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PRE FOUNDATION DIVISON IMO Stage-II Exam.-2017
CLASS-10 MATHEMATICS Q.1
Q.2
Q.3
If 3 is the least prime factor of number a and 7 is the least prime factor of number b, then the least prime factor of a + b, is (A) 2 (B) 3 (C) 5 (D) 10
Evaluate :
Q.6
(B)
2
(C) – 2
(D)
5
In the given figure (not drawn to scale) ABCD is a trapezium in which AB || DC and AD = BC. If P, Q, R, S be respectively the mid-points of BA, BD, CD and CA. Then PQRS is a
If , , are the roots of x2 – (k + 1) x +
Q.7
2
If (5 2 6 ) x 3 + (5 2 6 ) x x = _________. (A) 2
Q.4
Q.5
(B) Rectangle (D) Square
A right cone and a hemisphere lie on opposite sides of a common base of 2.5 m diameter and the cone is right angled at the vertex. If a cylinder circumscribe them in this position, approximate what additional space will be enclosed? (A) 7.12 m3 (B) 6.14 m3 3 (C) 6.69 m (D) 5.25 m3 The HCF of x3 + (a + b)x2 + (ab + 1) x + b and x3 + 2ax2 + (a2 + 1)x + a is (A) x2 + ax + 1 (B) x2 + bx + 1 2 (C) x + x + a (D) x2 + x + b
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2
3
= 10, then
(B) – 2 (C) –2 (D) All of these Q.8
If cot + tan = x and sec – cos = y, then (A) (B) (C) (D)
(A) Rhombus (C) Parallelogram
1 2 (k 2
+ k + 1) = 0, then ( – )2 ________. (A) –(k2 + 1) (B) k2 + 1 (C) 2k2 + k (D) k2 – 1
2 3 4 5 17 4 15
(A) 1
Date : 12-02-2017
Q.9
(x + 2y)2 = 2 x2 + y2 = 2 (x2y)2/3 – (xy2) 2/3 = 1 (x2y)1/3 + (xy2)1/3 = 1
In the given figure, the common tangents PR and QS intersect at the point T. A and B are centres of the two circles. Given that PAQ = 108º and PR = 8 cm, find
(a) RBS (b) The length of QS. (a) (A) 120º (B) 108º (C) 115º (D) 112º
(b) 8 cm 8 cm 6 cm 6 cm 1
Q.10
Q.11
Q.12
Q.13
The point A divides the join of P(–5, 1) and Q(3, 5) in the ratio k : 1. The values of k for which the area of ABC where B(1, 5) C(7, – 2) is 2 sq. units is 31 31 (A) 7, (B) –7, 9 9 31 31 (C) 7, – (D) –7, – 9 9
P(4, 3) and Q lies on the same straight line which is parallel to the y-axis. If Q is 3 units from the x-axis, the possible coordinates of Q are _________. (A) (–3, 0) (B) (3, 4) (C) (4, –3) (D) (3, 8)
Q.18
The coordinates of A, B, C are (6, 3), (–3, 5) and (4, –2) respectively and P is any point area ( PBC) (x, y). Find . area ( ABC)
If 2x – 3y = 7 and (a + b)x – (a + b – 3) y = 4a + b represent coincident lines, then a and b satisfy the equation (A) a + 5b = 0 (B) 5a + b = 0 (C) a – 5b = 0 (D) 5a – b = 0 A bag contains 11 white balls and some red balls. If the probability of drawing a red ball is double that of a white ball, find the number of red balls in the bag. (A) 22 (B) 33 (C) 11 (D) 0 2
If , are the roots of the equation ax + bx + 1 1 , are the roots of the c = 0 and equation px2 + qx + r = 0, then r = ________. (A) a + 2b (B) a + b + c (C) ab + bc + ca (D) abc
Q.14
Let Sn denote the sum of the first 'n' terms of S an A.P. S2n = 3Sn. Then the value of 3n is Sn equal to (A) 4 (B) 6 (C) 8 (D) 10
Q.15
Sum of the length, width and depth of a cuboid is 's' and its diagonal is 'd'. Find its surface area. (A) s2 + d2 (B) s2 – d2 2 2 (C) d – s (D) None of these
Q.16
Q.17
In a rectangle ABCD, P and Q are the midpoints of BC and AD respectively. If R is any point on PQ, then area (ARB) equals 1 (A) (area of ABCD) 2 1 (B) (area of ABCD) 3 1 (C) (area of ABCD) 4 (D) None of these
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Q.19
(A)
2x y 2 7
(B)
3x 2 y 2 7
(C)
xy 7
(D)
x y 2 7
A tree standing on a horizontal plane is leaning towards east. At two points situated at distance a and b exactly due west on it, the angles of elevation of the top of the tree are respectively and . Find the height of the top of the tree from the ground. tan tan (A) a b (b a ) tan tan (B) tan tan (C) (a – b)(tan + 2 tan) ab(tan tan ) (D) tan tan
Q.20
Find the number of coins, 1.5 cm in diameter and 0.2 cm thick, to be melted to form a right circular cylinder of height 10 cm and diameter 4.5 cm. (A) 435 (B) 231 (C) 450 (D) 520
Q.21
In the given figure, RTP and STQ are common tangents to the two circles with centres A and B of radii 3 cm and 5 cm respectively. If ST : TQ = 1 : 3 and RT = 4 cm, find 2
(a) The length of QT (b) The length of AB
(A) (B) (C) (D) Q.22
Q.23
Q.24
Q.25
Q.26
(a) 10 cm 16 cm 15 cm 12 cm
(b) 14 cm 25 cm 20 cm 18 cm
Q.27
The minute hand of a block is 21 cm long. Find the area described by the minute hand on the face of the clock between 7:00 AM and 7:05 AM. (A) 8.21 cm2 (B) 4.32 cm2 2 (C) 6.25 cm (D) 5.5 cm2
Q.28
In the given figure, ABCD is a rectangle with AB = 14 cm and BC = 7 cm. Taking DC, BC and AD as diameters, three semi-circles are drawn as shown in the figure. Find the area of the shaded region.
Two dice are thrown simultaneously. Find the probability of getting a multiple of 2 on the dice and a multiple of 3 on the other dice. (A) 1/3 (B) 7/36 (C) 1/6 (D) 11/36 If abx2 = (a – b)2(x + 1), then the value of 4 4 1+ + 2 is __________. x x ab (A) a b
2
a b (B) ab
a (C) a b
2
b (D) a b
(A) 59 cm2 (C) 60 cm2 Q.29
In ABC, P divides the side AB such that AP : PB = 1 : 2. Q is a point in AC such that PQ || BC. Find the ratio of the areas of APQ and trapezium BPQC. (A) 1 : 6 (B) 1 : 7 (C) 1 : 8 (D) None of these
Q.30
In a competitive examination, one mark is awarded for each correct answer while 1 mark is deducted for each wrong answer. 2 Jayanti answered 120 questions and got 90 marks. How many questions did the answer correctly? (A) 100 (B) 110 (C) 90 (D) 115
Q.31
Find a and b in order that x3 – 6x2 + ax + b may be exactly divisible by x2 – 3x + 2. (A) –7, 9 (B) 11, –6 (C) 8, 4 (D) 5, 4
2
2
The coefficient of x in a quadratic equation x2 + px + q = 0 was taken as 17 in place of 13 and its roots were found to be –2 and –15. The roots of the original equation are ______. (A) 2, 15 (B) 10, 3 (C) –10, –3 (D) –2, –15 P is the point (–5, 3) and Q is the point (–5, m). If the length of the straight line PQ is 8 units, then the possible values of 'm' are ________. (A) –5, 5 (B) –5, 11 (C) –5, –11 (D) 5, 11 How many odd integers beginning with 15 must be taken for their sum to be 975? (A) 27 (B) 25 (C) 23 (D) 21
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(B) 59.5 cm2 (D) 60.5 cm2
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Q.32
In the given figure, M = N = 46º. Express x in terms of a, b and c where a, b, c are lengths of LM, MN and NK respectively.
Step IV : Draw a circle with AO as radius. This circle cuts the circle drawn in step II at B and P. Step V : Joint AP. AP and AB are desired tangents drawn from A to the circle passing through B, C and D. (A) Only I (B) Only IV (C) Only III (D) Only V Q.36
ac bc ab (C) ac
(A)
Q.33
Q.34
Q.35
ab bc a a (D) b( a c )
(B)
– 90,000 in A man arranges to pay a debt of 40 monthly installments which are in A.P. When 30 installments are paid, he dies leaving one third of the debt unpaid. Find the value of second installments. – 1500 – 1800 (A) (B) – 1400 – 1325 (C) (D)
Marks 0-10 10-20 20-30 30-40 40-50 50-60 Total obtained No. of 10 a 25 30 b 10 100 students (A) 9, 16 (B) 10, 15 (C) 15, 13 (D) 8, 9
Q.37
Two years ago, a father was five times as old as his son. Two years later, his age will be 8 more than three times the age of the son. Find the present ages of father and son respectively. (A) 42 years, 10 years (B) 46 years, 12 years (C) 56 years, 18 years (D) 64 years, 20 years Let ABC be a right angled triangle in which AB = 3 cm, BC = 4 cm and B = 90º. BD is the perpendicular from B on AC. The circle through B, C, D is drawn. Given below are steps of construction of tangents from A to this circle. Identify the wrong step. Steps of construction : Step I : Draw ABC and perpendicular BD from B on AC. Step II : Draw a circle with BC as a diameter. This circle will pass through D. Step III : Let O be the mid point of BC. Join AO.
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Find the value of a and b respectively in the following frequency distribution table, if N = 100 and median is 32.
In the given figure points A, B, C and D are the centres of four circles that each have a radius of length 2 units. If a point is selected at random from the interior of square ABCD. What is the probability that the point will be chosen from the shaded region?
(A) 7/15 (C) 3/14
(B) 8/19 (D) 5/7
Q.38
X takes 3 hours more than Y to walk 30 km. But, if X doubles his pace, he is ahead of Y 1 by 1 hour. Find their speed of walking. 2 (A) 7 km/h, 4 km/h (B) 10/3 km/h, 5 km/h (C) 2 km/h, 3/2 km/h (D) 3 km/h, 7 km/h
Q.39
Find the square root of (ab – ac – bc)2 + 4abc(a + b). (A) ab – b – ca (B) ab + bc + ca (C) 1/2 (a + 2b + c) (D) (a – b + c)
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Q.40
The vertices of ABC are A(4, 6), B(1, 5) and C(7, 2). A line is drawn to intersect sides AB and AC at D and E respectively such that AD AE 1 = = . Find the area of ADE. BD CE 4 (A) 7/2 sq. units (B) 11/3 sq. units (C) 12 sq. units (D) None of these
Q.41
The coordinates of one end point of a diameter of a circle are (4, –1) and the coordinates of the centre of the circle are (1, – 3). Find the coordinates of the other end of the diameter. (A) (–2, –5) (B) (–2, 5) (C) (2, –5) (D) (2, 5)
Q.42
The value of
42 42 42 ...
Q.44
A man standing on the deck of a ship, which is 10 m above water level. He observes the angle of elevation of the top of a hill as 60º and the angle of depression of the base of the hill as 30º. Find the height of the hill (Ignore the height of the man). (A) 30 m (B) 42 m (C) 40 m (D) 35 m
Q.45
In the given figure (not drawn to scale), two circles with centres A and B of radii 3 cm and 4 cm respectively intersect at two points C and D such that AC and BC are tangents to the two circles. Find the length of the common chord CD.
is
6 6 6 ...
(A) 7/3 Q.43
(B) 6/8
(C) 5
(D) 8
Two circles of radii 5 cm and 3 cm and centres A and B touch internally. If the perpendicular bisector of segment AB meets the bigger circle in P and Q, find the length of PQ. (A) 5 7 cm
(B) 3 2 cm
(C) 10 6 cm
(D) 4 6 cm
(A) 3.2 cm (B) 2.4 cm (C) 4.8 cm (D) 5.6 cm
ACHIEVERS SECTION Q.46
Match the following : p. (1 – sin2)sec2 = i. 2 1 q. cos2 + + 1 = ii. 2sec2 2 1 cot 1 1 r. + = iii. cosec2sec2 1 sin 1 sin s. cosec2 + sec2 = iv. 1 (A) p iii, q iv, r ii, s i (B) p i, q ii, r iii, s iv (C) p iv, q ii, r i, s iii (D) p iv, q i, r ii, s iii
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Q.47
A peacock is sitting on the top of a pillar, which is 9 m high. From a point 27 m away from the bottom of the pillar, a snake is coming to its hole at the base of the pillar. Seeing the snake the peacock pounces on it. If their speeds are equal, at what distance from the hole is the snake caught? (A) 10 m (B) 11 m (C) 12 m (D) 13 m
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Q.48
Study the following statements and state 'T' for true and 'F' for false. (i) The common difference of an A.P., the sum of whose n terms is Sn, is Sn – 2 Sn–1 + Sn–2. (ii) If the sums of n terms of two arithmetic 3n 5 progressions are in the ratio , then 5n 7 3n 1 their nth terms are in the ratio . 5n 1 (iii) If Sn denote the sum of an terms of an A.P. with first term a and common Sx difference d such that is Skx
(A) (B) (C) (D) Q.49
independent of x, then d = 2a. (i) (ii) (iii) T T T F F F T F T F T F
Q.50
Read the statements carefully and select the correct option. Statement I : For any positive integer n, n3 – n divisible by 6. Statement II : If a and b are two odd positive integers such that a > b, then one of the two ab ab numbers and is odd and the 2 2 other is even. (A) Both Statement-1 and Statement-II are true. (B) Both Statement-1 and Statement-II are false. (C) Statement-1 is true and Statement-II is false. (D) Statement-1 is false and Statement-II is true.
In the given figure, ABC is a right angled triangle in which A = 90º, AB = 21 cm and AC = 28 cm. Semi-circles are described on AB, BC and AC as diameters. Find the area of the shaded region.
(A) 294 cm2 (B) 296 cm2 (C) 298 cm2 (D) None of these
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