CUET UG 2023 — Applied-Mathematics

The set of values of $x$ satisfying $26 \equiv 5 \pmod{x}$ is _______.

The increase in the sale of shawls during winters is an example of :

A die is thrown. If A is the event 'the number appearing is a multiple of 3 and B is the event 'the number appearing is even', then A and B are

Pipes A and B can fill a tank in 20 hours and 30 hours respectively and pipe C can empty the full tank in 40 hours. If all the pipes are opened together, how much time will be needed to make the tank full ?

Given the data for the sales of a product in a state is as follows: | Year | 2005 | 2006 | 2007 | 2008 | 2009 | |---|---|---|---|---|---| | Sales (In lakh Rs) | 150 | 130 | 160 | 170 | 200 | The equation of the straight-line trend by method of least squares is:

Which of the following statements are correct ? (A) $\text{var}(aX + b) = a^2 \text{var}(X)$ (B) $\text{var}(X) = E(X^2) - \{E(X)\}^2$ (C) $E(aX + b) = aE(X) + b$ (D) $E(X) = \sum_{i=1}^{n} p_i x_i^2$ Choose the correct answer from the options given below :

Matrix $A = \begin{bmatrix} 0 & a & 5 \\ 4 & b & -1 \\ c & 1 & 0 \end{bmatrix}$ is skew - symmetric, then the values of a, b c are :

The total cost function for $x$ units of a commodity is given by $C(x) = \frac{25x^3}{3} - 75x^2 + 48x + 34$. The output $x$ at which the marginal cost is minimum is :

For a manufacturer, total cost function is given by $C = \frac{x^2}{25} + 2x$. Which of the following are correct ? A. 2.6 is the marginal cost when 5 units are produced. B. $\frac{2x}{25} + 2$ is the marginal cost function. C. $\frac{x}{25} + 2$ is the average cost function. D. 2.4 is the marginal cost when 5 units are produced. E. $\frac{x}{25} + 1$ is the average cost function. Choose the correct answer from the options given below:

A matrix has 24 elements , which of the following cannot be the possible order of the matrix

A product costs the manufacturer ₹ 20 per unit. The demand function is given by $p(x) = 1000 - 20x$, then the quantity for maximum profit is :

Five litres of water is added to a certain quantity of pure milk costing Rs. 60 per litre. If by selling the mixture at the same price as before, a profit of 20% is made, then the amount of pure milk in the mixture is :

The sum of the minor and the cofactor of the element 6 in the determinant $\begin{vmatrix} 2 & 3 & 1 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{vmatrix}$ is :

If the matrix $\begin{bmatrix} 0 & 2 & 5x \\ -2 & 0 & 6 \\ 10 & -6 & y \end{bmatrix}$ is skew-symmetric matrix, then the value of $(y - 4x)$ is :

If $x = 6t^2$, $y = \frac{6}{t^2}$, then $\frac{d^2 y}{dx^2}$ is equal to :

A boat takes 6 hr 25 minutes to row upstream a certain distance with a speed which is 14.4 times that of the river current. The time taken by the boat to row down the same distance with same speed is:

A sum of Rs 60,000 invested at r percent compounded quarterly will provide payment at rupees 600 each at the end of every three months , then the value of r is :

If $A = \begin{bmatrix} -2 & 1 \\ 3 & 2 \end{bmatrix}$ and $B' = \begin{bmatrix} -1 & 1 \\ 0 & 2 \end{bmatrix}$, then $(A+2B)' =$

Let A be a square matrix. Then, (A) $A + A^T$ is a symmetric matrix (B) $A - A^T$ is a skew-symmetric matrix (C) $AA^T$ is a skew-symmetric matrix (D) $A^T A$ is a symmetric matrix Choose the correct answer from the options given below :

For a 3 $\times$ 3 matrix A = [a$_{ij}$], whose element a$_{ij}$ is given by $a_{ij} = \sqrt{i^2 + 3j}$ then what is element a$_{31}$?

A car costing ₹ 8,00,000 has scrap value of ₹ 3,00,000. If the book value of car at the end of fourth year is ₹ 6,00,000, then the useful life of the car is :

Given $\sum p_0 q_0 = 800$, $\sum p_0 q_1 = 1500$, $\sum p_1 q_1 = 1300$ and $\sum p_1 q_0 = 900$, where subscripts 0 and 1 are used for base year and current year respectively. The Paasche's index number is:

An asset costing Rs 2,00,000 has a useful life of 10 years and scrap value of Rs 40,000. Its book value at the end of year 6 by Straight Line Method, is :

Match List - I with list- II | List-I , Differential equation | List - II , Degree | |---|---| | A. $\left(\frac{dy}{dx}\right)^3 + yx = 0$ | 2 | | B. $e^{\frac{dy}{dx}} + y^2 + y'' = 0$ | 1 | | C. $Xyy'' + x(y')^2 - yy' = 0$ | Not defined | | D. $(Y'')^2 + y = 0$ | 3 | Choose the correct option below :

If the sum and product of the mean and variance of a binomial distribution are 18 and 72 respectively, then the probability of obtaining atmost one success is

A point $P(3a, -2b)$ lies in region $4x + 7y \leq (-3)$ which of the following options is true ?

The point on the curve $y^2 = 16x$ for which the y-coordinate is changing 2 times as fast as the x-coordinate is :

The following data is available for two independent sample: | | SAMPLE-1 | SAMPLE-2 | |---|---|---| | sample size | 8 | 10 | | sample mean | 191 | 199 | | sample standard deviation | 10 | 12 | Then the value of t-statistic is :

If $\begin{bmatrix} 2x + 3 & y + 1 \\ 2x - z & 3y \end{bmatrix} = \begin{bmatrix} 7 & -2 \\ -4 & -9 \end{bmatrix}$, then matrix $\begin{bmatrix} 3z + 1 & 4 \\ -2 & 4 \end{bmatrix}$ is equal to:

In a 1000 m race P beats Q by 100 m and in the same race Q beats R by 200 m. By what distance does P beat R?

Match List - I with List - II. | List - I | List - II | |----------|-----------| | (A) The common region determined by all the linear constraints of a L.P.P. is called | (I) corner point | | (B) A point in the feasible region which is the intersection of two boundary lines is called, | (II) non-negative | | (C) The feasible region for an LPP is always a | (III) feasible region | | (D) The constraints $x, y \geq 0$ describes that the variables involved in a LPP are | (IV) convex polygon | Choose the correct answer from the options given below :

Match list I with list II. 4 defective pens are mixed with 10 normal pens. 3 pens are drawn one by one with replacement , then the probability distribution of the number of defective pens is : | List-I | List - II | |---|---| | A. P(X=0) | 8/343 | | B. P(X=1) | 60/343 | | C. P(X=2) | 125/343 | | D. P(X=3) | 150/343 | Choose the correct option below :

If $y = 3e^{2x} + 2e^{3x}$, then which one of the following is true ?

A company produces two types of belts A & B with a profit of Rs 2 and Rs 1.50 respectively. Belt of type A needs twice as much time to make as belt type B . The company can produce at the most 1000 belts of type B per day . Material for 800 belts is available per day . At the most , 400 buckles for belt type A and 700 for belts type B are available . Then the appropriate LPP is :

If the sum of two positive numbers is 25 and their product is maximum when divided in the ratio of cubes of one and squares of the other, then the numbers are:

A company has issued a bond having a face value of Rs. 10,000 paying annual dividends at 8.5%. The bond will be redeemed at par at the end of 10 years, then the purchase price of this bond if the investor wishes a yield rate of 8% is : [Given $(1.08)^{-10} = 0.46319349$]

In a game of carrom of 100 points, A can give B 20 points and C 40 points, then points B can give C is:

Pipes A and B can fill a tank in 5 hours and 6 hours respectively. Another Pipe can empty the full tank in 30 hours. If all three pipes are opened together, then the tank will be filled in :

The corner point of the feasible region determined by a set of linear constraints are : (0,0) , (0,4), (2,5) , (6,3) and (6,0) then which of the following point lie in the feasible region ?

A vehicle has a scrap value of Rs 7,50,000 after 6 years of its purchase . If the annual depreciation charge is Rs 55,000, then the original cost of the vehicle is :

The number of all possible matrices of order 3 $\times$ 3 with each entry 2 or 3 is.

The present value of a perpetuity of Rs. 6,240 payable at the beginning of each year, if money is worth 10% effective, is :

From a sample of 5 items having values 2, 4, 6, 7, 6, the unbiased estimates of the population mean and the standard deviation are:

Match List - I with List - II. | List - I | List - II | |----------|-----------| | (A) The minimum value of $f(x) = 8x^2 - 4x + 7$ is | (I) 48 | | (B) The maximum value of $f(x) = x + \frac{1}{x}$, $x < 0$ is | (II) 13 | | (C) The maximum slope of the curve $y = -2x^3 + 6x^2 + 7x + 26$ is | (III) $-2$ | | (D) The minimum value of $f(x) = x^2 + \frac{128}{x}$ is | (IV) $\frac{13}{2}$ | Choose the correct answer from the options given below :

If the matrix $A = \begin{bmatrix} 0 & 2y & z \\ x & y & -z \\ x & -y & z \end{bmatrix}$ satisfies the equation $A^T A = I_3$, then $x^2 + y^2 + z^2$ is :

A company intends to create a sinking fund to replace at the end of 20$^{th}$ year assets costing Rs. 2,50,000. Then the value of the amount to be retained out of profits every year if the interest rate is 5% is : [Given $(1.05)^{20} = 2.6532$]

The effective rate that is equivalent to a nominal rate of 16% compounded semi-annually is :

The digit in the unit's place of $13^{37}$ is :

If Paasche's index number is 160 and Laspeyre's index number is 250, then Fisher's index number is :

A shopkeeper wants to check the average number of cars sold per call. Past record of sales is shown below: | Sale of cars (Units) | 0 | 1 | 2 | 3 | |---|---|---|---|---| | Probabilities | $\frac{1}{6}$ | $\frac{1}{2}$ | $\frac{3}{10}$ | $\frac{1}{30}$ | The expected number of cars sold is :

The price relatives and weights of a set of commodities are given as: | Commodity | A | B | C | |---|---|---|---| | Price Relative | 150 | 130 | 180 | | Weight | $x$ | $3x$ | $y$ | If the sum of weights is 30 and the index for the set is 144, then the values of $x$ and $y$ are:

A telephone exchange receives on an average 5 calls per minute. The probability of receiving 3 or less calls per minute is :

If the life expectancy of an asset is 10 years, scrap value Rs.5,000 and annual depreciation of Rs.3000 per annum, then original cost of the asset is:

Which constraints correctly represent the situation 'mixture of x and y must be at least 8 units' ?

A container contains 50 litres of milk. From this container 10 litres of milk was taken out and replaced by water. This process is repeated two more times. How much milk is now left in the container ?

Consider the following hypothesis test : $H_0 : \mu \leq 20$ $H_1 : \mu > 20$ A sample of 81 produced a sample mean of 20.55. The population standard deviation is 3. The value of the test statistic is :

If $f(x) = a \log x + \frac{b}{x} + x$ has its extreme values at $x = -1$ and $x = 3$, then $(a, b)$ is equal to:

The set of positive integers less than 50 forming the equivalence class of 6 modulo 9 is given by :

Rahul can run 34.4 m in the given time as Amit runs 50 m. By how much distance Rahul is away from Amit at the winning point, in a two km race ?

Match List I with List II | LIST I | LIST II | |---|---| | A. The variance of a Poisson distribution with mean $\lambda$ is | I. $\sqrt{\lambda}$ | | B. The standard deviation of a Poisson distribution with mean $\lambda$ is | II. 4 | | C. In a Poisson distribution, if mean is 4, then the standard deviation is | III. $\lambda$ | | D. In a Poisson distribution, if mean is 4, then the variance is | IV. 2 | Choose the correct answer from the options given below:

Match List I with List II | LIST I | LIST II | |---|---| | A. A special characteristic of a population is known as a: | I. statistic | | B. A special characteristic of a sample is known as a: | II. Confidence interval | | C. The uncertainty of a sampling process is expressed by: | III. Estimation | | D. The process by which one makes the inferences about a population based on the information obtained from a sample is known as: | IV. Parameter | Choose the correct answer from the options given below:

If the probability distribution of a discrete random variable $X$ is given as | X | 0 | 1 | 2 | 3 | 4 | |---|---|---|---|---|---| | P(X) | 0.5 | 2k | 3k | 3k | 2k | Then the value of $k$ is:

Match List I with List II | LIST I | LIST II | |---|---| | A. A matrix that has unequal number of rows and columns is called | I. Non-singular matrix | | B. A matrix whose determinant is non-zero is called | II. Null matrix | | C. A diagonal matrix whose diagonal elements are equal is called | III. Rectangular matrix | | D. A matrix that is both symmetric and skew-symmetric is | IV. Scalar matrix | Choose the correct answer from the options given below:

A, B and C entered in a partnership business. They invested their capitals in the ratio of $\frac{4}{3} : \frac{5}{2} : \frac{6}{5}$. After 5 months, B invested 40% more than what he had invested earlier. If the total profit at the end of one year was Rs 50,550, then how much profit did A earn?

If the cost function $C(x)$ of producing $x$ units of a commodity is given as $C(x) = x^3 - 60x^2 + 13x + 50$, then the level of output for which the marginal cost is minimum is

The minimum value of $Z = 30x + 10y$ subject to the constraints $x + 2y \leq 30, 3x + y \geq 30, 4x + 3y \geq 60, x, y \geq 0$ is

If the matrix $\begin{bmatrix} a & -2 & 5b \\ 2 & 0 & -15 \\ 15 & 3c & 0 \end{bmatrix}$ is skew-symmetric, then the value of $a^2 + b^2 + c^2$ is:

From a population having a mean of 20 and standard deviation 2, a random sample of size 64 is taken and its mean is found to be 19.5. The test statistic to test that the sample is taken from the population is

If a random variable X follows binomial distribution with mean 5 and variance $\frac{5}{2}$, then $P(X \leq 9)$ is :

Let $f : R \to R$ be defined such that $f(x) = 16x^2 - 16x + 12$ (A) Maximum value of $f(x)$ is 8 (B) Minimum value of $f(x)$ is 8 (C) Minimum value of $f(x)$ is 16 (D) No maximum value of $f(x)$ Choose the correct answer from the options given below :

Area of the region bounded by the curve $x^2 = 4y$, $x$ - axis and $x = 3$ is

Let $P = \begin{bmatrix} 5 & 2 \\ 7 & 4 \end{bmatrix}$, $Q = \begin{bmatrix} 2 & 5 \\ 3 & 8 \end{bmatrix}$, $R = \begin{bmatrix} 2 & -1 \\ 3 & 4 \end{bmatrix}$, then the matrix S such that QS - RP = 0 will be :

The objective function $z = 4x + 3y$ can be maximised subject to the constraints $3x + 4y \leq 24$, $8x + 6y \leq 48$, $x \leq 5$, $y \leq 6$, $x \geq 0$, $y \geq 0$ :

If $A' = \begin{bmatrix} -2 & 3 \\ 1 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} -1 & 0 \\ 2 & 3 \end{bmatrix}$ then $(3A + 2B)'$ is :

Consider the following data : | Year | 2012 | 2013 | 2014 | 2015 | 2016 | |------|------|------|------|------|------| | Sales (in ₹ crores) | 8 | 10 | 7 | 9 | 12 | The equation of the straight line trend by the method of least squares is :

If objective function $Z = 20x + 30y$ of an LPP is subject to the constraints $3x + 4y \geq 12$, $4x + y \geq 4$, $x \geq 0, y \geq 0$, then Z has : (A) Min at (0, 4) (B) Max at (0, 4) (C) Min at (4, 0) (D) Max at (4, 0) (E) Min at $\left(\frac{4}{13}, \frac{36}{13}\right)$ Choose the **correct** answer from the options given below :

The present value of a perpetual income of ₹ $x$ payable at the end of each 6 months is ₹ 1,80,000. If the money is worth 5% compounded semi-annually, then the value of $x$ is ₹ :

T- test : A t- test is a test of difference for parameter data $t = \frac{\overline{x1} - \overline{x2}}{s\sqrt{\frac{1}{n1} + \frac{1}{n2}}}$ Then read the following statements and choose the correct statements (A) the null hypothesis and the alternative hypothesis have the same viewpoint (B) In t- test testing the significance of mean value is done, when sample size is small (C) T - test for two independent groups when variance is equal (D) Testing is a process used by statisticians to accept or reject the hypothesis (E) if the value of test statistics is greater than the table values, we do not reject the null hypothesis Choose the correct answer from the options given below :

If $y = \log_e \left(\frac{x^3}{e^3}\right)$, then $\frac{d^2 y}{d x^2}$ is equal to

A discrete random variable X takes the values 0, 1, 2, 3, 4 and its mean is 1.6. If $P(X=1) = 0.4$, $P(X=4) = P(X=2)$ and $P(X=3) = 2P(X=2)$, then $P(X=0)$ is :

Match List - I with List - II. | List - I | List - II | |----------|-----------| | (A) In a binomial distribution, if $n = 10$, $q = 0.25$, then its mean is | (I) 12 | | (B) If the mean of a binomial distribution is 6 and its variance is 3, then p is | (II) 7.5 | | (C) In a binomial distribution, the probability of getting a success is $\frac{1}{4}$ and the standard distribution is 3, then its mean is | (III) 16 | | (D) If the mean and variance of a binomial distribution are 4 and 3 respectively, then the number of trials is | (IV) $\frac{1}{2}$ | Choose the correct answer from the options given below :

A simple random sample consists of four observations 7, 8, 10, 7. The point estimate of population standard deviation is :

The set of all positive integers less than 50 forming the equivalence class of 8 for modulo 11 is :

If Mr. Ravi borrows a sum of ₹ 1,50,000 at an interest rate of 10% (flat) for a tenure of 3 years, then his EMI based on above data is (approximately) ₹ :

The demand function of a monopolist is given by $p = 1500 - 2x - x^2$, then value of marginal revenue when $x = 20$ is :

$5^{100} (\mod 9) =$

The present value of a perpetuity of Rs. 2,500 payable at the end of each year, if money is worth 10% compounded annually, is :

Match List - I with List - II. Given that $\Sigma p_0 q_0 = 150$, $\Sigma p_0 q_1 = 80$, $\Sigma p_1 q_0 = 240$, $\Sigma p_1 q_1 = 200$. | List - I | List - II | |---|---| | (A) Laspeyre's index | (I) 160 | | (B) Paasche's index | (II) 200 | | (C) Fesher's index | (III) 205 | | (D) Dorbish and Bowley's | (IV) 250 | Choose the correct answer from the options given below :

If $C(x) = ax^2 - bx - c$ represents the total cost function then the slope of the tangent to the marginal cost curve at the point $(x, y)$ is :

The EMI (in Rs.) under the flat rate on a loan of Rs. 6,00,000 with 20% annual interest for 5 years is :

Two inlet pipes can fill a tank in 20 minutes and 24 minutes respectively . An outlet pipe can empty 30 liters of water per minute. If all three pipes working together can fill the tank in 15 minutes. The capacity of the tank is :

Match List - I with List - II. | List - I (Functions) | List - II (Maximum value) | |---|---| | (A) $f(x) = -x^2, x \in (-\infty, \infty)$ | (I) 8 | | (B) $f(x) = -x^2 + 1, x \in (-\infty, \infty)$ | (II) 7 | | (C) $f(x) = x + 1, x \in [0, 6]$ | (III) 1 | | (D) $f(x) = x^3, x \in [0, 2]$ | (IV) 0 | Choose the correct answer from the options given below :

The effective rate that is equivalent to a nominal rate of 12% compounded quarterly is :

If $x^3 + y^3 = xy$, then $\frac{dy}{dx}$ is equal to :

A shopkeeper has 10 litres of pure honey. He sells honey at the cost price of Rs. 300 per litre. After mixing some quantity of water in pure honey he sells the syrup of pure honey and water at Rs. 250 per litre. The quantity of water mixed in pure honey is

If $-\frac{1}{3x - 5} \leq 0$, then :

Mr. Ram took a loan of Rs. 4,00,000 at 10% annual interest rate and paid Rs. 20,000 as monthly instalment under flat rate system. What is the term of the loan ?

In binomial distribution with $n = 10$ and $P = \frac{1}{3}$, the probability of the event that unequal number of failures and successes occur is :

The standard deviation of a sampling distribution of a statistic is also known as :

If $t = e^{2x}$ and $\log_e t^2$, then $\frac{d^2y}{dx^2}$ is :

If X has a Poisson distribution such that $P(X = 1) = P(X = 2)$ then $P(X = 3)$ is :

If $x^{2/3} + y^{2/3} = a^{2/3}$, then $\frac{dy}{dx}$ is equal to :

Which of the following statements are correct ? (A) Index number are free from units (B) Index number represents specialised averages in percentage (C) $P_{01} = \frac{\Sigma P_0}{\Sigma P_1} \times 100$ (D) Index numbers are helpful in formulating and adopting appropriate economic policies Choose the **correct** answer from the options given below :

If A is square matrix of order 3 with $|A| = 3$, then $|4 \text{ adj} A|$ is equal to :

Match **List - I** with **List - II**. | | List - I (Matrix) | | List - II (Type) | |---|---|---|---| | (A) | $\begin{bmatrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{bmatrix}$ | (I) | Lower Triangular Matrix | | (B) | $\begin{bmatrix} 1 & 5 & 0 & -1 \end{bmatrix}$ | (II) | Row Matrix | | (C) | $\begin{bmatrix} 3 & 0 & 0 \\ 1 & -1 & 0 \\ 2 & 5 & 4 \end{bmatrix}$ | (III) | Diagonal Matrix | | (D) | $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 4 \end{bmatrix}$ | (IV) | Scalar Matrix | Choose the **correct** answer from the options given below :

For a given data of 5 observation $\sum y = 311$, $\sum x^2 = 10$ and $\sum xy = 90$. The equation of the trend line is :

Match **List - I** with **List - II**. | | List - I | | List - II | |---|---|---|---| | (A) | For break-even point | (I) | $< 0$ | | (B) | For maxima $\frac{d^2y}{dx^2}$ | (II) | $\frac{dy}{dx} = 0$ | | (C) | For points of maxima/minima | (III) | $R(x) - C(x)$ | | (D) | $P(x) = $ Profit function | (IV) | $R(x) = C(x)$ | Choose the **correct** answer from the options given below :

A simple random sample consists of five observations 2, 4, 7, 12, 15. The point estimate of the population mean is :

The random variable X has a probability distribution P(X) of the following form where k is a scalar and $P(X = x) = \begin{cases} k, & \text{if } x = 0 \\ 2k, & \text{if } x = 1 \\ 3k, & \text{if } x = 2 \\ 0, & \text{otherwise} \end{cases}$ then value of P(X < 2) = _______.

If the cash equivalent of a perpetuity of Rs.1200 payable at end of each half year is Rs.96,000, the annual rate of interest compounded half yearly is :

Mr. X takes a personal loan of Rs.10,00,000 at the rate of 12% per annum for 6 years. Calculate his EMI by using flat rate method

The feasible region for an LPP is shown in the figure given below: If objective is maximizing $Z = 22x + 18y$ find $(x, y)$ for the optimal $Z$.

A runs 5 times faster than B. If A gives B a start of 100 m, how far must the goal be so that A and B reach there at the same time.

The rise in prices before Christmas is an example of

Three partners Ramesh, Suresh and Mahesh shared the profit in a business in the ratio 5:4:7 respectively, they invested the money for 10 months, 8 months and 6 months respectively then ratio of their investment is :

A simple random sample consists of three observations 1, 3, 5. The point estimate of the population standard deviation is :

If A and B are events such that $P(A/B) = P(B/A)$, then :

If $(x+ 1) e^y = 1$ , then :

If $\begin{vmatrix} 2 & 4 \\ 5 & 1 \end{vmatrix} = \begin{vmatrix} 2x & 4 \\ 6 & x \end{vmatrix}$, then x is equal to :

Match List - I with list- II | List-I Equation of curves | List - II Slope of tangent at x = 2 | |---|---| | A. $Y = x^3 - x$ | 8 | | B. $Y = (x-2)^2$ | 2/3 | | C. $Y = 2x^2 + 3$ | 11 | | D. $Y = \sqrt{4x + 1} - 7$ | 0 | Choose the correct option below :

The feasible reason for the constraints $x \geq 0$, $x + y \leq 1$ and $x - y \leq 1$, is situated in : (A) I and II quadrant only (B) I Quadrant (C) II and IV Quadrant (D) IV Quadrant (E) I , II , III and IV Quadrant Choose the correct answer from the option given below :

Value of Z equals to 40x + 50y subject to constraints $3x + y \leq 9$, $x + 2y \leq 8$, $x, y \geq 0$ occurs at

Evaluate $3^{15} \mod(7)$

Mr. Jain takes a personal loan of rupees 10,00,000 at 12% rate of interest per annum for three years. His EMI by flat rate method is :

If the mean of a binomial distribution is 24 and its standard deviation is 4 , then the probability of getting success is :

The minimum value of $f(x) = 4x^3 - 48x + 105$ in the interval [1,3] is :

Consider the following hypothesis test : $H_0 : \mu \leq 24$, $H_a : \mu > 24$ A sample of 64 provided a sample mean of 24.3 . The population standard deviation is 2. The value of the test statistic is :

For testing the difference between the means of two samples, the following data is available : | | Size | Mean | Variance | |---|---|---|---| | Sample 1 | 5 | 40 | 101 | | Sample 2 | 7 | 30 | 60 | The value of the t-statistics is :

The quantity of water that must be added to 36 litres of pure milk at $1\frac{1}{2}$ litres for Rs. 75 so as to have mixture worth Rs. 40 a litre is :

If $x = \log t$ and $y = \frac{1}{t^2}$, then $\frac{d^2 y}{d x^2}$ is equal to

Corner points of the feasible region for an LPP are : (0, 2), (3, 0), (6, 0), (6, 8) and (0, 5). Let $z = 4x + 6y$ be the objective function. Then, Max z $-$ Min z is equal to :

The minimum value of $z = 3x + 6y$ subject to the constraints $2x + 3y \leq 180$, $x + y \geq 60$, $x \geq 3y$, $x \geq 0$, $y \geq 0$ is :

Which of the following is not true

Match **List - I** with **List - II**. | | List - I | | List - II | |---|---|---|---| | (A) | Left tailed test | (I) | $H_0 : \mu = 50$ | | (B) | Two tailed test | (II) | $H_1 : \mu > 50$ | | (C) | Null hypothesis | (III) | $H_1 : \mu < 50$ | | (D) | Right tailed test | (IV) | $H_1 : \mu \neq 50$ | Choose the **correct** answer from the options given below :

If $x\%$ of $y + y\%$ of $30 = 61\%$ of $xy$, then $x =$

Between 3 p.m. and 5 p.m. the average number of phone calls per minute coming into the helpline desk of a bank is 5. The probability that during one particular minute there will be only one phone call is :

Ram and Ankur make a partnership. Ram invests Rs. 35,000 for 6 months and Ankur invests some money for 7 months. Ankur claims $\frac{4}{7}$ of total profit. Then the money invested by Ankur is :

In the equation of trend line $y_t = a + bx$, a and b represent :

Jeep A and Jeep b are competing in a motor race. After starting together, Jeep B covers the target of 30 kilometer in 30 minutes 4 seconds. Jeep A covers the target in 30 minutes one second. By what distance will Jeep A beat Jeep B ?

A retailer has 250 kg of rice a part of which he sells at 10% profit. The remaining quantity of rice is of low quality and he sold it at 5% loss. Overall he made a profit of 7%. The quantity of rice sold at 5% loss is :

The value of a for which the function $f(x) = a^x$ is increasing on R are given by :

The effective rate equivalent to a nominal rate of 8% per annum compounded semi annually is :

Two positive numbers $x$ and $y$ such that $x + y = 60$ and $xy^3$ is maximum are :

The present value of a perpetuity of Rs 1,200 payable at the beginning of each year, if the money is worth 5% effective, is:

Let $f : R \to R$ be a function defined as $f(x) = 2x^3 - 21x^2 + 36x - 20$, then : (A) maximum value of $f(x)$ is $-3$ (B) minimum value of $f(x)$ is $-128$ (C) maximum value exists at $x = 6$ (D) minimum value exists at $x = 1$ Choose the **correct** answer from the options given below :

Which of the following is false about the central limit theorem

If $x = 2t^2 + 3, y = 3t^2 + 6t + 5$, then the value of $\frac{d^2y}{dx^2}$ is :

A swimmer's speed in swimming pool is 6 km/hr. He swims between two points in a river and returns back to the starting point. He took 24 minutes more to cover the distance upstream than downstream. If the speed of the stream is 4 km/hr, then the distance between two points is:

A discrete random variable X has the following probability distribution : | X: | 0 | 1 | 2 | 3 | 4 | 5 | |----|----|----|----|----|----|----| | P(X): | b | 3b | 5b | 3b | 4b | 6b | The value of b is :

The value of the integral $I = \int_{-1}^{1} (x + x^3 + x^5) dx$ is :

A motor boat can row at the speed of 12 kilometres per hour in Still water . if the river is flowing at 4 kilometer per hour and it takes 12 hours for a round trip , then the distance between the two places is

The minimum value of $ax + by$, where $xy = c^2$ and a, b, c are positive, is :

Pipe A can independently empty a half-filled tank in 3 hours whereas pipe B can alone fill the same tank in 4 hours, when the tank is completely empty. In how much time will the empty tank be completely filled, if both the pipes are opened together?

If a square matrix B satisfies $B^2 = I - B$ and $B^n = 5I - 8B$, then the value of n is :

The investment ratio of P, Q and R if their profit ratio is 3 : 5 : 7 respectively and their investment time period ratio is 4 : 5 : 6 respectively is :

Match **List - I** with **List - II**. | | List - I | | List - II | |---|---|---|---| | (A) | If $y = x^2 - 8$ and $\frac{dy}{dx} = 0$, then $x = ?$ | (I) | 1 | | (B) | If $p(x) = 3x + 1$, then $R(x)$ at $x = 2$ | (II) | 0 | | (C) | If $y = x^3$, then $\frac{dy}{dx}$ at $x = -1$ | (III) | 14 | | (D) | If $C(x) = 100 + 5x$, $R(x) = 102 + 3x$, then break-even point | (IV) | 3 | Choose the **correct** answer from the options given below :

A motor boat goes 20 km downstream and comes back to the starting point in 6 hours. If the speed of the boat in still water is 12 km/h, then the speed of the stream is :

Any function f(x) is an increasing function in [a,b] if : (A) $x_1, x_2 \in [a, b], f(x_1) \geq f(x_2)$ if $x_1 < x_2$ (B) $x_1, x_2 \in [a, b], f(x_1) \geq f(x_2)$ if $x_1 > x_2$ (C) (A) $x_1, x_2 \in [a, b], f(x_1) \leq f(x_2)$ if $x_1 < x_2$ (D) (A) $x_1, x_2 \in [a, b], f(x_1) < f(x_2)$ if $x_1 > x_2$

If $y = \log\left(\frac{x^5}{e^5}\right)$, then $\frac{d^2y}{dx^2}$ is,

Match List I with List II | LIST I | LIST II | |---|---| | A. A solution that does not satisfy all the constraints is called | I. Linear | | B. The objective function in an LPP is | II. Convex polygon | | C. Linear inequalities or equations on the variables of LPP are called | III. Infeasible solution | | D. The feasible region in an LPP, formed by the convex combinations of the corner points, is called | IV. Constraints | Choose the correct answer from the options given below:

The maximum value of $Z = 3x + y$ subject to the constraints $x + y \leq 30, 2x + y \leq 40, x, y \geq 0$ is

In an LPP if the objective function $z = ax + by$ has same maximum value on two corner points of the feasible region, then the number of points at which maximum value of $z$ occurs is :

If the probability distribution of X is : | X | 2 | 3 | 4 | 5 | 6 | |---|---|---|---|---|---| | P(X) | 1/15 | 2/15 | 3/15 | 4/15 | 5/15 | Then variance is equal to :

A person buys a flat for which he makes down payment of ₹ 7,50,000 and the balance is to be paid in 10 years by monthly instalments of ₹ 22,000 each. If the bank charges interest at the rate of 12% per annum, then the actual price of the flat using flat rate system is :

Which of the following statements are true ? (A) Central limit theorem states that the sampling distribution of the mean $(\bar{x})$ approaches a normal distribution as the sample size increases. (B) As per Central Limit Theorem, when the sample size increases, the mean $(\bar{x})$ for the data becomes closer to the mean of overall population. (C) The shape of t-distribution does not depend on degree of freedom. Choose the correct answer from the options given below :

The longest side of a triangle is 4 times the shortest side and the third side is 3 cm shorter than the longest side. If the perimeter of the triangle is at least 69 cm, then the minimum length of the shortest side is:

A book consisting of 2000 pages has 540 misprints distributed randomly throughout the book. The average number of misprints in one page of the book is:

The total cost of producing $x$ generators is given by TC = $x^3 - 60x^2 + 1500x + 2000$. The Marginal Cost (MC), when $x = 10$ units is:

Consider the following hypothesis test : $\mu_o : \mu \leq 26$ $\mu_a : \mu > 26$ A sample of 36 is provided with a sample mean of 25.75. The population standard deviation is 3. The value of the test statistic is :

Using simple average of relatives method, the price index for 2011, taking 2001 as base year, was found to be 127. If $\Sigma p_0 = 263$, then x and y from the following data are : | Commodities | A | B | C | D | E | F | |---|---|---|---|---|---|---| | Prices (in Rs.) in 2001 | 80 | 70 | x | 20 | 18 | 25 | | Prices (in Rs.) in 2011 | 100 | 87.50 | 61 | 22 | y | 32.50 |

The purchase price P of a Rs. 50,000, 6% bond, dividends payable semi-annually, redeemable at par in 10 years, if the yield rate is to be 5% compounded semi-annually. Then P is equal : [Given $(1.025)^{-20} = 0.61027094$]

Match List - I with List - II. | List - I | List - II | |----------|-----------| | (A) A special characteristic of a population is called | (I) Sample Size | | (B) The number of statistical individuals in a sample is called | (II) Statistic | | (C) A special characteristic of a sample is called | (III) Standard error | | (D) The standard deviation of the sampling distribution of a statistic is known as its | (IV) Parameter | Choose the correct answer from the options given below :

The corner points of the feasible region determined by the system of linear constraints are (0, 10), (5, 5), (15, 15), (0, 20). Let $Z = ax + by$, where a, b > 0. Condition on a and b so that the maximizing value of Z occurs at both the points (15, 15) and (0, 20) is:

A person takes a car loan of Rs 9,00,000 at the rate of 12% per annum for 5 years from a bank. The EMI under flat rate system is:

Ravi takes a loan of Rs. 40,000 at an interest of 12% per annum for a period of 4 year, then EMI by using flat method is :

Match List - I with list- II | List-I | List - II | |---|---| | A. the probability distribution is applied for discrete random variable | normal distribution | | B. A normal distribution is symmetric about | standard deviation | | C. this probability distribution is applied for continuous random variable | mean | | D. the shape of normal curve depend upon | Poisson distribution | Choose the correct option below :

The corner points of the feasible region determined by $x + y \leq 8$, $2x + y \geq 8$, $x \geq 0$, $y \geq 0$, are A(0, 8), B(4, 0) and C(8, 0). If the objective function Z = ax + by has its maximum value on the line segment AB, then the relation between a and b is :

A vehicle costing Rs. 10,50,000 has a final scrap value of Rs. 5,25,000. If annual depreciation charge is Rs. 75,000, then useful life of the vehicle is :

If $K \equiv 5 \pmod{11}$, then all the possible non-negative values of K are :

The point on the straight line $3x + 4y = 8$, which is closest to the origin is:

A die is rolled thrice. What is the probability of getting a number greater than 4 in the first and the second throw of dice and a number less than 4 in the third throw?

The variance of the Binomial Distribution $B\left(5, \frac{1}{4}\right)$ is:

The value of the integral $\int e^x \left(\frac{1}{x} - \frac{1}{x^2}\right) dx$ is :

The solution set of the inequation $\frac{7+5x}{4} \geq \frac{x}{3} - 10$, $x \in \mathbb{R}$ is :

An asset costing Rs. 50,000 has a useful life of 4 years. The estimated scrap value is Rs. 10,000. By using linear depreciation method, the book value at the end of the second year is :

The cost of type A cement is Rs 100 per kg and that of type B cement is Rs 120 per kg. If both are mixed in the ratio of 2:3, the price of the cement mixer per kg will be

If $y = \frac{\log x}{x^2}$, then $\frac{d^2y}{dx^2}$ is equal to

Given that p$_0$ and p$_1$ represent base and current prices of a commodity, q$_0$ and q$_1$ refer to the base and current quantities, then the formula used to calculate price relatives is :

If the mean of a binomial distribution is 12 and its standard deviation is 2, then the number of trials is :

Out of 1000 employees, 100 have to be selected for a survey . After being arranged in the alphabetical order each one is assigned a number from 1 to 1000. A number 4 is selected and then every 10th person is selected (i.e 4 , 14, 24 ... ) . Which form of sampling is this an example of ?

From the data given below the Laspeyre's price index for the year 2016 with year 2010 as base year is | Commodity | Price year 2010 | Price year 2016 | Quantity Year 2010 | Quantity Year 2016 | |---|---|---|---|---| | A | 1 | 2 | 10 | 13 | | B | 5 | 10 | 12 | 16 | | C | 6 | 10 | 15 | 18 |

In the formula for Z-test statistic $Z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}}$, mark the incorrect statement.

If the objective function for an L.P.P. is $z = 3x + 4y$ and the corner points for unbounded feasible region are (9, 0), (4, 3), (2, 5) and (0, 8), then the minimum value of $z$ occurs at :

A dishonest delivery man purchases 20 liters of pure milk and dilutes it by adding 4 liters of water. If he sells the diluted milk at Rs.50/L, then what is the cost price of pure milk?

If $x = 3at^2$, $y = 3at^4$ then $\frac{dy}{dx}$ is :

A monopolist's Demand function is $x = 70 - \frac{P}{2}$, the revenue at $x = 5$ will be :

A carpenter earns a profit of ₹ 50 and ₹ 80 on one chair and one table respectively. The requirement and availability of wood and labour are tabled as : | Required | Chair | Table | Available quantity | |----------|-------|-------|--------------------| | Wood | 3 | 5 | 150 | | Labour | 1 | 2 | 56 | The number of chairs and tables in appropriate units to be manufactured for maximum profit are, respectively :

If the cost function and the profit function for a company is given by $C = 10 - 0.3x^2$ and $P = 0.3x^2 + 2x - 10$ respectively , where X represent units of output, then the revenue function is given by :

The wholesale price index (or price relative) of rice in 2018 compared to 2014 is 150. If the cost of rice was Rs. 16 per kg in 2014, then per kg cost of rice in 2018 is :

The prices and the quantities of three commodities are given are : | Commodity | Price (₹) in Year 2006 | Price (₹) in Year 2009 | Quantities in Year 2006 | Quantities in Year 2009 | |-----------|------------------------|------------------------|--------------------------|--------------------------| | P | 100 | 90 | 12 | 10 | | Q | 80 | $x$ | 8 | 7 | | R | 60 | 50 | 4 | 6 | The Laspeyre's price index number for year 2009 with year 2006 as base is 200. The value of $x$ is :

The probability distribution of a random variable X is given below : | X | 0 | 1 | 2 | 3 | 4 | |---|---|---|---|---|---| | P(X) | 0.1 | 0.25 | 0.3 | 0.2 | 0.15 | Then, $\text{Var}\left(\frac{X}{2}\right)$ is :

A consumer in 2015, paid Rs 20 per kg for a particular variety of rice. The wholesale price index number for this variety of rice for the year 2018, with the year 2015 as the base year is 125. Then the cost per kg of rice in the year 2018 will be:

If $y = \log_3(\log_3 x)$, then $\frac{dy}{dx}$

If $x = at^2$ and $y = a^3 t^3$, then $\frac{d^2 y}{dx^2}$ will be :

The Solution set of the inequality three $3x + 4y \leq 12$ is :

If $57 \equiv x (\bmod 5)$, then the least positive value of $x$ is:

Mr. X invested Rs.20,000 in year 2022 for 5 years. If Compound annual growth rate for that investment turned out to be 12%, the end value of the investment will be : {Use $(1.12)^5 = 1.76$}

Which of the following is not a statistic ?

If the probability distribution of a random variable X is given as | $x_i$ | 0 | 1 | 2 | 3 | |---|---|---|---|---| | $p_i$ | $2k^2$ | $k^2$ | $3k^2$ | $k$ | Then the mean of X is

For the data : | Variable | Price - Base year | Price - Current year | Weights | |---|---|---|---| | X | 50 | 60 | 5 | | Y | 20 | 25 | 7 | | Z | 30 | 40 | 4 | The weighted aggregative index number is :

If the objective function $Z = px + qy$ ($p, q > 0$) of an LPP has minimum value 7p, at the corner points (2, 3) and (7, 0), then

Let us consider an annuity whose periodic payment is Rs. R payable at the end of each payment period for 'n' periods, interest paid r% per period or $i = \frac{r}{100}$, so the amount of obligation can be given as _______.

The price index of sugar in 2019 compared to 2016 is 120. If the cost of sugar was Rs.25 per kg in 2016 then the cost in 2019 is:

The assumption opposite of what is made in the null hypothesis is known as the _________ hypothesis.

Match List I with List II | LIST I | LIST II | |---|---| | A. The solution of $3x + 7 > 12$ is | I. $[-1, \infty)$ | | B. The solution of $\frac{3x+5}{2} \geq 1$ is | II. $\left[\frac{17}{8}, \infty\right)$ | | C. The solution of $2x + 5 < 7x + 9$ is | III. $\left(\frac{5}{3}, \infty\right)$ | | D. The solution of $6x - 5 \geq -2x + 12$ is | IV. $\left(-\frac{4}{5}, \infty\right)$ | Choose the correct answer from the options given below: