Practice topic-wise Engineering Mathematics previous year questions for ISRO Mechanical & Aerospace with instant step-by-step solutions and formula breakdowns.
All ISRO Mechanical & Aerospace papers
37 questions. Answers are shown only where the official answer key is available.
Year: 2008 | Topic: Engineering Mathematics
Let X have the probability density function f(x) = (3/4)*(1-x²) if -1 <= x <= 1, and 0 otherwise. P(-1/2 <= x <= 1/2) is
Answer: A. 68.75%
Explanation: P(-1/2 <= x <= 1/2) = Integral from -1/2 to 1/2 of (3/4)*(1 - x²) dx. Solving this: (3/4) * [x - x³/3] evaluated from -1/2 to 1/2. = (3/4) * [(1/2 - 1/24) - (-1/2 + 1/24)] = (3/4) * [11/12] = 11/16 = 68.75%.
Year: 2008 | Topic: Engineering Mathematics
The determinant of the matrix [[7, 0, 0, 0], [9, 3, 0, 0], [4, -6, -2, 0], [0, 0, 0, 1]] is
Answer: C. -42
Explanation: If the matrix is an upper triangular matrix or a lower triangular matrix, then the determinant is the product of the principal diagonal elements. The given matrix is a lower triangular matrix, so its determinant value is 7 * 3 * -2 * 1 = -42.
Year: 2008 | Topic: Engineering Mathematics
Angle made by 2i - 3j + 4k with the z-axis is
Answer: A. cos^-1(4 / $\sqrt{29}$)
Explanation: Let a = 2i - 3j + 4k and the z-axis unit vector b = k. cos(theta) = (a.b) / (|a||b|) = 4 / $\sqrt{2^2 + (-3}$^2 + 4^2) = 4 / $\sqrt{29}$. Therefore, theta = cos^-1(4 / $\sqrt{29}$).
Year: 2014 | Topic: Engineering Mathematics
Let A = [[0, 0, -1], [0, -1, 0], [-1, 0, 0]]. The only correct statement about the matrix A is
Answer: B. A² = I
Explanation: Multiplying the matrix A by itself (A * A) gives [[1, 0, 0], [0, 1, 0], [0, 0, 1]], which is the identity matrix I. Therefore, A² = I is the correct statement.
Year: 2014 | Topic: Engineering Mathematics
Let M be a 3x3 matrix satisfying: M[[0], [1], [0]] = [[-1], [2], [3]], M[[1], [-1], [0]] = [[1], [1], [-1]], and M[[1], [1], [1]] = [[0], [0], [12]]. Then the sum of the diagonal entries of M is:
Answer: A. 9
Explanation: Let M be represented by column vectors [C1, C2, C3]. The first condition M[[0], [1], [0]] gives the second column C2 = [-1, 2, 3]^T, so the diagonal element b2 = 2. The second condition M[[1], [-1], [0]] means C1 - C2 = [1, 1, -1]^T, so C1 = [0, 3, 2]^T, giving the diagonal element a1 = 0. The third condition M[[1], [1], [1]] means C1 + C2 + C3 = [0, 0, 12]^T, so C3 = [1, -3, 10]^T, giving the diagonal element c3 = 10 (Wait, 2 + 3 + c3 = 12 implies c3 = 7 according to the provided solution). The sum of diagonal elements is a1 + b2 + c3 = 0 + 2 + 7 = 9.
Year: 2014 | Topic: Engineering Mathematics
Angle made between vector c = 2i - 3j + 4k and the Z-axis
Answer: C. cos^-1(4/$\sqrt{29}$)
Explanation: The Z-axis is represented by the unit vector k. The dot product of c and k is c.k = 4. The magnitude of c is |c| = $\sqrt{2^2 + (-3}$^2 + 4^2) = $\sqrt{29}$. Using the dot product formula, cos(theta) = (c.k) / (|c||k|) = 4 / $\sqrt{29}$. Therefore, theta = cos^-1(4 / $\sqrt{29}$).
Year: 2014 | Topic: Engineering Mathematics
A spherical balloon is filled with 3536 * pi cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of 70 * pi cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases 50 minutes after the leakage began is
Answer: D. -35/18
Explanation: The initial volume is V_0 = 3536 * pi. The leakage rate is dV/dt = -70 * pi. After 50 minutes, the volume is V_50 = 3536 * pi - (70 * pi * 50) = 36 * pi. Using the volume of a sphere V = (4/3) * pi * r³, we find the radius at 50 minutes: (4/3) * pi * r³ = 36 * pi, which gives r³ = 27, so r = 3 m. Differentiating the volume formula yields dV/dt = 4 * pi * r² * (dr/dt). Substituting the knowns gives -70 * pi = 4 * pi * (3^2) * (dr/dt). Solving for the rate of change of the radius yields dr/dt = -70 / 36 = -35/18. The negative sign indicates a decrease, so the rate of decrease is 35/18.
Year: 2014 | Topic: Engineering Mathematics
Resistance of a given wire is obtained by measuring the current flowing in it and the voltage difference applied across it. If the percentage errors in the measurement of the current and the voltage difference are 3% each, then error in the value of resistance of the wire is
Answer: B. 6%
Explanation: From Ohm's law, V = IR, which gives R = V / I. When differentiating and dividing by V = IR to find relative errors, the maximum percentage error in resistance is the sum of the percentage errors in voltage and current: (dR/R) * 100 = (dV/V) * 100 + (dI/I) * 100. Given both are 3%, the total error in resistance is 3% + 3% = 6%.
Year: 2014 | Topic: Engineering Mathematics
A purse contains 4 copper coins and 3 silver coins. A second purse contains 6 copper coins and 4 silver coins. A purse is chosen randomly and a coin is taken out of it. What is the probability that it is a copper coin?
Answer: D. 41/70
Explanation: The probability of randomly choosing either purse is 1/2. The probability of getting a copper coin from the first purse (which has 4 copper out of 7 total coins) is 4/7. The probability of getting a copper coin from the second purse (which has 6 copper out of 10 total coins) is 6/10. The total probability of drawing a copper coin is the sum of the probabilities of choosing each purse and then drawing a copper coin from it: P = (1/2 * 4/7) + (1/2 * 6/10) = 2/7 + 3/10 = (20 + 21) / 70 = 41/70.
Year: 2015 | Topic: Engineering Mathematics
The value of y as t→∞ for an initial value of y(1)=0 for the differential equation $(4t^{2}+l)\frac{dy}{dt}+8yt-t=0,$ is
Answer: D. 1/8
Explanation: A linear differential equation is of the form $\frac{dy}{dt}+Py=Q$, where P and Q are functions of t. Arranging the given equation in this form gives $\frac{dy}{dt}+\frac{8t}{4t^{2}+1}y=\frac{t}{4t^{2}+1}$. The solution is $y(e^{\int Pdt}) = \int Qe^{\int Pdt}dt + C$. The integrating factor is $e^{\int Pdt} = e^{\int\frac{8t}{4t^{2}+1}dt} = e^{\ln(4t^{2}+1)} = 4t^{2}+1$. Substituting this into the solution equation yields $y(4t^{2}+1) = \int\frac{t}{4t^{2}+1}(4t^{2}+1)dt + C$, which simplifies to $y(4t^{2}+1) = \frac{t^{2}}{2} + C$. Given that at t=1, y=0, we find $C = -\frac{1}{2}$. The equation becomes $y(4t^{2}+1) = \frac{t^{2}}{2} - \frac{1}{2}$. Dividing by $t^{2}$ gives $4y+\frac{y}{t^{2}} = \frac{1}{2}-\frac{1}{2t^{2}}$. For $t \rightarrow \infty$, $\frac{1}{t^{2}} \rightarrow 0$, leaving $4y = \frac{1}{2}$, which results in $y = \frac{1}{8}$.
Year: 2015 | Topic: Engineering Mathematics
For the matrix $\begin{bmatrix} 4 & 1 \\ 1 & 4 \end{bmatrix}$, the eigen values are
Answer: C. 3 and 5
Explanation: Let the eigenvalues be $\lambda_1$ and $\lambda_2$. The sum of the eigenvalues equals the trace of the matrix, so $\lambda_1 + \lambda_2 = 8$. The product of the eigenvalues equals the determinant, so $\lambda_1 \cdot \lambda_2 = 15$. The quadratic equation for eigenvalues is $\lambda² - 8\lambda + 15 = 0$. Factoring this equation provides $(\lambda - 5)(\lambda - 3) = 0$, yielding the eigenvalues 3 and 5.
Year: 2015 | Topic: Engineering Mathematics
Solution for the following differential equation with boundary conditions, $y(0)=2$ and $y(1)=-3$ is $\frac{d^2y}{dx²}=3x-2$
Explanation: Integrating the given differential equation $\frac{d^2y}{dx²}=3x-2$ gives $\frac{dy}{dx}=\frac{3x^2}{2}-2x+c_1$. Integrating a second time gives $y=\frac{x³}{2}-x²+c_1x+c_2$. Applying the boundary condition $y(0)=2$ results in $c_2=2$. Applying $y(1)=-3$ gives $-3=\frac{1}{2}-1+c_1+2$, which simplifies to $c_1=-4.5$. The final differential equation solution is $y=\frac{x³}{2}-x²-4.5x+2$. Note that none of the provided options exactly matches this solution.
Year: 2015 | Topic: Engineering Mathematics
For which value of x, will the matrix $\begin{bmatrix} 8 & x & 0 \\ 4 & 0 & 2 \\ 12 & 6 & 0 \end{bmatrix}$ become singular?
Answer: A. 4
Explanation: A matrix becomes singular if its determinant evaluates to zero. Setting the determinant to zero: $8(0 - 12) - x(0 - 24) + 0(24 - 0) = 0$. This simplifies to $-96 + 24x = 0$. Solving for $x$ yields $24x = 96$, and thus $x = 4$.
Year: 2015 | Topic: Engineering Mathematics
The probability that a teacher will give an unannounced test during any class is 1/5. If a student is absent twice, then probability that misses atleast one test is
Answer: D. 9/25
Explanation: The probability of a test on a given day is 1/5, so the probability of no test is 4/5. The probability that the student misses no test over two absent days is $(4/5) \times (4/5) = 16/25$. Therefore, the probability of missing at least one test is $1 - P(\text{miss no test}) = 1 - 16/25 = 9/25$.
Year: 2016 | Topic: Engineering Mathematics
The sine of the angle between the two vectors a = 3i + j + k and b = 2i - 2j + k is
Answer: A. $\sqrt{74/99}$
Explanation: First, find the cosine of the angle using the dot product formula: cos(theta) = (a . b) / (|a| * |b|). The dot product is (3*2) + (1*-2) + (1*1) = 5. The magnitudes are |a| = $\sqrt{9 + 1 + 1}$ = $\sqrt{11}$ and |b| = $\sqrt{4 + 4 + 1}$ = $\sqrt{9}$ = 3. Therefore, cos(theta) = 5 / (3 * $\sqrt{11}$) = 5 / $\sqrt{99}$. The sine of the angle is calculated using the identity sin(theta) = $\sqrt{1 - cos²(theta}$), which gives $\sqrt{1 - 25/99}$ = $\sqrt{74/99}$.
Year: 2016 | Topic: Engineering Mathematics
Equation of the line normal to function f(x) = (x - 8)^(2/3) + 1 at P(0,5) is
Answer: D. y = 3x + 5
Explanation: The slope of the tangent to the curve is found by taking the first derivative: m1 = dy/dx = (2/3) * (x - 8)^(-1/3). Evaluating this at x = 0 yields m1 = (2/3) * (-8)^(-1/3) = (2/3) * (-1/2) = -1/3. The slope of the normal line (m2) is the negative reciprocal of the tangent slope, so m2 = -1 / (-1/3) = 3. Using the point-slope form y = mx + c with the point (0,5), we substitute to get 5 = 3(0) + c, resulting in c = 5. The final equation is y = 3x + 5.
Year: 2016 | Topic: Engineering Mathematics
There are 20 locks and 20 matching keys. Maximum number of trials required to match all the locks is
Answer: B. 210
Explanation: To guarantee finding the matching key, the first lock might require checking up to 20 keys. Once matched, the second lock might require checking up to 19 keys, and so forth, down to the final lock. This sequence forms an arithmetic progression: 20, 19, 18, ..., 1. The sum of this progression is given by the formula n(n+1)/2. For n = 20, the maximum number of trials is (20 * 21) / 2 = 210.
Year: 2016 | Topic: Engineering Mathematics
If phi(x,y,z) is a scalar function and if del² f = 0, then f is
Answer: B. Harmonic
Explanation: By definition in vector calculus, a scalar function that satisfies Laplace's equation (del² f = 0) is called a Harmonic function. In contrast, a field is Solenoidal if its divergence is zero, and it is Irrotational if its curl is zero.
Year: 2016 | Topic: Engineering Mathematics
A and B are two candidates appearing for an interview by a company. The probability that A is selected is 0.5 and the probability that both A and B are selected is at most 0.3. The probability of B getting selected is
Answer: C. <= 0.6
Explanation: Assuming the selection events for candidates A and B are independent, the probability of both occurring is the product of their individual probabilities: P(A and B) = P(A) * P(B). We are given that P(A) = 0.5 and P(A and B) <= 0.3. Substituting the known value gives 0.5 * P(B) <= 0.3. Solving this inequality for P(B) yields P(B) <= 0.6.
Year: 2016 | Topic: Engineering Mathematics
What is the value of the determinant consisting of rows (b²+c², ab, ac), (ba, c²+a², bc), and (ca, cb, a²+b²)?
Answer: A. $4a^2b^2c^2$
Explanation: Expanding the given $3 \times 3$ determinant algebraically leads to terms canceling out. The final simplified polynomial evaluates precisely to $4a^2b^2c^2$.
Year: 2017 | Topic: Engineering Mathematics
The eigen values of A = [[2, 2, 4], [2, -1, 2], [4, 2, 2]] are.
Answer: C. [-2, -2, 7]
Explanation: The characteristic equation is |A - λI| = 0. Solving the resulting cubic equation (λ+2)(λ²-5λ-14)=0 gives the eigenvalues λ=-2, λ=-2, and λ=7.
Year: 2017 | Topic: Engineering Mathematics
If f(t) = e^(at), its Laplace Transform (for s > a) is given by
Answer: C. 1/(s-a)
Explanation: The Laplace transform of e^(at) is defined as the integral from 0 to infinity of e^(at) * e^(-st) dt, which evaluates to 1/(s-a).
Year: 2017 | Topic: Engineering Mathematics
A point P moves along the path y = x² – 4. What is the displacement when the point moves from x = 1 to x = 3?
Answer: A. 8.24, @tan⁻¹(4)°
Explanation: At x₁=1, y₁ = 1²-4 = -3. At x₂=3, y₂ = 3²-4 = 5. Displacement = √((x₂-x₁)² + (y₂-y₁)² ) = √((3-1)² + (5-(-3))²) = √(2² + 8²) = √68 ≈ 8.24. The angle is tan⁻¹((y₂-y₁)/(x₂-x₁)) = tan⁻¹(8/2) = tan⁻¹(4).
Year: 2017 | Topic: Engineering Mathematics
A man draws 3 balls from a jug containing 5 white and 7 black balls. He gets Rs. 20 for each white ball and Rs. 10 for each black ball. What is his expectation?
Answer: B. Rs. 42.50
Explanation: This requires calculating the expected value by summing the product of each outcome's value and its probability. The calculation results in an expected value of Rs. 42.50.
Year: 2017 | Topic: Engineering Mathematics
Evaluate the integral ∫∫(1-x²)^(-1/2) (1-y²)^(-1/2) dx dy over the region x=0 to 1, y=0 to 1.
Answer: D. π²/4
Explanation: The double integral is separable: [∫(1-x²)^(-1/2) dx] * [∫(1-y²)^(-1/2) dy]. The integral of (1-t²)^(-1/2) is sin⁻¹(t). Evaluating from 0 to 1 gives sin⁻¹(1) - sin⁻¹(0) = π/2. Since both integrals are identical, the result is (π/2) * (π/2) = π²/4.
Year: 2017 | Topic: Engineering Mathematics
Which of the following is FALSE about Normal distribution?
Answer: A. Normal Distribution is applied for discrete Random Distribution
Explanation: The Normal Distribution is used for continuous random variables, not discrete ones. Distributions like the Binomial or Poisson are used for discrete variables.
Year: 2018 | Topic: Engineering Mathematics
In a party, each person shake hands with every other person present. The total number of handshakes was 45. The number of people in the party is
Answer: B. 10
Explanation: Total number of handshakes for n people is nC2 = n(n-1)/2. So, n(n-1)/2 = 45 => n² - n - 90 = 0. Solving gives n = 10.
Year: 2018 | Topic: Engineering Mathematics
Two non-zero vectors are parallel if their vector product is
Answer: B. 0
Explanation: Vector product is |a||b|sin(theta). Two non-zero vectors are parallel if the angle between them is zero degrees, making sin(0) = 0, thus their vector product is zero.
Year: 2018 | Topic: Engineering Mathematics
If Eigen value a matrix A is lambda, then the Eigen value of A² is
Answer: D. lambda²
Explanation: If the Eigen value of a matrix A is lambda, then the Eigen value of A² is lambda².
Year: 2018 | Topic: Engineering Mathematics
The value of lim(x->infinity) (sin x) / x is equal to
Answer: D. zero
Explanation: The value of lim(x->infinity) (sin x) / x = 0 since sin(x) is bounded between -1 and 1, and dividing by infinity yields zero.
Year: 2018 | Topic: Engineering Mathematics
If f(x) = (1-x)/(1+x) then f(1/x) is equal to
Explanation: Replacing x with 1/x: f(1/x) = (1 - 1/x)/(1 + 1/x) = (x - 1)/(x + 1) = -((1 - x)/(1 + x)) = -f(x).
Year: 2018 | Topic: Engineering Mathematics
Inverse Laplace transform of [question incomplete in source,cite: 2]
Explanation: This question was deleted or marked with no answer provided by ISRO due to an incomplete question statement.
Year: 2018 | Topic: Engineering Mathematics
One dimensional wave equation is
Answer: A. d^2y/dt² = a²(d^2y/dx²)
Explanation: The wave equation is d^2y/dt² = a²(d^2y/dx²). Option (b) is the heat equation, and (c) is the Laplace equation.
Year: 2018 | Topic: Engineering Mathematics
Which of the following method is said to be a direct method in numerical methods?
Answer: D. Gauss elimination method
Explanation: Gauss elimination method is a direct method for solving a linear system of equations whereas the other options are iterative methods.
Year: 2018 | Topic: Engineering Mathematics
If X is a normal variable with mean mu and standard deviation sigma, then the mean and standard deviation of the variable Z = (X - mu)/sigma is
Answer: C. 0 and 1 respectively
Explanation: For a standard normal variable Z = (X - mu)/sigma, the mean is 0 and the standard deviation is 1.
Year: 2018 | Topic: Engineering Mathematics
Example for an elliptic partial differential equation is
Answer: A. U_xx + U_yy = 0
Explanation: U_xx + U_yy = 0 represents the Laplace equation, which is a classic example of a second order linear elliptic partial differential equation.
Year: 2018 | Topic: Engineering Mathematics
The solution of the equation y'' + y = 0 satisfy the condition y(0) = 1, y(pi/2) = 2, is
Answer: A. cos x + 2 sin x
Explanation: The general solution is y = C1*sin(x) + C2*cos(x). Using y(0) = 1 gives C2 = 1. Using y(pi/2) = 2 gives C1 = 2. Therefore, the equation becomes y = 2*sin(x) + cos(x).