\[\text{The point on the curve } y = (x - 2)^2 \text{ at which the tangent is parallel to the chord joining the points } (2,0) \text{ and } (4,4) \[\text{ is:}\]

\[\text{1. } (3,-1)\]

\[\text{2. } (2,3)\]

\[\text{3. } (2,-3)\]

\[\text{4. } (3,1)\]

\[\text{A boat covers } 32 \text{ km upstream and } 36 \text{ km downstream in } 7 \text{ hours.}\]

\[\text{Also, it covers } 40 \text{ km upstream and } 48 \text{ km downstream in } 9 \text{ hours.}\]

\[\text{The speed of the boat in still water is:}\]

\[\text{1. } 12 \text{ km/hr}\]

\[\text{2. } 8 \text{ km/hr}\]

\[\text{3. } 9 \text{ km/hr}\]

\[\text{4. } 10 \text{ km/hr}\]

\[\text{Consider the linear programming problem (LPP):}\]

\[\text{Maximize } Z = 6x + 3y\]

\[\text{subject to the conditions}\]

\[4x + y \ge 80\]

\[x + 5y \ge 115\]

\[3x + 2y \le 150\]

\[x \ge 0,\; y \ge 0\]

\[\text{In reference to the above LPP, which of the following are correct?}\]

\[\text{(A) The feasible region is bounded.}\]

\[\text{(B) The corner points of the feasible region are } (15,20),\; (40,15) \text{ and } (0,75).\]

\[\text{(C) The maximum value of the objective function is } 285.\]

\[\text{(D) The LPP does not have optimal solution.}\]

\[\text{Choose the correct answer from the options given below:}\]

\[\text{1. (A), (B) and (D) only}\]

\[\text{2. (A), (B) and (C) only}\]

\[\text{3. (A) and (C) only}\]

\[\text{4. (B) and (D) only}\]

\[\text{The following data shows the percentage of rural and urban Indian households who have a high speed internet connection.}\]

\[\text{Based on the above data, which of the following statements are correct?}\]

\[\text{(A) A straight line trend by the method of least squares for rural Indians is } y_t = 3.4 + 0.7(x - 2022)\]

\[\text{(B) A straight line trend by the method of least squares for urban Indians is } y_t = 11.4 + 4.3(x - 2022)\]

\[\text{(C) The forecast for the year 2025 for urban group using the trend equation is } 22.2\%\]

\[\text{(D) The forecast for the year 2025 for rural group using the trend equation is } 5.5\%\]

\[\text{Choose the correct answer from the options given below:}\]

\[\text{1. (A), (B) and (D) only}\]

\[\text{2. (A), (C) and (D) only}\]

\[\text{3. (A) and (D) only}\]

\[\text{4. (B) and (C) only}\]

\[\text{In a game, a man wins ₹ 8 for getting a number greater than 3 and loses ₹ 3 otherwise, when a fair die is thrown.}\]

\[\text{The man decided to throw a die 4 times but to quit as and when he gets a number greater than 3.}\]

\[\text{If } X \text{ denotes the amount which the man wins or loses, then which of the following are correct?}\]

\[\text{(A) All the possible values of } X \text{ are } 8, 5, 2 \text{ and } -1.\]

\[\text{(B) The probability distribution of } X \text{ is:}\]

\[\text{(C) The mean value of } X \text{ is } \frac{75}{16}.\]

\[\text{(D) The variance of } X \text{ is } \frac{6615}{256}.\]

\[\text{Choose the correct answer from the options given below:}\]

\[\text{1. (A) and (B) only}\]

\[\text{2. (A), (B) and (C) only}\]

\[\text{3. (B), (C) and (D) only}\]

\[\text{4. (C) and (D) only}\]

\[\text{Shyam invested ₹ } 2,00,000 \text{ in 2019 for } 5 \text{ years.}\]

\[\text{If the compound annual growth rate (CAGR) for his investment is } 10\%,\]

\[\text{then the end balance of his investment is:}\]

\[\text{1. ₹ } 3,22,102\]

\[\text{2. ₹ } 3,49,900\]

\[\text{3. ₹ } 3,49,960\]

\[\text{4. ₹ } 3,60,490\]

\[\text{Which of the following are normal equations to fit a straight line trend } y = a + bx \text{ by the method of least squares?}\]

\[\text{1. } \sum y = a \sum x + b \sum x^2 , \quad \sum xy = a \sum x + b \sum y^2\]

\[\text{2. } \sum y = na + b \sum x , \quad \sum xy = a \sum x + b \sum x^2\]

\[\text{3. } \sum xy = na + b \sum x , \quad \sum xy = na + b \sum x^2\]

\[\text{4. } \sum x = na + b \sum y^2 , \quad \sum y = a \sum x + b \sum y^2\]

\[\text{A machine has been producing steel tubes of mean inner diameter of } 2 \text{ cm.}\]

\[\text{A sample of } 10 \text{ tubes gives an inner diameter of } 2.01 \text{ cm}\]

\[\text{and a standard deviation of } 0.063 \text{ cm.}\]

\[\text{Test the hypothesis that the machine is working in the proper order at } 5\% \text{ level of significance}\]

\[\text{and answer, which of the following statements are correct?}\]

\[\text{[Given that } t_{0.05} = 2.262 \text{]}\]

\[\text{(A) The value of the test statistic is } t = 0.476\]

\[\text{(B) The null hypothesis is rejected at } 5\% \text{ level of significance.}\]

\[\text{(C) The null hypothesis is accepted at } 5\% \text{ level of significance.}\]

\[\text{(D) The degree of freedom is } 10\]

\[\text{Choose the correct answer from the options given below:}\]

\[\text{1. (A), (B) and (D) only}\]

\[\text{2. (A), (C) and (D) only}\]

\[\text{3. (A) and (C) only}\]

\[\text{4. (C) and (D) only}\]

\[\text{If the random variable } X \text{ is normally distributed with mean } 16\]

\[\text{and standard deviation } 4,\]

\[\text{then the standard normal variable } Z \text{ corresponding to } X = 17 \text{ is}\]

\[\text{1. } 1.25\]

\[\text{2. } -1.25\]

\[\text{3. } -0.25\]

\[\text{4. } 0.25\]

\[\text{The fund created to accumulate money over the years to discharge a future obligation is known as:}\]

\[\text{1. Emergency Fund}\]

\[\text{2. Perpetuity}\]

\[\text{3. Sinking Fund}\]

\[\text{4. Equity}\]

\[\text{In an influenza epidemic, the number of cases diagnosed in a week were as follows:}\]

\[\text{The 3-day moving averages are:}\]

\[\text{1. } 10, 11, 12, 13, 9\]

\[\text{2. } 10, 11, 13, 12, 9\]

\[\text{3. } 10, 12, 14, 13, 11\]

\[\text{4. } 10, 12, 11, 13, 9\]

\[\text{The value of }\begin{bmatrix}1 & bc & bc(b+c) \\1 & ca & ca(c+a) \\1 & ab & ab(a+b)\end{bmatrix}\text{ is}\]

\[\text{1. } 1\]

\[\text{2. } 0\]

\[\text{3. } abc\]

\[\text{4. } (a+b+c)^2 abc\]

\[\text{The corner points of the bounded feasible region for a linear programming problem (LPP) are }(0,\tfrac{3}{2}),\; (1,2) \text{ and } (4,0).\]

\[\text{If the objective function is } Z = ax + by,\text{ where } a \text{ and } b \text{ are positive,}\]

\[\text{then the condition on } a \text{ and } b\text{ so that the maximum of } Z \text{ occurs at } (1,2)\text{ and } (4,0) \text{ is:}\]

\[\text{1. } a = 2b\]

\[\text{2. } 2a = b\]

\[\text{3. } 3a = 2b\]

\[\text{4. } 3a = 4b\]

\[\text{If a moneylender charges interest at the rate of 10 rupees per 100 rupees per half year,}\]

\[\text{payable in advance, then the effective rate of interest per annum is:}\]

\[\text{1. } 10\%\]

\[\text{2. } 11.1\%\]

\[\text{3. } 25\%\]

\[\text{4. } 23.45\%\]

\[\text{If } A \text{ is a square matrix of order } 3 \text{ and } |A| = 4,\]

\[\text{then the value of } |2A^{T}| \text{ is}\]

\[\text{1. } 8\]

\[\text{2. } -8\]

\[\text{3. } -32\]

\[\text{4. } 32\]

\[\text{Which of the following are correct?}\]

\[\text{(A) For a square matrix } A,\; \text{if } A^{3} = I,\; \text{then } A^{-1} = A^{2}.\]

\[\text{(B) The determinant of only a square matrix can be defined.}\]

\[\text{(C) If } A \text{ is a square matrix of order } 3,\; \text{then the number of minors of the matrix } A \text{ is } 3.\]

\[\text{(D) If } A \text{ and } B \text{ are two non-singular matrices of the same order,}\text{ then } (AB)^{-1} = B^{-1}A^{-1}.\]

\[\text{Choose the correct answer from the options given below:}\]

\[\text{1. (A), (B) and (D) only}\]

\[\text{2. (A), (B) and (C) only}\]

\[\text{3. (B) and (D) only}\]

\[\text{4. (C) and (D) only}\]

\[\text{Mr. X purchased a house for ₹ } 7,00,000 \text{ and made a down payment of ₹ } 1,50,000.\]

\[\text{He repays the balance in } 25 \text{ years by equal monthly installments at }9\% \text{ per annum compounded monthly.}\]

\[\text{The equated monthly installment (EMI) is:}\]

\[\text{[Given that } (1.0075)^{-300} = 0.106 \text{]}\]

\[\text{1. ₹ } 3625\]

\[\text{2. ₹ } 4614\]

\[\text{3. ₹ } 5614\]

\[\text{4. ₹ } 4714\]

Match List-I with List-II

\[\text{Choose the correct answer from the options given below:}\]

\[\text{1. (A)-(I), (B)-(II), (C)-(III), (D)-(IV)}\]

\[\text{2. (A)-(II), (B)-(I), (C)-(III), (D)-(IV)}\]

\[\text{3. (A)-(IV), (B)-(III), (C)-(I), (D)-(II)}\]

\[\text{4. (A)-(III), (B)-(IV), (C)-(I), (D)-(II)}\]

\[\text{If the integral } I = \int \frac{x^{2}}{\sqrt{1+x}} \, dx= \frac{1}{\alpha}(1+x)^{\alpha}- \frac{8\alpha}{15}(1+x)^{\alpha-1}+ 2(1+x)^{\alpha-2}+ C,\]\[\text{where } C \text{ is the constant of integration, then the value of } \alpha \text{ is:}\]

\[\text{(1) } 1\]

\[\text{(2) } \frac{1}{2}\]

\[\text{(3) } \frac{3}{2}\]

\[\text{(4) } \frac{5}{2}\]

\[\text{If } A =\begin{bmatrix}0 & a & 2 \\-2 & 0 & b \\-2 & 2 & c\end{bmatrix}\text{ is a skew symmetric matrix, then the value of } (a+b+c)^3 \text{ is:}\]

\[\text{1. } 1\]

\[\text{2. } 0\]

\[\text{3. } 8\]

\[\text{4. } 2\]

\[\text{In a } 500 \text{ m race, the ratio of speeds of two participants A and B is } 4:5.\]

\[\text{If A has a start of } 180 \text{ m, then the distance by which A wins is:}\]

\[\text{1. } 100 \text{ m}\]

\[\text{2. } 180 \text{ m}\]

\[\text{3. } 150 \text{ m}\]

\[\text{4. } 120 \text{ m}\]

\[\text{If the random variable } X \text{ follows the Poisson distribution such that }P(X = k) = P(X = k+1),\]

\[\text{then the mean value of } X \text{ is:}\]

\[\text{1. } k\]

\[\text{2. } k - 1\]

\[\text{3. } k^{2}\]

\[\text{4. } k + 1\]

\[\text{Match List-I with List-II}\]

\[\text{Choose the correct answer from the options given below:}\]

\[\text{1. (A)-(II), (B)-(I), (C)-(III), (D)-(IV)}\]

\[\text{2. (A)-(I), (B)-(II), (C)-(III), (D)-(IV)}\]

\[\text{3. (A)-(I), (B)-(II), (C)-(IV), (D)-(III)}\]

\[\text{4. (A)-(III), (B)-(II), (C)-(IV), (D)-(I)}\]

\[\text{Which of the following are correct?}\]

\[\text{(A) If } a \equiv b \pmod n,\text{ then } -a \equiv -b \pmod n\]

\[\text{(B) If } a + b = c,\text{ then } a(\text{mod } n) + b(\text{mod } n) \equiv (a+b+c)(\text{mod } n)\]

\[\text{(C) If } a \equiv b \pmod n,\text{ then } ka \equiv kb \pmod n,\; \forall k \in I\]

\[\text{(D) If } a \equiv b \pmod n,\text{ then } (a+k) \equiv (b+k) \pmod n,\; \forall k \in I\]

\[\text{Choose the correct answer from the options given below:}\]

\[\text{1. (A), (B) and (D) only}\]

\[\text{2. (A), (B) and (C) only}\]

\[\text{3. (A), (C) and (D) only}\]

\[\text{4. (B), (C) and (D) only}\]

\[\text{Irregular trends in a time series are caused by}\]

\[\text{(A) Epidemics}\]

\[\text{(B) Strikes}\]

\[\text{(C) Festival season}\]

\[\text{(D) Floods}\]

\[\text{Choose the correct answer from the options given below:}\]

\[\text{1. (A), (B) and (D) only}\]

\[\text{2. (A), (B) and (C) only}\]

\[\text{3. (A), (B), (C) and (D)}\]

\[\text{4. (C) and (D) only}\]

\[\text{Pipe A can fill a tank three times faster than pipe B.}\]

\[\text{If both pipes A and B are running together, they can fill the tank in } 12 \text{ minutes.}\]

\[\text{Then the time taken by pipe B alone to fill the tank is:}\]

\[\text{1. } 24 \text{ minutes}\]

\[\text{2. } 36 \text{ minutes}\]

\[\text{3. } 40 \text{ minutes}\]

\[\text{4. } 48 \text{ minutes}\]

\[\text{If } y = \left( \log \left( x + \sqrt{x^{2}+a^{2}} \right) \right)^{2}\text{ and } x \ne \frac{1-a^{2}}{2},\]

\[\text{then } (x^{2}+a^{2})\frac{d^{2}y}{dx^{2}} + x\frac{dy}{dx} \text{ is equal to:}\]

\[\text{1. } 0\]

\[\text{2. } 1\]

\[\text{3. } 2\]

\[\text{4. } 4\]

\[\text{The particular solution of the differential equation }x(1+y^{2})dx - y(1+x^{2})dy = 0,\]

\[\text{given that } y(0)=1 \text{ is:}\]

\[\text{1. } xy^{2} = x^{2} + 1\]

\[\text{2. } y^{2} = 2x^{2} + 1\]

\[\text{3. } y^{2} = 2x^{2} + xy + 1\]

\[\text{4. } x^{2} = 3y^{2} + 2xy + 1\]

\[\text{If, in a pair of consecutive positive integers, both numbers are greater than } 5\]

\[\text{and their sum is less than } 23,\text{ then the number of such pairs is:}\]

\[\text{1. Two}\]

\[\text{2. Three}\]

\[\text{3. Four}\]

\[\text{4. Five}\]

\[\text{A person invested ₹ } 2,50,000 \text{ in a fund.}\]

\[\text{At the end of the year, the value of the fund is ₹ } 3,00,000.\]

\[\text{If the market price is the same at the end of the year,}\]

\[\text{then the nominal rate of return is:}\]

\[\text{1. } 10\%\]

\[\text{2. } 15\%\]

\[\text{3. } 20\%\]

\[\text{4. } 12\%\]

\[\text{If } C \text{ is the original value of the asset, } S \text{ is the scrap value,}\]

\[n \text{ is the useful life of the asset and } D \text{ is the annual depreciation, then}\]

\[\text{1. } D = \frac{C+S}{n}\]

\[\text{2. } D = \frac{C-n}{S}\]

\[\text{3. } D = n\frac{C}{S}\]

\[\text{4. } D = \frac{C-S}{n}\]

\[\text{A container contains } 100 \text{ liters of milk.}\]

\[\text{From this container, } 10 \text{ liters of milk is taken out and replaced by water.}\]

\[\text{This process is repeated two more times.}\]

\[\text{The amount of milk left in the container now is:}\]

\[\text{1. } 85 \text{ liters}\]

\[\text{2. } 70 \text{ liters}\]

\[\text{3. } 67.1 \text{ liters}\]

\[\text{4. } 72.9 \text{ liters}\]

\[\text{The point estimate of the population standard deviation for the random sample }5,\,8,\,10,\,7,\,10,\,14 \text{ is:}\]

\[\text{1. } 1.5\]

\[\text{2. } 4.9\]

\[\text{3. } 3.1\]

\[\text{4. } 9\]