Y to XZ is the axis of symmetry of D XYZ.
31.	A line AB is drawn with A = (-4, -8) and B = (6, 2). P is a point on AB such that AP : AQ = 3 : 1. Find the coordinates of P.

Questions for CBSE' students

25.

Given below are three questions. Two of them have infinite solutions and two have a unique solution. State the two pairs.

3x-2y=4, 6x+2y=4 and 9x-6y=12.

26.

Find the value of k for which the given equation x²+5kx+16=0 has no real roots.

Reduce

x⁴-8x²+16
(x²-3x+2 )(x²+3x+2 )

to its lowest terms.

27.

The GCD of two polynomials P(x) P = 4x²(x² - 3x + 2) and Q(x) = 12x(x - 2) ( x² - 4)
is 4x( x-2) .Find the LCM of the polynomials P(x) and Q(x).

28.

Put the equation 4x²- 4x + 1 = 2x² - 3x + 4 in the form of a quadratic equation.