KCET · Maths · Determinants
The value of \(\left|\begin{array}{lll}x & p & q \\ p & x & q \\ p & q & x\end{array}\right|\) is
- A \(x(x-p)(x-q)\)
- B \((x-p)(x-q)(x+p+q)\)
- C \((p-q)(x-q)(x-p)\)
- D \(p q(x-p)(x-q)\)
Answer & Solution
Correct Answer
(B) \((x-p)(x-q)(x+p+q)\)
Step-by-step Solution
Detailed explanation
Given
\[
\begin{aligned}
&\left|\begin{array}{lll}
x & p & q \\
p & x & q \\
p & q & x
\end{array}\right| \\
=& x\left[x^{2}-q^{2}\right]-p[p x-p q]+q[p q-p x] \\
=& x^{3}-x q^{2}-p^{2} x+p^{2} q+p q^{2}-p q x \\
=& x(x-q)(x+q)-p^{2}(x-q)-p q(x-q) \\
=&(x-q)\left[x^{2}+x q-p^{2}-p q\right] \\
=&(x-q)(x-p)(x+p+q)
\end{aligned}
\]
\[
\begin{aligned}
&\left|\begin{array}{lll}
x & p & q \\
p & x & q \\
p & q & x
\end{array}\right| \\
=& x\left[x^{2}-q^{2}\right]-p[p x-p q]+q[p q-p x] \\
=& x^{3}-x q^{2}-p^{2} x+p^{2} q+p q^{2}-p q x \\
=& x(x-q)(x+q)-p^{2}(x-q)-p q(x-q) \\
=&(x-q)\left[x^{2}+x q-p^{2}-p q\right] \\
=&(x-q)(x-p)(x+p+q)
\end{aligned}
\]
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- Let \(\boldsymbol{f}: \boldsymbol{R} \rightarrow \boldsymbol{R}\) be defined by \(f(x)=3 x^2-5\) and \(g: R \rightarrow R\) by \(g(x)=\frac{x}{x^2+1}\), then \(g \circ f\) isKCET 2023 Easy
- \(f: R \rightarrow R\) defined by \(f(x)\) is equal to \(\left\{\begin{array}{c}2 x, x>3 \\ x^{2}, 1 < x \leq 3, \text { then } f(-2)+f(3)+f(4) \text { is } \\ 3 x, x \leq 1\end{array}\right.\)KCET 2021 Easy
- \(\frac{1}{2 \cdot 5}+\frac{1}{5 \cdot 8}+\) equal to
upto \(n\) terms isKCET 2008 Medium - If \(A, B\) and \(C\) are three independent events such that \(P(A)=P(B)=P(C)=P\), then \(P\) (at least two of \(A, B\) and \(C\) occur) is equal toKCET 2021 Easy
- If \(A M\) and \(G M\) of roots of a quadratic equation are 5 and 4 , respectively, then the quadratic equation isKCET 2024 Easy
- The points \(A(1,2), B(2,4)\) and \(C(4,8)\) form a/anKCET 2011 Easy
More PYQs from KCET
- Which of the following is the most stable complex ion?KCET 2026 Medium
- Electrophile that participates in nitration of benzene isKCET 2016 Medium
- Match the compounds given in List - I with the items given in List - II.
\begin{array}{|l|l|l|l|}\hline & \text{List - I} & & \text{List - II} \\\hline \text{I}. & \text{Benzenesulphonyl Chloride} & \text{(a)} & \text{Zwitterioin} \\\hline \text{II.} & \text{Sulphanilic acid} & \text{(b)} & \text{Hinsberg reagent} \\\hline \text{III.} & \text{Alkyl Diazonium salts} & \text{(c)} & \text{Dyes} \\\hline \text{IV.} & \text{Aryl Diazonium salts} & \text{(d)} & \text{Conversion to alcohols} \\\hline\end{array}KCET 2025 Medium - If the function \( f(x) \) defined by
\[
f(x)=\frac{x^{100}}{100}+\frac{x^{99}}{99}+\ldots \ldots+\frac{x^{2}}{2}+x+1, \text { then } f^{\prime}(0)=
\]KCET 2014 Easy - Identify the correctly matched pair/pairs of the germ layers and their derivatives.
A. Ectoderm-Epidermis
B. Endoderm-Dermis
C. Mesoderm-Muscles
D. Mesoderm-Notochord
E. Endoderm-Enamel of teethKCET 2009 Hard - Which of the following is an incorrect statement?KCET 2021 Easy