Chapter 7

If \(p+q+r=0=a+b+c\), then the value of the determinant \(\left|\begin{array}{lll}p a & q b & r c \\ q c & r a & p b \\ r b & p c & q a\end{array}\right|\) isa. 0 b. \(p a+q b+r c\) c. 1 d. none of these

Which of the following has/have value equal to zero? a. \(\left|\begin{array}{ccc}8 & 2 & 7 \\ 12 & 3 & 5 \\ 16 & 4 & 3\end{array}\right|\) b. \(\left|\begin{array}{ccc}1 / a & a^{2} & b c \\ 1 / b & b^{2} & a c \\ 1 / c & c^{2} & a b\end{array}\right|\) c. \(\left|\begin{array}{ccc}a+b & 2 a+b & 3 a+b \\ 2 a+b & 3 a+b & 4 a+b \\\ 4 a+b & 5 a+b & 6 a+b\end{array}\right|\) d. \(\left|\begin{array}{lll}2 & 43 & 6 \\ 7 & 35 & 4 \\ 3 & 17 & 2\end{array}\right|\)

\text { Solve for } x,\left|\begin{array}{lll} x^{2}-a^{2} & a^{2}-b^{2} & x^{2}-c^{2} \\ (x-a)^{3} & (x-b)^{3} & (x-c)^{3} \\ (x+a)^{3} & (x+b)^{3} & (x+c)^{2} \end{array}\right|=0, a \neq b \neq c \text {. }

If a determinant of order \(3 \times 3\) is formed by using the numbers 1 or \(-1\), then the minimum value of the determinant is a. \(-2\) b. \(-4\) c. 0 d. \(-8\)

If \(g(x)=\left|\begin{array}{lll}a^{-x} & e^{x \log _{e} a} & x^{2} \\ a^{-3 x} & e^{3 x \log _{e} n} & x^{4} \\ a^{-5 x} & e^{5 x \log _{e}^{a}} & 1\end{array}\right|\), then a. graphs of \(g(x)\) is symmetrical about origin b. graphs of \(g(x)\) is symmetrical about \(Y\)-axis c. \(\left.\frac{d^{4} g(x)}{d x^{4}}\right|_{K=0}=0\) d. \(f(x)=g(x) \times \log \left(\frac{a-x}{a+x}\right)\) is an odd function

If \(z=\left|\begin{array}{ccc}-5 & 3+4 i & 5-7 i \\ 3-4 i & 6 & 8+7 i \\ 5+7 i & 8-7 i & 9\end{array}\right|\), then \(z\) is a. purely real b. purely imaginary c. \(a+i b\), where \(a \neq 0, b \neq 0\) d. \(a+i b\), where \(b=4\)

If \(\alpha, \beta, \gamma\) are the roots of \(p x^{3}+q x^{2}+r=0\), then the value of the determinant \(\left|\begin{array}{ccc}\alpha \beta & \beta \gamma & \gamma \alpha \\ \beta \gamma & \gamma \alpha & \alpha \beta \\ \gamma \alpha & \alpha \beta & \beta \gamma\end{array}\right|\) is a. \(\vec{p}\) b. \(q\) c. 0 d. \(r\)

If \(\Delta=\left|\begin{array}{ccc}\sin \theta \cos \phi & \sin \theta \sin \phi & \cos \theta \\ \cos \theta \cos \phi & \cos \theta \sin \phi & -\sin \theta \\ -\sin \theta \sin \phi & \sin \theta \cos \phi & 0\end{array}\right|\) then a. \(\Delta\) is independent of \(\theta\) b. \(\Delta\) is independent of \(\phi\) c. \(\Delta\) is a constant d. \(\left.\frac{d \Delta}{d \theta}\right]_{\theta_{-\pi / 2}}=0\)

When the determinant \(\left|\begin{array}{ccc}\cos 2 x & \sin ^{2} x & \cos 4 x \\ \sin ^{2} x & \cos 2 x & \cos ^{2} x \\ \cos 4 x & \cos ^{2} x & \cos 2 x\end{array}\right|\) is expanded in powers of \(\sin x\), then the constant term in that expression is a. 1 b. 0 c. \(-1\) d. 2

If \(\phi(\alpha, \beta)=\left|\begin{array}{ccc}\cos \alpha & -\sin \alpha & 1 \\\ \sin \alpha & \cos \alpha & 1 \\ \cos (\alpha+\beta) & -\sin (\alpha+\beta) & 1\end{array}\right|\), then a. \(f(300,200)=f(400,200)\) b. \(f(200,400)=f(200,600)\) c. \(f(100,200)=f(200,200)\) d. none of these

The determinant \(\Delta=\left|\begin{array}{ccc}a^{2}+x & a b & a c \\ a b & b^{2}+x & b c \\ a c & b c & c^{2}+x\end{array}\right|\) is divisible a. \(x\) b. \(x^{2}\) \(\mathbf{C}, x^{3}\) d. none of these

Show that in general there are three values of \(t\) for which the following system of equations has a non-trivial solution: \((a-t) x+b y+c_{2}=0\) \(b x+(c-t) y+a z=0\) \(c x+a y+(b-t) z=0\) Express the product of these values of \(t\) in the form of a determinant.

If \(\left|\begin{array}{ccc}a & b-c & c+b \\ a+c & b & c-a \\ a-b & a+b & c\end{array}\right|=0\), then the line \(a x+b y+c=0\) passes through the fixed point which is a. \((1,2)\) b. \((I, 1)\) c. \((-2,1)\) d. \((1,0)\)

\(f(x)=\left|\begin{array}{ccc}a & -1 & 0 \\ a x & a & -1 \\ a x^{2} & a x & a\end{array}\right|\), then \(f(2 x)-f(x)\) is divisible by 1\. \(x\) b. \(a\) \(2 a+3 x\) d. \(x^{2}\)

Show that in general there are three values of \(t\) for which the following system of equations has a non-trivial solution: \((a-t) x+b y+c_{2}=0\) \(b x+(c-t) y+a z=0\) \(c x+a y+(b-t) z=0\) Express the product of these values of \(t\) in the form of a determinant.

If \(a x_{1}^{2}+b y_{1}^{2}+c z_{1}^{2}=a x_{2}^{2}+b y_{2}^{2}+c z_{2}^{2}=a x_{3}^{2}+b y_{3}^{2}+c z_{3}^{2}=d\), \(a x_{2} x_{3}+b y_{2} y_{3}+c z_{2} z_{3}=a x_{3} x_{1}+a x_{3} x_{1}+b y_{y} y_{1}+c z_{3} z_{1}\) \(=a x_{1} x_{2}+b y_{1} y_{2}+c z_{1} z_{2}=f\), then prove that \(\left|\begin{array}{lll}x_{1} & y_{1} & z_{1} \\ x_{2} & y_{2} & z_{2} \\\ x_{3} & y_{3} & z_{3}\end{array}\right|=(d-f)\left\\{\frac{(d+2 f)}{a b c}\right\\}^{1 / 2}\)

If [ ] denotes the greatest integer less than or equal to the real number under consideration, and \(-1 \leq x<0,0 \leq y<1\), \(1 \leq z<2\), then the value of the determinant \(\left|\begin{array}{ccc}x]+1 & {[y]} & {[z]} \\ {[x]} & {[y]+1} & {[z]} \\\ {[x]} & {[y]} & {[z]+1}\end{array}\right|\) is a. \([x]\) b. [y] c. \([z]\) d. none of these

). If \(\Delta(x)=\left|\begin{array}{ccc}x^{2}+4 x-3 & 2 x+4 & 13 \\ 2 x^{2}+5 x-9 & 4 x+5 & 26 \\ 8 x^{2}-6 x+1 & 16 x-6 & 104\end{array}\right|=a x^{3}+b x^{2}+c x+d\). then a. \(a=3\) b. \(b=0\) c. \(c=0\) d. None of these

If \(a x_{1}^{2}+b y_{1}^{2}+c z_{1}^{2}=a x_{2}^{2}+b y_{2}^{2}+c z_{2}^{2}=a x_{3}^{2}+b y_{3}^{2}+c z_{3}^{2}=d\), \(a x_{2} x_{3}+b y_{2} y_{3}+c z_{2} z_{3}=a x_{3} x_{1}+a x_{3} x_{1}+b y_{y} y_{1}+c z_{3} z_{1}\) \(=a x_{1} x_{2}+b y_{1} y_{2}+c z_{1} z_{2}=f\), then prove that \(\left|\begin{array}{lll}x_{1} & y_{1} & z_{1} \\ x_{2} & y_{2} & z_{2} \\\ x_{3} & y_{3} & z_{3}\end{array}\right|=(d-f)\left\\{\frac{(d+2 f)}{a b c}\right\\}^{1 / 2}\)

Let \(a, b, c \in R\) such that no two of them are equal and satisfy \(\left|\begin{array}{lll}2 a & b & c \\ b & c & 2 a \\ c & 2 a & b\end{array}\right|=0\), then equation \(24 a x^{2}+4 b x+c=0\) has a. at least one root in \([0,1]\) b. at least one root in \(\left[-\frac{1}{2}, \frac{1}{2}\right]\) c. at least one root in \([-1,0]\) d. at least two roots in \([0,2]\)

). If \(\Delta(x)=\left|\begin{array}{ccc}x^{2}+4 x-3 & 2 x+4 & 13 \\ 2 x^{2}+5 x-9 & 4 x+5 & 26 \\ 8 x^{2}-6 x+1 & 16 x-6 & 104\end{array}\right|=a x^{3}+b x^{2}+c x+d\). then a. \(a=3\) b. \(b=0\) c. \(c=0\) d. None of these

If \(f(\theta)=\left|\begin{array}{ccc}\sin ^{2} A & \cot A & 1 \\ \sin ^{2} B & \cos B & 1 \\ \sin ^{2} C & \cos C & 1\end{array}\right|\), then a. tan \(A+\tan B+C\) b. \(\operatorname{Cot} A \cot B \cot C\) c. \(\sin ^{2} A+\sin ^{2} B+\sin ^{2} C\) d. 0

Evaluate \(\left|\begin{array}{ccc}{\underline{\phantom{xx}}}^{x} C_{1} & { }^{x} C_{2} & { }^{x} C_{3} \\ { }^{y} C_{1} & { }^{y} C_{2} & { }^{y} C_{3} \\ { }^{2} C_{1} & { }^{2} C_{2} & { }^{2} C_{3}\end{array}\right| .\)

If \(\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}=a^{2}\) \(\left(x_{2}-x_{3}\right)^{2}+\left(y_{2}-y_{3}\right)^{2}=b^{2}\) \(\left(x_{3}-x_{1}\right)^{2}+\left(y_{3}-y_{1}\right)^{2}=c^{2}\) and \(k\left|\begin{array}{lll}x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\\ x_{3} & y_{3} & 1\end{array}\right|=(a+b+c)(b+c-a)(c+a-b)\) \(\times(a+b-c)\), then the value of \(k\) is a. b. 2 C. 4 d. none of these

Prove that \(\left|\begin{array}{ccc}-2 a & a+b & a+c \\ b+a & -2 b & b+c \\\ c+a & c+b & -2 c\end{array}\right|=4(b+c)(c+a)(a+b)\)

The value of the determinant \(\left|\begin{array}{lll}k a & k^{2}+a^{2} & 1 \\\ k b & k^{2}+b^{2} & 1 \\ k c & k^{2}+c^{2} & 1\end{array}\right|\) is a. \(k(a+b)(b+c)(c+a)\) b. \(k a b c\left(a^{2}+b^{2}+c^{2}\right)\) c. \(k(a-b)(b-c)(c-a)\) d. \(k(a+b-c)(b+c-a)(c+a-b)\)

If \(f(x)=\left|\begin{array}{ccc}3 & 3 x & 3 x^{2}+2 a^{2} \\ 3 x & 3 x^{2}+2 a^{2} & 3 x^{3}+6 a^{2} x \\ 3 x^{2}+2 a^{2} & 3 x^{3}+6 a^{2} x & 3 x^{4}+12 a^{2} x^{2}+2 a^{4}\end{array}\right|\), then a. \(f^{\prime}(x)=0\) b. \(y=f(x)\) is a straight line parallel to \(x\)-axis c. \(\int_{0}^{2} f(x) d x=32 a^{4}\) d. none of these

Prove that \(\left|\begin{array}{ccc}a x-b y-c z & a y+b x & c x+a z \\ a y+b x & b y-c z-a x & b z+c y \\ c x+a z & b z+c y & c z-a x-b y\end{array}\right|\) \(=\left(x^{2}+y^{2}+z^{2}\right)\left(a^{2}+b^{2}+c^{2}\right)(a x+b y+c z) .\)

If \(\left|\begin{array}{lll}1 & 1 & 1 \\ a & b & c \\ a^{3} & b^{3} & c^{3}\end{array}\right|=(a-b)(b-c)(c-a)(a+b+c)\), where \(a, b .\) care all different, then the determinant \(\left|\begin{array}{ccc}1 & 1 & 1 \\\ (x-a)^{2} & (x-b)^{2} & (x-c)^{2} \\ (x-b)(x-c) & (x-c)(x-a) & (x-a)(x-b)\end{array}\right|\) vanishes when a. \(a+b+c=0\) b. \(x=\frac{1}{3}(a+b+c)\) c. \(x=\frac{1}{2}(a+b+c)\) d. \(x=a+b+c\)

If \(\left|\begin{array}{lll}y z-x^{2} & z x-y^{2} & x y-z^{2} \\ x z-y^{2} & x y-z^{2} & y z-x^{2} \\ x y-z^{2} & y z-x^{2} & z x-y^{2}\end{array}\right|=\left|\begin{array}{ccc}r^{2} & u^{2} & u^{2} \\\ u^{2} & r^{2} & u^{2} \\ u^{2} & u^{2} & r^{2}\end{array}\right|\), then a. \(r^{2}=x+y+z\) b. \(r^{2}=x^{2}+y^{2}+z^{2}\) c. \(u^{2}=y z+z x+x y\) d. \(u^{2}=x y z\)

If \(\Delta(x)=\left|\begin{array}{lll}a_{1}+x & b_{1}+x & c_{1}+x \\ a_{2}+x & b_{2}+x & c_{2}+x \\ a_{3}+x & b_{3}+x & c_{3}+x\end{array}\right|\), show that \(\Delta^{\prime \prime}(x)=0\) and that \(\Delta(x)=\Delta(0)+S x\), where \(S\) denotes the sum of all the cofactors of all the elements in \(\Delta(0)\).

The determinant \(\left|\begin{array}{ccc}y^{2} & -x y & x^{2} \\ a & b & c \\\ a^{\prime} & b^{\prime} & c^{\prime}\end{array}\right|\) is equal to a. \(\left|\begin{array}{cc}b x+a y & c x+b y \\ b^{\prime} x+a^{\prime} y & c^{\prime} x+b^{\prime} y\end{array}\right|\) b. \(\left|\begin{array}{cc}a x+b y & b x+c y \\ a^{\prime} x+b^{\prime} y & b^{\prime} x+c^{\prime} y\end{array}\right|\) c. \(\left|\begin{array}{cc}b x+c y & a x+b y \\ b^{\prime} x+c^{\prime} y & a^{\prime} x+b^{\prime} y\end{array}\right|\) d. \(\left|\begin{array}{cc}a x+b y & b x+c y \\ a^{\prime} x+b^{\prime} y & b^{\prime} x+c^{\prime} y\end{array}\right|\)

If \(\left|\begin{array}{ccc}b^{2}+c^{2} & a b & a c \\ a b & c^{2}+a^{2} & b c \\ c a & c b & a^{2}+b^{2}\end{array}\right|=k a^{2} b^{2} c^{2}\), then the value of \(k\) is a. 2 b. 4 c. 0 d. none of these

If \(a, b\) and \(c\) are non-zero real numbers, then \(\Delta=\left|\begin{array}{ccc}b^{2} c^{2} & b c & b+c \\ c^{2} a^{2} & c a & c+a \\ a^{2} b^{2} & a b & a+b\end{array}\right|\) is equal to a. \(a b c\) b. \(a^{2} b^{2} c^{2}\) c. \(b c+c a+a b\) d. none of these

The values of \(k \in R\) for which the system of equations \(x+k y+3 z=0, k x+2 y+2 z=0,2 x+3 y+4 z=0\) admits of non-trivial solution is a. 2 b. \(5 / 2\) c. 3 d. \(5 / 4\)

The value of \(\left|\begin{array}{ccc}-1 & 2 & 1 \\ 3+2 \sqrt{2} & 2+2 \sqrt{2} & 1 \\ 3-2 \sqrt{2} & 2-2 \sqrt{2} & 1\end{array}\right|\) is equal to a. zero b. \(-16 \sqrt{2}\) c. \(-8 \sqrt{2}\) d. none of these

If determinent \(\left|\begin{array}{ccc}\cos (\theta+\phi) & -\sin (\theta+\phi) & \cos 2 \phi \\ \sin \theta & \cos \theta & \sin \phi \\\ -\cos \theta & \sin \theta & \cos \phi\end{array}\right|\) is a. positive b. independent of \(\theta\) c. independent of \(\phi\) d. none of these

If \(w\) is a complex cube root of unity, then value of \(\Delta=\left|\begin{array}{lll}a_{1}+b_{1} w & a_{1} w^{2}+b_{1} & c_{1}+b_{1} \bar{w} \\ a_{2}+b_{2} w & a_{2} w^{2}+b_{2} & c_{2}+b_{2} \bar{w} \\ a_{3}+b_{3} w & a_{3} w^{2}+b_{3} & c_{3}+b_{3} \bar{w}\end{array}\right|\) a. 0 b. \(-1\) c. 2 d. none of these

In triangle \(A B C\), if \(\left|\begin{array}{ccc}1 & 1 & 1 \\ \cot \frac{A}{2} & \cot \frac{B}{2} & \cot \frac{C}{2} \\ \tan \frac{B}{2}+\tan \frac{C}{2} & \tan \frac{C}{2}+\frac{A}{2} & \tan \frac{A}{2}+\tan \frac{B}{2}\end{array}\right|=0\), then the triangle must be a. equilateral b. isosceles c. obtuse angled d. none of these

In triangle \(A B C\), if \(\left|\begin{array}{ccc}1 & 1 & 1 \\ \cot \frac{A}{2} & \cot \frac{B}{2} & \cot \frac{C}{2} \\ \tan \frac{B}{2}+\tan \frac{C}{2} & \tan \frac{C}{2}+\frac{A}{2} & \tan \frac{A}{2}+\tan \frac{B}{2}\end{array}\right|=0\), then the triangle must be a. equilateral b. isosceles c. obtuse angled d. none of these

If \(a, b, c, d, e\) and \(f\) are in G.P., then the value of \(\left|\begin{array}{lll}a^{2} & d^{2} & x \\ b^{2} & e^{2} & y \\ c^{2} & f^{2} & z\end{array}\right|\) depends on a. \(x\) and \(y\) b. \(x\) and \(z\) c. \(y\) and \(z\) d. independent of \(x, y\) and \(z\)

The value of the determinant \(\left|\begin{array}{ccc}1 & 1 & 1 \\ { }^{m} C_{1} & { }^{n+1} C_{1} & { }^{m+2} C_{1} \\ { }^{m} C_{2} & { }^{m+1} C_{2} & { }^{m+2} C_{2}\end{array}\right|\) is equal to a. 1 b. \(-1\) c. 0 d. none of these

If \(x \neq y \neq z\) and \(\left|\begin{array}{lll}x & x^{2} & 1+x^{3} \\ y & y^{2} & 1+y^{3} \\ z & z^{2} & 1+z^{3}\end{array}\right|=0\), then the value of \(x y z\) is a. 1 b. 2 c. \(-1\) d. \(-2\)

If \(x \neq 0, y \neq 0, z \neq 0\) and \(\left|\begin{array}{ccc}1+x & 1 & 1 \\\ 1+y & 1+2 y & 1 \\ 1+z & 1+z & 1+3 z\end{array}\right|=0\), then \(x^{-1}+y^{-1}+z^{-1}\) is equal to a. \(-1\) b. \(-2\) c. \(-3\) d. none of these

If \(a_{1} b_{1} c_{1}, a_{2} b_{2} c_{2}\) and \(a_{3} b_{3} c_{3}\) are 3 -digit even natural numbers and \(\Delta=\left|\begin{array}{lll}c_{1} & a_{1} \cdot & b_{1} \\ c_{2} & a_{2} & b_{2} \\ c_{3} & a_{3} & b_{3}\end{array}\right|\), then \(\Delta\) is a. divisible by 2 but not necessarily by 4 b. divisible by 4 but not necessarily by 8 c. divisible by 8 d. nonc of these

The value of the determinant of \(n^{\text {lh }}\) order, being given by \(\left|\begin{array}{cccc}x & 1 & 1 & \cdots \\ 1 & x & 1 & \cdots \\ 1 & 1 & x & \cdots \\ \ldots & \cdots & \ldots & \cdots\end{array}\right|\) is a. \((x-1)^{-1}(x+n-1)\) b. \((x-1)^{n}(x+n-1)\) c. \((1-x)^{-1}(x+n-1)\) d. none of these

If \(a, b, c\) are positive and are the \(p^{\text {ll }}, q^{\text {Ht }}\) and \(r^{\text {th }}\) terms, respectively, of a G.P., then \(\Delta=\left|\begin{array}{lll}\log a & p & 1 \\ \log b & q & 1 \\ \log c & r & 1\end{array}\right|\) is a. 0 b. \(\log (a b c)\) c. \(-(p+q+r)\) d. none of these

If \(f(x)=\left|\begin{array}{ccc}m x & m x-p & m x+p \\ n & n+p & n-p \\ m x+2 n & m x+2 n+p & m x+2 n-p\end{array}\right|\), then \(y=f(x)\) represents a. a straight line parallel to \(x\)-axis b. a straight line parallel to \(y\)-axis c. parabola d. a straight line with negative slope

If \(\left|\begin{array}{lll}x & 3 & 6 \\ 3 & 6 & x \\ 6 & x & 3\end{array}\right|=\left|\begin{array}{lll}2 & x & 7 \\ x & 7 & 2 \\ 7 & 2 & x\end{array}\right|=\left|\begin{array}{ccc}4 & 5 & x \\ 5 & x & 4 \\ x & 4 & 5\end{array}\right|=0\), then \(\cdot x^{*}\) is equal to a. 0 b. \(-9\) c. 3 d. none of these

If \(\left|\begin{array}{ccc}x^{n} & x^{n+2} & x^{2 n} \\ 1 & x^{a} & a \\\ x^{n+5} & x^{a+6} & x^{2 n+5}\end{array}\right|=0, \forall x \in R\), where \(n \in N\), then value of ' \(a^{\prime}\) is a. \(n\) b. \(n-1\) c. \(n+1\) d. none of these