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Mathematical Tools is not a chapter JEE Main asks about by name, and that is precisely why students skip it and then bleed marks everywhere else. JEE Main physics is written in calculus. If differentiating a polynomial or reading the area under a curve still takes you a moment, that moment will cost you in kinematics, in electricity and in every numerical in between. Fix it now, once. Ten questions below. Give them ten honest minutes.
What this chapter carries in JEE Main
- JEE Main does not ask this chapter on its own. It asks it inside every other chapter.
- Calculus appears directly in kinematics, work and energy, and electromagnetic induction.
- Vectors, binomial approximation and small-angle approximation are used throughout the paper.
What the syllabus says
In the syllabus (no unit of its own): Nothing. NTA does not list a unit for this. Vectors and basic calculus run through every other chapter, so you still need them.
Taken out in 2024: the whole vector block was taken out of Kinematics in 2024 — scalars and vectors, addition and subtraction, zero vector, scalar and vector products, unit vector, resolution of a vector.
Taken from the NTA syllabus for JEE Main 2027. See the full syllabus, unit by unit.
Formulas you must know by heart
| Power rule | d/dx (xn) = n xn−1 |
| Exponential and log | d/dx (eax) = a eax, d/dx (ln x) = 1/x |
| Trigonometric derivatives | d/dx (sin x) = cos x, d/dx (cos x) = −sin x |
| Standard integrals | ∫ xn dx = xn+1/(n+1) + C, ∫ dx/x = ln x + C |
| A useful definite integral | ∫0π sin x dx = 2 |
| Maximum or minimum | set dy/dx = 0 |
| Binomial approximation | (1 + x)n ≈ 1 + nx when x ≪ 1 |
| Small angle approximation | sinθ ≈ tanθ ≈ θ, cosθ ≈ 1 |
| Scalar product in components | A·B = AxBx + AyBy + AzBz |
| Magnitude of a vector | |A| = √(Ax2 + Ay2 + Az2) |
| Vector product | |A × B| = AB sinθ |
The binomial approximation is the quiet workhorse of JEE Main. It is what lets you simplify a field at a large distance, a small change in time period, or a tiny correction to g, without ever touching a calculator.
Ten questions. Give yourself ten minutes.
Write your answers down before you scroll. A question you looked up is a question you have not answered.
Q1. If y = x3 − 6x2 + 9x, the smaller value of x at which dy/dx = 0 is
- (a) 1
- (b) 2
- (c) 3
- (d) 0
Q2. The value of ∫0π sin x dx is
- (a) 0
- (b) 1
- (c) 2
- (d) π
Q3. The derivative of e2x with respect to x is
- (a) e2x
- (b) 2e2x
- (c) e2x/2
- (d) 2x e2x−1
Q4. If A = 3î + 4ĵ and B = 4î − 3ĵ, then A·B is
- (a) 0
- (b) 24
- (c) 12
- (d) 25
Q5. The magnitude of the vector A = 2î + 3ĵ + 6k̂ is
- (a) 5
- (b) 7
- (c) 11
- (d) 9
Q6. For a small angle θ measured in radian, sinθ is approximately
- (a) 1
- (b) θ
- (c) θ2
- (d) 0
Q7. The derivative of ln x with respect to x is
- (a) x
- (b) 1/x
- (c) ln x
- (d) ex
Q8. For x much smaller than 1, (1 + x)n is approximately
- (a) 1 + nx
- (b) 1 − nx
- (c) nx
- (d) 1 + xn
Q9. The angle made by the vector A = î + ĵ with the x-axis is
- (a) 30°
- (b) 45°
- (c) 60°
- (d) 90°
Q10. ∫ dx/x is equal to
- (a) x2/2
- (b) ln x + C
- (c) −1/x2
- (d) 1
Answer key, with a hint for each
| Q1 | (a) | dy/dx = 3x2 − 12x + 9 = 3(x − 1)(x − 3), so x = 1 or 3. |
| Q2 | (c) | −cos x evaluated from 0 to π gives −(−1) − (−1) = 2. |
| Q3 | (b) | Chain rule. The 2 inside the exponent comes out in front. |
| Q4 | (a) | 3(4) + 4(−3) = 0. The two vectors are perpendicular. |
| Q5 | (b) | √(4 + 9 + 36) = √49. |
| Q6 | (b) | For small angles the sine, the tangent and the angle itself are all nearly equal, provided the angle is in radian. |
| Q7 | (b) | And its integral runs the other way: ∫dx/x = ln x. |
| Q8 | (a) | The binomial approximation. Everything beyond the first two terms is too small to matter. |
| Q9 | (b) | Equal components along x and y, so tanθ = 1. |
| Q10 | (b) | This is the one integral the power rule cannot handle, because n = −1 would divide by zero. |
Now mark yourself honestly
JEE Main marks the way it marks. Use the same rule here:
- +4 for a right answer
- −1 for a wrong one
- 0 for a blank
So ten questions here are out of 40. In the real paper, 25 Physics questions are out of 100.
If you scored 32 or more, the basics of this chapter are secure. Move to harder questions, not more of these.
If you scored below 24, do not feel bad about it. You have just found out something useful before the exam did. Go back to the formula table above, learn it properly, and sit these ten again tomorrow.
And if you guessed a few and they landed, count those as wrong. A guess that works once will not work in January.
The full chapter, on a clock
Ten questions tell you a little. Twenty-five tell you the truth.
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