Probability Engine · MTH112

Mathematics I (BSc CSIT, MTH112): the questions likely to come

84 analyzed questions from 7 past papers (2074-2081), grouped by syllabus unit — each with its probability, how often it's been asked, and where to study the answer.

Papers analyzed

2074-2081

Analyzed questions

across 7 syllabus units

Very likely units

high-probability topics

Units = 80% of marks

study these first

Model answers for this subject are being written. Every question links to its original paper so you can study from the source meanwhile.

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U2 · Q1/23 · 208110 marks

Derivatives and Applications of Derivatives

State Leibnitz's theorem for the nth derivative of a product. If (y = x^2 e^x), find (y_n).

20%

Occasional to appearAppeared in 1 of the last 1 board papers

Seen in

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MODEL ANSWERU2 · 10 marks

Leibnitz's Theorem

If $u$ and $v$ are functions of $x$ that are $n$ -times differentiable, then the $n$ th derivative of their product is

(uv)_n = \sum_{r=0}^{n} \binom{n}{r}\, u_{n-r}\, v_r = \binom{n}{0} u_n v + \binom{n}{1} u_{n-1} v_1 + \binom{n}{2} u_{n-2} v_2 + \cdots + \binom{n}{n} u\, v_n,

where $u_k = \dfrac{d^k u}{dx^k}$ and $v_k = \dfrac{d^k v}{dx^k}$ .

Finding $y_n$ for $y = x^2 e^x$

Take $u = e^x$ and $v = x^2$ (choose the polynomial as $v$ since its higher derivatives vanish).

Derivatives of $u = e^x$ : $\;u_{n} = e^x,\quad u_{n-1} = e^x,\quad u_{n-2} = e^x.$

Derivatives of $v = x^2$ : $\;v = x^2,\quad v_1 = 2x,\quad v_2 = 2,\quad v_3 = v_4 = \cdots = 0.$

By Leibnitz's theorem, only the first three terms survive:

y_n = \binom{n}{0} u_n v + \binom{n}{1} u_{n-1} v_1 + \binom{n}{2} u_{n-2} v_2.

Substituting:

y_n = e^x\, x^2 + n\, e^x (2x) + \frac{n(n-1)}{2}\, e^x (2).

\boxed{\,y_n = e^x\left[\,x^2 + 2nx + n(n-1)\,\right]\,}

AI-generated answer · unverifiedView in 2081 paper →

U2 · Question 1 of 23

Question Priority · U2ranked by appearance likelihood — study top-down

Derivatives and Applications of Derivatives

Analyzed next20%

★ TOP PICK

State Leibnitz's theorem for the nth derivative of a product. If (y = x^2 e^x), find (y_n).

10 marksSEEN IN

20%

Define the derivative of a function from first principles. Find the derivative of (\sin x) using the first-principle method.

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18%

State and prove the mean value theorem (Lagrange's). Verify it for (f(x) = \sqrt{x}) on [1,4].

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If (y = (\sin^{-1} x)^2), prove that ((1 - x^2)y_{n+2} - (2n+1)xy_{n+1} - n^2 y_n = 0).

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State Leibnitz's theorem. If (y = \sin^{-1} x), find the nth derivative (y_n) at x = 0.

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If (y = \tan^{-1} x), find (\frac{dy}{dx}).

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State and verify Rolle's theorem for (f(x) = x^2 - 5x + 6) on [2,3].

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Find the equation of the tangent to the curve (y = x^2 + 2x) at the point (1,3).

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State and prove that a differentiable function is continuous. Examine the continuity and differentiability of f(x) = |x| at x = 0.

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If (y = x^n \log x), find (y_n).

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State and verify Cauchy's mean value theorem for (f(x) = x^2, g(x) = x) on [1,2].

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Find the maximum and minimum values of (f(x) = \sin x + \cos x).

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If (y = e^{m\sin^{-1} x}), find (\frac{dy}{dx}).

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Differentiate (\tan^{-1}\left(\frac{2x}{1 - x^2}\right)) with respect to x.

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State and verify Rolle's theorem for (f(x) = \sin x) on ([0, \pi]).

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Find the points of inflection of (y = x^3 - 6x^2 + 9x).

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Find the radius of curvature of the curve (y = x^2) at the point (1,1).

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Differentiate (x^x) with respect to x.

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State and verify Lagrange's mean value theorem for (f(x) = x^2) on [2,4].

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Find the maximum and minimum values of (f(x) = x^3 - 3x + 2).

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If (y = e^{ax}\sin bx), find (\frac{dy}{dx}).

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State and verify Rolle's theorem for (f(x) = x^2 - 4x + 3) on [1,3].

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03The mock

Sit a probable paper

A full mock exam built from the most likely questions, mirroring the real paper's structure. Every slot is a real past question.

Most Probable Paper

Mirrors the real structure · 60 marks · based on 7 past papers

Section A: Long Answer QuestionsAttempt any TWO questions.

1.
State Leibnitz's theorem for the nth derivative of a product. If (y = x^2 e^x), find (y_n).
[10 marks]
Derivatives and Applications of DerivativesVery likelyfrom 2081 paper →
Asked once (2081); so far only in internal assessments, not the board; and its topic (Derivatives and Applications of Derivatives) appears in 100% of years.
2.
Define the derivative of a function from first principles. Find the derivative of (\sin x) using the first-principle method.
[10 marks]
Derivatives and Applications of DerivativesVery likelyfrom 2080 paper →
Asked once (2080); so far only in internal assessments, not the board; and its topic (Derivatives and Applications of Derivatives) appears in 100% of years.
3.
State and prove the mean value theorem (Lagrange's). Verify it for (f(x) = \sqrt{x}) on [1,4].
[10 marks]
Derivatives and Applications of DerivativesVery likelyfrom 2079 paper →
Asked once (2079); so far only in internal assessments, not the board; and its topic (Derivatives and Applications of Derivatives) appears in 100% of years.

Section B: Short Answer QuestionsAttempt any EIGHT questions.

1.
If (y = \tan^{-1} x), find (\frac{dy}{dx}).
[5 marks]
Derivatives and Applications of DerivativesVery likelyfrom 2081 paper →
Asked once (2081); so far only in internal assessments, not the board; and its topic (Derivatives and Applications of Derivatives) appears in 100% of years.
2.
State and verify Rolle's theorem for (f(x) = x^2 - 5x + 6) on [2,3].
[5 marks]
Derivatives and Applications of DerivativesVery likelyfrom 2081 paper →
Asked once (2081); so far only in internal assessments, not the board; and its topic (Derivatives and Applications of Derivatives) appears in 100% of years.
3.
Find the equation of the tangent to the curve (y = x^2 + 2x) at the point (1,3).
[5 marks]
Derivatives and Applications of DerivativesVery likelyfrom 2081 paper →
Asked once (2081); so far only in internal assessments, not the board; and its topic (Derivatives and Applications of Derivatives) appears in 100% of years.
4.
If (y = x^n \log x), find (y_n).
[5 marks]
Derivatives and Applications of DerivativesVery likelyfrom 2080 paper →
Asked once (2080); so far only in internal assessments, not the board; and its topic (Derivatives and Applications of Derivatives) appears in 100% of years.
5.
State and verify Cauchy's mean value theorem for (f(x) = x^2, g(x) = x) on [1,2].
[5 marks]
Derivatives and Applications of DerivativesVery likelyfrom 2080 paper →
Asked once (2080); so far only in internal assessments, not the board; and its topic (Derivatives and Applications of Derivatives) appears in 100% of years.
6.
Find the maximum and minimum values of (f(x) = \sin x + \cos x).
[5 marks]
Derivatives and Applications of DerivativesVery likelyfrom 2080 paper →
Asked once (2080); so far only in internal assessments, not the board; and its topic (Derivatives and Applications of Derivatives) appears in 100% of years.
7.
If (y = e^{m\sin^{-1} x}), find (\frac{dy}{dx}).
[5 marks]
Derivatives and Applications of DerivativesVery likelyfrom 2079 paper →
Asked once (2079); so far only in internal assessments, not the board; and its topic (Derivatives and Applications of Derivatives) appears in 100% of years.
8.
Differentiate (\tan^{-1}\left(\frac{2x}{1 - x^2}\right)) with respect to x.
[5 marks]
Derivatives and Applications of DerivativesVery likelyfrom 2078 paper →
Asked once (2078); so far only in internal assessments, not the board; and its topic (Derivatives and Applications of Derivatives) appears in 100% of years.
9.
State and verify Rolle's theorem for (f(x) = \sin x) on ([0, \pi]).
[5 marks]
Derivatives and Applications of DerivativesVery likelyfrom 2078 paper →
Asked once (2078); so far only in internal assessments, not the board; and its topic (Derivatives and Applications of Derivatives) appears in 100% of years.

04The receipts

Behind the numbers

The raw evidence the predictions are computed from: marks per unit per year, syllabus weights, trends, and coverage.

Show the heatmap, topic table and coverage analysis

The receipt: marks per unit, per year

Each row is a syllabus unit, each column an exam year, each cell the marks that unit earned that year. Click any cell to see the actual questions behind it.

Marks:nonefew → many

2074

2075

2077

2078

2079

2080

2081

Total

U2Derivatives and Applications of Derivatives

145

U4Integrals and Transcendental Functions, Techniques of Integration

130

U1Preliminaries: Functions, Limits and Continuity

U3Integration and Applications of Definite Integrals

U7Vectors and the Geometry of Space, Partial Derivatives

U5Infinite Sequences and Series

U6Conic Sections and Polar Coordinates

#	Syllabus unit	Probability	Appeared	Avg marks	Syllabus weight	Exam vs syllabus	Trend	Questions
1	U2Derivatives and Applications of Derivatives	Very likely100%	2074 2075 2077 2078 2079 2080 2081	20.7	20%9 lecture hrs	Over-examinedexam 28% · syllabus 20%	Steady	none repeat23 total
2	U4Integrals and Transcendental Functions, Techniques of Integration	Very likely100%	2074 2075 2077 2078 2079 2080 2081	18.6	18%8 lecture hrs	Over-examinedexam 25% · syllabus 18%	Steady	none repeat20 total
3	U1Preliminaries: Functions, Limits and Continuity	Very likely100%	2074 2075 2077 2078 2079 2080 2081	12.1	11%5 lecture hrs	Balancedexam 16% · syllabus 11%	Steady	none repeat13 total
4	U3Integration and Applications of Definite Integrals	Very likely100%	2074 2075 2077 2078 2079 2080 2081	12.1	20%9 lecture hrs	Balancedexam 16% · syllabus 20%	Fading	none repeat12 total
5	U7Vectors and the Geometry of Space, Partial Derivatives	Very likely100%	2074 2075 2077 2078 2079 2080 2081	10	9%4 lecture hrs	Balancedexam 13% · syllabus 9%	Steady	none repeat14 total
6	U5Infinite Sequences and Series	Occasional29%	2077 2079	5	13%6 lecture hrs	Under-examinedexam 2% · syllabus 13%	Steady	none repeat2 total
7	U6Conic Sections and Polar Coordinates	Occasional0%		0	9%4 lecture hrs	Under-examinedexam 0% · syllabus 9%	Steady	None

Study smart, not hard

Drag the slider: studying the top 4 units in priority order covers ~85% of all observed marks.

Units to study4/7

~80% line

Lecture time vs exam marks

Where the exam pays more than the curriculum spends: ● lectures vs ● exam marks, as a share of the whole course. A long teal-leading bar = high-yield unit.

U2Derivatives and Applications of Derivatives

20% of lectures → 28% of markshigh yield

U4Integrals and Transcendental Functions, Techniques of Integration

18% of lectures → 25% of markshigh yield

U1Preliminaries: Functions, Limits and Continuity

11% of lectures → 16% of marks

U3Integration and Applications of Definite Integrals

20% of lectures → 16% of marks

U7Vectors and the Geometry of Space, Partial Derivatives

9% of lectures → 13% of marks

U5Infinite Sequences and Series

13% of lectures → 2% of markslow yield

U6Conic Sections and Polar Coordinates

9% of lectures → 0% of markslow yield

Mathematics I (BSc CSIT, MTH112): the questions likely to come

Leibnitz's Theorem

Finding $y_n$ for $y = x^2 e^x$

Derivatives and Applications of Derivatives

Sit a probable paper

Most Probable Paper

Behind the numbers

The receipt: marks per unit, per year

Study smart, not hard

Lecture time vs exam marks

2081 B.S.