Matrices and Determinants Questions

MathematicsMatrices and Determinants115 Questions Found

MathematicsSolution of System of Linear Equations (Matrix Method and Cramer's Rule)JEE Advanced 1980Moderate

For what values of $m$ , does the system of equations $3 x + m y = m; 2 x - 5 y = 20$ has solution satisfying the conditions $x > 0, y > 0$ .

Answer:

m \in (- \infty, - 15/2) \cup (30, \infty)

MathematicsProperties of DeterminantsJEE Advanced 1981Easy

View in:English Hindi

Let $a, b, c$ be positive and not all equal. Show that the value of the determinant $a b c b c a c a b$ is negative.

Answer:The value of the determinant is

- (a + b + c) (a^{2} + b^{2} + c^{2} - ab - b c - c a) = - \frac{1}{2} (a + b + c) [(a - b)^{2} + (b - c)^{2} + (c - a)^{2}]

, which is negative as

a, b, c > 0

and not all equal.

MathematicsProperties of DeterminantsJEE Advanced 1981Easy

View in:English Hindi

The solution set of the equation $111 4 - 2 2 x 205 5 x^{2} = 0$ is .........

Answer:

{- 1, 2}

MathematicsProperties of DeterminantsJEE Advanced 1982Moderate

View in:English Hindi

Without expanding a determinant at any stage, show that $x^{2} + x 2 x^{2} + 3 x - 1 x^{2} + 2 x + 3 x + 1 3 x 2 x - 1 x - 2 3 x - 3 2 x - 1 = x A + B$ , where $A$ and $B$ are determinants of order $3$ not involving $x$ .

Answer:Proof by row operations

R_{2} \to R_{2} - 2 R_{1}

and

R_{3} \to R_{3} - R_{1}

MathematicsSolution of System of Linear Equations (Matrix Method and Cramer's Rule)JEE Advanced 1982Moderate

View in:English Hindi

Find all values of $λ$ , such that $x, y, z \neq = (0, 0, 0)$ and $(\hat{i} + \hat{j} + 3 \hat{k}) x + (3 \hat{i} - 3 \hat{j} + \hat{k}) y + (- 4 \hat{i} + 5 \hat{j}) z = λ (x \hat{i} + y \hat{j} + z \hat{k})$ where $\hat{i}, \hat{j}, \hat{k}$ are unit vectors along the coordinate axes.

Answer:

λ = 0, - 1

MathematicsProperties of DeterminantsJEE Advanced 1983Easy

View in:English Hindi

Given that $x = - 9$ is a root of $x 27 3 x 6 72 x = 0$ the other two roots are ......... and .........

Answer:

2

and

7

MathematicsProperties of DeterminantsJEE Advanced 1985Easy

View in:English Hindi

Show that $^{x} C_{r}^{y} C_{r}^{z} C_{r}^{x} C_{r + 1}^{y} C_{r + 1}^{z} C_{r + 1}^{x} C_{r + 2}^{y} C_{r + 2}^{z} C_{r + 2} =^{x} C_{r}^{y} C_{r}^{z} C_{r}^{x + 1} C_{r + 1}^{y + 1} C_{r + 1}^{z + 1} C_{r + 1}^{x + 2} C_{r + 2}^{y + 2} C_{r + 2}^{z + 2} C_{r + 2}$ .

Answer:Proof using

^{n} C_{r} +^{n} C_{r + 1} =^{n + 1} C_{r + 1}

MathematicsSolution of System of Linear Equations (Matrix Method and Cramer's Rule)JEE Advanced 1986Moderate

View in:English Hindi

Consider the system of linear equations in $x, y, z$ : $(sin 3 θ) x - y + z = 0, (cos 2 θ) x + 4 y + 3 z = 0, 2 x + 7 y + 7 z = 0$ . Find the values of $θ$ for which this system has nontrivial solutions.

Answer:

θ = nπ

θ = nπ + (- 1)^{n} π /6, n \in Z

MathematicsProperties of DeterminantsJEE Advanced 1989Easy

View in:English Hindi

Let $Δ_{a} = a - 1 (a - 1)^{2} (a - 1)^{3} n 2 n^{2} 3 n^{3} 6 4 n - 2 3 n^{2} - 3 n$ . Show that $\sum_{a = 1}^{n} Δ_{a} = c$ , a constant.

Answer:Proof by showing

\sum_{a = 1}^{n} Δ_{a} = 0

MathematicsProperties of DeterminantsJEE Advanced 1990Easy

View in:English Hindi

Let the three digit numbers $A 28, 3 B 9$ , and $62 C$ , where $A, B$ , and $C$ are integers between $0$ and $9$ , be divisible by a fixed integer $k$ . Show that the determinant $A 82 39 B 6 C 2$ is divisible by $k$ .

Answer:Proof by operation

R_{2} \to R_{2} + 10 R_{3} + 100 R_{1}

...12

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For what values of m, does the system of equations 3x+my=m;2x−5y=20 has solution satisfying the conditions x>0,y>0.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a,b,c be positive and not all equal. Show that the value of the determinant ​abc​bca​cab​​ is negative.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The solution set of the equation ​111​4−22x​2055x2​​=0 is .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Given that x=−9 is a root of ​x27​3x6​72x​​=0 the other two roots are ......... and .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Consider the system of linear equations in x,y,z: (sin3θ)x−y+z=0,(cos2θ)x+4y+3z=0,2x+7y+7z=0. Find the values of θ for which this system has nontrivial solutions.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let Δa​=​a−1(a−1)2(a−1)3​n2n23n3​64n−23n2−3n​​. Show that ∑a=1n​Δa​=c, a constant.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For what values of m, does the system of equations 3x+my=m;2x−5y=20 has solution satisfying the conditions x>0,y>0.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a,b,c be positive and not all equal. Show that the value of the determinant ​abc​bca​cab​​ is negative.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The solution set of the equation ​111​4−22x​2055x2​​=0 is .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Given that x=−9 is a root of ​x27​3x6​72x​​=0 the other two roots are ......... and .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Consider the system of linear equations in x,y,z: (sin3θ)x−y+z=0,(cos2θ)x+4y+3z=0,2x+7y+7z=0. Find the values of θ for which this system has nontrivial solutions.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let Δa​=​a−1(a−1)2(a−1)3​n2n23n3​64n−23n2−3n​​. Show that ∑a=1n​Δa​=c, a constant.

For what values of $m$ , does the system of equations $3 x + m y = m; 2 x - 5 y = 20$ has solution satisfying the conditions $x > 0, y > 0$ .

Let $a, b, c$ be positive and not all equal. Show that the value of the determinant $a b c b c a c a b$ is negative.

The solution set of the equation $111 4 - 2 2 x 205 5 x^{2} = 0$ is .........

Given that $x = - 9$ is a root of $x 27 3 x 6 72 x = 0$ the other two roots are ......... and .........

Consider the system of linear equations in $x, y, z$ : $(sin 3 θ) x - y + z = 0, (cos 2 θ) x + 4 y + 3 z = 0, 2 x + 7 y + 7 z = 0$ . Find the values of $θ$ for which this system has nontrivial solutions.

Let $Δ_{a} = a - 1 (a - 1)^{2} (a - 1)^{3} n 2 n^{2} 3 n^{3} 6 4 n - 2 3 n^{2} - 3 n$ . Show that $\sum_{a = 1}^{n} Δ_{a} = c$ , a constant.

For what values of $m$ , does the system of equations $3 x + m y = m; 2 x - 5 y = 20$ has solution satisfying the conditions $x > 0, y > 0$ .

Let $a, b, c$ be positive and not all equal. Show that the value of the determinant $a b c b c a c a b$ is negative.

The solution set of the equation $111 4 - 2 2 x 205 5 x^{2} = 0$ is .........

Given that $x = - 9$ is a root of $x 27 3 x 6 72 x = 0$ the other two roots are ......... and .........

Consider the system of linear equations in $x, y, z$ : $(sin 3 θ) x - y + z = 0, (cos 2 θ) x + 4 y + 3 z = 0, 2 x + 7 y + 7 z = 0$ . Find the values of $θ$ for which this system has nontrivial solutions.

Let $Δ_{a} = a - 1 (a - 1)^{2} (a - 1)^{3} n 2 n^{2} 3 n^{3} 6 4 n - 2 3 n^{2} - 3 n$ . Show that $\sum_{a = 1}^{n} Δ_{a} = c$ , a constant.