Q1
mcq
Which of the following cannot be the probability of an event?
- A. 1.5
- B. 3/5
- C. 25%
- D. 0.3
Correct Answer: A —
Solution
The probability of any event must lie between 0 and 1, inclusive (0 ≤ P(E) ≤ 1). Option (A) is 1.5, which is greater than 1, so it cannot be a probability. Option (B) 0.6, (C) 0.25, and (D) 0.3 are all between 0 and 1.
Q2
mcq
The mid-point of segment AB is the point P (0, 4). If the coordinates of B are (-2, 3) then the coordinates of A are
- A. (2, 5)
- B. (-2, -5)
- C. (2, 9)
- D. (-2, 11)
Correct Answer: A —
Solution
Using the mid-point formula, if P(x, y) is the midpoint of A(x1, y1) and B(x2, y2), then x = (x1+x2)/2 and y = (y1+y2)/2. Given P(0, 4) and B(-2, 3), let A be (x, y). 0 = (x - 2)/2 implies x = 2; 4 = (y + 3)/2 implies y + 3 = 8, so y = 5. Thus, A is (2, 5).
Q3
mcq
The point P which divides the line segment joining the point A (2, -5) and B (5, 2) in the ratio 2:3 lies in the quadrant.
- A. I
- B. II
- C. III
- D. IV
Correct Answer: D —
Solution
Using the section formula for points A(2, -5) and B(5, 2) in ratio m:n = 2:3, P(x, y) = ((mx2+nx1)/(m+n), (my2+ny1)/(m+n)). x = (2*5 + 3*2) / 5 = 16/5 = 3.2. y = (2*2 + 3*(-5)) / 5 = (4-15)/5 = -11/5 = -2.2. Since x > 0 and y < 0, the point lies in the IV quadrant.
Q4
mcq
The angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of the tower, is 45°. The height of the tower (in metres) is
- A. 15
- B. 30
- C. 30√3
- D. 10√3
Correct Answer: B —
Solution
Let h be the height of the tower. In the right-angled triangle formed, tan(45°) = height / base. 1 = h / 30, therefore h = 30 m.
Q5
mcq
A sphere of diameter 18 cm is dropped into a cylindrical vessel of diameter 36 cm partly filled with water. If the sphere is completely submerged, then the water level rises (in cm) by
- A. 3
- B. 4
- C. 5
- D. 6
Correct Answer: A —
Solution
The volume of the displaced water equals the volume of the sphere. Volume of sphere = (4/3)πr^3. Radius of sphere = 9 cm. Volume = (4/3)π(9)^3 = 972π. Let the rise in water level be h in the cylinder. Volume of cylinder water = πR^2h. Radius of cylinder = 18 cm. π(18)^2h = 972π. 324h = 972. h = 3 cm.
Q6
mcq
In figure 1, O is the centre of a circle, AB is a chord and AT is the tangent at A. If ∠AOB = 100°, then ∠BAT is equal to
- A. 100°
- B. 40°
- C. 50°
- D. 90°
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Q7
mcq
In figure 2, PA and PB are tangents to the circle with centre O. If ∠APB = 60°, then ∠OAB is
- A. 30°
- B. 60°
- C. 90°
- D. 15°
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Q8
mcq
The roots of the equation x² + x – (p + 1) = 0, where p is a constant are
- A. p, p+1
- B. –p, p+1
- C. p, -(p+1)
- D. –p, -(p + 1)
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Q9
mcq
In an AP, if a= 15, d = -3 and an = 0, then the value of n is
- A. 5
- B. 6
- C. 19
- D. 4
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Q10
mcq
The radii of two circles are 8 cm and 6 cm respectively. The diameter of the circle having area equal to the sum of the areas of the two circles (in cm) is
- A. 10
- B. 14
- C. 20
- D. 28
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Q11
short answer
A coin is tossed two times. Find the probability of getting both heads and both tails.
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Q12
short answer
Find the value of m so that the quadratic equation mx (5x – 6) + 9 = 0 has two equal roots.
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Q13
short answer
Find the value (s) of x for which the distance between the points P(x, 4) and Q (9, 10) is 10 units.
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Q14
short answer
Two cubes, each of side 4 cm are joined end to end. Find the surface area of the resulting cuboid.
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Q15
short answer
Draw a line segment of length 6 cm. Using compasses and ruler, find a point P on it which divides it in the ratio 3:4.
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Q16
short answer
Two concentric circles are of radii 7 cm and r cm respectively, where r > 7. A chord of the larger Circle, of length 48 cm, touches the smaller circle. Find the value of r.
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Q17
short answer
Find whether - 150 is a term of the AP 17, 12, 7, 2 …….?
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Q18
long answer
In figure 3, APB and CQD are semi-circles of diameter 7 cm each, while ARC and BSD are semi-circles of diameter 14 cm each. Find the perimeter of the shaded region. [Use π = 22/7]. OR Find the area of a quadrant of a circle, where the circumference of circle is 44 cm. [Use π = 22/7]
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Q19
long answer
If two vertices of an equilateral triangle are (3, 0) and (6, 0), find the third vertex. OR Find the value of k, if the points P (5, 4), Q (7, k) and R (9, -2) are collinear.
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Q20
long answer
From the top of a tower 100 m high, a man observes two cars on the opposite of the tower with angles of depression 30° and 45° respectively. Find the distance between the cars. Use [√3 = 1.73]
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Q21
long answer
Two dice are rolled once. Find the probability of getting such numbers on two dice, whose product is a perfect square. OR A game consists of tossing a coin 3 times and noting its outcomes each time. Hanif wins if he gets three heads or three tails, and loses otherwise. Calculate the probability that Hanif will lose the game.
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Q22
long answer
The radii of the circles ends of a bucket of height 15 cm are 14 cm and r cm (r < 14 cm). If the volume of bucket is 5390 cm³, then find the value of r. [Use π = 22/7]
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Q23
long answer
In fig. 4, a triangle ABC is drawn to circumscribe a circle of radius 2 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 4 cm and 3 cm respectively. If area of ∆ABC = 21 cm², then find the lengths of sides AB and AC.
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Q24
long answer
Find the value of the middle term of the following AP: -6, -2, 2 …………. 58. OR Determine the AP whose fourth term is 18 and the differences of the ninth term from the fifteenth term is 30.
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Q25
short answer
Find the roots of the following quadratic equation: 2√3x² – 5x + √3 = 0
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Q26
long answer
Find the area of the major segment APB, in Fig 5, of a circle of radius 35 cm and ∠AOB = 90° [Use π = 22/7]
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Q27
long answer
Draw a triangle PQR such that PQ = 5 cm, ∠P = 120° and PR = 6 cm. Construct another triangle Whose sides are 3/4 times the corresponding sides of ∆PQR?
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Q28
short answer
Find the point of y-axis which is equidistant from the points (-5, -2) and (3, 2).
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Q29
long answer
From a solid cylinder of height 20 cm and diameter 12 cm, a conical cavity of height 8 cm and radius 6 cm is hollowed out. Find the total surface area of the remaining solid. [Use π = 22/7]
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Q30
long answer
The length and breadth of a rectangular piece of paper are 28 cm and 14 cm respectively. A semi-circular portion is cut off from the breadth’s side and a semicircular portion is added on length’s side, as shown in Fig. 6. Find the area of the shaded region. [Use π = 22/7]
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Q31
long answer
From the top of a 15 m high building, the angle of elevation of the top of a cable tower is 60° and the angle of depression of its foot is 30°. Determine the
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Q32
short answer
Determine the AP whose fourth term is 18 and the differences of the ninth term from the fifteenth term is 30.
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Q32
long answer
A motor boat whose speed is 20 km/h is still water, takes 1 hour more to go 48 km upstream than to return downstream to the same spot. Find the speed of the steam.
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Q33
short answer
Prove that the lengths of tangents drawn from an external point to a circle are equal.
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Q34
long answer
Find the roots of the equation 1/(x+4) - 1/(x-7) = 11/30, x≠ -4, 7
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Q34
long answer
If the sum of first 4 terms of an AP is 40 and that of first 14 terms is 280. Find the sum of its first n terms.
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Q37
long answer
Find the sum of the first 30 positive integers divisible by 6.
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