Visual Lens G Strategy | Hexagon Area | Easy
Let ABCDEF be a regular hexagon and P and Q be the midpoints of AB and CD, respectively.
Then, the ratio of the areas of trapezium PBCQ and hexagon ABCDEF is _______
Answer & Explanation
Correct answer is B) 5 : 24
Let area of the hexagon is 6E, where E represents the area of any one of the six congruent central triangles, such as triangle OBC.
The area of trapezium PBCQ is obtained by decomposing it into simpler regions. It consists of the full area of triangle OBC along with suitable fractional parts of the triangles adjacent to the base BC.
This decomposition gives: Area(PBCQ) = 5E/4.
Therefore, the required ratio of areas is: (5E/4) ÷ 6E = 5/24.
Visual Lens G Strategy | Easy
Let f(x) = x/(2x − 1) and g(x) = x/(x − 1). Then, the domain of the function
h(x) = f(g(x)) + g(f(x)) is all real numbers except
A) −1, 1, and 1/2
B) −1/2, 1/2, and 1
C) 1/2 and 1
D) 1/2, 1, and 3/2
Answer & Explanation
Correct Answer A) −1, 1, and 1/2
| x value | f(x) | g(x) | f(g(x)) | g(f(x)) | h(x) | Reason |
| −1 | ✅ | ✅ | ❌ | g(−1) = 1/2 → f(1/2) undefined | ||
| −1/2 | ✅ | ✅ | ✅ | ✅ | ✅ | No denominator becomes zero |
| 1/2 | ❌ | f(1/2) undefined | ||||
| 1 | ✅ | ❌ | g(1) undefined | |||
| 3/2 | ✅ | ✅ | ✅ | ✅ | ✅ | All steps valid |
This confirms x is not valid on −1, 1, and 1/2
Visual Lens G Strategy | Triangle | Moderate
In a triangle ABC, points D and E are on the sides BC and AC, respectively. BE and AD intersect at point T such that AD : AT = 4 : 3, and BE : BT = 5 : 4. Point F lies on AC such that DF is parallel to BE. Then, BD : CD is
A) 9 : 4
B) 7 : 4
C) 11 : 4
D) 15 : 4
Answer & Explanation
Correct answer: C) 11:4
Given, AD : AT = 4 : 3 ⇒ AT : TD = 3 : 1 and BE : BT = 5 : 4 ⇒ BT : TE = 4 : 1
By the mass points method, masses are assigned inversely to the segments.
From AT : TD = 3 : 1 ⇒ Mass at A : Mass at D = 1 : 3.
From BT : TE = 4 : 1, ⇒ Mass at B : Mass at E = 1 : 4.
Choosing consistent masses,
A = 5/4, D = 15/4 and B = 1, E = 4, so mass at T = 5.
Since D lies on BC, Mass at D = Mass at B + Mass at C
15/4 = 1 + Mass at C ⇒ Mass at C = 11/4.
Therefore, BD : DC = Mass at C : Mass at B ⇒ (11/4) : 1 ⇒ 11 : 4.
Visual Lens G strategy | 1.2 / 0.8 Percentages | Moderate
A loan of Rs 1000 is fully repaid by two instalments of Rs 530 and Rs 594, paid at the end of first and second year, respectively. If the interest is compounded annually, then the rate of interest, in percentage, is _____
A) 10
B) 9
C) 8
D) 11
Answer & Explanation
Correct Answer is c) 8%
Let the annual rate of interest be r%. The present value of the two instalments must equal the loan amount.
1000 = 530 / (1 + r/100) + 594 / (1 + r/100)²
| Rate (%) | 530 ÷ (1+r) | 594 ÷ (1+r)² | Total PV | Match |
| 10 | ≈ 481.8 | ≈ 491.0 | ≈ 972.8 | ❌ |
| 9 | ≈ 486.2 | ≈ 500.2 | ≈ 986.4 | ❌ |
| 8 | ≈ 490.7 | ≈ 508.9 | ≈ 999.6 | ✅ |
| 11 | ≈ 477.5 | ≈ 482.2 | ≈ 959.7 | ❌ |
