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JEE Main Entrance Exam Preparation Model Test Paper

Time Allowed : 3 Hours Maximum Marks:

SECTION-A

Q1. For quadratic equation x2 – 2x + 1= 0, find the value of x + 1

𝑥?

Q2. In fig, if ABC is circumscribing a circle, find BC.

B

3cm

P Q

4cm

A R C

11 cm

Q3. If three coins are tossed simultaneously, then find the probability of

getting at least two heads.

Q4. Which term of the sequence 114, 109, 104… is the first negative

term?

Q5. Find k, so that the sum of the roots of the quadratic equation

3x2 + (2k+1)x – (k+5) = 0 is equal to the product of the roots.

Q6. If the surface area of two spheres are in the ratio 9:16, then find the

ratio of their volumes.

Q7. If 2P+1, 13, 5P-3 are three consecutive terms of an A.P, then find the

value of P.

Q8. If the length of the shadow of a Pole is √3 times the length of the

Pole, then find the angle of elevation of the sun?

SECTION-B

Q9. The diameter of a sphere is 42cm. It is melted and drawn into a

cylinder wire of 28cm diameter. Find the length of the wire.

Q10. If the pth term of an AP is 1

𝑞 and qth term is

1

𝑝, Show that the sum of pq

2

term is 1

2 (pq+1).

Q11. The minute hand of a clock is 12cm long. Find the area on the face of

the lock described by the minute hand between 8 A.M. and 8.35 A.M.

Q12. The roots and of the quadratic equation x2-5x+3(k-1)=0 are such

that -=11. Find the value of k.

Q13. In the Figure, O is the centre of the Circle, PT is the tangent and PAB

is the secant passing through Centre O. If PT=8cm and PA=4cm, find the radius of the circle.

I 8cm

B O A 4cm P

Q14. Find a relation between x & y such that the point (x,y) is equidistance

from the point (3,6) and (-3,4).

SECTION-C

Q15. Find the centre of a circle passing through the points (6,-6), (3,-7) and

(3,3)

Or

If the point R(x,y) is equidistant from the points P(a-b, a+b) and Q

(a+b, a-b) then prove that bx=ay.

Q16. A box contains 90 discs, which are numbered from 1 to 90. If one disc

is drawn at random from the box, then the probability that it bears

i) A two digit number

ii) A perfect square number

iii) A number divisible by 5.

Q17. Solve for x,

36x2 – 12ax + (a2-b2) = 0 by using quadratic formula.

Q18. The cost of ploughing a circular field at the rate of Rs. 0.25 per sq.m.

is Rs. 3850. Find the cost of fencing the field at Rs. 15 per m.

Q19. In an A.P., the sum of First n terms is 3𝑛2

2+

5𝑛

2, find its nth term and

25th term.

Q20. Find the area of the shaped region in the given figure, if PQ=24cm,

PR=7cm and O is the Centre of the circle.

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Q21. From the top of a 7m high building, the angle of elevation of the top of

a tower is 60° and the angle of depression of its foot is 45°. Determine

the height of the tower.

Q22. If PAB is a secant to a circle intersecting the circle at A, B and PT is a

tangent, then prove that PA x PB = PT2

Q23. A solid right circular cone of diameter 14cm and height 8cm is melted

to form a hollow sphere. If the external diameter of the sphere is

10cm, find the internal diameter of the sphere.

Q24. Find k, so that k2+4k+8, 2k2 + 3k + 6, 3k2 + 4k + 4 are in A.P.

OR

If the sum of m terms of an A.P. is to sum of n terms of same A.P. is

as 𝑚2

𝑛2, then Prove that ratio of its mth and nth term is 2m-1: 2n-1

SECTION-D

Q25. A motor boat whose speed in still water is 15 km/hr, goes 30 km down

stream and returns back to the starting point in a total time of 4hr & 30

min. Find the speed of the stream.

Q26. Prove that the Parellogram circumscribing a circle is a rhombus.

OR

A ∆ ABC is drawn to circumscribe a circle of radius 4cm such that the

segment BD and DC into which BC is divided by the point of contact

D are of lengths 8cm and 6cm respectively. Find the sides AB & AC.

Q27. Draw a triangle ABC with side BC=7cm, B = 45°, A = 105°, then

construct a triangle whose sides are 4

3 times the corresponding sides

of ∆𝐴𝐵𝐶.

Q28. The two opposite vertices of a square are (-1,2) and (3,2). Find the

coordinates of the other two vertices.

Q29. An aeroplane flying horizontally 1km above the ground is observed at

an elevation of 60°. After 10 seconds, is elevation is observed to be

30°. Find the speed of the aeroplane in km/hr.

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Q30. What is the probability that:

i) A non-leap year has 53 Tuesday?

ii) A leap year has 53 Wednesday?

iii) A leap year has 53 Friday and 53 Saturday?

Q31. A chord of a circle of radius 12cm subtends an angle of 120° at the

Centre. Find the area of the corresponding segment of the circle [use

π= 3.14 & √3 =1.73]

Q32. In an A.P. the first term is 2 and the sum of the first terms is one-

fourth of the sum of the next five terms. Show that the 20th term is

– 112.

Q33. A well of diameter 3m is dug 14m deep. The earth taken out of it has

been spread evenly all around it to a width of 4m to form an

embankment. Find the height of the embankment.

Q34. The height of a cone is 30cm. a small cone is cut off at the top by a

plane parallel to the base. If its volume be 1

27 of the volume of the

given cone, at which height above the base is the section made?