Duration: 2h

**Select the appropriate alternative.**

In ∆ABC and ∆PQR, in a one to one correspondence \[\frac{AB}{QR} = \frac{BC}{PR} = \frac{CA}{PQ}\]

∆PQR ~ ∆ABC

∆PQR ~ ∆CAB

∆CBA ~ ∆PQR

∆BCA ~ ∆PQR

Chapter: [1] Similarity

In a right angled triangle, if sum of the squares of the sides making right angle is 169 then what is the length of the hypotenuse?

15

13

5

12

Chapter: [2] Pythagoras Theorem

In a right angled triangle, if sum of the squares of the sides making right angle is 169 then what is the length of the hypotenuse?

15

13

5

12

Chapter: [2] Pythagoras Theorem

Choose the correct alternative answer for each of the following questions

(A) 1 cm^{3}

B) 0.001 cm^{3 }

(C) 0.0001 cm^{3}

(D) 0.000001 cm^{3}

Chapter: [7] Mensuration

Find the distance between the following pair of point.

\[W\left( \frac{- 7}{2} , 4 \right), X\left( 11, 4 \right)\]

Chapter: [4] Co-ordinate Geometry

Prove that:

\[\frac{\sin^2 \theta}{\cos\theta} + \cos\theta = \sec\theta\]

Chapter: [6] Trigonometry

∆PQR ~ ∆LTR. In ∆PQR, PQ = 4.2 cm, QR = 5.4 cm, PR = 4.8 cm. Construct ∆PQR and ∆LTR, such that \[\frac{PQ}{LT} = \frac{3}{4} .\]

Chapter: [5] Geometric Constructions

Seg RM and seg RN are tangent segments of a circle with centre O. Prove that seg OR bisects ∠MRN as well as ∠MON.

Chapter: [3] Circle

In the given figure, ∠QPR = 90°, seg PM ⊥ seg QR and Q–M–R, PM = 10, QM = 8, find QR.

Chapter: [1] Similarity [2] Pythagoras Theorem

The dimensions of a cuboid are 44 cm, 21 cm, 12 cm. It is melted and a cone of height 24 cm is made. Find the radius of its base.

Chapter: [7] Mensuration

Prove that:

\[\sec^4 \theta - \cos^4 \theta = 1 - 2 \cos^2 \theta\]

Chapter: [6] Trigonometry

In adjoining figure PQ ⊥ BC, AD⊥ BC then find following ratios.

(i) \[\frac{A\left( ∆ PQB \right)}{A\left( ∆ PBC \right)}\]

(ii) \[\frac{A\left( ∆ PBC \right)}{A\left( ∆ ABC \right)}\]

(iii) \[\frac{A\left( ∆ ABC \right)}{A\left( ∆ ADC \right)}\]

(iv) \[\frac{A\left( ∆ ADC \right)}{A\left( ∆ PQC \right)}\]

Chapter: [1] Similarity

In the given figure, M is the midpoint of QR. ∠PRQ = 90°. Prove that, PQ^{2 }= 4PM^{2 }– 3PR^{2}

^{}

Chapter: [2] Pythagoras Theorem

While landing at an airport, a pilot made an angle of depression of 20°. Average speed of the plane was 200 km/hr. The plane reached the ground after 54 seconds. Find the height at which the plane was when it started landing. (sin 20° = 0.342)

Chapter: [6] Trigonometry

From the top of a lighthouse, an observer looking at a ship makes angle of depression of 60°. If the height of the lighthouse is 90 metre, then find how far the ship is from the lighthouse.

Chapter: [6] Trigonometry

Solve the following example.

Find the height of an equilateral triangle having side 2*a*.

Chapter: [2] Pythagoras Theorem

In ∆PQR, PM = 15, PQ = 25 PR = 20, NR = 8. State whether line NM is parallel to side RQ. Give reason.

Chapter: [1] Similarity

In the adjoining figure, O is the centre of the circle. From point R, seg RM and seg RN are tangent segments touching the circle at M and N. If (OR) = 10 cm and radius of the circle = 5 cm, then

(1) What is the length of each tangent segment ?

(2) What is the measure of ∠MRO ?

(3) What is the measure of ∠ MRN ?

Chapter: [3] Circle

Draw a circle with radius 3.4 cm. Draw a chord MN of length 5.7 cm in it. construct tangents at point M and N to the circle.

Chapter: [5] Geometric Constructions

Find the co-ordinates of the points of trisection of the line segment AB with A(2, 7) and B(–4, –8).

Chapter: [4] Co-ordinate Geometry

In the given figure, the circles with centres P and Q touch each other at R. A line passing through R meets the circles at A and B respectively. Prove that – (1) seg AP || seg BQ,

(2) ∆APR ~ ∆RQB, and

(3) Find ∠ RQB if ∠ PAR = 35°

Chapter: [3] Circle

In the given figure shows a toy. Its lower part is a hemisphere and the upper part is a cone. Find the volume and surface area of the toy from the measures shown in the figure (\[\pi = 3 . 14\])

Chapter: [7] Mensuration

In ΔABC, D and E are points on the sides AB and AC respectively such that DE || BC

If AD = 8x − 7, DB = 5x − 3, AE = 4x − 3 and EC = (3x − 1), find the value of x.

Chapter: [1] Similarity

Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

Chapter: [3] Circle

A solid is in the form of a right circular cylinder, with a hemisphere at one end and a cone at the other end. The radius of the common base is 3.5 cm and the heights of the cylindrical and conical portions are 10 cm. and 6 cm, respectively. Find the total surface area of the solid. (Use n =`22/7`)

Chapter: [7] Mensuration

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.

Chapter: [5] Geometric Constructions

If A (–14, –10), B(6, –2) is given, find the coordinates of the points which divide segment AB into four equal parts.

Chapter: [4] Co-ordinate Geometry

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