CAT — Maxima and Minima

Step 1: Find the minimum value of f(x). Since the coefficient of x² is positive, the parabola opens upward. The vertex is at x = 2c. Substituting: Minimum value of f(x) = (2c)² − 4c(2c) + 8c = 4c² − 8c² + 8c = −4c² + 8c.

Step 2: Find the maximum value of g(x). Since the coefficient of x² is negative, the parabola opens downward. The vertex is at x = 3c/2. Substituting: Maximum value of g(x) = −(3c/2)² + 3c(3c/2) − 2c = −9c²/4 + 9c²/2 − 2c = 9c²/4 − 2c.

Step 3: Apply the given condition. −4c² + 8c > 9c²/4 − 2c. Multiplying both sides by 4: −16c² + 32c > 9c² − 8c → 25c² − 40c < 0 → 5c(5c − 8) < 0. Therefore, 0 < c < 8/5. Among the options, only 1/2 lies in this range.

Step 1: Rewrite the quadratic Expand: x² − 2x + 1 + 2kx + 11 = 0 → x² + (2k − 2)x + 12 = 0 Step 2: Use the condition for no real roots Discriminant < 0: (2k − 2)² − 48 < 0 (k − 1)² < 12 −2√3 < k − 1 < 2√3 Since 2√3 ≈ 3.464: k < 4.464 Largest integer: k = 4 Step 3: Minimize the expression k/(4y) + 9y = 1/y + 9y By AM-GM: 1/y + 9y ≥ 2√(1/y × 9y) = 2√9 = 6 Equality when 1/y = 9y → y = 1/3 Least possible value = 6

Let Price = k × w². Original price = k × 18² = 324k.

Step 1: Find the minimum possible total price
To minimize sum of squares, four distinct integer weights should be as close as possible.
Partition: 3, 4, 5, 6
Sum of squares = 9 + 16 + 25 + 36 = 86
Minimum total price = 86k

Step 2: Find the maximum possible total price
To maximize, make one piece as large as possible.
Partition: 1, 2, 3, 12
Sum of squares = 1 + 4 + 9 + 144 = 158
Maximum total price = 158k

Step 3: Use the given difference
(158 − 86)k = 72k = 288000 → k = 4000

Step 4: Find the price of the original stone
Original price = 324 × 4000 = Rs. 12,96,000

Step 1: Write the given condition. We have a + b + c + d = 46. We need to minimize (a − b)² + (a − c)² + (a − d)². Since the expression contains only differences with a, the values of b, c and d should be as close to a as possible.

Step 2: Use the sum condition. Let b = a + x, c = a + y, d = a + z. Then, a + (a + x) + (a + y) + (a + z) = 46 → 4a + (x + y + z) = 46. Since 46 ≡ 2 (mod 4) and 4a is always a multiple of 4, x + y + z ≡ 2 (mod 4).

Step 3: Rewrite the expression. The required expression becomes x² + y² + z². So, we need to minimize x² + y² + z² subject to x + y + z ≡ 2 (mod 4).

Step 4: Find the minimum possible value. The smallest possible values satisfying x + y + z = 2 are 1, 1, 0 (or any permutation). Then x² + y² + z² = 1² + 1² + 0² = 2. This is achievable. For example, a = 11, b = 12, c = 12, d = 11. Their sum = 11 + 12 + 12 + 11 = 46 and (a − b)² + (a − c)² + (a − d)² = (−1)² + (−1)² + 0² = 2. No smaller value is possible because the sum of three integer squares is non-negative, and a value of 1 cannot satisfy the required congruence condition.

Given:
4x² + 4y² − 4xy − 6y + 3 = 0
Find the value of
4x + 5y.

Step 1: Rearrange the expression
Write the equation as
4(x² − xy + y²) − 6y + 3 = 0
Now,
x² − xy + y²
= (x − y/2)² + 3y²/4
Substituting,
4(x − y/2)² + 3y² − 6y + 3 = 0

Step 2: Complete the square
Since
3y² − 6y + 3 = 3(y − 1)²,
the equation becomes
4(x − y/2)² + 3(y − 1)² = 0

Step 3: Find x and y
Both terms are perfect squares and are non-negative.
Their sum is zero only if each is zero.
So,
x − y/2 = 0
and
y − 1 = 0
Hence,
y = 1
and
x = 1/2

Step 4: Find the required value
4x + 5y
= 4 × 1/2 + 5 × 1
= 2 + 5
= 7

Answer: 7