In a parallelogram, two diagonals of parallelogram bisects each other, which concludes that mid-point of both diagonals are the same.Midpoint of AC =&nbsp;($\frac{1-2}{2}, \frac{1+8}{2}$)Let the coordinates of vertex D be (x, y)($\frac{x+3}{2}, \frac{y+4}{2}$) = ($\frac{1-2}{2}, \frac{1+8}{2}$)x = -4 and y = 5The answer is option D.

Let the number of balloons each child received be 2a, 2b, 2c and 2d2a + 2b + 2c + 2d = 20a + b + c + d = 10Each of them should get more than zero balloons.Therefore, total number of ways =&nbsp;$ (n-1)C_{r-1} = (10-1)C_{4-1} = 9C_3 = 84 $

Lemon juice : sugar syrup in the mixture is 1:1, i.e. 50% Lemon juice and 50% sugar syrup.In sugar syrup, 100% is sugar syrup.These two are mixed in the ratio 1:3.Lemon juice = $\frac{1(50\%)}{1+3}$Sugar syrup = $\frac{1(50\%) + 3(100\%)}{1+3}$ = $\frac{350}{4}$Required ratio = $50:350 = 1:7$

xy + yz = 19y(x + z) = 19Observe that 19 is prime and can’t be factorized further. Since x, y and z are all naturalnumbers, the value of (x + z) can’t be 1. (It should at least be 2). Therefore,&nbsp;y = 1&nbsp;and ( x + z) = 19yz + xz = 51z(y + x) = 51z(x + 1) = 5151 = 17 * 3 Case 1: z = 17, x + 1 = 3x = 2, y = 1, z = 17xyz = 2 * 1 * 17 = 34 Case 2: z = 3, x + 1 = 17x = 16, y = 1, z = 3xyz = 16 * 1 * 3 = 48Hence the minimum value of x * y * z = 34Hence, the answer is '34'

$b^2$ &lt; $4ac$ means that the discriminant is less than 0. Therefore, f(x)&gt;0 for all x if the coefficient of $x^2$ is positive, and f(x)&lt;0 for all x if the coefficient of $x^2$ is negative.We are given that f(m)&lt;0 and m is an integer.So the set containing values of m will either be empty if the coefficient of $x^2$ is positive, or it will be a set of all

<img src="https://lightyellow-stork-291267.hostingersite.com/wp-content/uploads/2025/11/6.image_zQ685bb.png">M - First meeting pointLet the speeds of trains A and B be 'a' and 'b', respectively.$\frac{x}{a} = \frac{D - x}{b}$It is given,$\frac{D}{a} = 10$ and $\frac{x}{b} = 9$$\frac{x}{\frac{D}{10}} = \frac{D - x}{\frac{x}{9}}$$\frac{10x}{D} = \frac{9D - 9x}{x}$$10x^2 = 9D^2 - 9Dx$$10x^2 + 9Dx - 9D^2 = 0$Solving, we get $x = \frac{3D}{5}$$\frac{x}{b} = 9$$\frac{3D}{5b} = 9$$\frac{D}{b} = 15$The total time taken by train B to travel from station Y to station X is 15 minutes.

It is given that average of three numbers is 13.Sum = $3 * 13 = 39$It is given, $\frac{39 + n}{4}$ is a odd number.Minimum value $\frac{39 + n}{4}$ can take such that n is a natural number is 11$\frac{39 + n}{4} = 11$n = 5

Let the number of persons standing ahead and behind of Pinky be 3a and 5a.Total number of persons = 3a + 5a + 1(including pinky) = 8a + 18a + 1 &lt; 3008a &lt; 299a &lt; 37.375Maximum value a can take is 37.The maximum possible number of persons standing ahead of Pinky = 3a = 3*37 = 111

It is given,$7C = 30P = 9A$ and Ankita bought $4C, 14P$ and $6A$.Let $7C = 30P = 9A = 630k$$C = 90k$, $P = 21k$, and $A = 70k$Cost price of $4C, 14P$ and $6A$ = $4(90k)+14(21k)+6(70k) = 1074k$Marked up price = $1074k + 1752$S.P = $\frac{1}{6}(1074k + 1752)$ + $\left(\frac{4}{5}\right)\left(\frac{5}{6}\right)(1074k + 1752)$ = $\frac{5}{6}(1074k + 1752)$S.P - C.P = profit$1460 - \frac{1074k}{6} = 744$$\frac{1074k}{6} = 716$$k = 4$Money spent on buying almonds = $420k = 420*4 = Rs 1680$The answer is option C.

In the question, it is given that&nbsp;the equation&nbsp;∣x+a∣+∣x−1∣=2∣x+a∣+∣x−1∣=2&nbsp;has an infinite number of solutions for any value of x. This is possible when x in |x+a| and x in |x-1| cancels out.Case 1:x + a &lt; 0, x - 1&nbsp;≥&nbsp;0- a - x + x - 1 = 2a = -3Case 2:x + a&nbsp;≥&nbsp;0 and x - 1 &lt; 0x + a - x + 1 = 2a = 1Largest value of a is 1.The answer is option C.

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cat 2022 Complete Paper Solution | Slot 1

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<div><p>Let ABCD be a parallelogram such that the coordinates of its three vertices A, B, C are (1, 1), (3, 4) and (&minus;2, 8), respectively. Then, the coordinates of the vertex D are</p></div>

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<div><p>The number of ways of distributing 20 identical balloons among 4 children such that each child gets some balloons but no child gets an odd number of balloons, is</p></div>

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<div><p>A mixture contains lemon juice and sugar syrup in equal proportion. If a new mixture is created by adding this mixture and sugar syrup in the ratio 1 : 3, then the ratio of lemon juice and sugar syrup in the new mixture is</p></div>

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<div>For natural numbers x,y, and z, if xy + yz = 19 and yz + xz = 51, then the minimum possible value of xyz is</div>

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<div><p>Let a,b,c be non-zero real numbers such that $b^2 \lt 4 a c$ and $f(x)=a x^2+b x+c$. If the set S consists of al integers m such that f(m) &lt; 0, then the set S must necessarily be</p></div>

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<div><p>The average of three integers is 13. When a natural number n is included, the average of these four integers remains an odd integer. The minimum possible value of n is</p></div>

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<div><p>The largest real value of a for which the equation |x + a| + |x &minus; 1| = 2 has an infinite number of solutions for x is</p></div>