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All questions of Engineering Mathematics for Computer Science Engineering (CSE) Exam

In an examination, 20% of the students have failed in Computer Networks, 30% of the students have failed in Computer Architecture and 10% in Computer Networks and Computer Architecture. If a student is selected at random, then what is the probability that the student has failed in at least one subject?
  • a)
    1/5
  • b)
    2/5
  • c)
    3/5
  • d)
    4/5
Correct answer is option 'B'. Can you explain this answer?

Mainak Kulkarni answered
Let the number of students be 100.
P(N)→ probability that students failed in Computer Networks

P(A)→ probability that students failed in Computer Architecture

P(N∩A)→ probability that the student has failed in both the subject

 
P(NUA)→ probability that the  student has failed in atleast one subject
P(NUA) = P(N) + P(A) − P(N∩A) 
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Matrix for which LU decomposition is not possible?
  • a)
  • b)
  • c)
  • d)
Correct answer is option 'D'. Can you explain this answer?

Rohit Mukherjee answered
An invertible matrix A has an LU decomposition provided that all it's leading submatrices have non-zero determinants.
All the matrices in the options are invertible because 

In option 4
 in this leading sub matrix A = [0]
Hence LU decomposition is not possible

As xx varies from −1 to +3, which one of the following describes the behaviour of the function f(x) = x3–3x2 + 1?
  • a)
    f(x)f(x) increases monotonically.
  • b)
    f(x)f(x) increases, then decreases and increases again.
  • c)
    f(x)f(x) decreases, then increases and decreases again.
  • d)
    f(x)f(x) increases and then decreases.
Correct answer is option 'B'. Can you explain this answer?

Athira Reddy answered
Person to person, as it depends on their individual experiences, perspectives, and attitudes. Some individuals may view success as achieving financial stability and career advancement, while others may define success as having strong personal relationships and a sense of fulfillment in their daily life. Additionally, cultural and societal influences can also shape one's definition of success. Ultimately, success is a subjective concept that is unique to each individual.

An anti-aircraft gun can take a maximum of 4 shots at an enemy plane moving away from it. The probabilities of hitting the plane at the first, second, third and fourth shot are 0.4, 0.3, 0.2 and 0.1 respectively. The probability that the gun hits the plane is_____
    Correct answer is between '0.69,0.7'. Can you explain this answer?

    Arka Bajaj answered
    Let p1 = 0.4, p2 = 0.3, p3 = 0.2 and p4 = 0.1
    P(the gun hits the plane) = P(the plane is hit at least once)
    = 1 – P(the plane is hit in none of the shots)
    = 1− (1−p1) (1−p2) (1−p3) (1−p4)
    = 1− (0.6 × 0.7 × 0.8 × 0.9)
    = (1− 0.3024) = 0.6976

    The system of equation, 5x + 2y + z = 3,7x + 10y + 2z = 7,2x + 8y + z = 4 has
    • a)
      One Solution
    • b)
      Two Solutions
    • c)
      No Solution
    • d)
      Infinite Solutions
    Correct answer is option 'D'. Can you explain this answer?

    Sandeep Majumdar answered
    The given system of equations are of the form AX = Bwhich represents the system of  non homogeneous equations.
    Augmented matrix is given by

    by applying row transformation R2 → R2 − (R1+R3), we get
    so, rank of matrix = rank of augmented matrix = 2 < number of variables
    ∴ System has infinite solutions
    Alternatively we can simply observe that second equation is redundant as it can be obtained from rest two. Hence we have 2 primary equations and 3 variables hence infinite solution.

     The probability that GATE CS/IT question has an error is 1/20 and 65 questions are made in such an exam. If the probability that at least 2 question has an error is [1−[a(19/20)64]] then what is the value of a (answer up to 2 decimal place)?
      Correct answer is '4.2'. Can you explain this answer?

      Megha Dasgupta answered
      Understanding the Problem
      The problem is about calculating the probability of errors in questions on a GATE CS/IT exam. Given that the probability of a question having an error is 1/20 (or 0.05), we can analyze the situation using the complementary probability.
      Calculating the Complementary Probability
      - The probability that a question does not have an error is 1 - (1/20) = 19/20.
      - For 65 questions, the probability that none of them have errors is (19/20)^65.
      - The probability that at least 2 questions have errors can be calculated as:
      P(at least 2 errors) = 1 - P(0 errors) - P(1 error)
      Calculating the Individual Probabilities
      1. No Errors:
      - P(0 errors) = (19/20)^65
      2. Exactly One Error:
      - For exactly one error, we choose 1 question to have an error and the rest not:
      - P(1 error) = C(65, 1) * (1/20)^1 * (19/20)^64
      - C(65, 1) = 65 (the number of ways to choose 1 question from 65)
      Therefore, P(1 error) = 65 * (1/20) * (19/20)^64.
      Final Calculation
      Combining these probabilities:
      P(at least 2 errors) = 1 - [(19/20)^65 + 65 * (1/20) * (19/20)^64].
      This can be factored as:
      P(at least 2 errors) = 1 - [a * (19/20)^64], where a = (19/20) + 65 * (1/20).
      Calculating a:
      - a = (19/20) + 65 * (1/20) = (19 + 65) / 20 = 84 / 20 = 4.2.
      Conclusion
      Thus, the value of "a" is 4.2, which matches the given correct answer.

      The simultaneous equations
      2x + ay + z = 20 
      x + 3y + 4z = b
      x + 2y + 3z = c
      has unique solution then what is the value of a, b and c respectively?
      • a)
        a ≠ ± 1, b and c can be any value
      • b)
        a ≠ ± 1, b ≠ 0, c ≠ 1
      • c)
        a ≠ -1, b and c can be any value
      • d)
        a, b and c can be any value
      Correct answer is option 'C'. Can you explain this answer?

      Kalyan Menon answered
      For given system of equations to have unique solution, the determinant of the coefficient matrix should be not equal to 0

       
      2 (9−8) − a(3−4) + (2−3) ≠ 0
      2 + a − 1 ≠ 0
      a ≠ −1
       
      Since Δ of coefficient matrix is not equal to 0 then b and c can take any value
      Note:
      If coefficient matrix’s Δ ≠ 0 then rank of coefficient matrix is always equal rank of augmented matrix so right-hand side of given equation can take any value

      From an urn containing 3 red and 2 white balls, a man is to draw 2 balls at random without replacement, being promised Rs. 20 for each red ball he draws and Rs. 10 for each white one. Find his expectation (In rupees).
        Correct answer is '32'. Can you explain this answer?

        Bijoy Sharma answered
        A man is to draw 2 balls at random without replacement.
        Probability to draw 2 red balls 
        Probability to draw 2 white balls 
        Probability to draw one red ball and one white ball = 
        Let X be the random variable and it shows the money he earn.

        Expectation =  

        A sample of 15 data is as follows: 17, 18, 17, 17, 13, 18, 5, 5, 6, 7, 8, 9, 20, 17, 3. the mode of the data is
        • a)
          4
        • b)
          13
        • c)
          17
        • d)
          20
        Correct answer is option 'C'. Can you explain this answer?

        Harshitha Sarkar answered
        Understanding Mode in Data Sets
        The mode of a data set is defined as the value that appears most frequently. To determine the mode, we need to count the occurrences of each number in the given sample.

        Data Sample
        The provided data sample is as follows:
        - 17
        - 18
        - 17
        - 17
        - 13
        - 18
        - 5
        - 5
        - 6
        - 7
        - 8
        - 9
        - 20
        - 17
        - 3

        Counting Frequencies
        Let's tally the occurrences of each number:
        - **3**: 1 time
        - **5**: 2 times
        - **6**: 1 time
        - **7**: 1 time
        - **8**: 1 time
        - **9**: 1 time
        - **13**: 1 time
        - **17**: 4 times
        - **18**: 2 times
        - **20**: 1 time

        Identifying the Mode
        From the frequency count, we can see that:
        - **17** appears 4 times, which is more than any other number.
        - The next highest frequencies are for **5** and **18**, which appear 2 times each.

        Conclusion
        Since **17** has the highest frequency, it is the mode of this data set. Therefore, the correct answer is:
        - **Option C: 17**
        This indicates that in this particular data sample, 17 is the most common value.

        A die is tossed thrice. Success is getting 2 or 5 on a toss. What is the variance of the number of success?
        • a)
          1/3
        • b)
          2/3
        • c)
          4/5
        • d)
          35
        Correct answer is option 'B'. Can you explain this answer?

        Meghana Rane answered
        Understanding the Problem
        When a die is tossed, we define success as rolling a 2 or a 5. This gives us two successful outcomes out of six possible outcomes on a die.
        Calculating Probability of Success
        - The probability of success (rolling a 2 or 5) is:
        - P(success) = 2 successful outcomes / 6 total outcomes = 1/3
        - The probability of failure (not rolling a 2 or 5) is:
        - P(failure) = 1 - P(success) = 1 - 1/3 = 2/3
        Setting Up the Experiment
        - The die is tossed three times, and we define the random variable X as the number of successes in these three tosses.
        - Since each toss is independent, X follows a binomial distribution with parameters n = 3 (number of trials) and p = 1/3 (probability of success).
        Calculating Variance
        - The formula for the variance of a binomial distribution is given by:
        - Variance = n * p * (1 - p)
        - Substituting the values:
        - Variance = 3 * (1/3) * (2/3)
        - Variance = 3 * (1/3) * (2/3) = 2/3
        Conclusion
        The variance of the number of successes when tossing a die three times, where success is defined as rolling a 2 or 5, is indeed 2/3. Thus, the correct answer is option 'B'.

        An open box is to be made out of square cardboard of 18 cm by cutting offs equal squares from the corners and turning up the sides. What is the maximum volume of the box in cm3?
        • a)
          423
        • b)
          432
        • c)
          412
        • d)
          424
        Correct answer is option 'B'. Can you explain this answer?

        Aman Menon answered
        Let x be the corner side which is cut off and v be volume.
        v = l × b × h
        v = (18 − 2x) × (18 − 2x) × x
        v = 4x3 − 72x2 + 324x
        v′ = 12x2 − 144x + 324v′
        v′′ = 24x − 144
        v′ = 12x2 − 144x + 324 = 0
        x2 − 12x + 27=0
        (x−3) (x−9) = 0
        (x=3) or (x = 9)
        atx = 3
        v′= 24x −144 = 24(3) −144 = −72<0
        at x = 3 we have volume maximum
        v = 4(3)3 − 72(2)2 + 324(2)
        ∴ v = 432

        Find the value of k for which the following system of equations will be consistent.
         2x – 5y = 10 and 6x – 15y = k
        • a)
          40
        • b)
          15
        • c)
          60
        • d)
          30
        Correct answer is option 'D'. Can you explain this answer?

        Maulik Iyer answered
        In the given system of equations, the ratio of the coefficients of x equals the ratio of coefficients of y.
        Therefore, they would be consistent only if this ratio equals the constant terms.
        That is, If 10/k = 2/6 = -5/-15
        Hence if k = 30, then the given system of equations is consistent.

        If a matrix A is given by f(x) = a0 + a1x + a2x2+…+ an−1xn−1 + anxn, then the determinant of A is
        • a)
        • b)
        • c)
        • d)
        Correct answer is option 'C'. Can you explain this answer?

        Gargi Menon answered
        matrix ‘A’ ⇒ f(x) = a0 + a1x + a2x2 +……….+an−1xn−1 + anxnf(x)
        if 2 × 2 matrix then characteristic equation  ⇒ aλ2 + bλ + c = 0
        ⇒ |A| = λ1.λ2. = c/a
        where ao = c;an = a2 = a 
        if 3 × 3 matrix → then characteristic equation ⇒ aλ3 + bλ2 + cλ + d = 0

        ∴ For n × n matrix, |A| = λ1.λ2.λ3….λn

        If  A =  then  the value of adj (adj (A))?
        • a)
        • b)
        • c)
        • d)
        Correct answer is option 'C'. Can you explain this answer?

        Tanishq Banerjee answered

        A is upper triangular matrix
        ∴ product of diagonal elements = |A|
         = 1 × 5 × 8 × 10 = 400

         
        Note:
        |A||A| is determinant of matrix A

        The chances of A, B, C becoming managers of a Testbook company are 4:2:3. The probability that the bonus scheme will be introduced if A, B, C become manager are 0.3, 0.5, 0.8 respectively.
        If the bonus scheme has introduced, what is the probability that A is appointed as the manager of Testbook (answer up to 2 decimal place)?
          Correct answer is between '0.26,0.27'. Can you explain this answer?

          Samarth Ghosh answered
          P(X)→ probability that bonus scheme is introduced
          P(A) → probability of A being a manager.

          P(B) → probability of B being a manager.

          P(C) → probability of C being a manager.
          P(X) = P (A ∩ X) + P (B ∩ X) + P (C ∩ X)
          P(X) = P(A)P(X/A) + P(B)P(X/B) + P(C)P(X/C)

          If the determinant of the matrix  is 26, then the determinant of the matrix  is
            Correct answer is '-26'. Can you explain this answer?

            Rajveer Sharma answered
            If two rows are interchanged in a determinant then the value of the determinant does not change but sign will change. In the given question, first and third rows are interchanged.

            Among the given two sets which is/are consistent:   
            S1 = {3x + ay + 4z = 0,bx + 2y + z = 0,5x + 7z + 9z = 0
            S2 = {2x + 6y= −11, 6x + 20y − 6z = −3, 6y − 18z = −1}
            • a)
              Only I
            • b)
              Only II
            • c)
              Both I and II
            • d)
              None of these
            Correct answer is option 'A'. Can you explain this answer?

            Nitya Choudhury answered
            Set S1 contains homogeneous equations, homogeneous equations are always consistent because it always satisfied a trivial solution i.e. x = y = z = 0
            For S2

            R2 → 3R2

             
            Rank Augmented matrix is 3 while rank of coefficient matrix has rank is 2
            Hence the S2 is inconsistent
            ∴ only S1 is consistent

             
            Binomial distribution B(n,p) can be approximated to a normal distribution N(np, np(1−p)) if ____
            • a)
              n is large and p and 1-p are almost equal.
            • b)
              n is small and p is almost 1.
            • c)
              n is large and p is almost 0.
            • d)
              n is small and p and 1-p are almost equal.
            Correct answer is option 'A'. Can you explain this answer?

            Jay Basu answered
            Explanation:

            Binomial Distribution vs. Normal Distribution:
            - Binomial distribution B(n,p) is used to model the number of successes in a fixed number of trials, where each trial has the same probability of success p.
            - Normal distribution N(np, np(1-p)) is a continuous probability distribution that is symmetric and bell-shaped.

            Conditions for Approximation:
            - When n is large and p and 1-p are almost equal, the binomial distribution can be approximated by a normal distribution.
            - This is because when n is large, the binomial distribution becomes more symmetric and bell-shaped, resembling a normal distribution.

            Reason for Option A:
            - Option A states that n is large and p and 1-p are almost equal, which satisfies the conditions for approximating the binomial distribution to a normal distribution.
            - In this scenario, the mean of the normal distribution is np and the standard deviation is sqrt(np(1-p)), which closely approximates the binomial distribution.

            Conclusion:
            - Therefore, option A is the correct choice as it correctly identifies the conditions under which the binomial distribution can be approximated by a normal distribution.

            If f(x) is differentiable and g’(x) ≠ 0 such that f(1) = 4, f(2) = 16, f’(x)= 8g’(x) and g(2) = 4 then what is the value of g(1) ?
            • a)
              3/2
            • b)
              5/2
            • c)
              7-2
            • d)
              9/2
            Correct answer is option 'B'. Can you explain this answer?

            Mohit Unni answered
            (x) is differentiable, then the function h(x) = f(x) + g(x) is differentiable.

            To prove this, we need to show that h(x) satisfies the definition of differentiability.

            First, we know that f(x) is differentiable, so we can write its derivative as f'(x). Similarly, g(x) is differentiable, so its derivative can be written as g'(x).

            Now, let's find the derivative of h(x) using the limit definition of the derivative:

            h'(x) = lim┬(Δx→0)⁡〖(h(x+Δx)-h(x))/Δx〗
            = lim┬(Δx→0)⁡〖((f(x+Δx) + g(x+Δx)) - (f(x) + g(x)))/Δx〗
            = lim┬(Δx→0)⁡〖(f(x+Δx) - f(x))/Δx + (g(x+Δx) - g(x))/Δx〗
            = lim┬(Δx→0)⁡〖(f(x+Δx) - f(x))/Δx〗 + lim┬(Δx→0)⁡〖(g(x+Δx) - g(x))/Δx〗
            = f'(x) + g'(x)

            Since f'(x) and g'(x) exist, the limit of the difference quotient exists and is equal to f'(x) + g'(x). Therefore, h(x) = f(x) + g(x) is differentiable.

            What is the determinant of the below-given matrix?
            • a)
              766
            • b)
              768
            • c)
              786
            • d)
              788
            Correct answer is option 'B'. Can you explain this answer?

            Madhurima Chakraborty answered
            Determinant of the given matrix is

             
            Δ = (a − b) (a − c) (a − d) (b − c) (b − d)(c − d)
            a = 3, b = 5, c = 7, d = 9
            ∴ Δ = 768

            Given f(x) = x3 + ax2 + bx + c and y-intercept for the function is 1. Also f has local extrema at x = -4 and x = 2. Then a + b + c = ____
              Correct answer is '-20'. Can you explain this answer?

              Tanishq Yadav answered
              First, let's find the equation of the function f(x) based on the given information:
              Given y-intercept is 1:
              When x = 0, f(x) = 1
              So, c = 1
              Local extrema at x = -4 and x = 2:
              Since f(x) has local extrema at x = -4 and x = 2, the derivative f'(x) will be zero at these points. This means:
              f'(-4) = 0 and f'(2) = 0
              ### Finding the derivative of f(x)
              Let's find the derivative of f(x) and set it to zero to find the values of a and b.
              f(x) = x^3 + ax^2 + bx + 1
              f'(x) = 3x^2 + 2ax + b
              ### Solving for a and b
              1. f'(-4) = 0
              3(-4)^2 + 2a(-4) + b = 0
              48 - 8a + b = 0
              -8a + b = -48
              2. f'(2) = 0
              3(2)^2 + 2a(2) + b = 0
              12 + 4a + b = 0
              4a + b = -12
              ### Solving the equations
              By solving equations -8a + b = -48 and 4a + b = -12 simultaneously, we get:
              a = -4 and b = -8
              ### Calculating a + b + c
              Now that we have a, b, and c values:
              a + b + c = -4 - 8 + 1 = -11
              Therefore, a + b + c = -11.

              The number of values of k for which the system of equations (k + 1) x + 8y = 4k, kx + (k + 3)y = 3k – 1 has infinitely many solutions, is
              • a)
                0
              • b)
                1
              • c)
                2
              • d)
                Infinite
              Correct answer is option 'B'. Can you explain this answer?

              Maulik Iyer answered
              (k + 1)x + 8y = 4k
              kx + (k + 3)y = 3k – 1
              The given system of linear equations has infinitely many solutions if ρ(A) = ρ(A|B) < n = 2
              To get the rank less than 2, one row should be dependent on another.
              nk = (k + 1), n(k + 3) = 8, n(3k – 1) = 4k
              by solving the above equations, we get
              n = 2, k = 1
              Hence for k = 1 only the system has infinitely many solutions.

              Find the value of x and y for the below given simultaneous equations that has no solutions?
              a + b + c = 18
              2a + 4b + 6c = 12
              2a + xc + 4b = y
              • a)
                x ≠ 6 and y ≠ 12
              • b)
                x = 6 and y = 12
              • c)
                x ≠ 6 and y = 12
              • d)
                x = 6 and y ≠ 12
              Correct answer is option 'D'. Can you explain this answer?

              Rajveer Sharma answered
              For given system of equations to have no solution, the determinant of the coefficient matrix should be equal to 0

              1 × (4x−24) − 1 × (2x−12) + 1(8−8) = 0
              2x − 12 = 0
              ∴ x = 6
              Augmented matrix is given by

              For given system of equations to have no solution, rank of augmented matrix should be greater than coefficient matrix
              ∴ y − 12 ≠ 0
              ∴ y ≠ 12

              What are the minimum and maximum value of the below-given function respectively?
              f(x) = 3x3 − 9x2 − 27x + 30
              • a)
                45 and 51
              • b)
                -51 and -45
              • c)
                51 ad -45
              • d)
                -51 and 45
              Correct answer is option 'D'. Can you explain this answer?

              Abhijeet Mehta answered
              f(x) = 3x3 − 9x2 − 27x + 30
              f′(x) = 9x2 − 18x − 27
              f′′(x) = 18x − 18
              Points at which maximum or minimum exists
              f′(x) = 0
              9x2 − 18x − 27 = 0
              x2 − 2x − 3 = 0
              (x − 3) (x + 1 ) = 0
              (x = 3) or (x = −1)
              Substitute the value in (x = 3) or (x = −1)
              f′′(−1) = 18(−1) − 18 = − 36 < 0
              ∴ at x = −1,maximum value exists
              f(−1) = 3(−1)3 −9(−1)2 −27(−1) + 30
              f (−1) = 45
              f′′ (3) = 18(3) − 18 = 36 > 0
              ∴ at x = 3, mimimum value exists
              f(3) = 3(3)3 − 9(3)2 − 27(3) + 30
              f (3) = −51

               
              Find C of Rolle’s theorem for f(x) = ex(sin x - cos x) in
              • a)
                0
              • b)
                3π/4
              • c)
                π
              • d)
                Does not exist
              Correct answer is option 'C'. Can you explain this answer?

              Aaditya Ghosh answered
              Given function f(x) = ex(sin x - cos x)
              f(x) is continuous in 3π/4
              f(x) is differentiable in 

               
              f(a) = f(b)
              According to Rolle’s theorem,
              there exists at least one value C ∈ (a,b)
              such that f’(c) = 0
              f’(c) = 0
              f’(x) = ex (sin x – cos x) + ex (cos x + sin x)
              f’(c) = ec (sin c – cos c) + ec (cos c + sin c) = 0
              ⇒ 2 ec sin c = 0

              Find the point at which maximum value of f(x) = 2x3−3x2 occurs in interval [-2, 2]?
              • a)
                1
              • b)
                0
              • c)
                -1
              • d)
                2
              Correct answer is option 'D'. Can you explain this answer?

              Mrinalini Menon answered
              f(x) = 2x3 − 3x2
              f′(x) = 6x2 − 6x
              f′′(x) = 12x − 6
              f′(x) = 6x2 − 6x = 0
              6x (x−1) = 0
              x = 0 or x = 1
              Now check f(x) at x = 0, x = 1 and the end points of given interval since at end point tangent cannot be drawn so, above concept cannot be used to find maximum or minimum value for end points.

              Which of the below-given statements is/are true?
              I. The eigenvalue of the lower triangular matrix is just the diagonal elements of the matrix.
              II. The product of the eigenvalue of a matrix is equal to its trace.
              III. If 1/λ is an eigenvalue of A’(inverse of A) then orthogonal of A also have 1/λ as its eigenvalue.
              • a)
                Only I
              • b)
                I and II
              • c)
                I and III
              • d)
                I, II and III
              Correct answer is option 'C'. Can you explain this answer?

              Navya Menon answered
              • The eigenvalue of the triangular (lower or upper) matrix are just the diagonal elements of matrix.
              • The product of the eigenvalue of a matrix is equal to its determinants.
              • If λ is the eigenvalue of matrix, then 1/λ is the eigenvalue of its inverse since orthogonal is equal to inverse matrix then it has 1/λ as its eigenvalue

              ​Example:
              Eigenvalues are 1, 4 and 6 (diagonal elements)
              Product of eigen value = determinants = 1 × 4 × 6 = 24
              Orthogonal matrix and Inverse of given matrix have eigenvalues: 1,1/4 and 1/6

              If a random variable X has a Poisson distribution with variance 6, then expectation [E(X + 3)2] equals _____.
                Correct answer is '87'. Can you explain this answer?

                Shubham Ghoshal answered
                Var(X) = E(X) = 6
                Var(X) = E(X2) - (E(X))2
                6 = E(X2) – 36
                ∴ E(X2) = 42
                [E(X + 3)2]
                = E(X2 + 6X + 9 )
                = E(X2) + 6E(X) + 9         
                = 42 + 6× 6 + 9
                = 87  

                In the given matrix one of the eigenvalues is 1. The eigenvectors corresponding to the eigenvalue 1 are
                • a)
                  {(4, 2, 1)| α ≠ 0, α ∈ ℝ}
                • b)
                  {(−4, 2, 1)| α ≠ 0, α ∈ℝ}
                • c)
                • d)
                Correct answer is option 'B'. Can you explain this answer?

                Palak Saini answered
                For a given matrix A if V is the eigen vector corresponding to the eigen value λ, then:
                AV = λV

                ∴ {α (−4, 2, 1) |α ≠ 0, αϵR} are the corresponding eigenvectors.

                The matrix  has one eigenvalue equal to 3. The sum of the other two eigenvalues is
                • a)
                  P
                • b)
                  P – 1
                • c)
                  P – 2
                • d)
                  P - 3
                Correct answer is option 'C'. Can you explain this answer?

                Avantika Shah answered
                Sum of the eigen values of matrix is = Sum of diagonal values present in the matrix
                ∴ 1 + 0 + P = 3 + λ2 + λ3
                ⇒ P + 1 = 3 + λ2 + λ3
                ⇒ λ2 + λ3 = P + 1 – 3 = P – 2

                What is the absolute difference of the eigenvalues for the matrix ad – bc = 6 and a + d = 7?
                • a)
                  7
                • b)
                  6
                • c)
                  5
                • d)
                  4
                Correct answer is option 'C'. Can you explain this answer?

                Nilanjan Chavan answered
                Let λ1 and λ2 be the two eigen values
                Product of eigen value is equal to determinant of matrix

                Sum of eigen value is equal to its trace
                 
                a + d = λ1 + λ2 = 7
                λ1 × (7 − λ1) = 6
                λ12 − 7λ + 6 = 0
                λ1 = 6 or λ2 = 1
                |λ1 − λ2| = 5

                3, k, 2, 8, m, 3
                The arithmetic mean of the list of numbers above is 4. If k and m are integers and k ≠ m, what is the median of the list?
                • a)
                  2
                • b)
                  2.5
                • c)
                  3
                • d)
                  3.5
                Correct answer is option 'C'. Can you explain this answer?

                Aditya Nair answered
                Mean = 4
                k ≠ m so k = m = 4 is out.
                {k, m} = {1, 7} or {2, 6} or {3, 5}
                for median of {3, k, 2, 8, m, 3}
                {1, 2, 3, 3, 7, 8} or {2, 2, 3, 3, 6, 8} or {2, 3, 3, 3, 5, 8}
                Median: (3 + 3) /2 = 3

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