MCSCCE
In a committee, 50 people speak French, 20 speak Spanish and 10 speak both Spanish and French. How many speak at least one of these languages?
A)
10
B)
60
C)
50
D)
80
Using the formula n(A U B) = n(A) + n(B) - n(A ∩ B) = 50 + 20 - 10 = 60.
MCSCCE
The length of the diagonals of a rhombus are 16 cm and 12 cm. Then the length of the side of the rhombus is:
A)
9 cm
B)
8 cm
C)
20 cm
D)
10 cm
The diagonals of a rhombus bisect each other perpendicularly. The half-diagonals are 8 cm and 6 cm. Using the Pythagorean theorem: side = sqrt(8^2 + 6^2) = sqrt(64 + 36) = sqrt(100) = 10 cm.
MCSCCE
A kite is attached to a string. If the kite is flying at a height of 45 m and the string makes an angle of 30° with the ground, then the length of the string is:
A)
60 m
B)
90 m
C)
120 m
D)
75 m
1. **Correct Option Analysis**: This problem can be modeled as a right-angled triangle where the height of the kite is the opposite side (45 m), the length of the string is the hypotenuse (L), and the angle with the ground is 30°. Using the trigonometric ratio for sine:
sin(θ) = Opposite / Hypotenuse
sin(30°) = 45 / L
Since sin(30°) = 1/2, we have:
1/2 = 45 / L
L = 45 * 2 = 90 m.
Therefore, option (b) is correct.
2. **Incorrect Options Analysis**:
- Option (a) 60 m: This would occur if the angle were approximately 41.8°.
- Option (c) 120 m: This does not correspond to the standard trigonometric ratios for this height and angle.
- Option (d) 75 m: This would occur if the angle were approximately 36.9°.
MCSCCE
The distance between the points A (6, 5) and B (0, – 3 ) is:
A)
10 units
B)
8 units
C)
14 units
D)
√40 units
Using the distance formula: d = √[(6 - 0)² + (5 - (-3))²] = √[6² + 8²] = √[36 + 64] = √100 = 10 units.
MCSCCE
Division of Rs 980 in the ratio 12 : 23 is -
A)
Rs. 335 & Rs. 644
B)
Rs. 336 & Rs. 634
C)
Rs. 336 & Rs. 644
D)
Rs. 335 & Rs. 544
Sum of the ratio terms = 12 + 23 = 35. The value of one unit = 980 / 35 = Rs. 28. First share = 12 × 28 = Rs. 336, and second share = 23 × 28 = Rs. 644.
MCSCCE
Which smallest number should be added to 80 so as to make it a perfect square?
A)
2
B)
3
C)
1
D)
4
The nearest perfect square greater than 80 is 9^2 = 81. The smallest number to be added to 80 is 81 - 80 = 1.
MCSCCE
You get a wage increase of 4% plus an extra £5 per week. Your present wages are £250 per week. What will your new wage be?
A)
£ 255
B)
£ 265
C)
£ 275
D)
£ 270
4% increase on £250 = (4 / 100) * 250 = £10. Total weekly wage increase = £10 + £5 = £15. New wage = £250 + £15 = £265.
MCSCCE
A school calculated that it had given merits to boys and girls in the ratio of 4:1. There were a total of 680 merits given. How many merits did the girls get?
A)
124
B)
136
C)
145
D)
152
Total merits = 680. Ratio (boys:girls) = 4:1. Total parts = 4+1 = 5. One part = 680 / 5 = 136. Therefore, girls got 136 merits.
MCSCCE
The times, in seconds, taken by 180 candidates in an Army Recruitment camp to run a 100 m race are tabulated below: Class [13.8–14, 14–14.2, 14.2–14.4, 14.4–14.6, 14.6–14.8, 14.8–15] with Frequencies [5, 7, 8, 68, 72, 20]. The number of athletes who completed the race in less than 14.6 seconds is:
A)
20
B)
88
C)
68
D)
140
Athletes completing the 100 m race in less than 14.6 seconds fall into the first 4 classes: 5 + 7 + 8 + 68 = 88 athletes.
MCSCCE
If the area of a circle is 616 cm², then its radius is:
A)
14 cm
B)
7 cm
C)
15.2 cm
D)
16.5 cm
Area of circle = pi * r² = 616 => (22/7) * r² = 616 => r² = (616 * 7) / 22 = 196 => r = 14 cm.
MCSCCE
The value of (2⁻¹ × 5⁻¹)⁻¹ ÷ 4⁻¹ is -
A)
2.5
B)
40
C)
4/10
D)
10/4
(2^-1 * 5^-1)^-1 / 4^-1 = (1/10)^-1 / (1/4) = 10 / (1/4) = 10 * 4 = 40 (option b).
MCSCCE
The time taken to cover 102 km at 34 km/hr is -
A)
6 hrs
B)
5 hrs
C)
4 hrs
D)
3 hrs
Time = Distance / Speed = 102 / 34 = 3 hours.
MCSCCE
A tank which has a leak in its bottom is filled in 20 hours. Had there been no leak in the bottom, it would have been filled in 15 hours. The leak can empty the tank in:
A)
20 hours
B)
60 hours
C)
75 hours
D)
15 hours
Rate with leak = 1/20. Rate without leak = 1/15. Rate of leak = (1/15) - (1/20) = (4-3)/60 = 1/60. The leak empties it in 60 hours.
MCSCCE
The product of the roots of the equation x² - 3x + 2 = 0 is -
A)
3
B)
-2/3
C)
2
D)
-3/2
For a quadratic equation ax² + bx + c = 0, the product of the roots is c/a. In x² - 3x + 2 = 0, the product is 2/1 = 2.
MCSCCE
From a point Q, the length of tangent to a circle is 2√7 cm and the distance of Q from the centre of the circle is 8 cm. The radius of the circle is:
A)
6 cm
B)
11 cm
C)
12 cm
D)
15 cm
Tangent (T) = 2*sqrt(7), Distance (D) = 8. Radius (R) = sqrt(D^2 - T^2) = sqrt(64 - 4*7) = sqrt(64 - 28) = sqrt(36) = 6 cm.
MCSCCE
If the common difference of an AP is 4, then what is a₈ – a₃ ?
A)
4
B)
8
C)
16
D)
20
In an Arithmetic Progression, a₈ - a₃ = (a + 7d) - (a + 2d) = 5d. With common difference d = 4, a₈ - a₃ = 5 * 4 = 20.
MCSCCE
A and B together can do a piece of work in 12 days, while B alone can finish it in 30 days. Then the work done by A in 1 day is -
A)
1/20
B)
1/12
C)
1/30
D)
1/60
A's 1-day work = (1/12) - (1/30) = (5 - 2)/60 = 3/60 = 1/20.
MCSCCE
A bus travelled a distance of 180 m in 12 seconds. The speed of bus is -
A)
15 km/hr
B)
15 m/s
C)
10 km/hr
D)
10 m/s
Speed = Distance / Time. Given Distance = 180 m and Time = 12 seconds. Speed = 180/12 = 15 m/s.
MCSCCE
In a Junior School there are 240 pupils and 35% have free school dinners. Work out the number of children who do not have free school dinners.
A)
156
B)
200
C)
167
D)
175
Percentage of children without free meals = 100% - 35% = 65%. Number of children = 0.65 * 240 = 156.
MCSCCE
From a point outside the circle, the number of tangents that can be drawn to the circle is:
A)
infinitely many
B)
1
C)
2
D)
0
1. Correct Option (c): From any point P located outside a circle, exactly two distinct tangent lines can be drawn to the circle. These two tangents touch the circle at two different points of tangency, and the lengths of these two tangent segments from point P to the points of contact are equal.
2. Incorrect Options:
- (a) Infinitely many: This is incorrect because a tangent must touch the circle at exactly one point; only two such lines can pass through a single external point.
- (b) 1: This is incorrect because a single tangent can only be drawn if the point is located ON the circumference of the circle.
- (d) 0: This is incorrect because zero tangents are drawn only if the point is located INSIDE the circle, as any line passing through an interior point will be a secant (intersecting the circle at two points) rather than a tangent.
MCSCCE
The expansion of a³ is -
A)
3 × a
B)
a + a + a
C)
3 × 3 × 3
D)
a × a × a
In algebra, a³ represents repeated multiplication of the base 'a' three times, which equals a × a × a.
MCSCCE
If radii of two concentric circles are 4 cm and 5 cm, then the length of each chord of one circle which is tangent to the other circle is:
A)
9 cm
B)
6 cm
C)
3 cm
D)
1 cm
Let chord length be 2x. In the right triangle formed with concentric radii: x = sqrt(5^2 - 4^2) = sqrt(25 - 16) = sqrt(9) = 3 cm. Total chord length = 2 * 3 = 6 cm (option b).
MCSCCE
If sec²θ ( 1 + sinθ) (1 – sinθ) = k, the value of k is:
A)
1
B)
– 1
C)
0
D)
2
Step-by-step derivation:
1. Start with the given expression: k = sec²θ (1 + sinθ)(1 – sinθ).
2. Use the algebraic identity (a + b)(a – b) = a² – b². Here, (1 + sinθ)(1 – sinθ) = 1² – sin²θ = 1 – sin²θ.
3. According to the fundamental trigonometric identity sin²θ + cos²θ = 1, we know that 1 – sin²θ = cos²θ.
4. Substitute this back into the expression: k = sec²θ × cos²θ.
5. Since secθ is the reciprocal of cosθ (secθ = 1/cosθ), then sec²θ × cos²θ = (1/cos²θ) × cos²θ = 1.
6. Therefore, k = 1.
Why other options are incorrect:
- Option b (-1), c (0), and d (2) are incorrect because the product of a trigonometric function and its reciprocal squared always results in 1, not any other constant.
MCSCCE
The head of English created the following table showing the number of pupils in each year group who had additional help in English. What is the percentage of pupils in all the year groups combined that are having additional tuition? (Year 7: 96 pupils, 15 help; Year 8: 108 pupils, 21 help; Year 9: 111 pupils, 16 help; Year 10: 98 pupils, 9 help; Year 11: 116 pupils, 15 help). Give your answer rounded to a whole number.
A)
13.5%
B)
14%
C)
15.2%
D)
16%
Total pupils across all year groups = 96 + 108 + 111 + 98 + 116 = 529. Total pupils receiving additional tuition = 15 + 21 + 16 + 9 + 15 = 76. Combined percentage = (76 / 529) * 100 = 14.3667% ≈ 14% rounded to the nearest whole number.
MCSCCE
The HCF of (x – 3)²( x + 4)² and (x – 1)(x + 4)(x – 3)² is:
A)
(x – 1)(x – 3)(x + 4)
B)
(x – 3)(x + 4)²
C)
(x + 4)(x – 3)²
D)
(x + 4)(x – 3)
The Highest Common Factor (HCF) is obtained by taking the lowest exponent of all common factors. Between (x – 3)²(x + 4)² and (x – 1)(x + 4)(x – 3)², the common factors are (x - 3) with power 2 and (x + 4) with power 1, giving HCF = (x + 4)(x – 3)².
