Chapter 1

Find all values of \(\theta\) lying between 0 and \(\frac{\pi}{2}\) which satisfy the equation $$ \left|\begin{array}{ccc} 1+\sin ^{2} \theta & \cos ^{2} \theta & 4 \sin 4 \theta \\ \sin ^{2} \theta & 1+\cos ^{2} \theta & 4 \sin 4 \theta \\ \sin ^{2} \theta & \cos ^{2} \theta & 1+4 \sin 4 \theta \end{array}\right|=0 $$

Using determinants, find the area of the tri angle whose vertices are \((-2,4),(2,-6)\) and \((5,4)\). Are the given points collinear?

\(\left|\begin{array}{ccc}2 b c-a^{2} & c^{2} & b^{2} \\ c^{2} & 2 c a-b^{2} & a^{2} \\ b^{2} & a^{2} & 2 a b-c^{2}\end{array}\right|\) is equal to: (a) \(3 a b c-a^{3}-b^{3}-c^{3}\) (b) \(\left(a^{3}+b^{3}+c^{3}-3 a b c\right)^{2}\) (c) \(\left(a^{2}+b^{2}+c^{2}-a b-b c-c a\right)^{3}\) (d) None of these

The value of determinant \(\left|\begin{array}{ccc}-1 & 1 & 1 \\ 1 & -1 & 1 \\\ 1 & 1 & -1\end{array}\right|\) is equal to: (a) \(-4\) (b) 0 (c) 1 (d) 4

Using determinants, find the area of the triangle whose vertices are \((1,4),(2,3)\) and \((-5,-3)\) Are the given points collinear?

Show that the points \(A(a, b+c), B(b, c+a)\) and \(C(c, a+b)\) are collinear.

If in the multiplication of \(\left|\begin{array}{cc}a & b \\ -b & a\end{array}\right|\) and \(\left|\begin{array}{cc}c & d \\ -d & c\end{array}\right|, A, B\) are the elements of the first row, then the elements of the second row will be: (a) \(-B, A\) (b) \(A, B\) (c) \(B, A\) \((\mathrm{d})-B,-A\)

What is the value of \(x\) if \(\left|\begin{array}{ccc}8 & -5 & 1 \\ 5 & x & 1 \\\ 6 & 3 & 1\end{array}\right|=2 ?\) (a) 2 (b) 8 (c) 5 (d) 9

Find the value(s) of \(p\), such that the area of the triangle with vertices \((5,4),(-2,6)\) and \((p, 4)\) is 35 square units.

Using determinants, show that the points \((3,8),(-4,2)\) and \((10,14)\) are collinear.

Let \(D\) be the determinant of the matrix \(\left[\begin{array}{cc}a & b \\ -b & -a\end{array}\right]\) and \(D^{\prime}\) the determinant of the cofactor of the elements of the matrix. Then which one of the following is correct? (a) \(D^{\prime}=D\) (b) \(D^{\prime}=D^{2}\) (c) \(D^{\prime}=D^{3}\) (d) \(D^{\prime}=1 / D\)

\(\left|\begin{array}{ccc}b+c & a & a \\ b & c+a & b \\ c & c & a+b\end{array}\right|\) is equal to: (a) \(a+b+c\) (b) \(4 a b c\) (c) \(a b c\) (d) 0

If the points \((a, b),\left(a^{\prime}, b^{\prime}\right)\) and \(\left(a-a^{\prime}, b-b^{\prime}\right)\) are collinear, show that \(a b^{\prime}=a^{\prime} b . \)

Vertices of a triangle \(A B C\) are \(A(1,3), B(0,0)\) and \(C(k, 0)\). Find the value of \(k\), such that area of triangle \(A B C\) is \(3 \mathrm{sq}\). units.

The value of the determinant \(\left|\begin{array}{ccc}1+\alpha & 1+\alpha z & 1+\alpha z^{2} \\ 1+\beta & 1+\beta z & 1+\beta z^{2} \\ 1+\gamma & 1+\gamma z & 1+\gamma z^{2}\end{array}\right|\) is: (a) \((\alpha-\beta)(\beta-\gamma)(\gamma-\alpha)\) (c) \(\alpha \beta \gamma\) (b) 0 (d) None of these

If \(\left|\begin{array}{ccc}(b+c)^{2} & a^{2} & a^{2} \\ b^{2} & (c+a)^{2} & b^{2} \\ c^{2} & c^{2} & (a+b)^{2}\end{array}\right|=\) (a) \(-1\) (b) 1 (c) 2 (d) \(-2\)

Find the area of the triangle whose vertices are \(A\left(a t_{1}^{2}, 2 a t_{1}\right), B\left(a t_{2}^{2}, 2 a t_{2}\right)\) and \(C\left(a t_{3}^{2}, 2 a t_{3}\right)\)

Let \(a, b, c\) be positive and not all equal. Show that the value of the determinant \(\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|\) is negative.

The value of \(\left|\begin{array}{cc}\log _{3} 512 & \log _{4} 3 \\ \log _{3} 8 & \log _{4} 9\end{array}\right| \times\left|\begin{array}{ll}\log _{2} 3 & \log _{8} 3 \\ \log _{3} 4 & \log _{3} 4\end{array}\right|\) is: (a) 7 (b) 10 (c) 13 (d) 17

If a point \(A\left(x_{1}, y_{1}\right), B\left(x_{2}, y_{2}\right), C\left(x_{3}, y_{3}\right)\) are the vertices of an equilateral triangle whose each side is equal to \(k\), then show that $$ \left|\begin{array}{lll} x_{1} & y_{1} & 2 \\ x_{2} & y_{2} & 2 \\ x_{3} & y_{3} & 2 \end{array}\right|^{2}=3 k^{4} $$

Evaluate \(\sum_{n=1}^{N} u_{n}\) if \(u_{n}=\left|\begin{array}{ccc}n & 1 & 5 \\\ n^{2} & 2 N+1 & 2 N+1 \\ n^{3} & 3 N^{2} & 3 N\end{array}\right|\)

If \(l, m\) and \(n\) are real numbers such that \(l^{2}+m^{2}+n^{2}=0\), then \(\left|\begin{array}{ccc}1+l^{2} & l m & n l \\\ \operatorname{lm} & 1+m^{2} & m n \\ \ln & m n & 1+n^{2}\end{array}\right|\) is equal to: (a) 0 (b) 1 (c) \(l+m+n+2\) (d) \(2(l+m+n)+3\) (e) \(\operatorname{lm} n-1\)

If the value of a third-order determinant is 11, the value of the square of the determinant formed by the cofactor will be: (a) 11 (b) 121 (c) 1331 (d) 14,641

Using determinants, find the value of \(k\) so that the points \((k, 2-2 k),(-k+1,2 k)\) and \((-4-k,\), \(6-2 k\) ) may be collinear.

Let \(\Delta=\left|\begin{array}{ccc}\sin x & \sin (x+h) & \sin (x+2 h) \\\ \sin (x+2 h) & \sin x & \sin (x+h) \\ \sin (x+h) & \sin (x+2 h) & \sin x\end{array}\right|\) Evaluate \(L_{h \rightarrow 0}{L t} \frac{\Delta}{h^{2}}\)

If \(\left|\begin{array}{lll}x & 3 & 6 \\ 3 & 6 & x \\ 6 & x & 3\end{array}\right|=\left|\begin{array}{lll}2 & x & 7 \\ x & 7 & 2 \\ 7 & 2 & x\end{array}\right|=\left|\begin{array}{lll}4 & 5 & x \\ 5 & x & 4 \\ x & 4 & 5\end{array}\right|=0\) then \(x\) is equal to: (a) 9 (b) \(-9\) (c) 0 (d) \(-1\)

\text { If }\left|\begin{array}{lll} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ c_{3} & b_{3} & c_{3} \end{array}\right|=5 \text {, then the value of } \(\left|\begin{array}{lll}b_{2} c_{3}-b_{3} c_{2} & a_{3} c_{2}-a_{2} c_{3} & a_{2} b_{3}-a_{3} b_{2} \\ b_{3} c_{1}-b_{1} c_{3} & a_{1} c_{3}-a_{3} c_{1} & a_{3} b_{1}-a_{1} b_{3} \\ b_{1} c_{2}-b_{2} c_{1} & a_{2} c_{1}-a_{1} c_{2} & a_{1} b_{2}-a_{2} b_{1}\end{array}\right|\) is: (a) 5 (b) 25 (c) 125 (d) 0

Find the value of \(x\) if area of \(\Delta\) is 35 sq. units with vertices \((x, 4),(2,-6)\) and \((5,4)\)

Let \([x]\) represent the greatest integer less than or equal to \(x\), then the value of the determinant \(\left|\begin{array}{ccc}{[e]} & {[\pi]} & {\left[\pi^{2}-6\right]} \\\ {[\pi]} & {\left[\pi^{2}-6\right]} & {[e]} \\ {\left[\pi^{2}-6\right]} & {[e]} & {[\pi]}\end{array}\right|\) is: (a) 8 (b) \(1 / 8\) (c) \(-8\) (d) None of these

If \(\Delta=\left|\begin{array}{lll}a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3}\end{array}\right|\) and \(A_{1}, B_{1}, C_{1}\) are cofactors of \(a_{1}, b_{1}, c_{1}\), respectively, then determinant \(\left|\begin{array}{lll}A_{1} & B_{1} & C_{1} \\ A_{2} & B_{2} & C_{2} \\ A_{3} & B_{3} & C_{3}\end{array}\right|=\) (a) \(\Delta\) (b) \(\Delta^{2}\) (c) \(\Delta^{3}\) (d) 0

Using properties of determinants, show that: $$ \left|\begin{array}{lll} 1 & a & a^{2}-b c \\ 1 & b & b^{2}-c a \\ 1 & c & c^{2}-a b \end{array}\right|=0 $$

Let \(m\) and \(p\) be two positive integers such that \(m \geq p+2 .\) Suppose $$ \Delta(m, p)=\left|\begin{array}{ccc} { }^{m} C_{p} & { }^{m} C_{p+1} & { }^{m} C_{p+2} \\ { }^{m+1} C_{p} & { }^{m+1} C_{p+1} & { }^{m+1} C_{p+2} \\ { }^{m+2} C_{p} & { }^{m+2} C_{p+1} & { }^{m+2} C_{p+2} \end{array}\right| $$

\(\left|\begin{array}{ccc}\left(a^{x}+a^{-x}\right)^{2} & \left(a^{x}-a^{-x}\right)^{2} & 1 \\ \left(b^{x}+b^{-x}\right)^{2} & \left(b^{x}-b^{-x}\right)^{2} & 1 \\ \left(c^{x}+c^{-x}\right)^{2} & \left(c^{x}-c^{-x}\right)^{2} & 1\end{array}\right|=\) (a) 0 (b) \(2 a b c\) (c) \(a^{2} b^{2} c^{2}\) (d) None of these

If \(A+B+C=\pi\), show that $$ \left|\begin{array}{ccc} \sin (A+B+C) & \sin B & \cos C \\ -\sin B & 0 & \tan A \\ \cos (A+B) & -\tan A & 0 \end{array}\right|=0 $$

If \(2 s=a+b+c\), prove that $$ \begin{aligned} &\left|\begin{array}{ccc} a^{2} & (s-a)^{2} & (s-a)^{2} \\ (s-b)^{2} & b^{2} & (s-b)^{2} \\ (s-c)^{2} & (s-c)^{2} & c^{2} \end{array}\right| \\ &=2 s^{3}(s-a)(s-b)(s-c) \end{aligned} $$

\(\left|\begin{array}{lll}1 / a & 1 & b c \\ 1 / b & 1 & c a \\ 1 / c & 1 & a b\end{array}\right|=\) (a) \(-2\) (b) \(x^{2}+2\) (c) 2 (d) None of these

Evaluate the determinant, $$ \left|\begin{array}{ccc} 0 & \sin \alpha & -\cos \alpha \\ -\sin \alpha & 0 & \sin \beta \\ \cos \alpha & -\sin \beta & 0 \end{array}\right| $$

If \(\alpha, \beta, \gamma\) are the roots of the equation \(x^{3}+a x^{2}+b=0\), then find the value of determinant \(\left|\begin{array}{lll}\alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta\end{array}\right|\)

The value of determinant $$ \left|\begin{array}{ccc} x+1 & x+2 & x+4 \\ x+3 & x+5 & x+8 \\ x+7 & x+10 & x+14 \end{array}\right| \text { is: } $$ (a) \(-2\) (b) \(x^{2}+2\) (c) 2 (d) None of these

If \(x, y\) and \(z\) (all positive) are the \(p^{\text {th }}, q^{\text {th }}\) and \(r^{\text {th }}\) terms, respectively, of a G.P. Prove that $$ \left|\begin{array}{lll} \log x & p & 1 \\ \log y & q & 1 \\ \log z & r & 1 \end{array}\right|=0 $$

If \(|A|=\left|\begin{array}{ccc}4 & 3 & 8 \\ 6 & 7 & 5 \\ 3 & 1 & 2\end{array}\right|\) then minor of \(a_{12}\) is (a) \(\begin{array}{ll}3 & 8 \\ 7 & 5\end{array} \mid\) (b) \(\begin{array}{ll}6 & 5 \\ 3 & 2\end{array} \mid\) (c) \(\left|\begin{array}{ll}7 & 5 \\ 1 & 2\end{array}\right|\) (d) \(\left|\begin{array}{ll}4 & 3 \\ 6 & 7\end{array}\right|\)

Find the value of determinant $$ \left|\begin{array}{ccc} 1 & 1 & 1 \\ { }^{n} C_{1} & { }^{n+1} C_{1} & { }^{n+2} C_{1} \\ { }^{n} C_{2} & { }^{n+1} C_{2} & { }^{n+2} C_{2} \end{array}\right| $$

If \(\left|\begin{array}{ccc}x & x+y & x+y+z \\ 2 x & 3 x+2 y & 4 x+3 y+2 z \\\ 3 x & 6 x+3 y & 10 x+6 y+3 z\end{array}\right|=64\) then find the value of \(x\) ?

$$ \left|\begin{array}{ccc} (x-2)^{2} & (x-1)^{2} & x^{2} \\ (x-1)^{2} & x^{2} & (x+1)^{2} \\ x^{2} & (x+1)^{2} & (x+2)^{2} \end{array}\right|= $$ (a) 0 (b) 8 (c) \(-8\) (d) 10

Without actual expansion, prove that: $$ \left|\begin{array}{ccc} 0 & 99 & -998 \\ -99 & 0 & 997 \\ 998 & -997 & 0 \end{array}\right|=0 $$

The value of determinant $$ \left|\begin{array}{ccc} 1 & 1 & 1 \\ { }^{n} c_{1} & { }^{m+1} c_{1} & { }^{m+2} c_{1} \\ { }^{m} c_{2} & { }^{m+1} c_{2} & { }^{m+2} c_{2} \end{array}\right|= $$ (a) 1 (b) \(-1\) (c) 0 (d) None of these

If \(a, b, c\) are in arithmetic progression, prove that \(\left|\begin{array}{lll}x+1 & x+2 & x+a \\ x+2 & x+4 & x+b \\\ x+3 & x+6 & x+c\end{array}\right|=0\)

If \(a \neq p, b \neq q, c \neq r\) and \(\left|\begin{array}{lll}p & b & c \\ a & q & c \\ a & b & r\end{array}\right|=0\) then \(\frac{p}{p-a}+\frac{q}{q-b}+\frac{r}{r-c}=\) (a) 0 (b) 2 (c) 1 (d) \(-1\)

Show that \(\left|\begin{array}{ccc}a & b & c \\ a+2 x & b+2 y & c+2 z \\ x & y & z\end{array}\right|=0\)

If \(\left|\begin{array}{ccc}a & 1 & 1 \\ 1 & b & 1 \\ 1 & 1 & c\end{array}\right|=0\), then \(\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=\) (a) 0 (b) 1 (c) 2 (d) 3