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JEE Mains Maths · Determinants

Determinants as Functions of x

Determinants whose entries depend on x or on an angle: simplifying them to a single function, then finding its greatest value, its roots, its derivative or a limit.

All Determinants notesJEE Mains Maths strategy

Why this matters

Fifteen PYQs, twelve of them multiple choice. Eight reduce a trigonometric determinant to one expression and then find its range or solve it; seven differentiate, integrate or take a limit of a determinant function. Two ideas cover the page.

Concept 1 of 2: Trigonometric determinants

Simplify first with row operations and identities like sin⁡2θ+cos⁡2θ=1\sin^2\theta+\cos^2\theta=1; the determinant usually collapses to a short expression in one trigonometric function. Then the range, the maximum or the number of solutions in an interval is an ordinary trigonometry question.

Definition

  • Simplify by row operations and identities before expanding.
  • The result is a function of θ\theta; find its range or roots.
  • sin⁡2θ∈[0,1]\sin^2\theta\in[0,1], asin⁡θ+bcos⁡θ∈[−a2+b2,a2+b2]a\sin\theta+b\cos\theta\in\left[-\sqrt{a^2+b^2},\sqrt{a^2+b^2}\right].

Range of a sinusoid

asin⁡θ+bcos⁡θ∈[−a2+b2, a2+b2]a\sin\theta+b\cos\theta\in\left[-\sqrt{a^2+b^2},\ \sqrt{a^2+b^2}\right]

Worked example

Find the range of f(θ)=∣1sin⁡θ1−sin⁡θ1sin⁡θ−1−sin⁡θ1∣f(\theta)=\begin{vmatrix}1&\sin\theta&1\\-\sin\theta&1&\sin\theta\\-1&-\sin\theta&1\end{vmatrix}.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2025 · 29 Jan 2025 · Q52Moderate

Example 1 · Determinants · Determinants as Functions of x

Let M and m respectively be the maximum and the minimum values of
f(x)=∣1+sin⁡2xcos⁡2x4sin⁡4xsin⁡2x1+cos⁡2x4sin⁡4xsin⁡2xcos⁡2x1+4sin⁡4x∣,x∈Rf(x) =\left| \begin{matrix} 1 +\sin^{2}x & \cos^{2}x & 4\sin4x \\ \sin^{2}x & 1 +\cos^{2}x & 4\sin4x \\ \sin^{2}x & \cos^{2}x & 1 + 4\sin4x \end{matrix} \right|,x \in R
Then M4−m4M^{4}-m^{4} is equal to :

Simplify before expanding

Expanding a trigonometric determinant straight away produces long products that are easy to get wrong. One row operation or identity usually removes most terms first.
Drill 7 more on trigonometric determinants

Concept 2 of 2: Derivatives, integrals and limits

A determinant of functions can be expanded into one function and then differentiated, integrated or taken to a limit. For a derivative there is a shortcut: differentiate one row at a time and add the determinants. A constant row or column survives unchanged.

Definition

  • ddxΔ=\frac{d}{dx}\Delta= sum of the determinants with one row differentiated.
  • Or expand to a single function first.
  • A row that is constant differentiates to zero, so that term drops.

Derivative of a determinant

ddx∣f1f2g1g2∣=∣f1′f2′g1g2∣+∣f1f2g1′g2′∣\frac{d}{dx}\begin{vmatrix}f_1&f_2\\g_1&g_2\end{vmatrix}=\begin{vmatrix}f_1'&f_2'\\g_1&g_2\end{vmatrix}+\begin{vmatrix}f_1&f_2\\g_1'&g_2'\end{vmatrix}

Worked example

f(x)=∣x1x22∣f(x)=\begin{vmatrix}x&1\\x^2&2\end{vmatrix}. Find f′(1)f'(1).
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2025 · 24 Jan 2025 · Q140Moderate

Example 2 · Determinants · Determinants as Functions of x

For some a,ba,b, let
f(x)=∣a+sin⁡xx1ba1+sin⁡xxba1b+sin⁡xx∣, x≠0f(x) =\left| \begin{matrix} a +\frac{\sin x}{x} & 1 & b \\ a & 1 +\frac{\sin x}{x} & b \\ a & 1 & b +\frac{\sin x}{x} \end{matrix} \right|,\ x \neq 0
lim⁡x→0f(x)=λ+μa+vb\lim_{x \rightarrow 0} f(x) = \lambda + \mu a + vb. Then (λ+μ+v)2(\lambda + \mu + v)^{2} is equal to:

Differentiate one row at a time

The derivative of a determinant is not the determinant of the derivatives. Differentiate each row separately and add, or expand first.
Drill 6 more on derivatives, integrals and limits

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

  • Trigonometric determinants

    Range of a sinusoid

    asin⁡θ+bcos⁡θ∈[−a2+b2, a2+b2]a\sin\theta+b\cos\theta\in\left[-\sqrt{a^2+b^2},\ \sqrt{a^2+b^2}\right]
  • Derivatives, integrals and limits

    Derivative of a determinant

    ddx∣f1f2g1g2∣=∣f1′f2′g1g2∣+∣f1f2g1′g2′∣\frac{d}{dx}\begin{vmatrix}f_1&f_2\\g_1&g_2\end{vmatrix}=\begin{vmatrix}f_1'&f_2'\\g_1&g_2\end{vmatrix}+\begin{vmatrix}f_1&f_2\\g_1'&g_2'\end{vmatrix}

Watch out for (2)

Test yourself on Determinants

20 past JEE Mains questions from this chapter, timed at 48 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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