Integration of a Determinant

Q. The number of elements in the set $\{A=\left(\begin{array}{cc}a& b\\ 0& d\end{array}\right):a,b,d\in \{-1,0,1\left\}\text{and}\right(I-A{)}^3=I-A^3\},$
where $I$ is $2\times 2$ identity matrix, is

Q. If $x\varphi \left(x\right)=\underset{5}{\overset{x}{\int }}(3t^2-2{\varphi }^{\prime }(t\left)\right)dt,x>-2,$ and $\varphi \left(0\right)=4,$ then $\varphi \left(2\right)$ is

Q. The sum of all the local minimum values of the twice differentiable function $f:R\to R$ defined by $f\left(x\right)=x^3-3x^2-\frac{3f^{\prime \prime }\left(2\right)}{2}x+f^{\prime \prime }\left(1\right)$ is:

1. $-22$
2. $5$
3. $-27$
4. $0$

Q. Let $f\left(x\right)=\frac{\text{sin}\left(\pi [x-\pi ]\right)}{1+{\left[x\right]}^2}$ where $[.]$ denotes the greatest integer function. Then $f\left(x\right)$ is

1. discontinuous at integral points
2. continuous everywhere but not differentiable
3. differentiable once but $f^{\prime \prime }\left(x\right),f^{\prime \prime \prime }\left(x\right),...$do not exist
4. differentiable for all$x$

Q. Let $f\left(x\right)=1+\underset{0}{\overset{1}{\int }}(x{e}^y+y{e}^x)f\left(y\right)dy$ where $x$ and $y$ are independent variables. If complete solution set of $x$ for which the function $h\left(x\right)=f\left(x\right)+3x$ is strictly increasing is $(-\infty ,k),$ and $[.]$ denotes the greatest integer function, then $\left[\frac{4}{3}{e}^k\right]$ equals to

1. $12$
2. $2$
3. $3$
4. $9$

Q. Let $f$ be a function satisfying the equation $f\left(x\right)+3\underset{-1}{\overset{1}{\int }}(xy-x^2{y}^2)f\left(y\right)dy=x^3.$ Then the value of $f\left(5\right)$ is

Q. If $f\left(x\right)=\left|\begin{array}{ccc}1& 2x& 3x^2\\ 3& a& 27\\ 1& 3& 9\end{array}\right|$ and ${\int }_0^{3}f\left(x\right)dx=0$, then a is equal to

1. 3
2. 6
3. 9
4. any real number

Q. If $f(x+y)=f\left(x\right)+f\left(y\right)\forall x,y\in R$ and $f\left(4\right)$ is the sum to infinite terms of the series $2,1,\frac{1}{2},\frac{1}{4},\dots .$, then the image of $y=\ln (x+3)$ with respect to $y=f\left(x\right)$ is:

1. $e^x+3$
2. $e^{x-3}$
3. $e^x-3$
4. $3-e^x$

Q. Let a matrix $A=[a_{ij}{]}_{m\times n}$ is defined as $a_{ij}=mn$ and $m^2-1\ge 0,n^2-9\le 0.$ Then matrix $A$ can be a

1. Null matrix
2. Scalar matrix
3. Row matrix
4. Column matrix

Q. lf $\left|\begin{array}{ccc}{x}^n& x^{n+2}& x^{n+3}\\ y^n& y^{n+2}& y^{n+3}\\ z^n& z^{n+2}& z^{n+3}\end{array}\right|$ $=(x-y)(y-z\left)\right(z-x)(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})$, then the value of $n$ is

1. $-2$
2. $-1$
3. $2$
4. $1$

Q. If $p{x}^4+q{x}^3+r{x}^2+sx+t=\left|\begin{array}{ccc}{x}^2+3x& x-1& x+3\\ x+1& 2-x& x-3\\ x-3& x+4& 3x\end{array}\right|$ then $t$ is equal to

1. 33
2. 0
3. 21
4. none

Q. Let $f$ be a real-valued function defined on the interval $(0,\infty )$ by $f\left(x\right)=\ln x+\underset{0}{\overset{x}{\int }}\sqrt{1+\sin t}dt$. Then which of the following statement(s) is/are true?

1. There exists $\alpha >1$ such that $|f^{\prime }\left(x\right)|<|f\left(x\right)|$ for all $x\in (\alpha ,\infty )$
2. $f^{\prime }\left(x\right)$exists for all $x\in (0,\infty )$ and $f^{\prime }$ is continuous on $(0,\infty )$, but not differentiable on $(0,\infty )$
3. There exists $\beta >0$ such that $|f\left(x\right)|+|f^{\prime }\left(x\right)|\le \beta$ for all $x\in (0,\infty )$
4. $f^{\prime \prime }\left(x\right)$exists for all $x\in (0,\infty )$

Q. If $f\left(x\right)=\left|\begin{array}{ccc}{x}^2-4x+6& 2x^2+4x+10& 3x^2-2x+16\\ x-2& 2x+2& 3x-1\\ 1& 2& 3\end{array}\right|$, then

1. $\underset{-3}{\overset{3}{\int }}\frac{x^2\sin x}{1+x^6}.f\left(x\right)dx=0$
2. $f\left(x\right)$ is a constant function
3. $f^{\prime }\left(x\right)$ is a constant function
   
4. $\underset{-3}{\overset{3}{\int }}\frac{x^2\sin x}{1+x^6}.f\left(x\right)dx=2$

Q.

$\text{Let}f\left(x\right)=\left|\begin{array}{ccc}a\left(x\right)& b\left(x\right)& c\left(x\right)\\ m\left(x\right)& n\left(x\right)& l\left(x\right)\\ g\left(x\right)& h\left(x\right)& k\left(x\right)\end{array}\right|\text{then}$

${\int }_0^{a}f\left(x\right)\text{is}\left|\begin{array}{ccc}{\int }_0^{a}a\left(x\right)& {\int }_0^{a}b\left(x\right)& {\int }_0^{a}c\left(x\right)\\ {\int }_0^{a}m\left(x\right)& {\int }_0^{a}n\left(x\right)& {\int }_0^{a}l\left(x\right)\\ {\int }_0^{a}g\left(x\right)& {\int }_0^{a}h\left(x\right)& {\int }_0^{a}k\left(x\right)\end{array}\right|$

1. False

2. True

Q.

$Letf\left(x\right)=\left|\begin{array}{ccc}a\left(x\right)& b\left(x\right)& c\left(x\right)\\ m\left(x\right)& n\left(x\right)& l\left(x\right)\\ g\left(x\right)& h\left(x\right)& k\left(x\right)\end{array}\right|then{\int }_0^{a}f\left(x\right)is\left|\begin{array}{ccc}{\int }_0^{a}a\left(x\right)& {\int }_0^{a}b\left(x\right)& {\int }_0^{a}c\left(x\right)\\ {\int }_0^{a}m\left(x\right)& {\int }_0^{a}n\left(x\right)& {\int }_0^{a}l\left(x\right)\\ {\int }_0^{a}g\left(x\right)& {\int }_0^{a}h\left(x\right)& {\int }_0^{a}k\left(x\right)\end{array}\right|$

1. F

2. T

Q. Paragraph for below question
नीचे दिए गए प्रश्न के लिए अनुच्छेद

If A is a square matrix, then determinant of A is represented by |A|. The value of |A| is equal to zero, if any two rows or columns are identical.

Let, $A=\left[\begin{array}{ccc}x& 3& 7\\ 4& x-1& 7\\ 4& 1& x+5\end{array}\right],$ then

यदि A एक वर्ग आव्यूह है, तब A के सारणिक को |A| द्वारा दर्शाया जाता है। |A| का मान शून्य है, यदि कोई भी दो पंक्तियाँ या दो स्तम्भ समरूप हैं।

माना, $A=\left[\begin{array}{ccc}x& 3& 7\\ 4& x-1& 7\\ 4& 1& x+5\end{array}\right],$ तब

Q. |A| is a polynomial of degree

प्रश्न - |A| एक बहुपद है जिसकी घात है

1. 1
2. 2
3. 3
4. 4

Q. $A=\left[\begin{array}{ccc}1& -1& 1\\ 2& 1& -3\\ 1& 1& 1\end{array}\right]$ and $B=\left[\begin{array}{ccc}4& 2& 2\\ -5& 0& \alpha \\ 1& -2& 3\end{array}\right]$ If $B$ is the adjoint of $A$ then $\alpha$ equals

1. $2$
2. $-1$
3. $-2$
4. $5$

Q. If the system of equations ax + y = 3, x + y = 2 and 2x – y = 1 is consistent, then the value of a is equal to

यदि समीकरणों का निकाय ax + y = 3, x + y = 2 तथा 2x – y = 1 संगत हैं, तब a का मान बराबर है

1. 0
2. 1
3. –1
4. 2

Q. If $\left|\begin{array}{ccc}x& 2& 3\\ 2& 3& x\\ 3& x& 2\end{array}\right|=\left|\begin{array}{ccc}1& x& 4\\ x& 1& 4\\ 4& 1& x\end{array}\right|=\left|\begin{array}{ccc}0& 5& x\\ 5& x& 0\\ x& 0& 5\end{array}\right|=0$ then the value x equals $\left(xϵR\right)$

1. 0
2. 5
3. -5
4. 8

Q. If $f(x+y)=f\left(x\right)+f\left(y\right)\forall x,y\in R$ and $f\left(4\right)$ is the sum to infinite terms of the series $2,1,\frac{1}{2},\frac{1}{4},\dots .$, then the image of $y=\ln (x+3)$ with respect to $y=f\left(x\right)$ is:

1. $e^x+3$
2. $e^{x-3}$
3. $e^x-3$
4. $3-e^x$

Q. Let $f$ be a function satisfying the equation $f\left(x\right)+3\underset{-1}{\overset{1}{\int }}(xy-x^2{y}^2)f\left(y\right)dy=x^3.$ Then the value of $f\left(5\right)$ is

Q. The anti-derivative of$(\sqrt{x}+\frac{1}{\sqrt{x}})$ equals to (where $C$ is the constant of integration)

1. $\left(\frac{1}{3}\right)x^{\frac{1}{3}}+\left(2\right)x^{\frac{1}{2}}+C$
2. $\left(\frac{2}{3}\right)x^{\frac{2}{3}}+\left(\frac{1}{2}\right)x^x+C$
3. $\left(\frac{2}{3}\right)x^{\frac{3}{2}}+\left(2\right)x^{\frac{1}{2}}+C$
4. $\left(\frac{3}{2}\right)x^{\frac{3}{2}}+\left(\frac{1}{2}\right)x^{\frac{1}{2}}+C$

Q. Paragraph for below question
नीचे दिए गए प्रश्न के लिए अनुच्छेद

If A is a square matrix, then determinant of A is represented by |A|. The value of |A| is equal to zero, if any two rows or columns are identical.

Let, $A=\left[\begin{array}{ccc}x& 3& 7\\ 4& x-1& 7\\ 4& 1& x+5\end{array}\right],$ then

यदि A एक वर्ग आव्यूह है, तब A के सारणिक को |A| द्वारा दर्शाया जाता है। |A| का मान शून्य है, यदि कोई भी दो पंक्तियाँ या दो स्तम्भ समरूप हैं।

माना, $A=\left[\begin{array}{ccc}x& 3& 7\\ 4& x-1& 7\\ 4& 1& x+5\end{array}\right],$ तब

Q. If |A| = 0, then one of the possible value of x is

प्रश्न - यदि |A| = 0, तब x का कोई एक संभावित मान है

1. Zero
   
   शून्य
2. 4
3. 1
4. –2

Q.

1. 
   

2. 
   

3. 

4. 
   

Q.

$f\left(x\right)=\left|\begin{array}{ccc}3& 1& 2\\ a^3& 2a& a^2\\ 3x^2& 2& 2x\end{array}\right|then{\int }_0^{a}f\left(x\right)is......$

__

Q. If $f\left(x\right)=\left|\begin{array}{ccc}1& 2x& 3x^2\\ 3& a& 27\\ 1& 3& 9\end{array}\right|$ and ${\int }_0^{3}f\left(x\right)dx=0$, then a is equal to

1. 3
2. 6
3. 9
4. any real number

Q. $\left|\begin{array}{ccc}0& p-q& p-r\\ q-p& 0& q-r\\ r-p& r-q& 0\end{array}\right|$ =

1. $0$
2. $(p-q)(q-r)(r-p)$
3. $pqr$
4. $3pqr$

Q. $A=\left[\begin{array}{ccc}1& -3& -4\\ -1& 3& 4\\ 1& -3& -4\end{array}\right]$ and $A^2=\lambda I$ then $\lambda =$

1. $\frac{1}{2}$
   
2. $-2$
3. $1$
4. $0$

Q. $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right],B=\left[\begin{array}{cc}2& 1\\ 3& 4\end{array}\right]$ then $\left|\right(B^T{A}^T{)}^{-1}|$ is equal to

1. $1$
2. $10$
3. $-1$
4. $\frac{1}{10}$

Q. If $pth,qth$ and $rth$ terms of an H.P. be $a,b,c$ respectively, then $\Delta =\left|\begin{array}{ccc}bc& ca& ab\\ p& q& r\\ 1& 1& 1\end{array}\right|$ is

1. $abc.pqr$
2. $-1$
3. $pa+qb+rc$
4. $0$