Solution Key - Vardaan Learning Institute

1. Which of the following numbers is a perfect square?

Ans: (b) 256
Reason: 16 × 16 = 256. (154, 32, 720 are not perfect squares).

2. The cube of a negative integer is always:

Ans: (b) Negative
Reason: (-m)³ = -m × -m × -m = -m³.

3. In the equation 2x - 3 = 7, the value of x is:

Ans: (c) 5
Reason: 2x = 7 + 3 ⇒ 2x = 10 ⇒ x = 5.

4. If 15 men can do a work in 20 days, how many days will 10 men take?

Ans: (a) 30 days
Reason: Inverse variation. 15 × 20 = 10 × x ⇒ 300 = 10x ⇒ x = 30.

5. The discount is always calculated on the:

Ans: (c) Marked Price

6. The value of (a + b)² is:

Ans: (d) a² + b² + 2ab

7. The sum of the interior angles of a quadrilateral is:

Ans: (c) 360°

8. Which of the following is a polynomial in one variable?

Ans: (b) x² + 2x + 3
Reason: It contains only one variable 'x' with non-negative integer powers.

9. The coordinates of the origin are:

Ans: (c) (0, 0)

10. The area of a trapezium is given by:

Ans: (a) 1/2 × (sum of parallel sides) × height

11. Find the square root of 1156 by prime factorization.

1156 = 2 × 2 × 17 × 17
√1156 = √(2² × 17²)
= 2 × 17 = 34

12. Simplify using laws of exponents: (4/9)³/² × (4/9)¹/²

Using law: aᵐ × aⁿ = aᵐ⁺ⁿ
= (4/9)^(3/2 + 1/2)
= (4/9)^(4/2)
= (4/9)² = 16/81

13. Find the value of 'x' if: 5x - 3 = 3x + 7.

Transposing 3x to LHS and -3 to RHS:
5x - 3x = 7 + 3
2x = 10
x = 10/2 = 5

14. Factorize: x² + 5x + 6

Find two numbers whose sum is 5 and product is 6 (Numbers are 2 and 3).
= x² + 2x + 3x + 6
= x(x + 2) + 3(x + 2)
= (x + 2)(x + 3)

15. If MP = ₹900 and Discount = 10%, find SP.

Discount Amount = 10% of 900 = (10/100) × 900 = ₹90
SP = MP - Discount
SP = 900 - 90 = ₹810

16. Find the smallest number by which 3087 must be multiplied to make it a perfect cube.

Prime factorization of 3087 = 3 × 3 × 7 × 7 × 7
Grouping: (7 × 7 × 7) and (3 × 3)
The factor 3 does not form a triplet. We need one more 3.
Smallest number to multiply = 3.

17. A train 360 m long (speed 45 km/hr) crosses a 140 m bridge. Find time.

Total distance = Train length + Bridge length = 360 + 140 = 500 m
Speed = 45 km/hr = 45 × (5/18) m/s = 12.5 m/s
Time = Distance / Speed = 500 / 12.5
Time = 40 seconds

18. Divide 6x² - 13x + 6 by 2x - 3.

Using long division or factorization:
Factorize numerator: 6x² - 9x - 4x + 6 = 3x(2x - 3) - 2(2x - 3) = (3x - 2)(2x - 3)
Divide: [(3x - 2)(2x - 3)] / (2x - 3)
Quotient = 3x - 2

19. Find CI on ₹12,000 for 2 years at 10% p.a.

A = P(1 + R/100)ⁿ
A = 12000 (1 + 10/100)² = 12000 (11/10)²
A = 12000 × 121 / 100 = 120 × 121 = ₹14,520
CI = Amount - Principal = 14520 - 12000
CI = ₹2,520

20. Adjacent angles of a parallelogram are in ratio 3:2. Find all angles.

Let angles be 3x and 2x.
Sum of adjacent angles = 180° ⇒ 3x + 2x = 180° ⇒ 5x = 180° ⇒ x = 36°
Angles: 3(36) = 108° and 2(36) = 72°
Opposite angles are equal.
Angles are: 108°, 72°, 108°, 72°

21. Area of rhombus = 240 cm², d1 = 16 cm. Find d2.

Area = (1/2) × d1 × d2
240 = (1/2) × 16 × d2
240 = 8 × d2
d2 = 240 / 8
Other diagonal = 30 cm

22. Find square root of 147.1369 OR subtract from 7581.

Choice A (147.1369):
Using long division method:
Pairs: 1, 47, 13, 69
Square root calculation yields: 12.13

Choice B (7581):
87² = 7569 and 88² = 7744.
7581 is slightly more than 87².
Remainder when finding square root of 7581 by division is 12.
Least number to subtract = 12
Perfect square = 7581 - 12 = 7569. Root = 87.

23. Difference between CI and SI on a sum at 10% for 3 years is ₹93. Find sum.

Let Sum (P) = ₹100.
SI = (100×10×3)/100 = ₹30
CI = 100(1 + 10/100)³ - 100 = 100(1.331) - 100 = 33.1
Difference = 33.1 - 30 = ₹3.1
If diff is 3.1, Sum is 100. If diff is 93, Sum = (100/3.1) × 93
Sum = 1000/31 × 93 = 1000 × 3
Sum = ₹3,000

24. Ages of Ram and Rahim ratio 4:3. 4 years later ratio 6:5. Find present ages.

Let present ages be 4x and 3x.
After 4 years: (4x + 4) / (3x + 4) = 6 / 5
Cross multiply: 5(4x + 4) = 6(3x + 4)
20x + 20 = 18x + 24
20x - 18x = 24 - 20 ⇒ 2x = 4 ⇒ x = 2
Ram's age = 4(2) = 8 years
Rahim's age = 3(2) = 6 years