Q1
mcq
The length of shadow of a tower on the plane ground is √3 times the height of the tower. The angle of elevation of sun is :
- A. 450
- B. 300
- C. 600
- D. 900
Correct Answer: B —
Solution
Let height of tower be h and length of shadow be s. Given s = h√3. In the right triangle formed by the tower and shadow, tan(θ) = height / shadow = h / (h√3) = 1/√3. Since tan(30°) = 1/√3, the angle of elevation is 30°.
Q2
mcq
If the area of a circle is equal to sum of the areas of two circles of diameters 10 cm and 24 cm, then the diameter of the larger circle (in cm) is :
- A. 34
- B. 26
- C. 17
- D. 14
Correct Answer: B —
Solution
Radii are r1 = 10/2 = 5 cm and r2 = 24/2 = 12 cm. Area of new circle = π(5)^2 + π(12)^2 = π(25 + 144) = 169π. Let radius be R, so πR^2 = 169π, which gives R = 13. Diameter = 2R = 26 cm.
Q3
mcq
If the radius of the base of a right circular cylinder is halved, keeping the height the same, then the ratio of the volume of the cylinder thus obtained to the volume of original cylinder is:
- A. 1 : 2
- B. 2 : 1
- C. 1 : 4
- D. 4 : 1
Correct Answer: C —
Solution
Original volume V1 = πr^2h. New radius r' = r/2. New volume V2 = π(r/2)^2h = π(r^2/4)h = V1/4. The ratio V2:V1 = (V1/4) : V1 = 1 : 4.
Q4
mcq
Two dice are thrown together. The probability of getting the same number on both dice is:
- A. 1/2
- B. 1/3
- C. 1/6
- D. 1/12
Correct Answer: C —
Solution
Total outcomes when two dice are thrown is 6 × 6 = 36. Favorable outcomes for same numbers are (1,1), (2,2), (3,3), (4,4), (5,5), (6,6), which are 6. Probability = 6/36 = 1/6.
Q5
mcq
The coordinates of the point P dividing the line segment joining the points A (1, 3) and B (4, 6) in the ratio 2 : 1
- A. (2,4)
- B. (3,5)
- C. (4,2)
- D. (5,3)
Correct Answer: B —
Solution
Using section formula x = (m1x2 + m2x1)/(m1 + m2) and y = (m1y2 + m2y1)/(m1 + m2). x = (2*4 + 1*1)/(2+1) = 9/3 = 3. y = (2*6 + 1*3)/(2+1) = 15/3 = 5. Point is (3, 5).
Q6
mcq
If the coordinates of one end of a diameter of a circle are (2, 3) and the coordinates of its centre are (-2, 5), then the coordinates of the other end of the diameter are :
- A. (-6,7)
- B. (6,-7)
- C. (6,7)
- D. (-6,-7)
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Q7
mcq
The sum of first 20 odd natural numbers is:
- A. 100
- B. 210
- C. 400
- D. 420
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Q8
mcq
If 1 is a root of the equations ay2 + ay + 3 = 0 and y2 + y + b = 0 then ab equals :
- A. 3
- B. - 7/2
- C. 6
- D. -3
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Q9
mcq
In Fig. 1, the sides AB, BC and CA of a triangle ABC, touch a circle at P, Q and R respectively. If PA = 4 cm, BP = 3 cm and AC = 11 cm, then the length of BC (in cm) is :
- A. 11
- B. 10
- C. 14
- D. 15
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Q10
mcq
In Fig 2, a circle touches the side DF of EDF at H and touches ED and EF produced at K and M respectively. If EK = 9 cm, then the perimeter of EDF (in cm) is:
- A. 18
- B. 13.5
- C. 12
- D. 9
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Q11
short answer
If a point A (0, 2) is equidistant from the points B (3, p) and C (p, 5), then find the value of p.
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Q12
short answer
A number is selected at random from first 50 natural numbers. Find the probability that it is a multiple of 3 and 4.
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Q13
short answer
The volume of a hemisphere is 2425 1/2 cm3. Find its curved surface area. [Use = 22/7]
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Q14
short answer
Tangents PA and PB are drawn from an external point P to two concentric circles with centre O and radii 8 cm and 5 cm respectively, as shown in Fig.3. If AP = 15 cm, then find the length of BP.
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Q15
long answer
In Fig.4, an isosceles triangle ABC, with AB = AC, circumscribes a circle. Prove that the point of contact P bisects the base BC. OR In Fig.5, the chord AB of the larger of the two concentric circles, with centre O, touches the smaller circle at C. Prove that AC = CB.
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Q16
short answer
In Fig.6, OABC is a square of side 7 cm. IF OAPC is a quadrant of a circle with centre O, then find the area of the shaded regions. [Use = 22/7]
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Q17
short answer
Find the sum of all three digit natural numbers, which are multiples of 7.
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Q18
short answer
Find the value(s) of k so that the quadratic equation 3x2 – 2kx + 12 = 0 has equal roots.
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Q19
short answer
A point P divides the line segment joining the points A (3,-5) and B (-4, 8) such that AP PB = K 1 . If P lies on the line x + y = 0, then fine the value of K.
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Q20
short answer
If the vertices of a triangle are (1,-3), (4, p) and (-9, 7) and its area is 15 sq.units, find the value(s) of p.
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Q21
long answer
Prove that the parallelogram circumscribing a circle is a rhombus. OR Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.
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Q22
long answer
From a solid cylinder of height 7n cm and base diameter 12 cm, a conical cavity of same height and same base diameter is hollowed out. Find the total surface area of the remaining solid. [Use = 22/7] OR A cylindrical bucket, 32 cm high and with radius of base 18 cm, is filled with sand. This bucket is emptied on the ground and a conical heap of sand is formed. If the height of the conical heap is 24 cm, then find the radius and slant height of the heap.
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Q22
long answer
From a solid cylinder of height 7 cm and base diameter 12 cm, a conical cavity of same height and same base diameter is hollowed out. Find the total surface area of the remaining solid. [Use π = 22/7]
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Q23
short answer
In Fig. 7, PQ and AB are respectively the arcs of two concentric circles of radii 7 cm and 3.5 cm and centre O. If ∠ POQ = 300, then find the area of the shaded region. [Use = 22/7]
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Q24
short answer
Solve for x : 4x2 – 4ax + (a2 – b2) = 0 OR Solve for x : 3x2− 2√6x + 2 = 0
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Q25
short answer
A kite is flying at a height of 45 m above the ground. The string attached to the kite is temporarily tide to a point on the ground. The inclination of the string with the ground is 600. Find the length of the string assuming that there is no slack in the string.
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Q26
long answer
Draw a triangle ABC with side BC = 6 cm, ∠ C = 30° and ∠ A = 105°. Then construct another triangle whose sides are 2/3 times the corresponding side of ABC.
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Q27
long answer
A cylindrical bucket, 32 cm high and with radius of base 18 cm, is filled with sand. This bucket is emptied on the ground and a conical heap of sand is formed. If the height of the conical heap is 24 cm, then find the radius and slant height of the heap.
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Q27
long answer
The 16th term of an AP is 1 more than twice its 8th term. If the 12th term of the AP is 47, then find its nth term.
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Q28
short answer
A card is drawn from a well shuffled deck of 52 cards. Find the probability of getting (i) a king of red colour (ii) a face and (iii) the queen of diamonds.
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Q29
long answer
A bucket is in the form of a frustum of a cone and it can hold 28.49 litres of water. If the radii of its circular ends are 28 cm and 21 cm, find the height of the bucket. [Use π = 22/7]
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Q30
long answer
The angle of elevation of the top of a hill at the foot of a tower is 60° and the angle of depression from the top of the tower to the foot of the hill is 30°. If the tower is 50 m high, find the height of the hill.
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Q31
long answer
Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.
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Q32
long answer
A shopkeeper buys some books for Rs. 80. If he had bought 4 more books for the same amount, each book would have cost Rs 1 less. Find the number of books he bought.
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Q33
short answer
Sum of first 20 terms of an AP is -240, and its first term is 7. Find its 24th term.
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Q34
long answer
A solid is in the shape of a cone standing on a hemisphere with both their radii being equal to 7 cm and the height of the cone is equal to its diameter. Find the volume of the solid. [Use π = 22/7]
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Q36
long answer
A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = AD + BC.
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Q38
long answer
The sum of two numbers is 9 and the sum of their reciprocals is 1/2. Find the numbers
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