question stringlengths 79 9.83k | answer stringlengths 33 9.39k |
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The contrapositive of the following statement,
<br/><br/>“If the side of a square doubles, then its area increases four times”, is :
Options:
[{"identifier": "A", "content": "If the side of a square is not doubled, then its area does not increase four times."}, {"identifier": "B", "content": "If the area of a square ... | ["D"]
Explanation:
Contrapositive of p $$ \to $$ q is $$ \sim $$q $$ \to $$ $$ \sim $$p.
<br><br>Here,
<br><br>Let
<br><br> p = Side of a square is doubles.
<br><br> q = Area of square increases four times.
<br><br>$$ \therefore $$ $$ \sim $$q $$ \to $$ $$ \sim... |
Contrapositive of the statement
<br/><br/>‘If two numbers are not equal, then their squares are not equal’, is :
Options:
[{"identifier": "A", "content": "If the squares of two numbers are equal, then the numbers are equal."}, {"identifier": "B", "content": "If the squares of two numbers are equal, then the numbers ar... | ["A"]
Explanation:
Let,
<br><br>p : two numbers are not equal
<br><br>q : squares of two numbers are not equal
<br><br>Contrapositive of p $$ \to $$ q is $$ \sim $$q $$ \to $$ $$ \sim $$p.
<br><br>$$ \therefore $$ $$ \sim $$q $$ \to $$ $$ \sim $$p means "If the squares of two numbers are equal, then the numbers are eq... |
Consider the following two statements :
<br/><br/><b>Statement p :</b>
<br/>The value of sin 120<sup>o</sup> can be derived by taking $$\theta = {240^o}$$ in the equation
<br/>2sin$${\theta \over 2} = \sqrt {1 + \sin \theta } - \sqrt {1 - \sin \theta } $$
<br/><br/><b>Statement q :</b>
<br/>The angles A, B, C and... | ["A"]
Explanation:
<b>Statement p :</b>
<br>sin 120<sup>o</sup> = cos 30<sup>o</sup> = $${{\sqrt 3 } \over 2}$$ $$ \Rightarrow $$ 2 sin 120<sup>o</sup> = $$\sqrt 3 $$
<br><br>So, $$\sqrt {1 + \sin {{240}^o}} - \sqrt {1 - \sin {{240}^o}} $$
<br><br> $$ = \sqrt {{{1 - \sqrt 3 } \over 2}} - \sqrt {{{1 + \sqrt 3 } \over... |
Consider the statement : "P(n) : n<sup>2</sup> – n + 41 is prime". Then which one of the following is true ?
Options:
[{"identifier": "A", "content": "P(5) is false but P(3) is true"}, {"identifier": "B", "content": "Both P(3) and P(5) are true"}, {"identifier": "C", "content": "P(3) is false but P(5) is true"}, {"ide... | ["B"]
Explanation:
P(n) : n<sup>2</sup> $$-$$ n + 41 is prime
<br><br>P(5) = 61 which is prime
<br><br>P(3) = 47 which is also prime |
Consider the following three statements :
<br/><br/>P : 5 is a prime number
<br/><br/>Q : 7 is a factor of 192
<br/><br/>R : L.C.M. of 5 and 7 is 35
<br/><br/>Then the truth value of which one of the following statements is true ?
Options:
[{"identifier": "A", "content": "(P $$ \\wedge $$ Q) $$ \\vee $$ ($$ \\sim $$ R... | ["B"]
Explanation:
It is obvious |
Contrapositive of the statement " If two numbers are not equal, then their squares are not equal." is :
Options:
[{"identifier": "A", "content": "If the squares of two numbers are equal, then the numbers are not equal"}, {"identifier": "B", "content": "If the squares of two numbers are equal, then the numbers are equa... | ["B"]
Explanation:
Let,
<br><br>p : two numbers are not equal
<br><br>q : squares of two numbers are not equal
<br><br>Contrapositive of p $$ \to $$ q is $$ \sim $$q $$ \to $$ $$ \sim $$p.
<br><br>$$ \therefore $$ $$ \sim $$q $$ \to $$ $$ \sim $$p means "If the squares of two numbers are equal, then the numbers are eq... |
The contrapositive of the statement "If you are
born in India, then you are a citizen of India", is :
Options:
[{"identifier": "A", "content": "If you are not a citizen of India, then you are\nnot born in India."}, {"identifier": "B", "content": "If you are born in India, then you are not a\ncitizen of India."}, {"id... | ["A"]
Explanation:
Let p = you are born in India.
<br><br>q = you are a citizen of India.
<br><br>$$ \therefore $$ $$ \sim $$ p = you are not born in India.
<br><br>$$ \sim $$ q = you are not a citizen of India.
<br><br>We know Contrapositive of p $$ \to $$ q is ~q $$ \to $$ ~p
<br><br>So contrapositive of statement w... |
Negation of the statement :
<br/><br/>$$\sqrt 5 $$ is an integer or 5 is an irrational is :
Options:
[{"identifier": "A", "content": "$$\\sqrt 5 $$ is not an integer and 5 is not irrational."}, {"identifier": "B", "content": "$$\\sqrt 5 $$ is irrational or 5 is an integer."}, {"identifier": "C", "content": "$$\\sqrt 5... | ["A"]
Explanation:
p = $$\sqrt 5 $$ is an integer.
<br><br>q : 5 is irrational
<br><br>$$ \sim $$$$\left( {p \vee q} \right)$$ $$ \equiv $$ $$ \sim $$p $$ \wedge $$ $$ \sim $$q
<br><br>= $$\sqrt 5 $$ is not an integer and 5 is not irrational. |
Consider the statement : <br/>‘‘For an integer n, if n<sup>3</sup> – 1 is even, then n is odd.’’<br/> The contrapositive statement of this statement is :
Options:
[{"identifier": "A", "content": "For an integer n, if n is even, then n<sup>3</sup> \u2013 1 is even."}, {"identifier": "B", "content": "For an integer n, ... | ["D"]
Explanation:
Let,
p : n<sup>3</sup>–1 is even,
<br>q : n is odd
<br><br>Contrapositive of p $$ \to $$ q = $$ \sim $$q $$ \to $$ $$ \sim $$p
<br><br>$$ \Rightarrow $$ If n is not odd then n<sup>3</sup> – 1 is not even.
<br><br>$$ \Rightarrow $$ For an integer n, if n is even, then n<sup>3</sup> – 1 is odd. |
Contrapositive of the statement :<br/>
‘If a function f is differentiable at a, then it is also continuous at a’, is:
Options:
[{"identifier": "A", "content": "If a function f is continuous at a, then it is not differentiable at a."}, {"identifier": "B", "content": "If a function f is not continuous at a, then it is d... | ["C"]
Explanation:
p = function is differentiable at a
<br><br>q = function is continuous at a
<br><br>Contrapositive of statements p $$ \to $$ q is
<br><br>$$ \sim $$q $$ \to $$ $$ \sim $$p
<br><br>$$ \therefore $$ Contrapositive statement is :
<br><br> If a function f is not continuous at a, then it is not different... |
The contrapositive of the statement <br/>"If I reach the
station in time, then I will catch the train" is :
Options:
[{"identifier": "A", "content": "If I will catch the train, then I reach the station\nin time."}, {"identifier": "B", "content": "If I do not reach the station in time, then I will\nnot catch the train.... | ["C"]
Explanation:
Let p denotes statement
<br><br>p : I reach the station in time.
<br><br>q : I will catch the train.
<br><br>Contrapositive of p $$ \to $$ q
is $$ \sim $$q $$ \to $$ $$ \sim $$p
<br><br>$$ \sim $$q $$ \to $$ $$ \sim $$p : If I will not catch the train, then I do not reach the station in time. |
The contrapositive of the statement "If you will work, you will earn money" is :
Options:
[{"identifier": "A", "content": "If you will not earn money, you will not work"}, {"identifier": "B", "content": "If you will earn money, you will work"}, {"identifier": "C", "content": "You will earn money, if you will not work"... | ["A"]
Explanation:
Contrapositive of p $$ \to $$ q is ~q $$ \to $$ ~p
<br><br>p : you will work
<br><br>q : you will earn money
<br><br>~q : you will not earn money
<br><br>~p : you will not work
<br><br>$$ \therefore $$ ~q $$ \to $$ ~p : If you will not earn money, you will not work |
Consider the following three statements :<br/><br/>(A) If 3 + 3 = 7 then 4 + 3 = 8<br/><br/>(B) If 5 + 3 = 8 then earth is flat.<br/><br/>(C) If both (A) and (B) are true then 5 + 6 = 17.<br/><br/>Then, which of the following statements is correct?
Options:
[{"identifier": "A", "content": "(A) is false, but (B) and (C... | ["B"]
Explanation:
Truth Table <br><br><style type="text/css">
.tg {border-collapse:collapse;border-spacing:0;}
.tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px;
overflow:hidden;padding:10px 5px;word-break:normal;}
.tg th{border-color:black;border-style:soli... |
Which of the following is the negation of the statement "for all M > 0, there exists x$$\in$$S such that x $$\ge$$ M" ?
Options:
[{"identifier": "A", "content": "there exists M > 0, such that x < M for all x$$\\in$$S"}, {"identifier": "B", "content": "there exists M > 0, there exists x$$\\in$$S such that x... | ["A"]
Explanation:
P : for all M > 0, there exists x$$\in$$S such that x $$\ge$$ M.<br><br>$$ \sim $$ P : there exists M > 0, for all x$$\in$$S<br><br>Such that x < M<br><br>Negation of 'there exists' is 'for all'. |
Consider the statement "The match will be played only if the weather is good and ground is not wet". Select the correct negation from the following :
Options:
[{"identifier": "A", "content": "The match will not be played and weather is not good and ground is wet."}, {"identifier": "B", "content": "If the match will no... | ["C"]
Explanation:
p : weather is good<br><br>q : ground is not wet<br><br>$$\sim$$ (p $$ \wedge $$ q) $$ \equiv $$ $$\sim$$ p $$ \vee $$ $$\sim$$ q<br><br>$$\equiv$$ weather is not good or ground is wet |
<p>Consider the following statements:</p>
<p>A : Rishi is a judge.</p>
<p>B : Rishi is honest.</p>
<p>C : Rishi is not arrogant.</p>
<p>The negation of the statement "if Rishi is a judge and he is not arrogant, then he is honest" is</p>
Options:
[{"identifier": "A", "content": "B $$\\to$$ (A $$\\vee$$ C)"}, {"identifi... | ["B"]
Explanation:
<p>$$\because$$ Given statement is</p>
<p>(A $$\wedge$$ C) $$\to$$ B</p>
<p>Then its negation is</p>
<p>$$\sim$$ {(A $$\wedge$$ C) $$\to$$ B}</p>
<p>or $$\sim$$ {$$\sim$$ (A $$\wedge$$ C) $$\vee$$ B}</p>
<p>$$\therefore$$ (A $$\wedge$$ C) $$\wedge$$ ($$\sim$$ B)</p>
<p>or ($$\sim$$ B) $$\wedge$$ (A ... |
<p>Consider the following statements:</p>
<p>P : Ramu is intelligent.</p>
<p>Q : Ramu is rich.</p>
<p>R : Ramu is not honest.</p>
<p>The negation of the statement "Ramu is intelligent and honest if and only if Ramu is not rich" can be expressed as:</p>
Options:
[{"identifier": "A", "content": "$$((P \\wedge(\\sim R)) ... | ["D"]
Explanation:
<p>P : Ramu is intelligent</p>
<p>Q : Ramu is rich</p>
<p>R : Ramu is not honest</p>
<p>Given statement, "Ramu is intelligent and honest if and only if Ramu is not rich"</p>
<p>$$ = (P \wedge \sim R) \Leftrightarrow \, \sim Q$$</p>
<p>So, negation of the statement is</p>
<p>$$ \sim [(P \wedge \sim... |
<p>Let</p>
<p>$$\mathrm{p}$$ : Ramesh listens to music.</p>
<p>$$\mathrm{q}$$ : Ramesh is out of his village.</p>
<p>$$\mathrm{r}$$ : It is Sunday.</p>
<p>$$\mathrm{s}$$ : It is Saturday.</p>
<p>Then the statement "Ramesh listens to music only if he is in his village and it is Sunday or Saturday" can be expressed as</p... | ["D"]
Explanation:
<p>p : Ramesh listens to music</p>
<p>q : Ramesh is out of his village</p>
<p>r : It is Sunday</p>
<p>s : It is Saturday</p>
<p>p $$\to$$ q conveys the same p only if q</p>
<p>Statement "Ramesh listens to music only if he is in his village and it is Sunday or Saturday"</p>
<p>$$p \Rightarrow (( \sim... |
<p>Consider the following statements:</p>
<p>P : I have fever</p>
<p>Q: I will not take medicine</p>
<p>$\mathrm{R}$ : I will take rest.</p>
<p>The statement "If I have fever, then I will take medicine and I will take rest" is equivalent to :</p>
Options:
[{"identifier": "A", "content": "$((\\sim P) \\vee \\sim Q) \\w... | ["C"]
Explanation:
<p>The given expression is</p>
<p>$$P \to \sim Q \wedge R$$</p>
<p>$$ \equiv ( \sim P) \vee ( \sim Q \wedge R)$$</p>
<p>$$ \equiv ( \sim P \vee \sim Q) \wedge ( \sim P \vee R)$$</p> |
<p>The number of ordered triplets of the truth values of $$p, q$$ and $$r$$ such that the truth value of the statement $$(p \vee q) \wedge(p \vee r) \Rightarrow(q \vee r)$$ is True, is equal to ___________.</p>
Options:
[] | 7
Explanation:
$$
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline \boldsymbol{p} & \boldsymbol{q} & \boldsymbol{r} & \boldsymbol{p} \vee \boldsymbol{q} & \boldsymbol{p} \vee \boldsymbol{r} & \begin{array}{c}
(\boldsymbol{p} \vee \boldsymbol{q}) \wedge \\
(\boldsymbol{p} \vee \boldsymbol{r})
\end{array} & \boldsymbol{q} \vee \... |
If $$A = \left[ {\matrix{
{5a} & { - b} \cr
3 & 2 \cr
} } \right]$$ and $$A$$ adj $$A=A$$ $${A^T},$$ then $$5a+b$$ is equal to :
Options:
[{"identifier": "A", "content": "$$4$$ "}, {"identifier": "B", "content": "$$13$$"}, {"identifier": "C", "content": "$$-1$$ "}, {"identifier": "D", "content": "... | ["D"]
Explanation:
$$A\left( {Adj\,\,A} \right) = A\,{A^T}$$
<br><br>$$ \Rightarrow {A^{ - 1}}A\left( {adj\,\,A} \right) = {A^{ - 1}}A\,{A^T}$$
<br><br>$$Adj\,\,A = {A^T}$$
<br><br>$$ \Rightarrow \left[ {\matrix{
2 & b \cr
{ - 3} & {5a} \cr
} } \right] = \left[ {\matrix{
{5a} & 3 \cr
{ ... |
If $$A = \left[ {\matrix{
2 & { - 3} \cr
{ - 4} & 1 \cr
} } \right]$$,
<br/><br/>then adj(3A<sup>2</sup> + 12A) is equal to
Options:
[{"identifier": "A", "content": "$$\\left[ {\\matrix{\n {51} & {63} \\cr \n {84} & {72} \\cr \n\n } } \\right]$$"}, {"identifier": "B", "content": "$$\\... | ["A"]
Explanation:
We have, $$A = \left[ {\matrix{
2 & { - 3} \cr
{ - 4} & 1 \cr
} } \right]$$
<br><br>$$ \therefore $$ A<sup>2</sup> = A.A = $$\left[ {\matrix{
2 & { - 3} \cr
{ - 4} & 1 \cr
} } \right]\left[ {\matrix{
2 & { - 3} \cr
{ - 4} & 1 \cr
} } \right... |
Let A be any 3 $$ \times $$ 3 invertible matrix. Then which one of the following is <b>not</b> always true ?
Options:
[{"identifier": "A", "content": "adj (A) = $$\\left| \\right.$$A$$\\left| \\right.$$.A<sup>$$-$$1</sup>"}, {"identifier": "B", "content": "adj (adj(A)) = $$\\left| \\right.$$A$$\\left| \\right.$$.A"... | ["D"]
Explanation:
We know, the formula
<br><br>A<sup>-1</sup> = $${{adj\left( A \right)} \over {\left| A \right|}}$$
<br><br>$$ \therefore $$ adj (A) = $$\left| \right.$$A$$\left| \right.$$.A<sup>$$-$$1</sup>
<br><br><b>So, Option (A) is true.</b>
<br><br>We know, the formula
<br><br>adj (adj (A)) = $${\left| A \rig... |
<p>Let $$B=\left[\begin{array}{lll}1 & 3 & \alpha \\ 1 & 2 & 3 \\ \alpha & \alpha & 4\end{array}\right], \alpha > 2$$ be the adjoint of a matrix $$A$$ and $$|A|=2$$. Then
$$\left[\begin{array}{ccc}\alpha & -2 \alpha & \alpha\end{array}\right] B\left[\begin{array}{c}\alpha \\ -2 \alph... | ["B"]
Explanation:
$$
B=\left[\begin{array}{lll}
1 & 3 & \alpha \\
1 & 2 & 3 \\
\alpha & \alpha & 4
\end{array}\right], \alpha>2
$$
<br/><br/>And $\operatorname{adj}(A)=B,|A|=2$
<br/><br/>$$
\begin{aligned}
& \Rightarrow|\operatorname{adj}(A)|=|B| \\\\
& \Rightarrow 2^2=(8-3 \alpha)-3(4-3 \alpha)+\alpha(-\alpha) \\\\
... |
<p>Let $$A=\left[\begin{array}{ccc}2 & 1 & 0 \\ 1 & 2 & -1 \\ 0 & -1 & 2\end{array}\right]$$. If $$|\operatorname{adj}(\operatorname{adj}(\operatorname{adj} 2 A))|=(16)^{n}$$, then $$n$$ is equal to :</p>
Options:
[{"identifier": "A", "content": "9"}, {"identifier": "B", "content": "8"}, {"iden... | ["C"]
Explanation:
We have,
<br/><br/>$$
\begin{aligned}
& |\mathrm{A}|=\left|\begin{array}{ccc}
2 & 1 & 0 \\
1 & 2 & -1 \\
0 & -1 & 2
\end{array}\right|=2(4-1)-1(2-0)+0 \\\\
& =6-2=4 \\\\
& \text { So, }|2 \mathrm{~A}|=2^3|\mathrm{~A}|=8 \times 4=32 \\\\
& \text { Now, }|\operatorname{adj}(\operatorname{adj}(\operato... |
<p>Let A be a $$3 \times 3$$ matrix and $$\operatorname{det}(A)=2$$. If $$n=\operatorname{det}(\underbrace{\operatorname{adj}(\operatorname{adj}(\ldots . .(\operatorname{adj} A))}_{2024-\text { times }}))$$, then the remainder when $$n$$ is divided by 9 is equal to __________.</p>
Options:
[] | 7
Explanation:
<p>$$\begin{aligned}
& |\mathrm{A}|=2 \\
& \underbrace{\operatorname{adj}(\operatorname{adj}(\operatorname{adj} \ldots . .(\mathrm{a})))}_{2024 \text { times }}=|\mathrm{A}|^{(\mathrm{n}-1)^{2024}} \\
& =|\mathrm{A}|^{2024} \\
& =2^{2^{2024}}
\end{aligned}$$</p>
<p>$$\begin{aligned}
& 2^{2024}=\left(2^2... |
<p>Let $$A$$ be a non-singular matrix of order 3. If $$\operatorname{det}(3 \operatorname{adj}(2 \operatorname{adj}((\operatorname{det} A) A)))=3^{-13} \cdot 2^{-10}$$ and $$\operatorname{det}(3\operatorname{adj}(2 \mathrm{A}))=2^{\mathrm{m}} \cdot 3^{\mathrm{n}}$$, then $$|3 \mathrm{~m}+2 \mathrm{n}|$$ is equal to ___... | 14
Explanation:
<p>$$|\operatorname{adj}(2 \operatorname{adj}(|A| A))|=3^{-13} \cdot 2^{-10}$$</p>
<p>Let $$|A| A=B \Rightarrow|B|=\| A|A|=|A|^3|A|=|A|^4$$</p>
<p>$$\begin{aligned}
\Rightarrow \quad & \operatorname{adj}(|A| A)=(\operatorname{adj} B) \\
\Rightarrow \quad & 2 \operatorname{adj}(|A| A)=(2 \operatorname{a... |
<p>Let $$\alpha \in(0, \infty)$$ and $$A=\left[\begin{array}{lll}1 & 2 & \alpha \\ 1 & 0 & 1 \\ 0 & 1 & 2\end{array}\right]$$. If $$\operatorname{det}\left(\operatorname{adj}\left(2 A-A^T\right) \cdot \operatorname{adj}\left(A-2 A^T\right)\right)=2^8$$, then $$(\operatorname{det}(A))^2$$ is equa... | ["A"]
Explanation:
<p>$$\begin{aligned}
& \left|\operatorname{adj}\left(A-2 A^T\right) \cdot \operatorname{adj}\left(2 A-A^T\right)\right|=2^8 \\
& P=A-2 A^{\top} \\
& Q=2 A^T-A \Rightarrow Q^T=2 A^T-A=-P \\
& |\operatorname{adj}(P) \operatorname{adj}(Q)| \Rightarrow|P Q|=-2^4 \\
& \Rightarrow|P|(-|P|)=-2^4 \Rightarro... |
<p>Let $$A=\left[\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right]$$ and $$B=I+\operatorname{adj}(A)+(\operatorname{adj} A)^2+\ldots+(\operatorname{adj} A)^{10}$$.
Then, the sum of all the elements of the matrix $$B$$ is:</p>
Options:
[{"identifier": "A", "content": "$$-$$110"}, {"identifier": "B", "content": ... | ["D"]
Explanation:
<p>$$\begin{aligned}
& \operatorname{adj}(A)=\left[\begin{array}{ll}
1 & -2 \\
0 & 1
\end{array}\right] \\
& (\operatorname{adj} A)^2=\left[\begin{array}{ll}
1 & -4 \\
0 & 1
\end{array}\right] \\
& (\operatorname{adj} A)^3=\left[\begin{array}{cc}
1 & -6 \\
0 & 1
\end{array}\right] \\
& (\operatornam... |
<p>Let $$\alpha \beta \neq 0$$ and $$A=\left[\begin{array}{rrr}\beta & \alpha & 3 \\ \alpha & \alpha & \beta \\ -\beta & \alpha & 2 \alpha\end{array}\right]$$. If $$B=\left[\begin{array}{rrr}3 \alpha & -9 & 3 \alpha \\ -\alpha & 7 & -2 \alpha \\ -2 \alpha & 5 & -2 \beta\e... | ["D"]
Explanation:
<p>$$A=\left[\begin{array}{ccc}
\beta & \alpha & 3 \\
\alpha & \alpha & \beta \\
-\beta & \alpha & 2 \alpha
\end{array}\right], B=\left[\begin{array}{ccc}
3 \alpha & -9 & 3 \alpha \\
-\alpha & 7 & -2 \alpha \\
-2 \alpha & 5 & -2 \beta
\end{array}\right]$$</p>
<p>Cofactor of $$A$$-matrix is</p>
<p>$$... |
<p>If $$A$$ is a square matrix of order 3 such that $$\operatorname{det}(A)=3$$ and $$\operatorname{det}\left(\operatorname{adj}\left(-4 \operatorname{adj}\left(-3 \operatorname{adj}\left(3 \operatorname{adj}\left((2 \mathrm{~A})^{-1}\right)\right)\right)\right)\right)=2^{\mathrm{m}} 3^{\mathrm{n}}$$, then $$\mathrm{m}... | ["B"]
Explanation:
<p>$$\begin{aligned}
& |A|=3 \\
& \left|\operatorname{adj}\left(-4 \operatorname{adj}\left(-3 \operatorname{adj}\left(3 \operatorname{adj}(2 A)^{-1}\right)\right)\right)\right| \\
& =\left|-4 \operatorname{adj}\left(-3 \operatorname{adj}\left(3 \operatorname{adj}\left((2 A)^{-1}\right)\right)\right)... |
The number of $$3 \times 3$$ non-singular matrices, with four entries as $$1$$ and all other entries as $$0$$, is :
Options:
[{"identifier": "A", "content": "$$5$$ "}, {"identifier": "B", "content": "$$6$$ "}, {"identifier": "C", "content": "at least $$7$$ "}, {"identifier": "D", "content": "less than $$4$$ "}] | ["C"]
Explanation:
$$\left[ {\matrix{
1 & {...} & {...} \cr
{...} & 1 & {...} \cr
{...} & {...} & 1 \cr
} } \right]\,\,$$ are $$6$$ non-singular matrices because $$6$$
<br><br>blanks will be filled by $$5$$ zeros and $$1$$ one.
<br><br>Similarly, $$\left[ {\matrix{
{...} &am... |
<p>Let $$A=\left[\begin{array}{lll}
1 & a & a \\
0 & 1 & b \\
0 & 0 & 1
\end{array}\right], a, b \in \mathbb{R}$$. If for some <br/><br/>$$n \in \mathbb{N}, A^{n}=\left[\begin{array}{ccc}
1 & 48 & 2160 \\
0 & 1 & 96 \\
0 & 0 & 1
\end{array}\right]
$$ then $$n+a+b$$ is equ... | 24
Explanation:
<p>$$A = \left[ {\matrix{
1 & 0 & 0 \cr
0 & 1 & 0 \cr
0 & 0 & 1 \cr
} } \right] + \left[ {\matrix{
0 & a & a \cr
0 & 0 & b \cr
0 & 0 & 0 \cr
} } \right] = I + B$$</p>
<p>$${B^2} = \left[ {\matrix{
0 & a & a \cr
0 & 0 & b \cr
0 & 0 & 0 \cr
} } \right] +... |
If $$a>0$$ and discriminant of $$\,a{x^2} + 2bx + c$$ is $$-ve$$, then
<br/>$$\left| {\matrix{
a & b & {ax + b} \cr
b & c & {bx + c} \cr
{ax + b} & {bx + c} & 0 \cr
} } \right|$$ is equal to
Options:
[{"identifier": "A", "content": "$$+ve$$ "}, {"identifier": "B", "content"... | ["C"]
Explanation:
We have $$\left| {\matrix{
a & b & {ax + b} \cr
b & c & {bx + c} \cr
{ax + b} & {bx + c} & 0 \cr
} } \right|$$
<br><br>By $$\,\,\,{R_3} \to {R_3} - \left( {x{R_1} + {R_2}} \right);$$
<br><br>$$ = \left| {\matrix{
a & b & {ax + b} \cr
b & c... |
If $$1,$$ $$\omega ,{\omega ^2}$$ are the cube roots of unity, then
<p>$$\Delta = \left| {\matrix{
1 & {{\omega ^n}} & {{\omega ^{2n}}} \cr
{{\omega ^n}} & {{\omega ^{2n}}} & 1 \cr
{{\omega ^{2n}}} & 1 & {{\omega ^n}} \cr
} } \right|$$ is equal to</p>
Options:
[{"identifier"... | ["B"]
Explanation:
$$\Delta = \left| {\matrix{
1 & {{\omega ^n}} & {{\omega ^{2n}}} \cr
{{\omega ^n}} & {{\omega ^{2n}}} & 1 \cr
{{\omega ^{2n}}} & 1 & {{\omega ^n}} \cr
} } \right|$$
<br><br>$$ = 1\left( {{\omega ^{3n}} - 1} \right) - {\omega ^n}\left( {{\omega ^{2n}} - {\ome... |
If $${a_1},{a_2},{a_3},.........,{a_n},......$$ are in G.P., then the value of the determinant
<p>$$\left| {\matrix{
{\log {a_n}} & {\log {a_{n + 1}}} & {\log {a_{n + 2}}} \cr
{\log {a_{n + 3}}} & {\log {a_{n + 4}}} & {\log {a_{n + 5}}} \cr
{\log {a_{n + 6}}} & {\log {a_{n + 7}}} &... | ["D"]
Explanation:
$$\left| {\matrix{
{\log {a_n}} & {\log {a_{n + 1}}} & {\log {a_{n + 2}}} \cr
{\log {a_{n + 3}}} & {\log {a_{n + 4}}} & {\log {a_{n + 5}}} \cr
{\log {a_{n + 6}}} & {\log {a_{n + 7}}} & {\log {a_{n + 8}}} \cr
} } \right|$$
<br><br>$$ = \left| {\matrix{
{\lo... |
If $${a_1},{a_2},{a_3},........,{a_n},.....$$ are in G.P., then the determinant
$$$\Delta = \left| {\matrix{
{\log {a_n}} & {\log {a_{n + 1}}} & {\log {a_{n + 2}}} \cr
{\log {a_{n + 3}}} & {\log {a_{n + 4}}} & {\log {a_{n + 5}}} \cr
{\log {a_{n + 6}}} & {\log {a_{n + 7}}} & {\log {... | ["B"]
Explanation:
As $$\,\,\,\,{a_1},{a_2},{a_3},.........$$ are in $$G.P.$$
<br><br>$$\therefore$$ Using $${a_n} = a{r^{n - 1}},\,\,\,$$ we get the given determinant,
<br><br>as $$\,\,\,\,\,\,\,\left| {\matrix{
{\log a{r^{n - 1}}} & {\log a{r^n}} & {\log a{r^{n + 1}}} \cr
{\log a{r^{n + 2}}} & ... |
If $${a^2} + {b^2} + {c^2} = - 2$$ and
<p>f$$\left( x \right) = \left| {\matrix{
{1 + {a^2}x} & {\left( {1 + {b^2}} \right)x} & {\left( {1 + {c^2}} \right)x} \cr
{\left( {1 + {a^2}} \right)x} & {1 + {b^2}x} & {\left( {1 + {c^2}} \right)x} \cr
{\left( {1 + {a^2}} \right)x} & {\left( {1... | ["D"]
Explanation:
Applying, $${C_1} \to {C_1} + {C_2} + {C_3}\,\,\,$$ we get
<br><br>$$f\left( x \right) = \left| {\matrix{
{1 + \left( {{a^2} + {b^2} + {c^2} + 2} \right)x} & {\left( {1 + {b^2}} \right)x} & {\left( {1 + {c^2}} \right)x} \cr
{1 + \left( {{a^2} + {b^2} + {c^2} + 2} \right)x} & {1 ... |
If $$D = \left| {\matrix{
1 & 1 & 1 \cr
1 & {1 + x} & 1 \cr
1 & 1 & {1 + y} \cr
} } \right|$$ for $$x \ne 0,y \ne 0,$$ then $$D$$ is :
Options:
[{"identifier": "A", "content": "divisible by $$x$$ but not $$y$$"}, {"identifier": "B", "content": "divisible by $$y$$ but not $$x$$"... | ["D"]
Explanation:
Given, $$D = \left| {\matrix{
1 & 1 & 1 \cr
1 & {1 + x} & 1 \cr
1 & 1 & {1 + y} \cr
} } \right|$$
<br><br>Apply $$\,\,\,{R^2} \to {R_2} - {R_1}$$ $$\,\,\,\,$$
<br><br>and $$\,\,\,\,$$ $$R \to {R_3} - {R_1}$$
<br><br>$$\therefore$$ $$\,\,\,\,\,D = \left| {\ma... |
Let $$a, b, c$$ be such that $$b\left( {a + c} \right) \ne 0$$ if
<p>$$\left| {\matrix{
a & {a + 1} & {a - 1} \cr
{ - b} & {b + 1} & {b - 1} \cr
c & {c - 1} & {c + 1} \cr
} } \right| + \left| {\matrix{
{a + 1} & {b + 1} & {c - 1} \cr
{a - 1} & {b - 1} & {... | ["B"]
Explanation:
$$\left| {\matrix{
a & {a + 1} & {a - 1} \cr
{ - b} & {b + 1} & {b - 1} \cr
c & {c - 1} & {c + 1} \cr
} } \right| + \left| {\matrix{
{a + 1} & {b + 1} & {c - 1} \cr
{a - 1} & {b - 1} & {c + 1} \cr
{{{\left( { - 1} \right)}^{n + 2}... |
If $$\alpha ,\beta \ne 0,$$ and $$f\left( n \right) = {\alpha ^n} + {\beta ^n}$$ and
$$$\left| {\matrix{
3 & {1 + f\left( 1 \right)} & {1 + f\left( 2 \right)} \cr
{1 + f\left( 1 \right)} & {1 + f\left( 2 \right)} & {1 + f\left( 3 \right)} \cr
{1 + f\left( 2 \right)} & {1 + f\left( 3 \... | ["A"]
Explanation:
Consider
<br><br>$$\left| {\matrix{
3 & {1 + f\left( 1 \right)} & {1 + f\left( 2 \right)} \cr
{1 + f\left( 1 \right)} & {1 + f\left( 2 \right)} & {1 + f\left( 3 \right)} \cr
{1 + f\left( 2 \right)} & {1 + f\left( 3 \right)} & {1 + f\left( 4 \right)} \cr
} } \... |
If A = $$\left[ {\matrix{
{ - 4} & { - 1} \cr
3 & 1 \cr
} } \right]$$,
<br/><br/>then the determinant of the matrix (A<sup>2016</sup> − 2A<sup>2015</sup> − A<sup>2014</sup>) is :
Options:
[{"identifier": "A", "content": "2014"}, {"identifier": "B", "content": "$$-$$ 175"}, {"identifier": "C", ... | ["D"]
Explanation:
Given,
<br><br>$$A = \left[ {\matrix{
{ - 4} & { - 1} \cr
3 & 1 \cr
} } \right]$$
<br><br>$${A^2} = \left[ {\matrix{
{ - 4} & { - 1} \cr
3 & 1 \cr
} } \right]\left[ {\matrix{
{ - 4} & { - 1} \cr
3 & 1 \cr
} } \right]$$
<br><br>$$ = \left[ {... |
The number of distinct real roots of the equation,
<br/><br/>$$\left| {\matrix{
{\cos x} & {\sin x} & {\sin x} \cr
{\sin x} & {\cos x} & {\sin x} \cr
{\sin x} & {\sin x} & {\cos x} \cr
} } \right| = 0$$ in the interval $$\left[ { - {\pi \over 4},{\pi \over 4}} \right]$$ is :... | ["C"]
Explanation:
Given,
<br><br>$$\left| {\matrix{
{\cos x} & {\sin x} & {\sin x} \cr
{\sin x} & {\cos x} & {\sin x} \cr
{\sin x} & {\sin x} & {\cos x} \cr
} } \right| = 0$$
<br><br>R<sub>1</sub> $$ \to $$ R<sub>1</sub> $$-$$ R<sub... |
If
<br/><br/>$$S = \left\{ {x \in \left[ {0,2\pi } \right]:\left| {\matrix{
0 & {\cos x} & { - \sin x} \cr
{\sin x} & 0 & {\cos x} \cr
{\cos x} & {\sin x} & 0 \cr
} } \right| = 0} \right\},$$
<br/><br/>then $$\sum\limits_{x \in S} {\tan \left( {{\pi \over 3} + x} \right)} $$ i... | ["C"]
Explanation:
Given,
<br><br> $$\left| {\matrix{
0 & {\cos x} & { - \sin x} \cr
{\sin x} & 0 & {\cos x} \cr
{\cos x} & {\sin x} & 0 \cr
} } \right|$$ = 0
<br><br>$$ \Rightarrow $$$$\,\,\,$$ 0 (0 $$-$$ cosx sinx) $$-$$ cosx (0... |
Let $$A$$ be a matrix such that $$A.\left[ {\matrix{
1 & 2 \cr
0 & 3 \cr
} } \right]$$ is a scalar matrix and |3A| = 108.
<br/>Then A<sup>2</sup> equals :
Options:
[{"identifier": "A", "content": "$$\\left[ {\\matrix{\n 4 & { - 32} \\cr \n 0 & {36} \\cr \n\n } } \\right]$$"}, {"ident... | ["D"]
Explanation:
According to questions, <br/><br>
A. $$\left[ {\matrix{
1 & 2 \cr
0 & 3 \cr
} } \right]$$ = $$\left[ {\matrix{
\lambda & 0 \cr
0 & \lambda \cr
} } \right]$$<br><br>
$$ \Rightarrow $$ A = $$\left[ {\matrix{
\lambda & 0 \cr
0 & \lambda \cr
}... |
If $$\left| {\matrix{
{x - 4} & {2x} & {2x} \cr
{2x} & {x - 4} & {2x} \cr
{2x} & {2x} & {x - 4} \cr
} } \right| = \left( {A + Bx} \right){\left( {x - A} \right)^2}$$
<br/><br/>then the ordered pair (A, B) is equal to :
Options:
[{"identifier": "A", "content": "(4, 5)"}, {"ident... | ["D"]
Explanation:
$$\left| {\matrix{
{x - 4} & {2x} & {2x} \cr
{2x} & {x - 4} & {2x} \cr
{2x} & {2x} & {x - 4} \cr
} } \right|$$
<br><br>Applying c<sub>1</sub> $$ \to $$ c<sub>1</sub> + c<sub>2</sub> + c<sub>3</sub>
<br><br>$$ = \,\,\,\,\left| {\matrix{
{5x - 4} & {2x}... |
If $$A = \left[ {\matrix{
{{e^t}} & {{e^{ - t}}\cos t} & {{e^{ - t}}\sin t} \cr
{{e^t}} & { - {e^{ - t}}\cos t - {e^{ - t}}\sin t} & { - {e^{ - t}}\sin t + {e^{ - t}}co{\mathop{\rm s}\nolimits} t} \cr
{{e^t}} & {2{e^{ - t}}\sin t} & { - 2{e^{ - t}}\cos t} \cr
} } \right]$$
<br... | ["A"]
Explanation:
$$A = \left[ {\matrix{
{{e^t}} & {{e^{ - t}}\cos t} & {{e^{ - t}}\sin t} \cr
{{e^t}} & { - {e^{ - t}}\cos t - {e^{ - t}}\sin t} & { - {e^{ - t}}\sin t + {e^{ - t}}co{\mathop{\rm s}\nolimits} t} \cr
{{e^t}} & {2{e^{ - t}}\sin t} & { - 2{e^{ - t}}\cos t} \cr
} ... |
A value of $$\theta \in \left( {0,{\pi \over 3}} \right)$$, for which
<br/>$$\left| {\matrix{
{1 + {{\cos }^2}\theta } & {{{\sin }^2}\theta } & {4\cos 6\theta } \cr
{{{\cos }^2}\theta } & {1 + {{\sin }^2}\theta } & {4\cos 6\theta } \cr
{{{\cos }^2}\theta } & {{{\sin }^2}\theta } &... | ["B"]
Explanation:
$$\left| {\matrix{
{1 + {{\cos }^2}\theta } & {{{\sin }^2}\theta } & {4\cos 6\theta } \cr
{{{\cos }^2}\theta } & {1 + {{\sin }^2}\theta } & {4\cos 6\theta } \cr
{{{\cos }^2}\theta } & {{{\sin }^2}\theta } & {1 + 4\cos 6\theta } \cr
} } \right| = 0$$<br><br>
R... |
If $$B = \left[ {\matrix{
5 & {2\alpha } & 1 \cr
0 & 2 & 1 \cr
\alpha & 3 & { - 1} \cr
} } \right]$$ is the inverse of a 3 × 3 matrix A, then the sum of all values of $$\alpha $$ for which
det(A) + 1 = 0, is :
Options:
[{"identifier": "A", "content": "2"}, {"identifier": "B",... | ["D"]
Explanation:
Given |A| + 1 = 0
<br><br>$$ \Rightarrow $$ |A| = -1
<br><br>$$\left| B \right| = \left| {{A^{ - 1}}} \right| = {1 \over {\left| A \right|}} = - 1$$<br><br>
$$\left| {\matrix{
5 & {2\alpha } & 1 \cr
0 & 2 & 1 \cr
\alpha & 3 & { - 1} \cr
} } \right| $$ = -1
... |
The sum of the real roots of the equation
<br/>$$\left| {\matrix{
x & { - 6} & { - 1} \cr
2 & { - 3x} & {x - 3} \cr
{ - 3} & {2x} & {x + 2} \cr
} } \right| = 0$$, is equal to :
Options:
[{"identifier": "A", "content": "- 4"}, {"identifier": "B", "content": "0"}, {"identifier":... | ["B"]
Explanation:
x(-3x $$ \times $$ (x + 2) - 2x(x - 3)) + (– 6) (2(x + 2) + 3 (x – 3)) + (–1) (4x + 3 (–3x))<br><br>
$$ \Rightarrow $$ – 5x<sup>3</sup> + 30x –30 + 5x = 0<br><br>
$$ \Rightarrow $$ x<sup>3</sup> – 7x + 6 = 0<br><br>
$$ \therefore $$ sum of roots = 0 |
If $${\Delta _1} = \left| {\matrix{
x & {\sin \theta } & {\cos \theta } \cr
{ - \sin \theta } & { - x} & 1 \cr
{\cos \theta } & 1 & x \cr
} } \right|$$ and
<br/>$${\Delta _2} = \left| {\matrix{
x & {\sin 2\theta } & {\cos 2\theta } \cr
{ - \sin 2\theta } & {... | ["B"]
Explanation:
$${\Delta _1} = \left| {\matrix{
x & {\sin \theta } & {\cos \theta } \cr
{ - \sin \theta } & { - x} & 1 \cr
{\cos \theta } & 1 & x \cr
} } \right|$$<br><br>
= x(–x<sup>2</sup> –1) – sin$$\theta $$(–xsin$$\theta $$ – cos$$\theta $$) + cos$$\theta $$(–sin$$\the... |
Let $$\alpha $$ and $$\beta $$ be the roots of the equation
x<sup>2</sup> + x + 1 = 0. Then for y $$ \ne $$ 0 in R,<br/>
$$$\left| {\matrix{
{y + 1} & \alpha & \beta \cr
\alpha & {y + \beta } & 1 \cr
\beta & 1 & {y + \alpha } \cr
} } \right|$$$
is equal to
Options:
[{"ident... | ["C"]
Explanation:
$$\alpha $$ and $$\beta $$ are the roots of the equation
x<sup>2</sup> + x + 1 = 0.
<br><br>$$ \therefore $$ $$\alpha $$ = $$\omega $$ and $$\beta $$ = $${\omega ^2}$$
<br><br>$$\left| {\matrix{
{y + 1} & \alpha & \beta \cr
\alpha & {y + \beta } & 1 \cr
\beta & 1... |
Let the number 2,b,c be in an A.P. and<br/>
A = $$\left[ {\matrix{
1 & 1 & 1 \cr
2 & b & c \cr
4 & {{b^2}} & {{c^2}} \cr
} } \right]$$. If det(A) $$ \in $$ [2, 16], then c
lies in the interval :
Options:
[{"identifier": "A", "content": "[2, 3)"}, {"identifier": "B", "content": ... | ["B"]
Explanation:
2, b, c are in AP.
<br><br>Let common difference = d
<br><br>$$ \therefore $$ b = 2 + d and c = 2 + 2d
<br><br>|A| = $$\left[ {\matrix{
1 & 1 & 1 \cr
2 & b & c \cr
4 & {{b^2}} & {{c^2}} \cr
} } \right]$$
<br><br>C<sub>2</sub> = C<sub>2</sub> - C<sub>1</sub>
<... |
If A = $$\left[ {\matrix{
1 & {\sin \theta } & 1 \cr
{ - \sin \theta } & 1 & {\sin \theta } \cr
{ - 1} & { - \sin \theta } & 1 \cr
} } \right]$$;
<br/><br/>then for all $$\theta $$ $$ \in $$ $$\left( {{{3\pi } \over 4},{{5\pi } \over 4}} \right)$$, det (A) lies in the interva... | ["A"]
Explanation:
$$\left| A \right| = \left| {\matrix{
1 & {\sin \theta } & 1 \cr
{ - \sin \theta } & 1 & {\sin \theta } \cr
{ - 1} & { - \sin \theta } & 1 \cr
} } \right|$$
<br><br>= 2(1 + sin<sup>2</sup>$$\theta $$)
<br><br>$$\theta $$ $$ \in $$ $$\left( {{{3\pi } \over 4},... |
If $$\left| {\matrix{
{a - b - c} & {2a} & {2a} \cr
{2b} & {b - c - a} & {2b} \cr
{2c} & {2c} & {c - a - b} \cr
} } \right|$$
<br/><br/> = (a + b + c) (x + a + b + c)<sup>2</sup>, x $$ \ne $$ 0,
<br/><br/>then x is equal to :
Options:
[{"identifier": "A", "content": "\u... | ["A"]
Explanation:
$$\left| {\matrix{
{a - b - c} & {2a} & {2a} \cr
{2b} & {b - c - a} & {2b} \cr
{2c} & {2c} & {c - a - b} \cr
} } \right|$$
<br><br>R<sub>1</sub> $$ \to $$ R<sub>1</sub> + R<sub>2</sub> + R<sub>3</sub>
<br><br>$$ = \left| {\matrix{
{a + b + c} & {a + b... |
Let A = $$\left[ {\matrix{
2 & b & 1 \cr
b & {{b^2} + 1} & b \cr
1 & b & 2 \cr
} } \right]$$ where b > 0.
<br/><br/>Then the minimum value of $${{\det \left( A \right)} \over b}$$ is -
Options:
[{"identifier": "A", "content": "$$\\sqrt 3 $$"}, {"identifier": "B", "conten... | ["D"]
Explanation:
A = $$\left[ {\matrix{
2 & b & 1 \cr
b & {{b^2} + 1} & b \cr
1 & b & 2 \cr
} } \right]$$ (b > 0)
<br><br>$$\left| A \right|$$ = 2(2b<sup>2</sup> + 2 $$-$$ b<sup>2</sup>) $$-$$ b(2b $$-$$ b) + 1(b<sub>2</sub> $$-$$ b<sub>2</sub> $$-$$ 1)
<br><br>$$\left|... |
Let d $$ \in $$ R, and
<br/><br/>$$A = \left[ {\matrix{
{ - 2} & {4 + d} & {\left( {\sin \theta } \right) - 2} \cr
1 & {\left( {\sin \theta } \right) + 2} & d \cr
5 & {\left( {2\sin \theta } \right) - d} & {\left( { - \sin \theta } \right) + 2 + 2d} \cr
} } \right],$$
<br/><br... | ["C"]
Explanation:
$$\det A = \left| {\matrix{
{ - 2} & {4 + d} & {\sin \theta - 2} \cr
1 & {\sin \theta + 2} & d \cr
5 & {2\sin \theta - d} & { - \sin \theta + 2 + 2d} \cr
} } \right|$$
<br><br>(R<sub>1</sub> $$ \to $$ R<sub>1</sub> + R<sub>3</sub> $$-$$ 2R<sub>2</sub>)
<b... |
If $$\Delta $$ = $$\left| {\matrix{
{x - 2} & {2x - 3} & {3x - 4} \cr
{2x - 3} & {3x - 4} & {4x - 5} \cr
{3x - 5} & {5x - 8} & {10x - 17} \cr
} } \right|$$ =
<br/><br/>Ax<sup>3</sup> + Bx<sup>2</sup> + Cx + D, then B + C is equal to :
Options:
[{"identifier": "A", "content": "-... | ["B"]
Explanation:
$$\Delta $$ = $$\left| {\matrix{
{x - 2} & {2x - 3} & {3x - 4} \cr
{2x - 3} & {3x - 4} & {4x - 5} \cr
{3x - 5} & {5x - 8} & {10x - 17} \cr
} } \right|$$
<br><br>R<sub>2</sub> $$ \to $$ R<sub>2</sub> – R<sub>1</sub>
<br>R<sub>3</sub> $$ \to $$ R<sub>3</sub> –... |
Let $$\theta = {\pi \over 5}$$ and $$A = \left[ {\matrix{
{\cos \theta } & {\sin \theta } \cr
{ - \sin \theta } & {\cos \theta } \cr
} } \right]$$. <br/><br/> If B = A + A<sup>4</sup>
, then det (B) :
Options:
[{"identifier": "A", "content": "lies in (1, 2)"}, {"identifier": "B", "content": "lies... | ["A"]
Explanation:
$$A = \left[ {\matrix{
{\cos \theta } & {\sin \theta } \cr
{ - \sin \theta } & {\cos \theta } \cr
} } \right]$$
<br><br>A<sup>2</sup> = $$\left[ {\matrix{
{\cos \theta } & {\sin \theta } \cr
{ - \sin \theta } & {\cos \theta } \cr
} } \right]$$$$\left[ {\matrix... |
Let m and M be respectively the minimum and maximum values of
<br/><br/>$$\left| {\matrix{
{{{\cos }^2}x} & {1 + {{\sin }^2}x} & {\sin 2x} \cr
{1 + {{\cos }^2}x} & {{{\sin }^2}x} & {\sin 2x} \cr
{{{\cos }^2}x} & {{{\sin }^2}x} & {1 + \sin 2x} \cr
} } \right|$$
<br/><br/>Then the... | ["A"]
Explanation:
$$\left| {\matrix{
{{{\cos }^2}x} & {1 + {{\sin }^2}x} & {\sin 2x} \cr
{1 + {{\cos }^2}x} & {{{\sin }^2}x} & {\sin 2x} \cr
{{{\cos }^2}x} & {{{\sin }^2}x} & {1 + \sin 2x} \cr
} } \right|$$
<br><br>R<sub>1</sub> $$ \to $$ R<sub>1</sub> – R<sub>2</sub>, R<sub>2... |
If the minimum and the maximum values of the function $$f:\left[ {{\pi \over 4},{\pi \over 2}} \right] \to R$$, defined by <br/>
$$f\left( \theta \right) = \left| {\matrix{
{ - {{\sin }^2}\theta } & { - 1 - {{\sin }^2}\theta } & 1 \cr
{ - {{\cos }^2}\theta } & { - 1 - {{\cos }^2}\theta } & 1 ... | ["B"]
Explanation:
Given <br>$$f\left( \theta \right) = \left| {\matrix{
{ - {{\sin }^2}\theta } & { - 1 - {{\sin }^2}\theta } & 1 \cr
{ - {{\cos }^2}\theta } & { - 1 - {{\cos }^2}\theta } & 1 \cr
{12} & {10} & { - 2} \cr
} } \right|$$
<br><br>C<sub>1</sub> $$ \to $$ C<sub>1</... |
If a + x = b + y = c + z + 1, where a, b, c, x, y, z<br/>
are non-zero distinct real numbers, then
<br/>$$\left| {\matrix{
x & {a + y} & {x + a} \cr
y & {b + y} & {y + b} \cr
z & {c + y} & {z + c} \cr
} } \right|$$ is equal to :
Options:
[{"identifier": "A", "content": "y(b \u2... | ["B"]
Explanation:
$$\left| {\matrix{
x & {a + y} & {x + a} \cr
y & {b + y} & {y + b} \cr
z & {c + y} & {z + c} \cr
} } \right|$$
<br><br>C<sub>3</sub> $$ \to $$ C<sub>3</sub> – C<sub>1</sub>
<br><br>= $$\left| {\matrix{
x & {a + y} & a \cr
y & {b + y} &... |
Let A be a 3 $$\times$$ 3 matrix with det(A) = 4. Let R<sub>i</sub> denote the i<sup>th</sup> row of A. If a matrix B is obtained by performing the operation R<sub>2</sub> $$ \to $$ 2R<sub>2</sub> + 5R<sub>3</sub> on 2A, then det(B) is equal to :
Options:
[{"identifier": "A", "content": "64"}, {"identifier": "B", "con... | ["A"]
Explanation:
$$A = \left[ {\matrix{
{{R_{11}}} & {{R_{12}}} & {{R_{13}}} \cr
{{R_{21}}} & {{R_{22}}} & {{R_{23}}} \cr
{{R_{31}}} & {{R_{32}}} & {{R_{33}}} \cr
} } \right]$$<br><br>$$2A = \left[ {\matrix{
{2{R_{11}}} & {2{R_{12}}} & {2{R_{13}}} \cr
{2{R_{21... |
The value of $$\left| {\matrix{
{(a + 1)(a + 2)} & {a + 2} & 1 \cr
{(a + 2)(a + 3)} & {a + 3} & 1 \cr
{(a + 3)(a + 4)} & {a + 4} & 1 \cr
} } \right|$$ is :
Options:
[{"identifier": "A", "content": "$$-$$2"}, {"identifier": "B", "content": "0"}, {"identifier": "C", "content": "(... | ["A"]
Explanation:
Given, $$\Delta $$ = $$\left| {\matrix{
{(a + 1)(a + 2)} & {a + 2} & 1 \cr
{(a + 2)(a + 3)} & {a + 3} & 1 \cr
{(a + 3)(a + 4)} & {a + 4} & 1 \cr
} } \right|$$
<br><br>R<sub>2</sub> $$ \to $$ R<sub>2</sub> $$-$$ R<sub>1</sub> and R<sub>3</sub> $$ \to $$ R<sub>... |
If x, y, z are in arithmetic progression with common difference d, x $$\ne$$ 3d, and the determinant of the matrix $$\left[ {\matrix{
3 & {4\sqrt 2 } & x \cr
4 & {5\sqrt 2 } & y \cr
5 & k & z \cr
} } \right]$$ is zero, then the value of k<sup>2</sup> is :
Options:
[{"identifier... | ["A"]
Explanation:
$$\left| {\matrix{
3 & {4\sqrt 2 } & x \cr
4 & {5\sqrt 2 } & y \cr
5 & k & z \cr
} } \right| = 0$$<br><br>$${R_1} \to {R_1} + {R_3} - 2{R_2}$$<br><br>$$ \Rightarrow $$ $$\left| {\matrix{
0 & {4\sqrt 2 - k - 10\sqrt 2 } & 0 \cr
4 & {5\sqrt... |
If 1, log<sub>10</sub>(4<sup>x</sup> $$-$$ 2) and log<sub>10</sub>$$\left( {{4^x} + {{18} \over 5}} \right)$$ are in arithmetic progression for a real number x, then the value of the determinant $$\left| {\matrix{
{2\left( {x - {1 \over 2}} \right)} & {x - 1} & {{x^2}} \cr
1 & 0 & x \cr
x &... | 2
Explanation:
1, $$lo{g_{10}}({4^x} - 2),\,lo{g_{10}}\left( {{4^x} + {{18} \over 5}} \right)$$ in AP.<br><br>$$ \therefore $$ 2$$ \times $$$$lo{g_{10}}({4^x} - 2) = 1 + \,lo{g_{10}}\left( {{4^x} + {{18} \over 5}} \right)$$ <br><br>$$lo{g_{10}}{({4^x} - 2)^2} = \,lo{g_{10}}\left( {10.\left( {{4^x} + {{18} \over 5}} \r... |
The solutions of the equation $$\left| {\matrix{
{1 + {{\sin }^2}x} & {{{\sin }^2}x} & {{{\sin }^2}x} \cr
{{{\cos }^2}x} & {1 + {{\cos }^2}x} & {{{\cos }^2}x} \cr
{4\sin 2x} & {4\sin 2x} & {1 + 4\sin 2x} \cr
} } \right| = 0,(0 < x < \pi )$$, are
Options:
[{"identifier": "... | ["D"]
Explanation:
By using C<sub>1</sub> $$ \to $$ C<sub>1</sub> $$-$$ C<sub>2</sub> and C<sub>3</sub> $$ \to $$ C<sub>3</sub> $$-$$ C<sub>2</sub> we get<br><br>$$\left| {\matrix{
1 & {{{\sin }^2}x} & 0 \cr
{ - 1} & {1 + {{\cos }^2}x} & { - 1} \cr
0 & {4\sin 2x} & 1 \cr
} } \r... |
Let I be an identity matrix of order 2 $$\times$$ 2 and P = $$\left[ {\matrix{
2 & { - 1} \cr
5 & { - 3} \cr
} } \right]$$. Then the value of n$$\in$$N for which P<sup>n</sup> = 5I $$-$$ 8P is equal to ____________.
Options:
[] | 6
Explanation:
$$P = \left[ {\matrix{
2 & { - 1} \cr
5 & { - 3} \cr
} } \right]$$<br><br>$$\left| {\matrix{
{2 - \lambda } & { - 1} \cr
5 & { - 3 - \lambda } \cr
} } \right| = 0$$<br><br>$$ \Rightarrow $$ $$\lambda$$<sup>2</sup> + $$\lambda$$ $$-$$ 1 = 0<br><br>$$ \Rightarrow $$... |
Let a, b, c, d in arithmetic progression with common difference $$\lambda$$. If $$\left| {\matrix{
{x + a - c} & {x + b} & {x + a} \cr
{x - 1} & {x + c} & {x + b} \cr
{x - b + d} & {x + d} & {x + c} \cr
} } \right| = 2$$, then value of $$\lambda$$<sup>2</sup> is equal to _______... | 1
Explanation:
$$\left| {\matrix{
{x + a - c} & {x + b} & {x + a} \cr
{x - 1} & {x + c} & {x + b} \cr
{x - b + d} & {x + d} & {x + c} \cr
} } \right| = 2$$<br><br>$${C_2} \to {C_2} - {C_3}$$<br><br>$$ \Rightarrow \left| {\matrix{
{x - 2\lambda } & \lambda & {x + a} ... |
The number of distinct real roots <br/><br/>of $$\left| {\matrix{
{\sin x} & {\cos x} & {\cos x} \cr
{\cos x} & {\sin x} & {\cos x} \cr
{\cos x} & {\cos x} & {\sin x} \cr
} } \right| = 0$$ in the interval $$ - {\pi \over 4} \le x \le {\pi \over 4}$$ is :
Options:
[{"identifie... | ["B"]
Explanation:
$$\left| {\matrix{
{\sin x} & {\cos x} & {\cos x} \cr
{\cos x} & {\sin x} & {\cos x} \cr
{\cos x} & {\cos x} & {\sin x} \cr
} } \right| = 0, - {\pi \over 4} \le x \le {\pi \over 4}$$<br><br>Apply : $${R_1} \to {R_1} - {R_2}$$ & $${R_2} \to {R_2} - {R_3}... |
Let $$f(x) = \left| {\matrix{
{{{\sin }^2}x} & { - 2 + {{\cos }^2}x} & {\cos 2x} \cr
{2 + {{\sin }^2}x} & {{{\cos }^2}x} & {\cos 2x} \cr
{{{\sin }^2}x} & {{{\cos }^2}x} & {1 + \cos 2x} \cr
} } \right|,x \in [0,\pi ]$$. Then the maximum value of f(x) is equal to ______________.
... | 6
Explanation:
$$\left| {\matrix{
{ - 2} & { - 2} & 0 \cr
2 & 0 & { - 1} \cr
{{{\sin }^2}x} & {{{\cos }^2}x} & {1 + \cos 2x} \cr
} } \right|\left( \matrix{
{R_1} \to {R_1} - {R_2} \hfill \cr
\& \,{R_2} \to {R_2} - {R_3} \hfill \cr} \right)$$<br><br>= $$ - 2({\cos ^2}x... |
Let $$A = \left( {\matrix{
{[x + 1]} & {[x + 2]} & {[x + 3]} \cr
{[x]} & {[x + 3]} & {[x + 3]} \cr
{[x]} & {[x + 2]} & {[x + 4]} \cr
} } \right)$$, where [t] denotes the greatest integer less than or equal to t. If det(A) = 192, then the set of values of x is the interval :
Opt... | ["B"]
Explanation:
$$\left| {\matrix{
{[x + 1]} & {[x + 2]} & {[x + 3]} \cr
{[x]} & {[x + 3]} & {[x + 3]} \cr
{[x]} & {[x + 2]} & {[x + 4]} \cr
} } \right| = 192$$<br><br>R<sub>1</sub> $$\to$$ R<sub>1</sub> $$-$$ R<sub>3</sub> & R<sub>2</sub> $$\to$$ R<sub>2</sub> $$-$$ R<s... |
If $${a_r} = \cos {{2r\pi } \over 9} + i\sin {{2r\pi } \over 9}$$, r = 1, 2, 3, ....., i = $$\sqrt { - 1} $$, then<br/> the determinant $$\left| {\matrix{
{{a_1}} & {{a_2}} & {{a_3}} \cr
{{a_4}} & {{a_5}} & {{a_6}} \cr
{{a_7}} & {{a_8}} & {{a_9}} \cr
} } \right|$$ is equal to :
... | ["C"]
Explanation:
$${a_r} = {e^{{{i2\pi r} \over 9}}}$$, r = 1, 2, 3, ......, a<sub>1</sub>, a<sub>2</sub>, a<sub>3</sub>, ..... are in G.P.<br><br>$$\left| {\matrix{
{{a_1}} & {{a_2}} & {{a_3}} \cr
{{a_n}} & {{a_5}} & {{a_6}} \cr
{{a_7}} & {{a_8}} & {{a_9}} \cr
} } \right| = ... |
<p>Let $$A = \left[ {\matrix{
1 & { - 2} & \alpha \cr
\alpha & 2 & { - 1} \cr
} } \right]$$ and $$B = \left[ {\matrix{
2 & \alpha \cr
{ - 1} & 2 \cr
4 & { - 5} \cr
} } \right],\,\alpha \in C$$. Then the absolute value of the sum of all values of $$\alpha$$ fo... | ["A"]
Explanation:
<p>Given,</p>
<p>$$A = \left[ {\matrix{
1 & { - 2} & \alpha \cr
\alpha & 2 & { - 1} \cr
} } \right]$$</p>
<p>and $$B = \left[ {\matrix{
2 & \alpha \cr
{ - 1} & 2 \cr
4 & { - 5} \cr
} } \right]$$</p>
<p>$$AB = \left[ {\matrix{
1 & { - 2} & \alpha \cr
\alpha &... |
<p>Let p and p + 2 be prime numbers and let</p>
<p>$$
\Delta=\left|\begin{array}{ccc}
\mathrm{p} ! & (\mathrm{p}+1) ! & (\mathrm{p}+2) ! \\
(\mathrm{p}+1) ! & (\mathrm{p}+2) ! & (\mathrm{p}+3) ! \\
(\mathrm{p}+2) ! & (\mathrm{p}+3) ! & (\mathrm{p}+4) !
\end{array}\right|
$$</p>
<p>Then the sum o... | 4
Explanation:
<p>$$\Delta = \left| {\matrix{
{p!} & {(p + 1)!} & {(p + 2)!} \cr
{(p + 1)!} & {(p + 2)!} & {(p + 3)!} \cr
{(p + 2)!} & {(p + 3)!} & {(p + 4)!} \cr
} } \right|$$</p>
<p>$$ = p!\,.\,(p + 1)!\,.\,(p + 2)!\left| {\matrix{
1 & {(p + 1)} & {(p + 1)(p + 2)} \cr
1 & {(p + 2)} & {(p + ... |
<p>Let $$\mathrm{A_1,A_2,A_3}$$ be the three A.P. with the same common difference d and having their first terms as $$\mathrm{A,A+1,A+2}$$, respectively. Let a, b, c be the $$\mathrm{7^{th},9^{th},17^{th}}$$ terms of $$\mathrm{A_1,A_2,A_3}$$, respective such that $$\left| {\matrix{
a & 7 & 1 \cr
{2b} &a... | 495
Explanation:
$a=A+6 d$
<br/><br/>
$$
\begin{aligned}
& b=A+8 d+1 \\\\
& c=A+16 d+2 \\\\
& \left|\begin{array}{ccc}
a & 7 & 1 \\
26 & 17 & 1 \\
c & 17 & 1
\end{array}\right|=-70 \\\\
& \Rightarrow\left|\begin{array}{ccc}
A+6 d & 7 & 1 \\
2 A+16 d+2 & 17 & 1 \\
A+16 d+2 & 17 & 1
\end{array}\right|=-70 \\\\
& R_{3} \... |
<p>Let $$\mathrm{D}_{\mathrm{k}}=\left|\begin{array}{ccc}1 & 2 k & 2 k-1 \\
n & n^{2}+n+2 & n^{2} \\
n & n^{2}+n & n^{2}+n+2\end{array}\right|$$. If $$\sum_\limits{k=1}^{n} \mathrm{D}_{\mathrm{k}}=96$$, then $$n$$ is equal to _____________.</p>
Options:
[] | 6
Explanation:
$$
\begin{aligned}
& \sum_{k=1}^n D_k=\left|\begin{array}{ccc}
\sum 1 & 2 \sum k & 2 \sum k-\sum 1 \\
n & n^2+n+2 & n^2 \\
n & n^2+n & n^2+n+2
\end{array}\right| \\\\
& =\left|\begin{array}{ccc}
n & n(n+1) & n^2 \\
n & n^2+n+2 & n^2 \\
n & n^2+n & n^2+n+2
\end{array}\right| \\\\
& =\left|\begin{array}{c... |
<p>$$\left|\begin{array}{ccc}x+1 & x & x \\ x & x+\lambda & x \\ x & x & x+\lambda^{2}\end{array}\right|=\frac{9}{8}(103 x+81)$$, then $$\lambda, \frac{\lambda}{3}$$ are the roots of the equation :</p>
Options:
[{"identifier": "A", "content": "$$4 x^{2}+24 x-27=0$$"}, {"identifier": "B", "conte... | ["B"]
Explanation:
$$\left|\begin{array}{ccc}x+1 & x & x \\ x & x+\lambda & x \\ x & x & x+\lambda^{2}\end{array}\right|=\frac{9}{8}(103 x+81)$$
<br/><br/>Put $x=0$
<br/><br/>$$
\begin{aligned}
& \left|\begin{array}{ccc}
1 & 0 & 0 \\
0 & \lambda & 0 \\
0 & 0 & \lambda^2
\end{array}\right|=\frac{9}{8} \times 81 \\\\
& ... |
<p>The values of $$\alpha$$, for which $$\left|\begin{array}{ccc}1 & \frac{3}{2} & \alpha+\frac{3}{2} \\ 1 & \frac{1}{3} & \alpha+\frac{1}{3} \\ 2 \alpha+3 & 3 \alpha+1 & 0\end{array}\right|=0$$, lie in the interval</p>
Options:
[{"identifier": "A", "content": "$$(-2,1)$$\n"}, {"identifier": "B... | ["C"]
Explanation:
<p>$$\left|\begin{array}{ccc}
1 & \frac{3}{2} & \alpha+\frac{3}{2} \\
1 & \frac{1}{3} & \alpha+\frac{1}{3} \\
2 \alpha+3 & 3 \alpha+1 & 0
\end{array}\right|=0$$</p>
<p>$$\begin{aligned}
& \Rightarrow(2 \alpha+3)\left\{\frac{7 \alpha}{6}\right\}-(3 \alpha+1)\left\{\frac{-7}{6}\right\}=0 \\
& \Rightar... |
<p>$$\text { Let } A=\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & \alpha & \beta \\
0 & \beta & \alpha
\end{array}\right] \text { and }|2 \mathrm{~A}|^3=2^{21} \text { where } \alpha, \beta \in Z \text {, Then a value of } \alpha \text { is }$$</p>
Options:
[{"identifier": "A", "content": "9"}, {"... | ["D"]
Explanation:
<p>$$\begin{aligned}
& |\mathrm{A}|=\alpha^2-\beta^2 \\
& |2 \mathrm{~A}|^3=2^{21} \Rightarrow|\mathrm{A}|=2^4 \\
& \alpha^2-\beta^2=16 \\
& (\alpha+\beta)(\alpha-\beta)=16 \Rightarrow \alpha=4 \text { or } 5
\end{aligned}$$</p> |
<p>For $$\alpha, \beta \in \mathbb{R}$$ and a natural number $$n$$, let $$A_r=\left|\begin{array}{ccc}r & 1 & \frac{n^2}{2}+\alpha \\ 2 r & 2 & n^2-\beta \\ 3 r-2 & 3 & \frac{n(3 n-1)}{2}\end{array}\right|$$. Then $$2 A_{10}-A_8$$ is</p>
Options:
[{"identifier": "A", "content": "$$4 \\alpha+2 \... | ["A"]
Explanation:
<p>$$A_r=\left|\begin{array}{ccc}
r & 1 & \frac{n^2}{2}+\alpha \\
2 r & 2 & n^2-\beta \\
3 r-2 & 3 & \frac{n(3 n-1)}{2}
\end{array}\right|$$</p>
<p>$${A_r} = 2\left| {\matrix{
r & 1 & {{{{n^2}} \over 2} + \alpha } \cr
{2r} & 2 & {{{{n^2}} \over 2} - \beta } \cr
{3r - 2} & 3 & {{{n(3n - ... |
Let $$A = \left( {\matrix{
1 & { - 1} & 1 \cr
2 & 1 & { - 3} \cr
1 & 1 & 1 \cr
} } \right).$$ and $$10$$ $$B = \left( {\matrix{
4 & 2 & 2 \cr
{ - 5} & 0 & \alpha \cr
1 & { - 2} & 3 \cr
} } \right)$$. if $$B$$ is
<p>the inverse of matri... | ["A"]
Explanation:
Given that $$10B$$ $$\,\,\, = \left[ {\matrix{
4 & 2 & 2 \cr
{ - 5} & 0 & \alpha \cr
1 & { - 2} & 3 \cr
} } \right]$$
<br><br>$$ \Rightarrow B = {1 \over {10}}\left[ {\matrix{
4 & 2 & 2 \cr
{ - 5} & 0 & \alpha \cr
1 & { - 2}... |
Let $$A = \left( {\matrix{
0 & 0 & { - 1} \cr
0 & { - 1} & 0 \cr
{ - 1} & 0 & 0 \cr
} } \right)$$. The only correct
<p>statement about the matrix $$A$$ is</p>
Options:
[{"identifier": "A", "content": "$${A^2} = 1$$ "}, {"identifier": "B", "content": "$$A=(-1)I,$$ where $$I$$ is... | ["A"]
Explanation:
$$A = \left[ {\matrix{
0 & 0 & { - 1} \cr
0 & { - 1} & 0 \cr
{ - 1} & 0 & 0 \cr
} } \right]$$
<br><br>clearly $$\,\,\,A \ne 0.\,$$ Also $$\,\,\left| A \right| = - 1 \ne 0$$
<br><br>$$\therefore$$ $${A^{ - 1}}\,\,$$ exists, further
<br><br>$$\left( { - 1} \r... |
If $${A^2} - A + 1 = 0$$, then the inverse of $$A$$ is :
Options:
[{"identifier": "A", "content": "$$A+I$$ "}, {"identifier": "B", "content": "$$A$$ "}, {"identifier": "C", "content": "$$A-I$$ "}, {"identifier": "D", "content": "$$I-A$$"}] | ["D"]
Explanation:
Given $${A^2} - A + I = 0$$
<br><br>$${A^{ - 1}}{A^2} - {A^{ - 1}}A + {A^{ - 1}}.I = {A^{ - 1}}.0$$
<br><br>(Multiplying $$\,\,\,{A^{ - 1}}$$ on both sides)
<br><br>$$ \Rightarrow A - 1 + {A^{ - 1}} = 0$$
<br><br>or $${A^{ - 1}} = 1 - A$$ |
If $$A$$ is a $$3 \times 3$$ non-singular matrix such that $$AA'=A'A$$ and
<br/>$$B = {A^{ - 1}}A',$$ then $$BB'$$ equals:
Options:
[{"identifier": "A", "content": "$${B^{ - 1}}$$ "}, {"identifier": "B", "content": "$$\\left( {{B^{ - 1}}} \\right)'$$"}, {"identifier": "C", "content": "$$I+B$$ "}, {"identifier": "D", ... | ["D"]
Explanation:
$$BB' = B\left( {{A^{ - 1}}A'} \right)' = B\left( {A'} \right)'\left( {{A^{ - 1}}} \right)' = BA\left( {{A^{ - 1}}} \right)'$$
<br><br>$$ = \left( {{A^{ - 1}}A'} \right)\left( {A\left( {{A^{ - 1}}} \right)'} \right)$$
<br><br>$$ = {A^{ - 1}}A.A'.\left( {{A^{ - 1}}} \right)'\,\,\,\,\,\,$$ $$\left\{ {... |
Let A be a 3 $$ \times $$ 3 matrix such that A<sup>2</sup> $$-$$ 5A + 7I = 0
<br/><br/><b>Statement - I :</b>
<br/><br/>A<sup>$$-$$1</sup> = $${1 \over 7}$$ (5I $$-$$ A).
<br/><br/><b>Statement - II :</b>
<br/><br/>The polynomial A<sup>3</sup> $$-$$ 2A<sup>2</sup> $$-$$ 3A + I can be reduced to 5(A $$-$$ 4I).
<br/><... | ["C"]
Explanation:
Given,
<br><br>A<sup>2</sup> $$-$$ 5A + 7I = 0
<br><br>$$ \Rightarrow $$ A<sup>2</sup> $$-$$ 5A = $$-$$ 7I
<br><br>$$ \Rightarrow $$ AAA<sup>$$-$$1</sup> $$-$$ 5AA<sup>$$-$$1</sup> = $$-$$ 7IA<sup>$$-$$1</sup>
<br><br>$$ \Rightarrow $$ AI $$-$$ 5I =... |
Suppose A is any 3$$ \times $$ 3 non-singular matrix and ( A $$-$$ 3I) (A $$-$$ 5I) = O where I = I<sub>3</sub> and O = O<sub>3</sub>. If $$\alpha $$A + $$\beta $$A<sup>-1</sup> = 4I, then $$\alpha $$ + $$\beta $$ is equal to :
Options:
[{"identifier": "A", "content": "8"}, {"identifier": "B", "content": "7"}, {"ident... | ["A"]
Explanation:
Given,
<br><br>( A $$-$$ 3I) (A $$-$$ 5I) = O
<br><br>$$ \Rightarrow $$ A<sup>2</sup> - 8A + 15I = O
<br><br>Multiplying both sides by A<sup>- 1</sup>, we get,
<br><br>A<sup>- 1</sup>A.A - 8A<sup>- 1</sup>A + 15A<sup>- 1</sup>I = A<sup>- 1</sup>O
<br><br>$$ \Rightarrow $$ A - 8I + 15A<sup>- 1</sup>... |
If $$A = \left[ {\matrix{
{\cos \theta } & { - \sin \theta } \cr
{\sin \theta } & {\cos \theta } \cr
} } \right]$$, then the matrix A<sup>–50</sup> when $$\theta $$ = $$\pi \over 12$$, is equal to :
Options:
[{"identifier": "A", "content": "$$\\left[ {\\matrix{\n { {{\\sqrt 3 } \\over 2}} &... | ["C"]
Explanation:
(A<sup>$$-$$50</sup>) = (A<sup>$$-$$1</sup>)<sup>50</sup>
<br><br>We know,
<br><br>A<sup>$$-$$1</sup> = $${{adjA} \over {\left| A \right|}}$$
<br><br>$$\left| A \right|$$ = cos<sup>2</sup>$$\theta $$ + sin<sup>2</sup>$$\theta $$ = 1
<br><br>cofactor of A = $$\left[ {\matrix{
{\cos \theta } &... |
If $$\left[ {\matrix{
1 & 1 \cr
0 & 1 \cr
} } \right]\left[ {\matrix{
1 & 2 \cr
0 & 1 \cr
} } \right]$$$$\left[ {\matrix{
1 & 3 \cr
0 & 1 \cr
} } \right]$$....$$\left[ {\matrix{
1 & {n - 1} \cr
0 & 1 \cr
} } \right] = \left[ {\matrix{
1 ... | ["C"]
Explanation:
Given<br><br>
$$\left[ {\matrix{
1 & 1 \cr
0 & 1 \cr
} } \right]\left[ {\matrix{
1 & 2 \cr
0 & 1 \cr
} } \right]$$$$\left[ {\matrix{
1 & 3 \cr
0 & 1 \cr
} } \right]$$....$$\left[ {\matrix{
1 & {n - 1} \cr
0 & 1 \cr
} } \... |
Let P = $$\left[ {\matrix{
3 & { - 1} & { - 2} \cr
2 & 0 & \alpha \cr
3 & { - 5} & 0 \cr
} } \right]$$, where $$\alpha $$ $$ \in $$ R. Suppose Q = [ q<sub>ij</sub>] is a matrix satisfying PQ = kl<sub>3</sub> for some non-zero k $$ \in $$ R. <br/>If q<sub>23</sub> = $$ - {k \ove... | 17
Explanation:
As $$PQ = kI \Rightarrow Q = k{P^{ - 1}}I$$<br><br>now $$Q = {k \over {|P|}}(adjP)I $$
<br><br>$$\Rightarrow Q = {k \over {(20 + 12\alpha )}}\left[ {\matrix{
- & - & - \cr
- & - & {( - 3\alpha - 4)} \cr
- & - & - \cr
} } \right]\left[ {\matrix{
... |
If $$A = \left[ {\matrix{
0 & { - \tan \left( {{\theta \over 2}} \right)} \cr
{\tan \left( {{\theta \over 2}} \right)} & 0 \cr
} } \right]$$ and <br/>$$({I_2} + A){({I_2} - A)^{ - 1}} = \left[ {\matrix{
a & { - b} \cr
b & a \cr
} } \right]$$, then $$13({a^2} + {b^2})$$ is equal... | 13
Explanation:
$$A = \left[ {\matrix{
0 & { - \tan {\theta \over 2}} \cr
{\tan {\theta \over 2}} & 0 \cr
} } \right]$$<br><br>$$ \Rightarrow I + A = \left[ {\matrix{
1 & { - \tan {\theta \over 2}} \cr
{\tan {\theta \over 2}} & 1 \cr
} } \right]$$<br><br>$$ \Rightarrow I - A... |
Let $$A = \left[ {\matrix{
1 & 2 \cr
{ - 1} & 4 \cr
} } \right]$$. If A<sup>$$-$$1</sup> = $$\alpha$$I + $$\beta$$A, $$\alpha$$, $$\beta$$ $$\in$$ R, I is a 2 $$\times$$ 2 identity matrix then 4($$\alpha$$ $$-$$ $$\beta$$) is equal to :
Options:
[{"identifier": "A", "content": "5"}, {"identifier": ... | ["D"]
Explanation:
$$A = \left[ {\matrix{
1 & 2 \cr
{ - 1} & 4 \cr
} } \right],|A| = 6$$<br><br>$${A^{ - 1}} = {{adjA} \over {|A|}} = {1 \over 6}\left[ {\matrix{
4 & { - 2} \cr
1 & 1 \cr
} } \right] = \left[ {\matrix{
{{2 \over 3}} & { - {1 \over 3}} \cr
{{1 \over 6}... |
<p>Let $$X = \left[ {\matrix{
0 & 1 & 0 \cr
0 & 0 & 1 \cr
0 & 0 & 0 \cr
} } \right],\,Y = \alpha I + \beta X + \gamma {X^2}$$ and $$Z = {\alpha ^2}I - \alpha \beta X + ({\beta ^2} - \alpha \gamma ){X^2}$$, $$\alpha$$, $$\beta$$, $$\gamma$$ $$\in$$ R. If $${Y^{ - 1}} = \left[ {\m... | 100
Explanation:
<p>$$\because$$ $$X = \left[ {\matrix{
0 & 1 & 0 \cr
0 & 0 & 1 \cr
0 & 0 & 0 \cr
} } \right]$$</p>
<p>$$\therefore$$ $${X^2} = \left[ {\matrix{
0 & 0 & 1 \cr
0 & 0 & 0 \cr
0 & 0 & 0 \cr
} } \right]$$</p>
<p>$$\therefore$$ $$Y = \alpha I + \beta X + \gamma {X^2}\left[ ... |
<p>The number of matrices $$A=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$$, where $$a, b, c, d \in\{-1,0,1,2,3, \ldots \ldots, 10\}$$, such that $$A=A^{-1}$$, is ___________.</p>
Options:
[] | 50
Explanation:
<p>$$\because$$ $$A = \left[ {\matrix{
a & b \cr
c & d \cr
} } \right]$$ then $${A^2} = \left[ {\matrix{
{{a^2} + bc} & {b(a + d)} \cr
{c(a + d)} & {bc + {d^2}} \cr
} } \right]$$</p>
<p>For A<sup>$$-$$1</sup> must exist $$ad - bc \ne 0$$ ...... (i)</p>
<p>and $$A = {A^{ - 1}} \R... |
<p>Let $$A = \left[ {\matrix{
{{1 \over {\sqrt {10} }}} & {{3 \over {\sqrt {10} }}} \cr
{{{ - 3} \over {\sqrt {10} }}} & {{1 \over {\sqrt {10} }}} \cr
} } \right]$$ and $$B = \left[ {\matrix{
1 & { - i} \cr
0 & 1 \cr
} } \right]$$, where $$i = \sqrt { - 1} $$. If $$\mathrm{M=A^T ... | ["C"]
Explanation:
$$
\begin{aligned}
& \mathrm{AA}^{\mathrm{T}}=\left[\begin{array}{cc}
\frac{1}{\sqrt{10}} & \frac{3}{\sqrt{10}} \\
\frac{-3}{\sqrt{10}} & \frac{1}{\sqrt{10}}
\end{array}\right]\left[\begin{array}{cc}
\frac{1}{\sqrt{10}} & \frac{-3}{\sqrt{10}} \\
\frac{3}{\sqrt{10}} & \frac{1}{\sqrt{10}}
\end{array}\... |
<p>Let $$\mathrm{A}$$ be a $$2 \times 2$$ matrix with real entries such that $$\mathrm{A}'=\alpha \mathrm{A}+\mathrm{I}$$, where $$\alpha \in \mathbb{R}-\{-1,1\}$$. If $$\operatorname{det}\left(A^{2}-A\right)=4$$, then the sum of all possible values of $$\alpha$$ is equal to :</p>
Options:
[{"identifier": "A", "conten... | ["D"]
Explanation:
We have, $A^T=\alpha A+I$, where $A$ is $2 \times 2$ matrix and $\alpha \in R-\{-1,1\}$
<br/><br/>$$
\begin{aligned}
\left(A^T\right)^T & =\alpha A^T+I \\\\
A & =\alpha A^T+I \\\\
A & =\alpha(\alpha A+I)+I \left[\because A^T=\alpha A+I\right]\\\\
A & =\alpha^2 A+(\alpha+1) I \\\\
A & \left(1-\alpha... |
<p>If $$A=\left[\begin{array}{cc}1 & 5 \\ \lambda & 10\end{array}\right], \mathrm{A}^{-1}=\alpha \mathrm{A}+\beta \mathrm{I}$$ and $$\alpha+\beta=-2$$, then $$4 \alpha^{2}+\beta^{2}+\lambda^{2}$$ is equal to :</p>
Options:
[{"identifier": "A", "content": "12"}, {"identifier": "B", "content": "10"}, {"identifie... | ["D"]
Explanation:
$$
\begin{aligned}
& \mathrm{A}=\left[\begin{array}{cc}
1 & 5 \\
\lambda & 10
\end{array}\right] \\\\
& \Rightarrow|\mathrm{A}-x \mathrm{I}|=0 \\\\
& \Rightarrow\left|\begin{array}{cc}
1-x & 5 \\
\lambda & 10-x
\end{array}\right|=0 \\\\
& \Rightarrow(1-x)(10-x)-5 \lambda=0 \\\\
& \Rightarrow 10-11 x... |
Consider the matrix $f(x)=\left[\begin{array}{ccc}\cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1\end{array}\right]$.
<br/><br/>Given below are two statements :
<br/><br/>Statement I : $ f(-x)$ is the inverse of the matrix $f(x)$.
<br/><br/>Statement II : $f(x) f(y)=f(x+y)$.
<br/><br/>I... | ["C"]
Explanation:
<p>$$\begin{aligned}
& f(-x)=\left[\begin{array}{ccc}
\cos x & \sin x & 0 \\
-\sin x & \cos x & 0 \\
0 & 0 & 1
\end{array}\right] \\
& f(x) \cdot f(-x)=\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right]=I
\end{aligned}$$</p>
<p>Hence statement- I is correct</p>
<p>Now, c... |
<p>Let $$B=\left[\begin{array}{ll}1 & 3 \\ 1 & 5\end{array}\right]$$ and $$A$$ be a $$2 \times 2$$ matrix such that $$A B^{-1}=A^{-1}$$. If $$B C B^{-1}=A$$ and $$C^4+\alpha C^2+\beta I=O$$, then $$2 \beta-\alpha$$ is equal to</p>
Options:
[{"identifier": "A", "content": "16"}, {"identifier": "B", "content": "... | ["B"]
Explanation:
<p>$$\begin{aligned}
& B=\left[\begin{array}{ll}
1 & 3 \\
1 & 5
\end{array}\right] \\
& A B^{-1}=A^{-1} \\
& \Rightarrow A^2=B
\end{aligned}$$</p>
<p>Also, $$B C B^{-1}=A$$</p>
<p>$$\begin{aligned}
\Rightarrow C & =B^{-1} A B \\
\Rightarrow C^4 & =\left(B^{-1} A B\right)\left(B^{-1} A B\right)\left(... |
<p>Let $$A$$ be a $$2 \times 2$$ symmetric matrix such that $$A\left[\begin{array}{l}1 \\ 1\end{array}\right]=\left[\begin{array}{l}3 \\ 7\end{array}\right]$$ and the determinant of $$A$$ be 1 . If $$A^{-1}=\alpha A+\beta I$$, where $$I$$ is an identity matrix of order $$2 \times 2$$, then $$\alpha+\beta$$ equals _____... | 5
Explanation:
<p>Let $$A=\left[\begin{array}{ll}a & b \\ b & c\end{array}\right]$$</p>
<p>$$|A|=1 \Rightarrow a c-b^2=0 \quad \text{... (i)}$$</p>
<p>$$\text { Given }\left[\begin{array}{ll}
a & b \\
b & c
\end{array}\right]\left[\begin{array}{l}
1 \\
1
\end{array}\right]=\left[\begin{array}{l}
3 \\
7
\end{array}\rig... |
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