4
4 Q.1. (A) Solve the following : (Any 4)
1. If A - B - C and d(A, C) = 17, d(B, C) = 6.5, then d(A, B) = ? 2. In DPQR, ∠P = 70º, ∠Q = 65º, then find ∠R.
3. The length of hypotenuse of a right angled triangle is 15. Find the length of median of its hypotenuse.
4. In DPQR, ÐPRQ = 45°, ÐPQS = 100°, S-Q-R. Find ÐQPR.
5. Measures of opposite angles of a parallelogram are (60 – x)° and (3x – 4)°. Find the value of x.
6. ABCD is a rectangle. AB = 7 cm, BC = 24 cm. Find AC. Q.1. (B) Solve the following : (Any 2)
1. In the adjoining figure, if line q line r and line p is their transversal and if ∠a = 80°, find the measures of ∠f and ∠g.
2. In DPQR, ∠Q = 90, PQ = 12, QR = 5 and QS is a median. Find l(QS).
3. In IJKL, side IJ side KL, ∠I = 108º°,° ∠K = 53º° then find the measures of ∠J and ∠L.
B C D A P R S Q 45° 100° Q R S P MATHEMATICS - PART II CHAPTERS : 1, 2, 3, 4 Marks : 40 Std. X b f c g a p q r e d h
MAHESH TUTORIALS
(STATE BOARD)
Duration : 2 hrs.1. In the figure seg XY seg BC, then which of the following statement is true? (A) AB AC=AXAY (B) AXXB =AYAC (C)AX YC=AYXB (D) AB YC=ACXB
2. Find perimeter of a square if its diagonal is 10 2cm. (A) 10 cm (B) 40 2
(C) 20 cm (D) 40 cm 3. ∠QPR = 60°
\ ∠AOB = ...
(A) 60° (B) 90°
(C) 120° (D) Can not be found 4. AB is tangent at B.
AB = 12, AP = 6 \ PQ = ... .
(A) 18 (B) 6
(C) 12 (D) 20
Q.2. (B) Solve the following : (Any 2)
1. Construct a tangent to a circle with centre P and radius 3.2 cm at any point M on it.
2. In the adjoining figure, chord AB = chord CD, prove that, arc AC = arc BD
3. In ∆RST, ∠S = 90o, ∠T = 30o , RT = 12 cm. Find RS and ST.
Q.3. (A) Solve the following activity : (Any 2)
1. In ∆PQR, seg PM is a median. Angle bisectors of ∠PMQ and ∠PMR intersect side PQ and side PR in point X and Y respectively. Prove that XY QR.
Proof : In ∆PMQ, ray MX bisects ∠PMQ ...(Given)
∴ = ...(i) (Angle bisector property of a triangle)
In ∆PMR, ray MY bisects ∠PMR ...(Given)
∴ = ...(ii) (Angle bisector property of a triangle)
But, PM
MQ=PMMR ...(
∴M is midpoint of seg QR, ∴ MQ = MR)
= ...[From (i) and (ii)] ∴ seg XY side QR ...
2.
From the information given in the figure,
Prove that : PM = PN =
3a
Proof : MQ = QR = RN = a ...(Given) Point Q is the midpoint of seg MR ...(i)
∴ In ∆PMR, seg PQ is a median ...[From (i), Definition] ∴ PM2 + = + 2QM2 ...(Apollonius theorem)
∴ PM2 + a2 = +
∴ PM2 = – a2
∴ PM2 = 3a2
∴ ...(Taking square roots)
Similarly we can prove, PN = ∴ PM = = 3
a
3.
In the adjoining fig. circles with centres X,
Y touch each other at Z.
A secant passing through Z meets the
circles at A and B respectively.
Prove that, Radius XA radius YB.
Fill in the blanks and complete the proof.
4
9 Construction : Draw segments XZ and
Proof :
By theorem of touching circles, points X, Z, Y are
∠XZA ≅ ...(Vertically Opposite angles)
Let ∠XZA = ∠BZY = a ...(i)
seg XA ≅ seg XZ
∴ ∠XAZ = = a ...(ii) (Isosceles triangle theorem)
seg YB ≅
∴ ∠BZY = = a ...(iii) (Isosceles triangle theorem)
m∠XAZ = m∠YBZ = a ...[From (i), (ii) and (iii)]
∴ Radius XA radius YB
Q.3. (B) Solve the following : (Any 2)
1. In the adjoining figure, point B is the point of contact and point O is the centre of the circle.
Seg OE ^ Seg AD. If AB = 12, AC = 8, then find (i) AD (ii) DC and (iii) DE
2. Seg PM is a median of DPQR. If PQ = 40, PR = 42 and PM = 29, find QR. 3. In ∆ABC, AP ⊥ BC, BQ ⊥ AC, B-P-C, A-Q-C,
then prove that CPA ~ ∆CQB.
If AP = 7, BQ = 8, BC = 12 then find AC.
Q.4. Solve the following : (Any 3)
1. In ∆ABC, Ray BD bisects ∠ABC and Ray CE bisects ∠ACB. If seg AB ≅ seg AC, then prove that ED BC.
2. Walls of two buildings on either side of a street are parallel to each other. A ladder 5.8 m long is placed on the street such that its top just reaches the window of a building at the height if 4 m. On turning the ladder over to the other side of the street, its top touches the window of the other building at a heitht 4.2 m. Find the width of the street.
3. In the adjoining figure, ABCD is a parallelogoam. It circscribs the circle with centre T. Point E, F, G, H are touching points. AE = 4.5, EB = 5.5, find AD.
4. Draw a circle of radius 3.4 cm and centre E. Take a point F on the circle. Take another point A such that E-F-A and FA - 4.1 cm. Draw tangents to the circle from point A.
Q.5 Solve the following : (Any 1)
1. In the adjoining figure, each of segments PA, QB, RC and SD is perpendicular to line l. If AB = 6, BC = 9, CD = 12 and PS = 36, then determine PQ, QR and RS.
2. In the adjoining figure, point A is a common point of contact of two externally touching circles and line l is a common tangent to both circles touching at B and C.
Line m is another common tangent at A and it intersects BC at D.
Prove that (i) ∠BAC = 90º (ii) Point D is the midpoint of seg BC.
Q.6 Solve the following : (Any 1)
1. ABCD is a parallelogram. Point E is on side BC, line DE intersects Ray AB in point T.
Prove that : DE × BE = CE × TE.
2. In the adjoining figure, seg YZ and seg XT are altitudes of ∆WXY, which intersect each other at point P.
Prove that,
(i) WZPT is cyclic.
(ii) Points X, Z, T, Y are concyclic
4 3
