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Questions and Answers
What is the sum of EA + EB + EC + ED?
What is the sum of EA + EB + EC + ED?
The integral of Sin 3 (2n + 1) x from - to 0 equals 1.
The integral of Sin 3 (2n + 1) x from - to 0 equals 1.
False
What is the value of A if A = [[0, 2x - 1, x], [1 - 2x, 0, 2x], [-x, -2x, 0]]?
What is the value of A if A = [[0, 2x - 1, x], [1 - 2x, 0, 2x], [-x, -2x, 0]]?
A = (2x + 1)/2
In the context of the linear programming problem, the constraint that is NOT present is ______.
In the context of the linear programming problem, the constraint that is NOT present is ______.
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Match the following expressions with their corresponding values:
Match the following expressions with their corresponding values:
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What is the vector form of the component of vector a along vector b?
What is the vector form of the component of vector a along vector b?
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If A is a square matrix of order 3 and A = -2, then adj(2A) equals -28.
If A is a square matrix of order 3 and A = -2, then adj(2A) equals -28.
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What is the probability that the problem will be solved by three students with chances of $ rac{1}{2}$, $ rac{1}{3}$, and $ rac{1}{4}$ respectively?
What is the probability that the problem will be solved by three students with chances of $ rac{1}{2}$, $ rac{1}{3}$, and $ rac{1}{4}$ respectively?
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The general solution of the differential equation $ydx - xdy = 0$ is of the form $y = cx$ where 'c' is a ______.
The general solution of the differential equation $ydx - xdy = 0$ is of the form $y = cx$ where 'c' is a ______.
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Match the following mathematical concepts with their descriptions:
Match the following mathematical concepts with their descriptions:
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What is the value of adj(2A) if A is a square matrix of order 3 and A = -2?
What is the value of adj(2A) if A is a square matrix of order 3 and A = -2?
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The events of the students solving the problem are dependent.
The events of the students solving the problem are dependent.
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If the vector a = 4i + 6j, what is the coefficient of j?
If the vector a = 4i + 6j, what is the coefficient of j?
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Which of the following expressions correctly defines the function f(x) for determining its increasing intervals?
Which of the following expressions correctly defines the function f(x) for determining its increasing intervals?
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The function f(x) = x^3 + x has a critical point.
The function f(x) = x^3 + x has a critical point.
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What is the maximum profit if the profit function is defined by P(x) = 72 + 42x - x^2?
What is the maximum profit if the profit function is defined by P(x) = 72 + 42x - x^2?
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To find the value of k in the probability distribution P(X), the equation to solve is ______.
To find the value of k in the probability distribution P(X), the equation to solve is ______.
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Match the following functions with their types of evaluation:
Match the following functions with their types of evaluation:
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Which value of k satisfies the probability distribution P(X)?
Which value of k satisfies the probability distribution P(X)?
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Evaluate the integral ∫ (1 / (1 - x^3)) dx over the interval (0,1).
Evaluate the integral ∫ (1 / (1 - x^3)) dx over the interval (0,1).
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The solution to the differential equation ye^y dx = (xe^y + y^2) dy is straightforward.
The solution to the differential equation ye^y dx = (xe^y + y^2) dy is straightforward.
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What is the value of $y$ when $ rac{dy}{dx} = rac{y^2}{a + bx}$?
What is the value of $y$ when $ rac{dy}{dx} = rac{y^2}{a + bx}$?
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The area of the region defined by $0 \leq y \leq x^2 + 1$ and $0 \leq x \leq 2$ can be found using basic arithmetic without integration.
The area of the region defined by $0 \leq y \leq x^2 + 1$ and $0 \leq x \leq 2$ can be found using basic arithmetic without integration.
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What is an equivalence relation?
What is an equivalence relation?
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The function $f(x) = \frac{x}{1 + x}$ is defined for $x \in \mathbb{R}$ with $-1 < x < 1$. This function is _____ and onto.
The function $f(x) = \frac{x}{1 + x}$ is defined for $x \in \mathbb{R}$ with $-1 < x < 1$. This function is _____ and onto.
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Match the following equations to their solutions:
Match the following equations to their solutions:
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Which of the following statements is true concerning the coordinates of the image of the point $(1, 6, 3)$ with respect to the line given?
Which of the following statements is true concerning the coordinates of the image of the point $(1, 6, 3)$ with respect to the line given?
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What percentage of forms does Sonia process?
What percentage of forms does Sonia process?
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The shortest distance between two planes can be found without solving for the points where they intersect.
The shortest distance between two planes can be found without solving for the points where they intersect.
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Oliver has the highest error rate among the three employees.
Oliver has the highest error rate among the three employees.
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What is the combined error rate of the employees when averaging their contributions?
What is the combined error rate of the employees when averaging their contributions?
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In tug of war, Team A pulls with force F1 = 6iˆ + 0 ˆj kN. The magnitude of this force is ______ kN.
In tug of war, Team A pulls with force F1 = 6iˆ + 0 ˆj kN. The magnitude of this force is ______ kN.
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Match the employees to their error rates:
Match the employees to their error rates:
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Team B's force can be represented as -4i + 4j kN.
Team B's force can be represented as -4i + 4j kN.
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What is the direction of the resultant force if Team A pulls with the strongest force?
What is the direction of the resultant force if Team A pulls with the strongest force?
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What is the value of λ for which the vectors $2\hat{i} - \hat{j} + 2\hat{k}$ and $3\hat{i} + \lambda \hat{j} + \hat{k}$ are perpendicular?
What is the value of λ for which the vectors $2\hat{i} - \hat{j} + 2\hat{k}$ and $3\hat{i} + \lambda \hat{j} + \hat{k}$ are perpendicular?
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The function $f(x) = x + x$ is differentiable everywhere.
The function $f(x) = x + x$ is differentiable everywhere.
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What is the set of all points where the function $f(x) = x + x$ is differentiable?
What is the set of all points where the function $f(x) = x + x$ is differentiable?
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The direction cosines of a line must satisfy the condition that ______.
The direction cosines of a line must satisfy the condition that ______.
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Which of the following statements is true for the polynomial function $f(x)$ given its derivative $\frac{d}{dx}f(x) = (x-1)^3 (x-3)^2$?
Which of the following statements is true for the polynomial function $f(x)$ given its derivative $\frac{d}{dx}f(x) = (x-1)^3 (x-3)^2$?
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Match the following statements about the function f with their descriptions:
Match the following statements about the function f with their descriptions:
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If f : {1, 2, 3, 4} → {x, y, z, p} is defined by f = {(1, x), (2, y), (3, z)}, is f a bijective function?
If f : {1, 2, 3, 4} → {x, y, z, p} is defined by f = {(1, x), (2, y), (3, z)}, is f a bijective function?
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The relation defined by f = {(1, x), (2, y), (3, z)} is a bijective function.
The relation defined by f = {(1, x), (2, y), (3, z)} is a bijective function.
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Find the value of $\sin^{-1}\left(\cos\left(\frac{33\pi}{5}\right)\right)$.
Find the value of $\sin^{-1}\left(\cos\left(\frac{33\pi}{5}\right)\right)$.
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Study Notes
Section A - Multiple Choice Questions
- Question 1: If A = [aij] is a 2x2 square matrix where aij = 1 when i ≠ j and aij = 0 when i = j, then A² = [0 1; 1 0]
- Question 2: The inverse of a product of invertible matrices is the product of their inverses in reverse order ( (AB)⁻¹ = B⁻¹A⁻¹ ). One incorrect option is (A+B)⁻¹ = B⁻¹ + A⁻¹
- Question 3: The area of a triangle with vertices (-3, 0), (3, 0), and (0, k) is 9 square units; k = ±6
- Question 4: A function f(x) = kx, for x < 0 and f(x) = 3, for x ≥ 0 is continuous at x = 0, then k = 3
Section B - Very Short Answer (VSA)
- Question 21: Find the value of sin⁻¹(cos(33π/5)). OR Find the domain of sin⁻¹(x² - 4)
- Question 22: Find the intervals where the function f(x) = xeˣ is increasing.
- Question 23: Given f(x) = 1 / (4x² + 2x + 1), find the maximum value of f(x) for x ∈ R. OR Find the maximum profit for a profit function P(x) = 72 + 42x - x² where x is the number of units and P is the profit in rupees.
- Question 24: Evaluate ∫ (1/(2-x)) dx
- Question 25: Determine whether the function f(x) = x² + x has any critical points.
Section C - Short Answer (SA)
- Question 26: Find the integral of (2x² + 3) / (x² (x² + 9)) dx for x ≠ 0.
-
Question 27: The random variable X has a probability distribution P(X) with k, 2k, 3k for x=0, 1, 2 respectively; 0 elsewhere.
- Determine k.
- Find P(X<2)
- Find P(X>2)
Section D - Long Answer (LA)
- Question 32: Sketch the region defined by 0 ≤ y ≤ x² + 1, 0 ≤ y ≤ x+1, 0 ≤ x ≤ 2 and find the area using integration.
- Question 33: Show that R = {(a,b),(c,d)} ∈ N x N | ad = bc is an equivalence relation on N x N. Also, find the equivalence class of (2,6), i.e., [(2,6)]? OR Show that the function f(x) = x / (1 + x), x ∈ R is one-one and onto function.
- Question 34: Solve the following system of linear equations using the matrix method.
- Question 35: Find the coordinates of the image of point (1, 6, 3) with respect to the line. OR An aeroplane...finds the shortest possible distance.
Section E - Case Study/Passage Based
-
Question 36: (Passage about error rates of employees processing forms).
- Find the probability that Sonia processed the form and committed an error.
- Find the total probability of committing an error in processing the form.
- If a randomly selected form has an error, find the probability it was not processed by Jayant.
-
Question 37: (Tug-of-war passage with forces)
- What is the magnitude of Team A's force?
- Which team will win the game?
- What is the magnitude of the resultant force exerted by the teams?
-
Question 38: (Plant growth passage)
- Find the rate of growth of the plant with respect to the number of days of exposure.
- What is the rate of growth of the plant on days 1, 2, and 3?
- How tall will the plant be after two days?
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Description
Test your knowledge on various mathematics topics including linear algebra, vector analysis, and probability. This quiz consists of multiple questions that cover concepts like matrices, vectors, equations, and their applications. Challenge yourself and see how well you understand these mathematical principles!
