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Grade 1 – Logic & Math Quiz Submit Results: Correct Answers:

ISC Class 12 Mathematics Paper Analysis ISC Class 12 Mathematics Paper Analysis The ISC Class 12 Mathematics paper has been analyzed question by question. Each question is classified into Easy, Moderate, or Difficult based on its complexity, concept depth, and required calculations. Section A: (Questions 1 – 14) Overview: Section A contains fundamental concepts, formula-based questions, and direct applications. Question No. Easy Moderate Difficult 1.(a) – 1.(k) 7 3 2 2 – 6 3 2 1 7 – 14 4 3 1 Section B: (Questions 15 – 18) Overview: Section B consists of more application-based questions, focusing on vectors, 3D geometry, probability, and calculus. Question No. Easy Moderate Difficult 15 – 16 1 1 0 17 – 18 0 1 1 Section C: (Questions 19 – 22) Overview: Section C contains high-weightage application-based problems requiring deep understanding and long calculations. Question No. Easy Moderate Difficult 19 – 20 1 1 0 21 – 22 0 0 2 Overall Paper Difficulty Category Percentage Easy 50% Moderate 30% Difficult 20% Final Verdict: The paper was balanced with a mix of direct and application-based questions. It can be classified as Moderate in difficulty. By: Sachin Sharma Founder www.mathstudy.in & www.udgamwelfarefoundation.com

Math Study Welcome to Math Study This is a sample content section where you can put your text and other information. By: Sachin Sharma | Founder – www.mathstudy.in & www.udgamwelfarefoundation.com SECTION A – 65 MARKS Question 1 In subparts (i) to (xi) choose the correct options and in subparts (xii) to (xv), answer the questions as instructed. (i) If \(A = \begin{bmatrix} 0 & a \\ 0 & 0 \end{bmatrix}\), then \(A^{16}\) is: Unit matrix Null matrix Diagonal matrix Skew matrix (ii) Which of the following is a homogeneous differential equation? \((4x^2 + 6y + 5) dy – (3y^2 + 2x + 4) dx = 0\) \((xy) dx – (x^3 + y^3) dy = 0\) \((x^3 + 2y^2) dx + 2xy dy = 0\) \(y^2 dx + (x^2 – xy – y^2) dy = 0\) (iii) Consider the graph of the function \(f(x)\) shown below: Statement 1: The function \(f(x)\) is increasing in \(\left(\frac{1}{2}, 2 \right)\). Statement 2: The function \(f(x)\) is strictly increasing in \(\left(\frac{1}{2},1 \right)\). Which of the following is correct with respect to the above statements? Statement 1 is true and Statement 2 is false. Statement 2 is true and Statement 1 is false. Both the statements are true. Both the statements are false. (iv) Evaluate the integral: \[\int_{0}^{1} \frac{x^4 – 1}{x^2 + 1} \, dx\] \(\frac{2}{3}\) \(\frac{1}{3}\) \(-\frac{2}{3}\) \(0\) (v) Assertion: Consider the two events \(A\) and \(B\) such that \(n(A) = n(B)\) and \[P\left(\frac{A}{B}\right) = P\left(\frac{B}{A}\right).\] Reason: The events \(A\) and \(B\) are mutually exclusive. Both Assertion and Reason are true and Reason is the correct explanation for Assertion. Both Assertion and Reason are true but Reason is not the correct explanation for Assertion. Assertion is true and Reason is false. Assertion is false and Reason is true. (vi) The existence of a unique solution of the system of equations \(x + y = \lambda\) and \(5x + ky = 2\) depends on: \(\lambda\) only \(\frac{\lambda}{k} = 1\) Both \(k\) and \(\lambda\) \(k\) only (vii) A cylindrical popcorn tub of radius 10 cm is being filled with popcorn at the rate of 314 cm\(^3\) per minute. The level of the popcorn in the tub is increasing at the rate of: 1 cm/minute 0.1 cm/minute 1.1 cm/minute 0.5 cm/minute (viii) If \[f(x) = \begin{cases} x + 2, & x < 0 \\ – x^2 – 2, & 0 \leq x < 1 \\ x, & x \geq 1 \end{cases}\] then the number of points of discontinuity of \(f(x)\) is/are: 1 3 2 0 (ix) Assertion: If Set A has \(m\) elements, Set B has \(n\) elements and \(n < m\), then the number of one-one function(s) from \(A \to B\) is zero. Reason: A function \(f: A \to B\) is defined only if all elements in Set A have an image in Set B. Both Assertion and Reason are true and Reason is the correct explanation for Assertion. Both Assertion and Reason are true but Reason is not the correct explanation for Assertion. Assertion is true and Reason is false. Assertion is false and Reason is true. (x) Let \(X\) be a discrete random variable. The probability distribution of \(X\) is given below: \[\begin{array}{|c|c|c|c|} \hline X & 30 & 10 & -10 \\ \hline P(X) & \frac{1}{5} & \frac{3}{10} & \frac{1}{2} \\ \hline \end{array}\] Then \(E(X)\) will be: 1 4 2 30 (xi) Statement 1: If \(A\) is an invertible matrix, then \((A^2)^{-1} = (A^{-1})^2\). Statement 2: If \(A\) is an invertible matrix, then \(|A^{-1}| = |A|^{-1}\). Statement 1 is true and Statement 2 is false. Statement 2 is true and Statement 1 is false. Both the statements are true. Both the statements are false. (xii) Write the smallest equivalence relation from the set \(A\) to \(A\), where \(A = \{1,2,3\}\). (xiii) For what value of \(x\), is \[A = \begin{bmatrix} 0 & 1 & -2 \\ -1 & 0 & 3 \\ x & -3 & 0 \end{bmatrix}\] a skew-symmetric matrix? (xiv) Three critics review a book. Odds in favour of the book are 5:2, 4:3 and 3:4 respectively for the three critics. Find the probability that all critics are in favour of the book. (xv) Evaluate: \[\int \frac{5}{\sqrt{2x+7}} \, dx\] Question 2 Find the point on the curve \(y = 2x^2 – 6x – 4\) at which the tangent is parallel to the \(x\)-axis. Question 3 Find the value of \(\tan^{-1}x – \cot^{-1}x\), if \[(\tan^{-1}x)^2 – (\cot^{-1}x)^2 = \frac{5\pi}{8}\] Question 4 If \(x^y = e^{x-y}\), prove that \[\frac{dy}{dx} = \frac{\log x}{(1+\log x)^2}\] OR If \[f(x) = \log(1 + x) + \frac{1}{1+x}\] show that \(f(x)\) attains its minimum value at \(x = 0\). Question 5 Three shopkeepers Gaurav, Rizwan and Jacob use carry bags made of polythene, handmade paper and newspaper. The number of polythene bags, handmade bags and newspaper bags used by Gaurav, Rizwan and Jacob are (20, 30, 40), (30, 40, 20) and (40, 20, 30) respectively. One polythene bag costs Rs. 1, one handmade bag is Rs. 5 and one newspaper bag costs Rs. 2. Gaurav, Rizwan and Jacob spend Rs. A, Rs. B and Rs. C respectively on these carry bags. Using the concepts of matrices and determinants, answer the following questions: Represent the above information in Matrix form. Find the values of Rs. A, Rs. B and Rs. C. Question 6 Differentiate \[\sin^{-1} \left( \frac{2x+1 \cdot 3^x}{1 + (36)^x} \right)\] with respect to \(x\). OR Show that \[\tan^{-1} x + \tan^{-1} y = C\] is the general solution of the differential equation \[(1 + x^2) dy + (1 + y^2) dx = 0.\] Question 7 If \(x + y + z = 0\) then show that \[\begin{vmatrix} 1 & 1 & 1 \\ x & y & z \\ x^3 & y^3 & z^3 \end{vmatrix} = 0\] using properties of determinants. Question 8 Evaluate: \[\int \frac{\cos x}{3\cos x – 5} \, dx\] OR Evaluate: \[\int (\log x)^2 \, dx\] Question 9 If \[x = \tan \left( \frac{1}{a} \log y \right)\] then show that \[(1 + x^2) \frac{d^2y}{dx^2} + (2x

Click the link below to download the Differentiation Cheat Sheet for Class 12 and for JEE Students Download the PDF Meaning of Differentiation and its different type Definition of Differentiation Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to a variable. Mathematically, the derivative of a function \( f(x) \) is defined as: \[ f'(x) = \lim\limits_{h \to 0} \frac{f(x+h) – f(x)}{h} \] This definition expresses the instantaneous rate of change of \( f(x) \) at a given point \( x \). Instantaneous Rate of Change The derivative \( f'(x) \) represents the slope of the tangent line to the curve at \( x \). Formulas and Properties of Differentiation 1. Basic Derivatives \[ \frac{d}{dx} (c) = 0 \quad \text{(where \( c \) is a constant)} \] \[ \frac{d}{dx} (x) = 1 \] \[ \frac{d}{dx} (x^n) = n x^{n-1}, \quad n \in \mathbb{R} \] \[ \frac{d}{dx} (\sqrt{x}) = \frac{1}{2\sqrt{x}} \] \[ \frac{d}{dx} \left(\frac{1}{x}\right) = -\frac{1}{x^2} \] 2. Trigonometric Derivatives \[ \frac{d}{dx} (\sin x) = \cos x \] \[ \frac{d}{dx} (\cos x) = -\sin x \] \[ \frac{d}{dx} (\tan x) = \sec^2 x, \quad x \neq \frac{\pi}{2} + n\pi \] \[ \frac{d}{dx} (\cot x) = -\csc^2 x, \quad x \neq n\pi \] 3. Logarithmic and Exponential Derivatives \[ \frac{d}{dx} (\ln x) = \frac{1}{x}, \quad x > 0 \] \[ \frac{d}{dx} (e^x) = e^x \] \[ \frac{d}{dx} (a^x) = a^x \ln a, \quad a > 0 \] 4. Differentiation Rules (i) Sum and Difference Rule \[ \frac{d}{dx} [f(x) \pm g(x)] = f'(x) \pm g'(x) \] (ii) Product Rule \[ \frac{d}{dx} [u(x) v(x)] = u’ v + u v’ \] (iii) Quotient Rule \[ \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{u’ v – u v’}{v^2}, \quad v(x) \neq 0 \] (iv) Chain Rule If \( y = f(g(x)) \), then: \[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \] Examples of Differentiation Example 1: Derivative of a Polynomial Function Let \( f(x) = x^2 + 3x + 5 \). Using the first principles: \[ f'(x) = \lim\limits_{h \to 0} \frac{(x+h)^2 + 3(x+h) + 5 – (x^2 + 3x + 5)}{h} \] Expanding: \[ = \lim\limits_{h \to 0} \frac{x^2 + 2xh + h^2 + 3x + 3h + 5 – x^2 – 3x – 5}{h} \] \[ = \lim\limits_{h \to 0} \frac{2xh + h^2 + 3h}{h} \] \[ = \lim\limits_{h \to 0} (2x + h + 3) \] \[ = 2x + 3 \] Thus, \[ \frac{d}{dx}(x^2 + 3x + 5) = 2x + 3 \] Example 2: Derivative of Trigonometric Functions Let \( f(x) = \sin x \), then using first principles: \[ f'(x) = \lim\limits_{h \to 0} \frac{\sin(x+h) – \sin x}{h} \] Using the identity \( \sin(A+B) = \sin A \cos B + \cos A \sin B \), \[ = \lim\limits_{h \to 0} \frac{\sin x \cos h + \cos x \sin h – \sin x}{h} \] \[ = \lim\limits_{h \to 0} \frac{\sin x (\cos h -1) + \cos x \sin h}{h} \] Since \( \lim\limits_{h \to 0} \frac{\sin h}{h} = 1 \) and \( \lim\limits_{h \to 0} \frac{\cos h -1}{h} = 0 \), \[ = \cos x \] Thus, \[ \frac{d}{dx}(\sin x) = \cos x \] Applications of Differentiation Finding Tangents and Normals: The slope of the tangent line at \( x \) is \( f'(x) \), and the normal’s slope is \( -\frac{1}{f'(x)} \). Maxima and Minima: If \( f'(x) = 0 \) and \( f”(x) > 0 \), it’s a local minimum; if \( f”(x) < 0 \), it's a local maximum. Rate of Change: \( f'(x) \) gives the instantaneous rate of change of a quantity. “` Mathematics E-Books JEE Mains Advance DPP Complex Numbers Mathematics workbook class 1st CAT Mathematics sample papers with solution Class 12 mathematics NCERT Solution DPP For JEE Mains Advance Trigonometry HOTS & Important Questions Mathematics class 12 Class 12 mathematics workbook Chapterwise Test Mathematics Class 12 Mathematics formula book for JEE Mathematics workbook class 2nd NCERT Exemplar solution class 12 mathematics Objective Type Question Bank for Mathematics class 12

Click the link below to download the Multiplication of Matrices Assignment Class X ICSE Board: Download the PDF Multiplication of Matrices Assignment – Class X ICSE Board Multiplication of Matrices Assignment – Class X ICSE Board Welcome to MathStudy.in. Today, we are delving into an essential algebraic operation: the multiplication of matrices. This assignment aims to solidify your understanding of matrix multiplication and its applications. As you work through the problems, focus on the process and the logic behind each step. This skill is fundamental not only for your ICSE exams but for higher mathematics and various practical applications in science, engineering, and economics. Introduction to Matrix Multiplication Unlike simple multiplication, matrix multiplication involves a series of calculations. The product of two matrices A and B is possible only when the number of columns in A is equal to the number of rows in B. The resulting matrix, AB, will have the same number of rows as A and the same number of columns as B. The element in the ith row and jth column of matrix AB is the dot product of the ith row of A and the jth column of B. Assignment Questions Calculate the product of A = [2 4; 3 1] and B = [1 2; 3 4]. If A = [a b; c d] and B = [e f; g h], express the product AB in terms of a, b, c, d, e, f, g, and h. Given matrices P = [1 2; 3 4] and Q = [5 6; 7 8], find both PQ and QP. Discuss the properties you observe about matrix multiplication from these calculations. Use matrix multiplication to solve a system of equations: 2x + 3y = 5 and 4x – y = 3. Explore the application of matrix multiplication in representing and solving real-life problems. For example, consider a business scenario where you need to calculate the total cost based on the unit costs and quantities of products. Reflection Questions What did you find most challenging about learning matrix multiplication, and how did you overcome it? In what ways do you think understanding matrix multiplication can be beneficial outside of mathematics? How can the skills you’ve developed in this assignment apply to other areas of study or real-world problems? Matrix multiplication opens the door to understanding more complex mathematical concepts and solving various types of problems. We hope this assignment has provided you with a solid foundation and sparked your interest in further exploring this topic. Remember, practice is key to mastery. For more assignments and resources, continue exploring MathStudy.in. Happy learning! Mathematics E-Books JEE Mains Advance DPP Complex Numbers Mathematics workbook class 1st CAT Mathematics sample papers with solution Class 12 mathematics NCERT Solution DPP For JEE Mains Advance Trigonometry HOTS & Important Questions Mathematics class 12 Class 12 mathematics workbook Chapterwise Test Mathematics Class 12 Mathematics formula book for JEE Mathematics workbook class 2nd NCERT Exemplar solution class 12 mathematics Objective Type Question Bank for Mathematics class 12 Know about different houses of birth chart First House Second House Third House Fourth House Fifth House Sixth House Seventh House Eighth House Ninth House Tenth House Eleventh House Twelfth House Know about different planets in astrology Sun Moon Ketu Rahu Saturn Jupiter Mars Venus Mercury

Click the link below to download the Subtraction of Matrices Assignment Class X ICSE Board: Download the PDF Subtraction of Matrices Assignment – Class X ICSE Board Subtraction of Matrices Assignment – Class X ICSE Board Welcome to MathStudy.in, where we offer a range of free assignments designed to help Class X ICSE Board students master key concepts in mathematics. This assignment focuses on the subtraction of matrices, a foundational topic that enables students to understand more complex algebraic structures and operations. By the end of this assignment, you will have practiced subtracting matrices of various sizes and applied this knowledge to solve real-world problems. Understanding Matrix Subtraction Matrix subtraction is similar to matrix addition and requires that both matrices involved have the same dimensions. The difference of two matrices, A and B, is found by subtracting the corresponding elements from each matrix. If A = [aij] and B = [bij], then their difference, C, is calculated as C = A – B = [aij – bij]. Assignment Questions Subtract the following matrices: 1. A = [2 5; 3 8], B = [1 0; 4 2] 2. C = [-3 7; 5 -6], D = [4 3; -2 8] Given matrices P = [1 2 3; 4 5 6; 7 8 9] and Q = [9 8 7; 6 5 4; 3 2 1], find P – Q and then Q – P. Discuss the results. Consider a scenario where Matrix X represents the initial quantities of two products in a store, and Matrix Y represents the quantities sold by the end of the day. Calculate the remaining quantities in the store. Use X = [50 30; 20 40] and Y = [15 10; 5 20]. Apply matrix subtraction to find the error matrix E given the theoretical matrix T = [3 5; 2 1] and the experimental matrix E = [2 4; 3 3]. Discuss the significance of the error matrix in scientific experiments. Reflection Questions How does the concept of matrix subtraction enhance your understanding of matrices in general? Can you think of a situation in your daily life where matrix subtraction could be applied? Describe the matrices involved. How might the skill of subtracting matrices be useful in other subjects or fields of study? We hope this assignment not only strengthens your grasp of matrix subtraction but also stimulates your interest in exploring how matrices are used in various real-world contexts. For more resources and assignments, visit MathStudy.in. Happy learning! Mathematics E-Books JEE Mains Advance DPP Complex Numbers Mathematics workbook class 1st CAT Mathematics sample papers with solution Class 12 mathematics NCERT Solution DPP For JEE Mains Advance Trigonometry HOTS & Important Questions Mathematics class 12 Class 12 mathematics workbook Chapterwise Test Mathematics Class 12 Mathematics formula book for JEE Mathematics workbook class 2nd NCERT Exemplar solution class 12 mathematics Objective Type Question Bank for Mathematics class 12