# 80. Determinants Problems Part 3

1. Show without expanding that

$$\begin{vmatrix}1 & a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2\end{vmatrix} = \begin{vmatrix}1 & bc & b + c \\ 1 & ca & c + a \\ 1 & ab & a + b\end{vmatrix}$$

2. Show without expanding that

$$\begin{vmatrix}x^2 + x & x + 1 & x - 2 \\ 2x^2 + 3x - 1 & 3x & 3x - 3 \\ x^2 + 2x + 3 & 2x - 1 & 2x - 1\end{vmatrix} = aA + B$$

where $$A$$ and $$B$$ are determinants of 3rd order not involving $$x.$$

3. If $$D_r = \begin{vmatrix}r & x & \frac{n(n + 1)}{2} \\ 2r - 1 & y & \frac{n(3n - 1)}{2} \\ 3r - 2 & z & \frac{n(3n - 1)}{2}\end{vmatrix}$$

show that $$\sum_{r = 1}^n D_r = 0$$

4. Without expansing the determinant show that the value of

$$\begin{vmatrix}-5 & 3 + 5i & \frac{3}{2} - 4i \\ 3 - 5i & 8 & 4 + 5i \\ \frac{3}{2} + 4i & 4 - 5i & 9\end{vmatrix}$$ is real.

5. Prove that $$\begin{vmatrix}-2a & a + b & b + c \\ b + a & -2b & b + c \\ c + a & c + b & -2c\end{vmatrix} = 4(a + b)(b + c)(c + a)$$

6. $$f_r(x), g_r(x), h_r(x)$$ where $$r = 1,2,3$$ are polynomials in $$x$$ such that $$f_r(a)=g_r(a)=h_r(a), r=1,2,3$$ and

$$F(x) = \begin{vmatrix}f_1(x) & f_2(x) & f_3(x) \\ g_1(x) & g_2(x) & g_3(x) \\ h_1(x) & h_2(x) & h_3(x)\end{vmatrix}$$

then find $$F'(x)$$

7. Let $$\alpha$$ be a repeated root of a quadratic equation $$f(x) = 0$$ and $$A(x), B(x), C(x)$$ be polynomials of degree $$3, 4, 5$$ respectively. Show that

$$\Delta(x) = \begin{vmatrix}A(x) & B(x) & C(x) \\ A(\alpha) & B(\alpha) & C(\alpha) \\ A'(\alpha) & B'(\alpha) & C'(\alpha)\end{vmatrix}$$

is divisible by $$f(x),$$ where prime denotes a derivative.

8. Prove that $$\begin{vmatrix}\cos(\theta + \alpha) & cos(\theta + \beta) & \cos(\theta + \gamma) \\ \sin(\theta + \alpha) & sin(\theta + \beta) & \sin(\theta + \gamma) \\ \sin(\beta - \gamma) & \sin(\gamma - \alpha) & \sin(\alpha - \beta)\end{vmatrix}$$ is independent of $$\theta$$.

9. If $$f, g, h$$ are differential functions of $$x$$ and

$$\Delta = \begin{vmatrix}f & g & h \\ (xf)' & (xg)' & (xh)' \\ (x^2f)'' & (x^2g)'' & (x^h)''\end{vmatrix}$$

prove that $$\Delta' = \begin{vmatrix}f & g & h \\ f' & g' & h' \\ (x^2f'')' & (x^2g'')' & (x^2h'')'\end{vmatrix}$$

10. If $$f(x) = \begin{vmatrix}x^n & \sin x & \cos x \\ n! & \sin\frac{n\pi}{2} & \cos\frac{n\pi}{2} \\ a & a^2 & a^3\end{vmatrix},$$

then show that $$\frac{d^nf(x)}{dx^n} = 0,$$ where $$x = 0$$

11. Prove that $$\begin{vmatrix}\cos(A-P) & \cos(A-Q) & \cos(A-R) \\ \cos(B-P) & \cos(B-Q) & \cos(B-R) \\ \cos(C-P) & \cos(C-Q) & \cos(C-R)\end{vmatrix} = 0$$

12. Show that $$\begin{vmatrix}2bc - a^2 & c^2 & b^2 \\ c^2 & 2bc - b^2 & a^2 \\ b^2 & a^2 & 2bc - c^2\end{vmatrix} = (a^3 + b^3 + c^3 - 3abc)^2$$

13. Show that $$\begin{vmatrix}1 & \cos(\beta - \alpha) & \cos(\gamma - \alpha) \\ \cos(\alpha - \beta) & 1 & \cos(\gamma - \beta) \\ \cos(\alpha - \gamma) & \cos(\beta - \gamma) & 1\end{vmatrix} = 0$$

14. For what value of $$m$$ does the system of equation $$3x + my = m$$ and $$2x - 5y = 20$$ has a solution satisfying the consdition $$x > 0, y>0.$$

15. PRove that he system of equation $$3x - y + 4z = 3; x + 2y -3z = -2; 6x + 5y +\lambda z = -3$$ has at least one solution for any real $$\lambda$$. Find the set of solutions when $$\lambda = -5$$

16. For what value of $$p$$ and $$q$$, the system of equations $$2x + py + 6z = 8; x + 2y + qz = 5; x + y + 3z = 4$$ has (i) no solution (ii) a unique solution, and (iii) infinite solutions.

17. Let $$\lambda$$ and $$\alpha$$ be real. Find the set of all valus of $$\lambda$$ for which the system:

$$x\lambda + y\sin\alpha - z\cos\alpha = 0; x + y\cos\alpha + z\sin\alpha = 0$$ $$-x + y\sin\alpha - z\cos\alpha = 0$$

has a non-trivial solution. For $$\lambda = 1,$$ find all values of $$\alpha$$

18. Evaluate $$\begin{vmatrix}a & b + c & a^2 \\ b & c + a & b^2 \\ c & a + b & c^2\end{vmatrix}$$

19. Find the value of the determinant $$\begin{vmatrix}\sqrt{13} + \sqrt{3} & 2\sqrt{5} & \sqrt{5} \\ \sqrt{15} + \sqrt{26} & 5 & \sqrt{10} \\ 3 + \sqrt{65} & \sqrt{15} & 5\end{vmatrix}$$

20. Evaluate the determinant $$\begin{vmatrix}x & x(x^2 + 1) & x + 1 \\ y & y(y ^2 + 1) & y + 1 \\ z & z(z^2 + 1) & z + 1\end{vmatrix}$$

21. If $$x, y, z$$ are respectively $$l$$ th, $$2m$$ th, $$2n$$ th terms of an H.P., then find the value of

$$\begin{vmatrix} yz & zx & xy \\ l & 2m & 3n \\ 1 & 1 & 1\end{vmatrix}$$

22. Show that $$\begin{vmatrix}1 & a^2 & a^3 \\ 1& b^2 & b^3 \\ 1 & c^2 & c^3\end{vmatrix} = (ab + bc + ca)\begin{vmatrix}1 & a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2\end{vmatrix}$$

23. Evaluate $$\begin{vmatrix}(b + c)^2 & a^2 & bc \\ (c + a)^2 & b^2 & ca \\ (a + b)^2 & c^2 & ab\end{vmatrix}$$

24. Prove that $$\begin{vmatrix}x^2 & x^2 - (y - z)^2 & yz \\ y^2 & y^2 - (z - x)^2 & zx \\ z^2 & z^2 - (x - y)^2 & xy\end{vmatrix} = (x - y)(y - z)(z - x)(x + y + z)(x^2 + y^2 + z^2)$$

25. If $$a_1b_1c_1, a_2b_2c_2, a_3b_3c_3$$ are three $$3$$ digit numbers such that each of them is divisible by $$k,$$ then prove that the determinant $$\begin{vmatrix}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{vmatrix}$$ is divisible by $$k.$$

26. If $$a_i, b_i, c_i \in R(i = 1,2,3)$$ and $$x\in R,$$ show that

$$\begin{vmatrix}a_1 + b_1x & a_1x + b_1 & c_1 \\ a_2 + b_2x & a_2x + b_2 & c_2 \\ a_3 + b_3x & a_3x + b_3 & c_3\end{vmatrix} = (1 - x^2)\begin{vmatrix}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{vmatrix}$$

27. If $$a,b,c$$ are the roots of the equation $$px^3 + qx^2 + rx + s = 0,$$ then find the value of $$\begin{vmatrix}1 + a & 1 & 1 \\ 1 & 1 + b & 1 \\ 1 & 1 & 1 + c\end{vmatrix}$$

28. If $$a<b<c$$, prove that

$$\begin{vmatrix}1 & a & a^4 \\ 1 & b & b^4 \\ 1 & c & c^4\end{vmatrix} > 0$$

29. If $$a, b, c$$ are distinct and $$\begin{vmatrix}a & a^3 & a^4 - 1 \\ b & b^3 & b^4 - 1 \\ c& c^3 & c^4 -1\end{vmatrix} = 0,$$ show that

$$abc(ab + bc + ca) = a + b + c$$

30. Show that if $$x_1, x_2, x_3 \neq 0,$$

$$\begin{vmatrix}x_1 + a_1b_1 & a_1b_2 & a_1b_3 \\ a_2b_1 & x_2 + a_2b_2 & a_2b_3 \\ a_3b_1 & a_3b_2 & x_3 + a_3b_3\end{vmatrix} = x_1x_2x_3\left(1 + \frac{a_1b_1}{x_1} + \frac{a_2b_2}{x_2} + \frac{a_3b_3}{c_3}\right)$$

31. Show that $$\begin{vmatrix}\frac{1}{a + x} & \frac{1}{a + y} & 1 \\ \frac{1}{b + x} & \frac{1}{b + y} & 1 \\ \frac{1}{c + x} & \frac{1}{c + y} & 1\end{vmatrix} = \frac{(a-b)(b - c)(a - c)(x - y)}{(a + x)(b + x)(c + x)(a + y)(b + y)(c + y)}$$

32. Show that $$\begin{vmatrix}a^2 & bc & ac + c^2 \\ a^2 + ab & b^2 & ac \\ ab & b^2 + bc & c^2\end{vmatrix} = 4a^2b^2c^2$$

33. Show that $$\begin{vmatrix}1 + a^2 - b^2 & 2ab & -2b \\ 2ab & 1 - a^2 + b^2 & 2a \\ 2b & -2a & 1 - a^2 - b^2\end{vmatrix} = (1 + a^2 + b^2)^3$$

34. If $$a, b, c$$ are sides of a triangle, show that

$$\begin{vmatrix}a^2 & (s - a)^2 & (s - a)^2 \\ (s - b)^2 & b^2 & (s - b)^2 \\ (s - c)^2 & (s - c)^2 & c^2\end{vmatrix} = \frac{1}{2}P^2A^2$$

where $$P$$ denotes the perimeter of the triangle, $$A$$ its area and $$s = \frac{P}{2}.$$

35. Show that $$\begin{vmatrix}(x - a)^2 & ab & ac \\ ba & (x - b)^2 & bc \\ ca & cb & (x - c)^2\end{vmatrix} = \\ x^2(x - 2a)(x - 2b)(x - 2c)\left(x + \frac{a^2}{x - 2a} + \frac{b^2}{x - 2b} + \frac{c^2}{x - 2c}\right)$$

36. If $$x, y, z$$ are unequal and $$\begin{vmatrix}x^3 & (x + a)^3 & (x - a)^3 \\ y^3 & (y + a)^3 & (y - a)^3 \\ z^3 & (z + a)^3 & (z - a)^3\end{vmatrix} = 0$$

prove that $$a^2(x + y + z) = 3xyz$$

37. Show that $$\begin{vmatrix}1 - x& a & a^2 \\ a & a^2 - x & a^3 \\ a^2 & a^3 & a^4 - x\end{vmatrix} = x^2(1 + a ^2 + a^3)- x^3$$

38. If $$y = \sin px$$ and $$y_n = \frac{d^ny}{dx^n},$$ find the value of

$$\begin{vmatrix}y & y_1 & y_2 \\ y_3 & y_4 & y_5 \\ y_6 & y_7 & y_8\end{vmatrix}$$

39. Evaluate $$\begin{vmatrix}\cos^2\theta & \cos\theta\sin\theta & -\sin\theta \\ \cos\theta\sin\theta & \sin^2\theta & \cos\theta \\ \sin\theta & -\cos\theta & 0\end{vmatrix}$$

40. Evaluate $$\begin{vmatrix}\cos\alpha & \sin\alpha\cos\beta & \sin\alpha\sin\beta \\ -\sin\alpha & \cos\alpha\cos\beta & \cos\alpha\sin\beta \\ 0 & -\sin\beta & \cos\beta\end{vmatrix}$$

41. Solve the equation $$\begin{vmatrix}a^2 + x & ab & ac \\ ab & b^2 + x & bc \\ ac & bc & c^2 + x\end{vmatrix} = 0$$

42. Solve the equation for $$x$$

$$\begin{vmatrix}{}^xC_r & {}^{n - 1}C_r & {}^{n - 1}C_{r - 1} \\ {}^{x + 1}C_r & {}^nC_r & {}^nC_{r - 1} \\ {}^{x + 2}C_r & {}^{n + 1}C_r & {}^{n + 1}C_{r - 1}\end{vmatrix} = 0~\forall~n, r > 1$$

43. Solve the equation $$\begin{vmatrix}u + a^2x & w' + abx & v' + acx \\ w' + abx & v + b^2x & u' + bcx \\ v' + acx & u' + bcx & w + c^2x\end{vmatrix} = 0$$ expressing the result by means of determinants.

44. If $$f(a, b) = \frac{f(b) - f(a)}{b - a}$$ and $$f(a, b, c) = \frac{f(b , c) - f(a, b)}{c - a},$$ show that

$$f(a,b,c) = \begin{vmatrix}f(a) & f(b) & f(c) \\ 1 & 1 & 1 \\ a & b& c\end{vmatrix} \div \begin{vmatrix}1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2\end{vmatrix}$$

45. If $$A, B, C$$ are the angles of a $$\triangle ABC,$$ then prove that

$$\begin{vmatrix}e^{2iA} & e^{-iC} & e^{-iB} \\ e^{-iC} & e^{2iB} & e^{-iA} \\ e^{-iB} & e^{-iA} & e^{2iC}\end{vmatrix}$$ is purely real.

46. If $$A, B, C$$ are the angles of a $$\Delta ABC$$ such that $$A\geq B \geq C$$, find the minimum value of $$\Delta,$$ where

$$\Delta = \begin{vmatrix}\sin^2A & \sin A\cos A & \cos^2A \\ \sin^2B & \sin B\cos B & \cos^2B \\ \sin^2C & \sin C\cos C & \cos^2C\end{vmatrix}$$

Also, show that $$\Delta = \frac{1}{4}[\sin(2A - 2B) + \sin(2B - 2C) + \sin(2C - 2A)]$$

47. Evaluate $$\begin{vmatrix}a^2 & a & 1 \\ \cos nx & \cos (n + 1)x & \cos (n + 2)x \\ \sin nx & \sin (n + 1)x & \sin (n + 2)x\end{vmatrix}$$

48. If $$0 < x < \frac{\pi}{2},$$ find the values of $$x$$ for which

$$\begin{vmatrix}1 + \sin^2x & \cos^2x & 4\sin 2x \\ \sin^2x & 1 + \cos^2x & 4\sin 2x \\ \sin^2x & \cos^2x & 1 + 4\sin 2x\end{vmatrix}$$ has maximum value.

49. If $$A, B, C$$ are the angles of a triangle, show that

$$\begin{vmatrix}-1 & \cos C & \cos B \\ \cos C & -1 & \cos A \\ \cos B & \cos A & 1\end{vmatrix} = 0$$

50. If $$A, B, C$$ are the angles of an isosceles triangle, evaluate

$$\begin{vmatrix}1 & 1& 1 \\ 1 + \sin A & 1 + \sin B & 1 + \sin C \\ \sin A + \sin^2A & \sin B + \sin^2 B & \sin C + \sin^2 C\end{vmatrix}$$