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FACULTY RECRUITMENT TEST CATEGORY-D

Formal School Education/IX, X / NSEJS/NTSE/ Foundation for KVPY, Senior Olympiads, JEE Main & JEE Advanced

MATHEMATICS

PAPER A Time: 1 Hour Maximum Marks: 40

Name:................................................................................. Subject:...........................................

Marks:

Instructions

Attempt all questions. PAPER-A has two parts I and II. Each question of partI carries 2 marks and that of

partII carries 4 marks.

Calculators and log tables are not permitted.

PARTI

1. Evaluate (yz)logylogz (zx)

logzlogx (xy)logxlogy.

2. If cos2A = sin A tan A, then find the value of cos

3A + cos

2A.

3. In the given figure, P and Q are point of

contact. O is the incentre. Line BO produced,

meets PQ at G. Find the value of AGB.

A

P

G O

B C Q

4. In right triangle ABC, AC is divided by the point O, which is also centre of semicircle in such a way that AO = 15 units, CO = 20 units. Find the radius of semicircle.

A

O

B C

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FACREC-1112-P2-A-MA-2

5. In the circle O is the centre. COB = ,

AOD = . Prove that APD = 2

A

B

C

D

P O

6. In the given right angled triangle, prove that a + b = c + d,

where d is the diameter of circle.

A

B

c

C

O a

b

7. In the given square ABCD, PBC = PCB = 15.

Prove that APD is an equilateral triangle.

D

C B

A

P

8. The sides of a right triangle are 9, 12 and 15 cm long. Find the sum of the square of the medians.

9. For any two independent events, A and B, P (A) = 0.3, P (B) = 0.4, then find the value of P (A B).

10. If x is real, then find the range of values of the expression y = 2

2

x 14x 9

x 2x 3

.

PARTII 1. If x is a positive real number different from unity such that logax, logbx and logcx are in A.P., then

prove that c2 = a

log bca .

2. If each pair of the three equations x

2 + p1x + q1 = 0, x

2 + p2x + q2 = 0 and x

2 + p3x + q3 = 0 have a

common root, then prove that 2 2 21 2 3 1 2 3 1 2 2 3 3 1p p p 4 q q q 2 p p p p p p .

3. The angular elevation of a tower CD at a place A due South of it is 60 and at a place B due West of

A, the elevation is 30. If AB = 3 km, then find the height of the tower

4. In the given figure, ABC = 2ACB and AB = DC.

Also, AD is the bisector of BAC. Find ABC.

A

B C D 5. From the vertex of angle of a triangle a straight line is drawn perpendicular to the base, then the

rectangle by the sides of the triangle is equal to the rectangle contained by the perpendicular and the diameter of the circumcircle. Prove it.