BSC CSIT MATH I IMPORTANT QUESTION CHAPTER WISE

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Unit 1: Function of One Variable

2080- Section A - 3a: As dry air moves upward, it expands and cools. If the ground temperature is 20°C and the temperature ata  height of 1 km is 10C, express the temperature T(in °C) as a function of height h(in kilometer), assuming that the linear model is appropriate.

(b) Draw a graph of the function in part(a). What does the slope represent?

(c) What is the temperature at a height of 2.5 km

 

• 2080 - Section B - 9: Sketch the graph and find the domain and range of f(x) = 2^x-1. (Exponential function graph & domain/range)
• 2079 - Group A - 1a: If a function is defined by f(x) = {1+x if x<=-1, {x² if x > -1, evaluate f(-3), f(-1) and f(0) and sketch the graph. (Piecewise function evaluation and graph)

 

As dry air moves upward, it expands and cools. If the ground temperature is 20°C and the temperature at height of 1 km is 10C, express the temeperature T(in °C) as a function of height h(in kilometer), assuming that the linear model is appropriate.

b

(b) Draw a graph of the function in part(a). What does the slope represent?

c

(c) What is the temperature at a height of 2.5 km?

 

• 2078 - Group A - 1a: If f(x) = √x and g(x) = √(3-x) then find fog and its domain and range. (Composite function, domain/range)

2078 - Group A - 1b:

A rectangular storage container with an open top has a volume of 20m3. The length of its base is twice its width. Material for the base costs Rs.10 per square meter material for the sides costs Rs 4 per square meter. Express the cost of materials as a function of the width of the base.

• 2077 - Group A - 1a: If f(x) = x^2then find (f(2+h) - f(2))/h (Function evaluation and difference quotient - related to derivative concept, but placed here due to function manipulation focus).
• 2077 - Group A - 1b: Dry air is moving upward... (Linear Model word problem - repeated)
• • 2077 - Group B - 5: If f(x) = x^2 – 1, g(x) = 2x + 1, find fog and gof and domain of fog. (Composite function and domain)
• • 2075 - Group A - 1a: A function is defined by f(x) = |x|, calculate f(-3), f(4), and sketch the graph. (Absolute value function evaluation and graph)
• • 2075 - Group B - 5: If f(x) = √(2-x) and g(x) = √x, Find fog and fof. (Composite functions)
• • 2074 - Group A - 1a: A function is defined by f(x) = {x+2 if x<0, 1-x if x>0, Evaluate f(-1), f(3) and sketch graph. (Piecewise function, evaluation & graph - like 2079-1a) 2074 - Group B - 5: If f(x) = √x and g(x) = √(3-x), find gof and gog. (Composite functions - repeated concept)

Unit 2: Limits and Continuity

• 2080 - Section A - 1b: Estimate the value of limx→0(√(x^2 + 9) – 3 )/ x^2 (Limit evaluation)

• 2079 - Group A - 1b: Prove that lim x→0 |x|/x does not exist. (Limit existence proof/demonstration)

 2078 - Group B - 6: Find the equation of tangent at (1, 2) to the curve y = 2x^2

• 2077 - Group B - 6: Define continuity of a function at a point x = a. Show that the function f(x) = √(1−x^2) is continuous on the interval [-1, 1]. (Definition of continuity and proving continuity on interval)

• 2075 - Group A - 1b: Prove that the limx→2 |x–2|/(x–2) doesn’t exist. (Limit non-existence - similar to 2079-1b)

• 2075 - Group B - 6: Define continuity on an interval. Show that the function f(x) = 1 – √(1−x^2) on the continuous on the interval [1,-1]. (Continuity definition and proof on interval - very similar to 2077-B-6)

 • 2074 - Group A - 1b: Prove that lim x→0 |x|/x the does not exist. (Limit non-existence - again)

• 2074 - Group B - 6: Use Continuity to evaluate the limit, lim x→4  (5 + √x)/(√5 + x). (Limit evaluation using continuity property)

• 2074 - Group B - 14: Define limit of a function. Find limit lim x→∞  (x – √x). (Definition of limit, limit at infinity)

Unit 3: Derivatives

 • 2080 - Section B - 7: Find y’ if x^3 + y3 = 6xy.  (Implicit differentiation)

 • 2079 - Group B - 6: Find the equation of the tangent at (1,3) to the curve y = 2(x^2)+ 1. (Tangent line equation)

 • 2078 - Group B - 6: Find the equation of tangent at (1, 2) to the curve y = 2(x^2)+ 1. (Tangent line equation - similar to 2079-B-6)

 • 2077 - Group A - 1c: Find the equation of the tangent to the parabola y = x^2+ x + 1 at (0, 1). (Tangent line equation - again)

• 2079 - Group B - 7: State Rolle’s theorem and verify the theorem for f(x) = (x^2) – 9, x ε[-3,3] (Rolle's Theorem - statement & verification)

• 2078 - Group B - 7: State Rolle’s theorem and verify the Rolle’s theorem for f(x) = (x^2) – 3x + 2 in [0, 3] (Rolle's Theorem - statement & verification - repeated function type)

• 2077 - Group B - 7: State Rolle’s theorem and verify the Rolle’s theorem for f(x) = (x^3) – (x^2) – 6x + 2 in [0, 3] (Rolle's Theorem - statement & verification - different function)

• 2075 - Group B - 7: Verify Mean value theorem of f(x) = (x^3) – 3x + 2 for [-1, 2]. (Mean Value Theorem Verification)

2074 - Group B - 7: Verify Mean Value Theorem by f(x) = x^3 – 3x + 3 for [-1, 2] (Mean Value Theorem Verification - similar function type)

Unit 4: Applications of Derivatives

 • 2080 - Section B - 6:  Find where the function f(x) = 3×4 – 4×3 – 12 x2 + 5 is increasing and where it is decreasing.

(Increasing/Decreasing Intervals - curve sketching aspect)

 • 2079 - Group A - 2a: Sketch the curve y=x2 +1 with the guidelines of sketching.

 (Curve sketching - parabola)

• 2079 - Group B - 8: Starting with x1= 1, find the third  approximate x3 to the root of the equation x³ – x – 5 = 0 (Newton's method - root finding)

• 2078 - Group B - 8: Use Newton’s method to find 6√2 correct five decimal places. (Newton's Method - root finding, implicit equation)

• 2077 - Group A - 2b: Sketch the curve y = 1/(x-3) (Curve sketching - rational function, asymptotes implied, though asymptotes mentioned more explicitly in Unit 2 of syllabus)

• 2077 - Group B - 8: Find the third approximation x3 to the root of the equation f(x) = x^3 – 2x – 7, setting x1 = 2.

2. (Newton's method - root finding - repeated type)

 • 2075 - Group B - 8:

Stating with x1 = 2, find the third approximation x3 to the root of the equation x^3 – 2^x – 5 = 0

 

 

2074 - Group B - 8: Sketch the curve y = (x^3) + x (Curve sketching - polynomial)

Unit 5: Antiderivatives

 • 2080 - Section B - 4: Integrate 0∫1 x2√(x3 + 1) dx . (Definite integral - u-substitution)

• 2079 - Group A - 3a: Estimate the area between the curve y=x2 and  the lines x=0 and x=1, using rectangle method, with four sub intervals.(Rectangular approximation of area - basic definite integral concept introduction)

 • 2078 - Group A - 2a: Using rectangular, estimate the area under the parabola y = x2 from 0 to 1. (Rectangular area approximation - again)

• 2077 - Group A - 3a:Show that the converges ∫1∞1x2dx∫∞11x2dx and diverges ∫1∞1xdx (Improper integral convergence/divergence)

• 2075 - Group B - 9: Evaluate ∫∞0x3√1–x4∫0∞x31–x4 dx

2074 - Group B - 9: Determine whether the integral ∫∞1(1x)dx∫1∞(1x)dx  is convergent or divergent

Unit 6: Applications of Antiderivatives

 • 2080 - Section A - 2a: The area of the parabola y = x2 from (1,1) to (2,4) is rotated about the y-axis. Find the area of the resulitng surface.

 • 2079 - Group A - 3b (Part 1 & 2):

A particle moves a line so that its velocity v at time t is

(1) Find the displacement of the particle during the fine period 1 ≤ t ≤ 4

(2) Find the distance travelled during this time period.

 

• 2079 - Group B - 10: Use Trapezoidal rule to approximate the integral 1 ∫² dx/x, with n=5. (Trapezoidal Rule approximation)

• 2078 - Group A - 2b (Part 1 & 2):

A particle moves along a line so that its velocity v at time t is

v = t2 – t + 6

1. Find the displacement of the particle during the time period 1 ≤ t ≤ 4.
2. Find the distance travelled during this time period.

 

• 2078 - Group A - 3a: Find the area of the region bounded by y = x^2 and y = 2x – x^2

 • 2078 - Group A - 3b: search online 

• 2078 - Group B - 10:  Find the volume of the solid obtained by rotating about the y-axis the region between y = x and y = x^2.

 • 2077 - Group A - 3c: A particle moves in a straight line and has acceleration given by a(t) = 6t2 + 1. Its initial velocity is 4m/sec and its initial displacement is s(0) = 5cm. Find its position function s(t).

 • 2077 - Group B - 10: Find the volume of the solid obtained by rotating about the y-axis the region between y = x and y = x2.

• 2075 - Group A - 2b: Estimate the area between the curve y = x^2and the lines y = 1 and y = 2. (Area between curves/regions - may be interpreted as needing integral)

 • 2075 - Group A - 3b: Find the volume of a sphere of radius r.

• 2075 - Group B - 10: Find the volume of the resulting solid which is enclosed by the curve y = x and y = x2 is rotated about the x-axis.

 

• 2074 - Group A - 2: Estimate the area between the curve y2 = x and the lines x = 0 and x = 2.

 

 • 2074 - Group A - 3c: Find the volume of a sphere of radius r.

• 2074 - Group B - 10: Find the length of the arc of the semi cubical parabola y2 = x2 between the points (1,1) and (4,8).

Unit 7: Ordinary Differential Equations

• 2080 - Section A - 2b:Find the solution of the equation y2dy = x2dx that satisfies the initial condition y(0) = 2.

• 2080 - Section B - 8: Show that y = x – 1/x is a solution of the differential equation xy’ + y = 2x. (Verifying a solution to a given ODE)

• 2079 - Group A - 4a: Define initial value problem. Solve y’’+y’ -6y =0, y(0)=1, y’(0)=0 (Second order linear homogeneous ODE - initial value problem)

• 2078 - Group A - 4a: Solve y’ = x2/y2, y(0) = 2

 • 2078 - Group A - 4b: Solve the initial value problem: y’’ + y’ – 6y = 0, y(0) = 1, y'(0) = 0 (Second order linear homogeneous ODE - initial value problem - repeated equation form as 2079-A-4a)

 • 2078 - Group B - 11: Solve: y’ + 2xy – 1 = 0 (First-order linear ODE)

 • 2077 - Group B - 11: Solve: y’’ + y’ = 0, y(0) = 5, y(π/4) = 3 (Second order linear homogeneous ODE - boundary value problem)

• 2075 - Group B - 11: Find the solution of y” + 4y’ + 4 = 0. (Second order linear homogeneous ODE - repeated roots)

 • 2074 - Group A - 3b: Define Initial Value Problem. Solve that value problem of  y2+5y=1y2+5y=1, y(0) = 2

2074 - Group B - 11: Find the solution of y”+6y’+9=0, y(0)=2, y'(0)=1 (Second order linear homogeneous ODE - repeated roots again)

Unit 8: Infinite Sequence and Series

• 2080 - Section B - 5: Find the Maclaurin series expansion of f(x) = e^x at x =0. (Maclaurin series)

 • 2080 - Section B - 10: Determine whether the series n=1∑∞ n2 / (5n2 + 4) converges or diverges.

 • 2079 - Group A - 4b: Find the Taylor’s series expansion for cosx at x=0. (Taylor/Maclaurin Series - cos x)

• 2079 - Group B - 12:

What is sequence? Is the sequence

an=n√5+nan=n5+n

convergent?

 

 • 2078 - Group B - 12:

What is sequence? Is the sequence

an=n√5+nan=n5+n

convergent?

 • 2077 - Group A - 4b: Find the Maclaurin series for cos x and prove that it represents cos x for all x. (Maclaurin series for cos x, proof of representation)

• 2077 - Group B - 12: search online

 

• 2075 - Group A - 3a: Find the Maclaurin series for cos x and prove that it represents cos x for all x. (Maclaurin series cos x and representation - repeated question)

• 2075 - Group B - 12: search online

• 2074 - Group A - 4a: search online

• 2074 - Group B - 12: search online

Unit 9: Plane and Space Vectors

 • 2079 - Group B - 11:

Find the derivative of (r(t)) = t^2i – te(-t)j + sin(2t)k and find the unit tangent vector at t = 0

 

• 2078 - Group B - 9:

Find the derivatives of r(t) = (1 + t2)i – te-tj + sin 2tk and find the unit tangent vector at t=0.

 

• 2077 - Group B - 9:

Find the derivatives of r(t) = (1 + t2)i – te-tj + sin 2tk and find the unit tangent vector at t=0.

 

 • 2079 - Group B - 13: Find the angle between the vectors a = (2, 2, -1) and b = (1, 3, 2) .

• 2078 - Group B - 13: Find a vector perpendicular to the plane that passes through points P, Q, R. (Vector perpendicular to plane - cross product)

• 2077 - Group B - 13: Find a vector perpendicular to the plane that passes through points P, Q, R. (Vector perpendicular to plane - cross product - repeated)

• 2075 - Group B - 13: If a = (4, 0, 3) and b = (-2, 1, 5) find |a|, a – b and 2a + b (Vector operations - magnitude, addition/subtraction, scalar multiplication)

• 2074 - Group B - 13: Define cross product of two vectors... find vector b×a and a×b for given vectors. (Cross product calculation, definition)

Unit 10: Partial Derivatives and Multiple Integrals

• 2079 - Group A - 2b:

If z=xy2 + y3 , x= sint, y=cost, find dz/dt at t=0

 

• 2079 - Group B - 14:  Find the partial derivative fxx and fyy of f(x,y)= x2 + x3y2 – y2 + xy, at (1,2).

 

 • 2079 - Group B - 15:

Evaluate

0∫3  1∫2 x2y dxdy

 

 • 2078 - Group B - 14: Find the partial derivative of f(x, y) = x2 + 2x3y2 – 3y2 + x + y at (1. 2)

 • 2078 - Group B - 15:  Find the local maximum and minimum values, saddle points of f(x,y) = x4 + y4 – 4xy + 1

 

• 2077 - Group A - 3b: If f(x,y)=xyx2+y2f(x,y)=xyx2+y2, does f(x, y) exist, as (x, y) → (0, 0)?

 

• 2077 - Group A - 4a: Evaluate ∫23∫π20(y+y2cosx)dxdy

 

 • 2077 - Group B - 14: Find the partial derivative of f(x, y) = x3 + 2x3y3 – 3y2 + x + y, at (2,1)

 • 2077 - Group B - 15: Find the local maximum and minimum values, saddle points of f(x,y) = x4 + y4 – 4xy + 1

 • 2075 - Group A - 4b: Calculate ∫R∫f(x,y)dA∫R∫f(x,y)dA for f(x,y) = 100 – 6x2y and R:0≤x≤2,−1≤y≤1R:0≤x≤2,−1≤y≤1

 • 2075 - Group B - 14: (search online)Find ∂z∂xand∂z∂y∂z∂xand∂z∂y if z is defined as a function of x and y by the equation  x3+y3+z3+6xyz=1x3+y3+z3+6xyz=1.

 (Implicit partial differentiation)

 • 2075 - Group B - 15:Find the extreme values of the function f(x,y)=x2+2y2f(x,y)=x2+2y2 on the circle x2+y2=1x2+y2=1.

 

 

 • 2074 - Group A - 4b: Calculate ∫R∫f(x,y)dA∫R∫f(x,y)dA for f(x,y) = 100 – 6x2y and R:0≤x≤2,−1≤y≤1

 

Analysis and Observations from Past Papers (2074-2080)

 Based on the chapter-wise arrangement, we can observe the following trends:

 • Emphasis on Integral Calculus (Units 5 & 6): These units consistently have a significant number of questions, both in Section A and Section B. Applications of antiderivatives (Area, Volume, Arc Length) are particularly frequent.

 • Differential Calculus Applications (Unit 4): Curve sketching (basic types), Newton's Method are consistently asked. Optimization problems also mentioned in syllabus but Newton's method appears more in papers than explicit "word problem" optimization, though linear model applications appear under function modeling (Unit 1).

• Ordinary Differential Equations (Unit 7): Solving first and second-order linear ODEs (homogeneous more than non-homogeneous seen in these papers) is a recurring theme. Initial Value Problems are common.

 • Infinite Series and Sequences (Unit 8): Convergence and divergence of series, tests for convergence (divergence test, comparison test, ratio test are relevant based on questions). Maclaurin/Taylor series expansion, especially for cos(x) and e^x, is a very frequently repeated question type.

• Vectors (Unit 9): Vector operations (dot product, cross product, magnitudes), equations of lines/planes mentioned in syllabus, but questions focus more on dot/cross product calculations, finding perpendicular vectors, angle between vectors and derivatives of vector functions and tangent vectors.

• Partial Derivatives and Multiple Integrals (Unit 10): Partial derivatives calculation, local maxima/minima/saddle points, double integral evaluation over rectangular regions are typical questions. Limits of multivariable functions and continuity sometimes tested.

• Limits and Continuity (Unit 2): Proving limit non-existence using the definition or properties, continuity proofs. Limit evaluation (simple types, not very complex limit manipulations consistently in these papers) and using continuity for limit evaluation appear.

• Functions (Unit 1): Function evaluation for piecewise and composite functions, finding domains/ranges, sketching basic function graphs, linear models from word problems are regularly tested.

Derivatives (Unit 3): Tangent line equations, implicit differentiation, Rolle's Theorem and Mean Value Theorem (statement and verification), derivative calculations in the context of vector functions and rates of change. L'Hopital's rule is mentioned in syllabus but hasn't prominently featured in these past papers (only one example question seen indirectly in 2080 Section A 1b requiring limit calculation where L'Hopital's rule could be used though likely simpler methods also apply for that specific question).

 

Repetitive Question Types and Direct Question Repeats:

 • Maclaurin Series for cos(x) and e^x: Asked multiple times, and even with the "prove it represents..." part sometimes.

• Newton's Method: Root-finding using Newton's method appears each year.

• Rolle's Theorem/Mean Value Theorem verification: Statement and verification for specific functions are repeated with variations in functions.

•Vector function derivative and tangent vector calculation: Very similar questions (even using almost identical vector functions in consecutive years).

 • Improper Integral Convergence/Divergence: Repeated.

 • Series Convergence/Divergence tests: Recurring theme. • Area between curves, Volume of revolution: Very frequent applications of integrals.

• Double integral evaluation over rectangular regions: Common.

 • Questions directly repeated: Some questions are exactly repeated from previous years, like the double integral evaluation in 2074 & 2075, the sequence convergence question in 2078 & 2079, and function definition for vector derivatives in 2077, 2078, 2079.

 

Guess Paper for This Year (Based on Past Trends) Considering the syllabus coverage and the trends observed in the past question papers,

 

 Section A (Attempt any TWO or THREE - Based on Exam Pattern Variability, be prepared for both formats. Assuming likely 3 based on most papers provided):

 1. (Function & Limit/Continuity Combination):

 ○ (a) Function Modeling: A word problem involving modeling a relationship with a linear, polynomial, or rational function. Could be similar to the dry air temperature problem or a cost/volume optimization setup.

 ○ (b) Limit Evaluation (or Non-Existence): A question requiring you to evaluate a limit using algebraic manipulation or L'Hopital's Rule OR prove that a limit (like |x|/x type) does not exist.

 2. (Applications of Integrals: Area/Volume/Surface Area):

 ○ (a) Area between curves OR Volume of Solid of Revolution: Find the area of the region bounded by given curves OR find the volume of the solid generated by rotating a region around the x or y-axis. Focus on parabolas, lines, basic polynomial and rational functions as boundaries. Be ready for washer/disk and cylindrical shell methods for volumes, and area between two functions by integration. Possibly Surface area of revolution could be asked again (like in 2080), so briefly review that.

3. (Ordinary Differential Equations):

 (a) Second-Order Linear Homogeneous ODE (with Initial/Boundary Conditions): Solve a second-order linear homogeneous differential equation with constant coefficients. Be prepared for both initial value problems (given y(0) and y'(0)) and possibly boundary value problems (given y(0) and y(L) or y'(L)). Focus on finding roots of characteristic equation (distinct real, repeated, complex).

 (b) Separable or First-Order Linear ODE: Solve a first-order ODE that is either separable OR linear. Practice recognizing and solving these types.

4. (Maclaurin/Taylor Series & Convergence):

 (a) Maclaurin Series Expansion: Find the Maclaurin series expansion for a common function like cos(x), sin(x), e^x, or possibly ln(1+x) or (1-x). 

(b) Series Convergence Test: Determine whether a given infinite series converges or diverges. Use appropriate tests like the divergence test, integral test, comparison test, ratio test, or root test. Likely a series that requires divergence test, comparison test, or ratio test.

 Section B (Attempt any TEN - Shorter Questions):

• Function Evaluation & Domain/Range: Question similar to evaluating piecewise functions, composite functions, or finding domain/range of basic function types (square root, rational, exponential, log).

 

 • Continuity: Define continuity or check continuity at a point/interval for a simple function (like √ function or rational function - maybe similar type to questions asked about √(1-x^2) before, or continuity using limit properties).

• Tangent Line Equation: Find the equation of the tangent line to a given curve at a given point (parabola likely, similar to repeated tangent questions).

 • Rolle's Theorem or Mean Value Theorem (Statement and/or Verification): State one of these theorems and potentially verify it for a polynomial function on a given interval (functions from past papers or similar complexity are good practice).

• Newton's Method: Find an approximation to the root of an equation using Newton's method .

 • Improper Integral Convergence/Divergence: Determine convergence or divergence of a given improper integrall. Review p-integrals.

 • Trapezoidal Rule Approximation: Approximate a definite integral using the Trapezoidal Rule with a small number of subintervals (n=4 or n=5, or similar).

 • Vector Operations (Dot Product/Cross Product): Calculate the dot product, cross product, or angle between two vectors. Find a vector perpendicular to a plane defined by points (using cross product).

 

 • Derivative of Vector Function & Unit Tangent Vector: Find the derivative of a given vector function and calculate the unit tangent vector at a specified point (t=0 likely). Practice differentiating components. Vector functions similar to

Find the derivative of (r(t)) = t^2i – te(-t)j + sin(2t)k and find the unit tangent vector at t = 0 or simpler.

 • Sequence Convergence: Determine if a given sequence is convergent or divergent. Check for limit as n→∞. Sequence examples similar to an= n/√(5 + n) or simpler.

• Series Convergence (Simpler Tests): Test for convergence of a series using the divergence test, comparison test or possibly integral test. Simpler series for testing fundamental understanding of convergence

. • Partial Derivatives: Calculate first-order and/or second-order partial derivatives of a given function of two variables, evaluated at a point. Practice functions like polynomials, rational functions in x and y.

• Double Integral Evaluation: Evaluate a double integral over a rectangular region. Straightforward integration, order of integration might be given or need to be chosen.

 Applications of Derivatives (Increasing/Decreasing Intervals or Curve Sketching Aspect