8666 + Mcqs in Mathematics Page 25 McqOptions

1201.	If \[a\ne p,\] \[b\ne q,\] \[c\ne r\]and \[\left\| \begin{matrix} p & b & c \\ a & q & c \\ a & b & r \\ \end{matrix} \right\|=0\] then the value of \[\frac{p}{p-a}+\frac{q}{q-b}+\frac{r}{r-c}\] is equal to
A.	\[-1\]
B.	\[1\]
C.	\[-2\]
D.	\[2\]
Answer» E.

Discussion

1202.	What is the value of the determinant \[\left\| \begin{matrix} 1 & bc & a(b+c) \\ 1 & ca & b(c+a) \\ 1 & ab & c(a+b) \\ \end{matrix} \right\|?\]
A.	\[0\]
B.	\[abc\]
C.	\[ab+bc+ca\]
D.	\[abc\,(a+b+c)\]
Answer» B. \[abc\]

Discussion

1203.	If \[f(x)=\left\| \begin{matrix} {{x}^{n-1}} & \cos x & \frac{1}{x+3} \\ 0 & \cos \frac{n\pi }{2} & \frac{{{(-1)}^{n}}n!}{{{3}^{n+1}}} \\ \alpha & {{\alpha }^{3}} & {{\alpha }^{5}} \\ \end{matrix} \right\|\] then \[\frac{{{d}^{n}}}{d{{x}^{n}}}\,\,{{[f(x)]}_{x=0}}=\]
A.	\[1\]
B.	\[-1\]
C.	\[0\]
D.	None of these
Answer» D. None of these

Discussion

1204.	If adj \[B=A,\] \[\left\| P \right\|=\left\| Q \right\|=1,\] then adj \[({{Q}^{-1}}B{{P}^{-1}})\]is
A.	\[PQ\]
B.	\[QAP\]
C.	\[PAQ\]
D.	\[P{{A}^{-1}}Q\]
Answer» D. \[P{{A}^{-1}}Q\]

Discussion

1205.	If \[y=\left\| \begin{matrix} \sin x & \cos x & \sin x \\ \cos x & -\operatorname{sinx} & \cos x \\ x & 1 & 1 \\ \end{matrix} \right\|,\]then \[\frac{dy}{dx}\] is
A.	\[1\]
B.	\[2\]
C.	\[3\]
D.	0
Answer» B. \[2\]

Discussion

1206.	The value of determinant \[\left\| \begin{matrix} {{\sin }^{2}}13{}^\circ & {{\sin }^{2}}77{}^\circ & \tan 135{}^\circ \\ {{\sin }^{2}}77{}^\circ & \tan 135{}^\circ & {{\sin }^{2}}13{}^\circ \\ \tan 135{}^\circ & {{\sin }^{2}}13{}^\circ & {{\sin }^{2}}77{}^\circ \\ \end{matrix} \right\|\]is
A.	\[-1\]
B.	\[0\]
C.	\[1\]
D.	\[2\]
Answer» C. \[1\]

Discussion

1207.	Let \[\Delta =\] \[\left\| \begin{matrix} \sin x & \sin (x+h) & \sin (x+2h) \\ \sin (x+2h) & \sin x & \sin (x+h) \\ \sin (x+h) & \sin (x+2h) & \sin x \\ \end{matrix} \right\|\]Then, \[\underset{h\to 0}{\mathop{\lim }}\,\,\,\left( \frac{\Delta }{{{h}^{2}}} \right)\] is
A.	\[9si{{n}^{2}}x\cos x\]
B.	\[3{{\cos }^{2}}x\]
C.	\[\sin x{{\cos }^{2}}x\]
D.	None of these
Answer» C. \[\sin x{{\cos }^{2}}x\]

Discussion

1208.	If A is a square matrix such that \[{{A}^{2}}=I\] where I is the identity matrix, then what is \[{{A}^{-1}}\] equal to?
A.	\[A+1\]
B.	Null matrix
C.	A
D.	Transpose of A
Answer» D. Transpose of A

Discussion

1209.	If the value of the determinant \[\left\| \begin{matrix} a & 1 & 1 \\ 1 & b & 1 \\ 1 & 1 & c \\ \end{matrix} \right\|\] is positive, where \[a\ne b\ne c,\] then the value of abc
A.	Cannot be less than 1
B.	Is greater than \[-8\]
C.	Is less than \[-8\]
D.	Must be greater than 8
Answer» C. Is less than \[-8\]

Discussion

1210.	If A is a square matrix of order 3 with \[\left\| A \right\|\ne 0,\] then which one of the following is correct?
A.	\[\left\| adj\,A \right\|=\left\| A \right\|\]
B.	\[\left\| adj\,A \right\|={{\left\| A \right\|}^{2}}\]
C.	\[\left\| adj\,A \right\|={{\left\| A \right\|}^{3}}\]
D.	\[{{\left\| adj\,A \right\|}^{2}}=\left\| A \right\|\]
Answer» C. \[\left\| adj\,A \right\|={{\left\| A \right\|}^{3}}\]

Discussion

1211.	If \[\alpha .\beta .\gamma \in R,\]then the determinant\[\Delta =\left\| \begin{matrix} {{({{e}^{i\alpha }}+{{e}^{-i\alpha }})}^{2}} & {{({{e}^{i\alpha }}-{{e}^{-i\alpha }})}^{2}} & 4 \\ {{({{e}^{i\beta }}+{{e}^{-i\beta }})}^{2}} & {{({{e}^{i\beta }}-{{e}^{-i\beta }})}^{2}} & 4 \\ {{({{e}^{i\gamma }}+{{e}^{-i\gamma }})}^{2}} & {{({{e}^{i\gamma }}-{{e}^{-i\gamma }})}^{2}} & 4 \\ \end{matrix} \right\|\] is
A.	Independent of \[\alpha ,\beta \] and \[\gamma \]
B.	Dependent on \[\alpha ,\beta \] and \[\gamma \]
C.	Independent of \[\alpha ,\beta \] only
D.	Independent of \[\alpha ,\gamma \] only
Answer» B. Dependent on \[\alpha ,\beta \] and \[\gamma \]

Discussion

1212.	If \[f(x),\,\,g(x)\] and \[h(x)\] are three polynomials of degree 2 and \[\Delta (x)=\left\| \begin{matrix} f(x) & g(x) & h(x) \\ f'(x) & g'(x) & h'(x) \\ f''(x) & g''(x) & h''(x) \\ \end{matrix} \right\|,\]then \[\Delta (x)\] is a polynomial of degree
A.	2
B.	3
C.	At most 2
D.	At most 3
Answer» D. At most 3

Discussion

1213.	The maximum and minimum value of \[(3\times 3)\]determinant whose elements belongs to \[\{0,1\}\] is
A.	\[1,-1\]
B.	\[2,-2\]
C.	\[4,-4\]
D.	None of these
Answer» C. \[4,-4\]

Discussion

1214.	If the system of equations \[\lambda {{x}_{1}}+{{x}_{2}}+{{x}_{3}}=1,\] \[{{x}_{1}}+\lambda {{x}_{2}}+{{x}_{3}}=1,\] \[{{x}_{1}}+{{x}_{2}}+\lambda {{x}_{3}}=1\] is consistent, then \[\lambda \] can be
A.	\[5\]
B.	\[-2/3\]
C.	\[-3\]
D.	None of these
Answer» E.

Discussion

1215.	If \[\left\| \begin{matrix} {{x}^{n}} & {{x}^{n+2}} & {{x}^{2n}} \\ 1 & {{x}^{a}} & a \\ {{x}^{n+5}} & {{x}^{a+6}} & {{x}^{2n+5}} \\ \end{matrix} \right\|=0\,\forall \,x\,\in R,\] where \[n\in N\] then value of 'a' is
A.	\[n\]
B.	\[n-1\]
C.	\[n+1\]
D.	None of these
Answer» D. None of these

Discussion

1216.	The value of the determinant \[\left\| \begin{matrix} {{\cos }^{2}}54{}^\circ & {{\cos }^{2}}36{}^\circ & \cot 135{}^\circ \\ {{\sin }^{2}}53{}^\circ & \cot 135{}^\circ & {{\sin }^{2}}37{}^\circ \\ \cot 135{}^\circ & co{{s}^{2}}25{}^\circ & {{\cos }^{2}}65{}^\circ \\ \end{matrix} \right\|\] is equal to
A.	\[-2\]
B.	\[-1\]
C.	\[0\]
D.	\[1\]
Answer» D. \[1\]

Discussion

1217.	If \[\omega \] is the cube root of unity, then what is one root of the equation \[\left\| \begin{matrix} {{x}^{2}} & -2x & -2{{\omega }^{2}} \\ 2 & \omega & -\omega \\ 0 & \omega & 1 \\ \end{matrix} \right\|=0?\]
A.	\[1\]
B.	\[-2\]
C.	\[2\]
D.	\[\omega \]
Answer» C. \[2\]

Discussion

1218.	If \[A=\left[ \begin{matrix} 1 & 2 \\ 0 & 3 \\ \end{matrix} \right]\] is a \[2\times 2\] matrix and \[f(x)={{x}^{2}}-x+2\] is a polynomial, then what is \[f(A)\]?
A.	\[\left[ \begin{matrix} 1 & 7 \\ 1 & 7 \\ \end{matrix} \right]\]
B.	\[\left[ \begin{matrix} 2 & 6 \\ 0 & 8 \\ \end{matrix} \right]\]
C.	\[\left[ \begin{matrix} 2 & 6 \\ 0 & 6 \\ \end{matrix} \right]\]
D.	\[\left[ \begin{matrix} 2 & 6 \\ 0 & 7 \\ \end{matrix} \right]\]
Answer» C. \[\left[ \begin{matrix} 2 & 6 \\ 0 & 6 \\ \end{matrix} \right]\]

Discussion

1219.	If D is determinant of order 3 and D' is the determinant obtained by replacing the elements of D by their cofactors, then which one of the following is correct?
A.	\[D'={{D}^{2}}\]
B.	\[D'={{D}^{3}}\]
C.	\[D'=2{{D}^{2}}\]
D.	\[D'=3{{D}^{3}}\]
Answer» B. \[D'={{D}^{3}}\]

Discussion

1220.	If \[{{e}^{i\theta }}=\cos \theta +i\sin \theta ,\] then the value of \[\left\| \begin{matrix} 1 & {{e}^{i\pi /3}} & {{e}^{i\pi /4}} \\ {{e}^{-i\pi /3}} & 1 & {{e}^{i2\pi /3}} \\ {{e}^{-i\pi /4}} & {{e}^{-i2\pi /3}} & 1 \\ \end{matrix} \right\|\]is
A.	\[-2+\sqrt{2}\]
B.	\[2-\sqrt{2}\]
C.	\[-2-\sqrt{2}\]
D.	1
Answer» D. 1

Discussion

1221.	If \[a\ne b\ne c\] are all positive, then the value of the determinant \[\left\| \begin{matrix} a & b & c \\ b & c & a \\ c & a & b \\ \end{matrix} \right\|\] is
A.	Non-negative
B.	Non-positive
C.	Negative
D.	Positive
Answer» D. Positive

Discussion

1222.	Matrix \[{{M}_{r}}\] is defined as \[{{M}_{r}}=\left( \begin{matrix} r & r-1 \\ r-1 & r \\ \end{matrix} \right),\]\[r\in N\]. The value of \[det({{M}_{1}})+\det \,({{M}_{2}})+\det \,({{M}_{3}})+....+\det \,({{M}_{2014}})\] is
A.	\[2013\]
B.	\[2014\]
C.	\[{{(2013)}^{2}}\]
D.	\[{{(2014)}^{2}}\]
Answer» E.

Discussion

1223.	Suppose \[\Delta =\left\| \begin{matrix} {{a}_{1}} & {{b}_{1}} & {{c}_{1}} \\ {{a}_{2}} & {{b}_{2}} & {{c}_{2}} \\ {{a}_{3}} & {{b}_{3}} & {{c}_{3}} \\ \end{matrix} \right\|\] and\[\Delta '=\left\| \begin{matrix} {{a}_{1}}+p{{b}_{1}} & {{b}_{1}}+q{{c}_{1}} & {{c}_{1}}+r{{a}_{1}} \\ {{a}_{2}}+p{{b}_{2}} & {{b}_{2}}+q{{c}_{2}} & {{c}_{2}}+r{{a}_{2}} \\ {{a}_{3}}+p{{b}_{3}} & {{b}_{3}}+q{{c}_{3}} & {{c}_{3}}+r{{a}_{3}} \\ \end{matrix} \right\|\]. Then
A.	\[\Delta '=\Delta \]
B.	\[\Delta '=\Delta \,\,(1-pqr)\]
C.	\[\Delta '=\Delta \,\,(1+p+q+r)\]
D.	\[\Delta '=\Delta \,\,(1+pqr)\]
Answer» E.

Discussion

1224.	If A is a square matrix of order n, then adj (adj A) is equal to
A.	\[\|A{{\|}^{n-1}}A\]
B.	\[\|A{{\|}^{n}}A\]
C.	\[\|A{{\|}^{n-2}}A\]
D.	None of these
Answer» D. None of these

Discussion

1225.	In a third order determinant, each element of the first column consists of sum of two terms, each element of the second column consists of sum of three terms and each element of the third column consists of sum of four terms. Then it can be decomposed into n determinants, where n has the value
A.	\[1\]
B.	\[9\]
C.	\[16\]
D.	\[24\]
Answer» E.

Discussion

1226.	If \[f(x)=a+bx+c{{x}^{2}}\]and \[\alpha ,\beta ,\lambda \] are roots of the equation \[{{x}^{3}}=1,\]then \[\left\| \begin{matrix} a & b & c \\ b & c & a \\ c & a & b \\ \end{matrix} \right\|\] is equal to
A.	\[f(\alpha )+f(\beta )+f(\lambda )\]
B.	\[f(\alpha )f(\beta )+f(\beta )f(\lambda )+f(\gamma )+f(\alpha )\]
C.	\[f(\alpha )f(\beta )f(\gamma )\]
D.	\[-f(\alpha )f(\beta )f(\gamma )\]
Answer» E.

Discussion

1227.	For all values of A, B, C and P, Q, R the value of the determinant\[{{(x+a)}^{3}}\left\| \begin{matrix} \cos (A-P) & \cos (A-Q) & \cos (A-R) \\ \cos (B-P) & \cos (B-Q) & \cos (B-R) \\ \cos (C-P) & \cos (C-Q) & \cos (C-R) \\ \end{matrix} \right\|\] is
A.	\[1\]
B.	\[0\]
C.	\[2\]
D.	None of these
Answer» C. \[2\]

Discussion

1228.	If \[f(x)=\left\| \begin{matrix} \cos x & x & 1 \\ 2\sin x & {{x}^{2}} & 2x \\ \tan x & x & 1 \\ \end{matrix} \right\|,\] then \[\underset{x\to 0}{\mathop{\lim }}\,\left[ \frac{f'(x)}{x} \right]\] is
A.	\[2\]
B.	\[-2\]
C.	\[1\]
D.	\[-1\]
Answer» C. \[1\]

Discussion

1229.	If \[C=2cos\theta ,\] then the value of the determinant\[\Delta =\left[ \begin{matrix} C & 1 & 0 \\ 1 & C & 1 \\ 6 & 1 & C \\ \end{matrix} \right]\] is
A.	\[\frac{2{{\sin }^{2}}2\theta }{\sin \theta }\]
B.	\[8{{\cos }^{3}}\theta -4\cos \theta +6\]
C.	\[\frac{2\sin 2\theta }{\sin \theta }\]
D.	\[8{{\cos }^{3}}\theta +4\cos \theta +6\]
Answer» C. \[\frac{2\sin 2\theta }{\sin \theta }\]

Discussion

1230.	If \[A=\left[ \begin{matrix} 1 & 2 & -1 \\ -1 & 1 & 2 \\ 2 & -1 & 1 \\ \end{matrix} \right],\] then det(adj (adj A)) is
A.	\[{{(14)}^{4}}\]
B.	\[{{(14)}^{3}}\]
C.	\[{{(14)}^{2}}\]
D.	\[{{(14)}^{1}}\]
Answer» B. \[{{(14)}^{3}}\]

Discussion

1231.	If \[{{A}^{-1}}=\left[ \begin{matrix} 1 & -2 \\ -2 & 2 \\ \end{matrix} \right],\] what is det(A)?
A.	\[2\]
B.	\[-2\]
C.	\[\frac{1}{2}\]
D.	\[-\frac{1}{2}\]
Answer» E.

Discussion

1232.	The determinant \[\left\| \begin{matrix} x & \sin \theta & \cos \theta \\ -\sin \theta & -x & 1 \\ \cos \theta & 1 & x \\ \end{matrix} \right\|\] is independent of
A.	x only
B.	\[\theta \] only
C.	x and \[\theta \] both
D.	None of these
Answer» C. x and \[\theta \] both

Discussion

1233.	Let \[f(x)=\left\| \begin{matrix} n & n+1 & n+2 \\ ^{n}{{P}_{n}} & ^{n+1}{{P}_{n+1}} & ^{n+2}{{P}_{n+2}} \\ ^{n}{{C}_{n}} & ^{n+1}{{C}_{n+1}} & ^{n+2}{{C}_{n+2}} \\ \end{matrix} \right\|,\] where the symbols have their usual meanings. The \[f(x)\] is divisible by
A.	\[{{n}^{2}}+n+1\]
B.	\[(n+1)!\]
C.	\[(2n+1)!\]
D.	None of the above
Answer» B. \[(n+1)!\]

Discussion

1234.	If [ ] denotes the greatest integer less than or equal to the real number under consideration and \[-1\le x
A.	\[[z]\]
B.	\[[y]\]
C.	[x]
D.	None of these
Answer» B. \[[y]\]

Discussion

1235.	If \[[a]\] denotes the integral part of a and \[x={{a}_{3}}y+{{a}_{2}}z,\] \[y={{a}_{1}}z+{{a}_{3}}z\] and \[z={{a}_{2}}x+{{a}_{1}}y,\]where x, y, z are not all zero. If \[{{a}_{1}}=m-[m],\] m being a non-integral constant, then \[{{a}_{1}}{{a}_{2}}{{a}_{3}}\] is
A.	\[>1\]
B.	\[>-1\]
C.	\[<1\]
D.	\[<-1\]
Answer» C. \[<1\]

Discussion

1236.	Let \[x
A.	Non-negative
B.	Non-positive
C.	Negative
D.	Positive
Answer» D. Positive

Discussion

1237.	For what values of k, does the system of linear equation \[x+y+z=2,\] \[2x+y-z=3,\] \[3x+2y+kz=4\] have a unique solution?
A.	\[k=0\]
B.	\[-1<k<1\]
C.	\[-2<k<2\]
D.	\[k\ne 0\]
Answer» E.

Discussion

1238.	Let \[\lambda \] and \[\alpha \] be real. The set of all values of x for which the system of linear equations \[\lambda x+(\sin \alpha )y+(cos\alpha )z=0\] \[x+(cos\alpha )y+(sin\alpha )z=0\] \[-x+(\sin \alpha )-(\cos \alpha )z=0\] has a non-trivial solution, is
A.	\[\left[ 0,\,\sqrt{2} \right]\]
B.	\[\left[ -\sqrt{2},0 \right]\]
C.	\[\left[ -\sqrt{2},\sqrt{2} \right]\]
D.	None of these
Answer» D. None of these

Discussion

1239.	If A is an orthogonal matrix of order 3 and \[B=\left[ \begin{matrix} 1 & 2 & 3 \\ -3 & 0 & 2 \\ 2 & 5 & 0 \\ \end{matrix} \right],\] then which of the following is/are correct? 1. \[\|AB\|=\pm 47\] 2. \[AB=BA\] Select the correct answer using the code given below:
A.	1 only
B.	2 only
C.	Both 1 and 2
D.	Neither 1 nor 2
Answer» D. Neither 1 nor 2

Discussion

1240.	If \[\left\| \begin{matrix} {{x}^{2}}+x & 3x-1 & -x+3 \\ 2x+1 & 2+{{x}^{2}} & {{x}^{3}}-3 \\ x-3 & {{x}^{2}}+4 & 3x \\ \end{matrix} \right\|\]\[={{a}_{0}}+{{a}_{1}}x+{{a}_{2}}{{x}^{2}}+....+{{a}_{7}}{{x}^{7}},\] then the value of \[{{a}_{0}}\] is
A.	\[25\]
B.	\[24\]
C.	\[23\]
D.	\[21\]
Answer» E.

Discussion

1241.	If \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\ne -2\] and\[f(x)=\left\| \begin{matrix} (1+{{a}^{2}})x & (1+{{b}^{2}})x & (1+{{c}^{2}})x \\ (1+{{a}^{2}})x & (1+{{b}^{2}})x & (1+{{c}^{2}})x \\ (1+{{a}^{2}})x & (1+{{b}^{2}})x & (1+{{c}^{2}})x \\ \end{matrix} \right\|\]then \[f(x)\] is a polynomial of degree
A.	\[1\]
B.	0
C.	\[3\]
D.	\[2\]
Answer» E.

Discussion

1242.	For what value of p, is the system of equations : \[{{p}^{3}}x+{{(p+1)}^{3}}y={{(p+2)}^{3}}\] \[px+(p+1)y=p+2\] \[x+y=1\] consistent?
A.	\[p=0\]
B.	\[p=1\]
C.	\[p=-1\]
D.	For all \[p>1\]
Answer» D. For all \[p>1\]

Discussion

1243.	Consider the system of linear equations \[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}z+{{d}_{1}}=0,\] \[{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}z+{{d}_{2}}=0,\] \[{{a}_{3}}x+{{b}_{3}}y+{{c}_{3}}z+{{d}_{3}}=0,\] Let us denote by \[\Delta (a,b,c)\] the determinant \[\left\| \begin{matrix} {{a}_{1}} & {{b}_{1}} & {{c}_{1}} \\ {{a}_{2}} & {{b}_{2}} & {{c}_{2}} \\ {{a}_{3}} & {{b}_{3}} & {{c}_{3}} \\ \end{matrix} \right\|\], if \[\Delta \,(a,b,c)\#0,\] then the value of x in the unique solution of the above equations is
A.	\[\frac{\Delta (b,c,d)}{\Delta (a,b,c)}\]
B.	\[\frac{-\Delta (b,c,d)}{\Delta (a,b,c)}\]
C.	\[\frac{\Delta (a,c,d)}{\Delta (a,b,c)}\]
D.	\[-\frac{\Delta (a,b,d)}{\Delta (a,b,c)}\]
Answer» B. \[\frac{-\Delta (b,c,d)}{\Delta (a,b,c)}\]

Discussion

1244.	If \[\left. \begin{align} & f(x)=sin\text{ }x,when\,\,x\,\,is\,\,rational \\ & \,\,\,\,\,\,\,\,\,\,=\,\cos x,when\,\,x\,\,is\,\,irrational \\ \end{align} \right\}\]
A.	Discontinuous at \[x=n\pi +\pi /4\]
B.	Continuous at \[x=n\pi +\pi /4\]
C.	Discontinuous at all x
D.	None of these
Answer» C. Discontinuous at all x

Discussion

1245.	If \[{{I}_{n}}=\frac{{{d}^{n}}}{d{{x}^{n}}}({{x}^{n}}\log \,x)\], then \[{{I}_{n}}-n{{I}_{n-1}}=\]
A.	n
B.	\[n-1\]
C.	\[n!\]
D.	\[\left( n-1 \right)!\]
Answer» E.

Discussion

1246.	Suppose\[f(x)={{e}^{ax}}+{{e}^{bx}}\], where\[a\ne b\], and that \[f{{\,}^{n}}(x)-2f'(x)-15f(x)=0\] for all x. Then the product ab is
A.	25
B.	9
C.	-15
D.	-9
Answer» D. -9

Discussion

1247.	Let \[3f(x)-2f(1/x)=x,\] then \[f'(2)\] is equal to
A.	\[\frac{2}{7}\]
B.	\[\frac{1}{2}\]
C.	\[2\]
D.	\[\frac{7}{2}\]
Answer» C. \[2\]

Discussion

1248.	What is the value of k for which the following function f(x) is continuous for all x? \[f(x)=\left\{ \begin{align} & \frac{{{x}^{3}}-3x+2}{{{(x-1)}^{2}}},for\,\,x\ne 1 \\ & k\,\,\,\,\,\,\,\,,for\,\,x=1 \\ \end{align} \right.\]
A.	3
B.	2
C.	1
D.	-1
Answer» B. 2

Discussion

1249.	If \[u=f({{x}^{3}}),v=g({{x}^{2}}),f'(x)=\cos x\] and \[g'(x)=\sin x,\] then \[\frac{du}{dv}=\]
A.	\[\frac{1}{2}x\cos {{x}^{3}}\cos ec\,{{x}^{2}}\]
B.	\[\frac{3}{2}x\cos {{x}^{3}}\cos ec\,{{x}^{2}}\]
C.	\[\frac{1}{2}x\sec {{x}^{3}}\sin \,{{x}^{2}}\]
D.	\[\frac{3}{2}x\sec {{x}^{3}}\cos ec\,{{x}^{2}}\]
Answer» C. \[\frac{1}{2}x\sec {{x}^{3}}\sin \,{{x}^{2}}\]

Discussion

1250.	If the functions \[f(x)\] and \[g(x)\] are continuous in [a, b] and differentiable in (a, b), then the equation \[\left\| \begin{matrix} f(a) & f(b) \\ g(a) & g(b) \\ \end{matrix} \right\|=(b-a)\left\| \begin{matrix} f(a) & f'(x) \\ g(a) & g'(x) \\ \end{matrix} \right\|\] has in the interval [a, b]
A.	At least one root
B.	Exactly one root
C.	At most one root
D.	No root
Answer» B. Exactly one root

Discussion

1210.	If A is a square matrix of order 3 with \[\left\| A \right\|\ne 0,\] then which one of the following is correct?
A.	\[\left\| adj\,A \right\|=\left\| A \right\|\]
B.	\[\left\| adj\,A \right\|={{\left\| A \right\|}^{2}}\]
C.	\[\left\| adj\,A \right\|={{\left\| A \right\|}^{3}}\]
D.	\[{{\left\| adj\,A \right\|}^{2}}=\left\| A \right\|\]
Answer» C. \[\left\| adj\,A \right\|={{\left\| A \right\|}^{3}}\]

Explore topic-wise MCQs in Joint Entrance Exam - Main (JEE Main).

If \[a\ne p,\] \[b\ne q,\] \[c\ne r\]and \[\left| \begin{matrix} p & b & c \\ a & q & c \\ a & b & r \\ \end{matrix} \right|=0\] then the value of \[\frac{p}{p-a}+\frac{q}{q-b}+\frac{r}{r-c}\] is equal to

What is the value of the determinant \[\left| \begin{matrix} 1 & bc & a(b+c) \\ 1 & ca & b(c+a) \\ 1 & ab & c(a+b) \\ \end{matrix} \right|?\]

If \[f(x)=\left| \begin{matrix} {{x}^{n-1}} & \cos x & \frac{1}{x+3} \\ 0 & \cos \frac{n\pi }{2} & \frac{{{(-1)}^{n}}n!}{{{3}^{n+1}}} \\ \alpha & {{\alpha }^{3}} & {{\alpha }^{5}} \\ \end{matrix} \right|\] then \[\frac{{{d}^{n}}}{d{{x}^{n}}}\,\,{{[f(x)]}_{x=0}}=\]

If adj \[B=A,\] \[\left| P \right|=\left| Q \right|=1,\] then adj \[({{Q}^{-1}}B{{P}^{-1}})\]is

If \[y=\left| \begin{matrix} \sin x & \cos x & \sin x \\ \cos x & -\operatorname{sinx} & \cos x \\ x & 1 & 1 \\ \end{matrix} \right|,\]then \[\frac{dy}{dx}\] is

The value of determinant \[\left| \begin{matrix} {{\sin }^{2}}13{}^\circ & {{\sin }^{2}}77{}^\circ & \tan 135{}^\circ \\ {{\sin }^{2}}77{}^\circ & \tan 135{}^\circ & {{\sin }^{2}}13{}^\circ \\ \tan 135{}^\circ & {{\sin }^{2}}13{}^\circ & {{\sin }^{2}}77{}^\circ \\ \end{matrix} \right|\]is

Let \[\Delta =\] \[\left| \begin{matrix} \sin x & \sin (x+h) & \sin (x+2h) \\ \sin (x+2h) & \sin x & \sin (x+h) \\ \sin (x+h) & \sin (x+2h) & \sin x \\ \end{matrix} \right|\]Then, \[\underset{h\to 0}{\mathop{\lim }}\,\,\,\left( \frac{\Delta }{{{h}^{2}}} \right)\] is

If A is a square matrix such that \[{{A}^{2}}=I\] where I is the identity matrix, then what is \[{{A}^{-1}}\] equal to?

If the value of the determinant \[\left| \begin{matrix} a & 1 & 1 \\ 1 & b & 1 \\ 1 & 1 & c \\ \end{matrix} \right|\] is positive, where \[a\ne b\ne c,\] then the value of abc

If A is a square matrix of order 3 with \[\left| A \right|\ne 0,\] then which one of the following is correct?

If \[f(x),\,\,g(x)\] and \[h(x)\] are three polynomials of degree 2 and \[\Delta (x)=\left| \begin{matrix} f(x) & g(x) & h(x) \\ f'(x) & g'(x) & h'(x) \\ f''(x) & g''(x) & h''(x) \\ \end{matrix} \right|,\]then \[\Delta (x)\] is a polynomial of degree

The maximum and minimum value of \[(3\times 3)\]determinant whose elements belongs to \[\{0,1\}\] is

If the system of equations \[\lambda {{x}_{1}}+{{x}_{2}}+{{x}_{3}}=1,\] \[{{x}_{1}}+\lambda {{x}_{2}}+{{x}_{3}}=1,\] \[{{x}_{1}}+{{x}_{2}}+\lambda {{x}_{3}}=1\] is consistent, then \[\lambda \] can be

If \[\left| \begin{matrix} {{x}^{n}} & {{x}^{n+2}} & {{x}^{2n}} \\ 1 & {{x}^{a}} & a \\ {{x}^{n+5}} & {{x}^{a+6}} & {{x}^{2n+5}} \\ \end{matrix} \right|=0\,\forall \,x\,\in R,\] where \[n\in N\] then value of 'a' is

If \[\omega \] is the cube root of unity, then what is one root of the equation \[\left| \begin{matrix} {{x}^{2}} & -2x & -2{{\omega }^{2}} \\ 2 & \omega & -\omega \\ 0 & \omega & 1 \\ \end{matrix} \right|=0?\]

If \[A=\left[ \begin{matrix} 1 & 2 \\ 0 & 3 \\ \end{matrix} \right]\] is a \[2\times 2\] matrix and \[f(x)={{x}^{2}}-x+2\] is a polynomial, then what is \[f(A)\]?

If D is determinant of order 3 and D' is the determinant obtained by replacing the elements of D by their cofactors, then which one of the following is correct?

If \[{{e}^{i\theta }}=\cos \theta +i\sin \theta ,\] then the value of \[\left| \begin{matrix} 1 & {{e}^{i\pi /3}} & {{e}^{i\pi /4}} \\ {{e}^{-i\pi /3}} & 1 & {{e}^{i2\pi /3}} \\ {{e}^{-i\pi /4}} & {{e}^{-i2\pi /3}} & 1 \\ \end{matrix} \right|\]is

If \[a\ne b\ne c\] are all positive, then the value of the determinant \[\left| \begin{matrix} a & b & c \\ b & c & a \\ c & a & b \\ \end{matrix} \right|\] is

Matrix \[{{M}_{r}}\] is defined as \[{{M}_{r}}=\left( \begin{matrix} r & r-1 \\ r-1 & r \\ \end{matrix} \right),\]\[r\in N\]. The value of \[det({{M}_{1}})+\det \,({{M}_{2}})+\det \,({{M}_{3}})+....+\det \,({{M}_{2014}})\] is

If A is a square matrix of order n, then adj (adj A) is equal to

In a third order determinant, each element of the first column consists of sum of two terms, each element of the second column consists of sum of three terms and each element of the third column consists of sum of four terms. Then it can be decomposed into n determinants, where n has the value

If \[f(x)=a+bx+c{{x}^{2}}\]and \[\alpha ,\beta ,\lambda \] are roots of the equation \[{{x}^{3}}=1,\]then \[\left| \begin{matrix} a & b & c \\ b & c & a \\ c & a & b \\ \end{matrix} \right|\] is equal to

For all values of A, B, C and P, Q, R the value of the determinant\[{{(x+a)}^{3}}\left| \begin{matrix} \cos (A-P) & \cos (A-Q) & \cos (A-R) \\ \cos (B-P) & \cos (B-Q) & \cos (B-R) \\ \cos (C-P) & \cos (C-Q) & \cos (C-R) \\ \end{matrix} \right|\] is

If \[f(x)=\left| \begin{matrix} \cos x & x & 1 \\ 2\sin x & {{x}^{2}} & 2x \\ \tan x & x & 1 \\ \end{matrix} \right|,\] then \[\underset{x\to 0}{\mathop{\lim }}\,\left[ \frac{f'(x)}{x} \right]\] is

If \[C=2cos\theta ,\] then the value of the determinant\[\Delta =\left[ \begin{matrix} C & 1 & 0 \\ 1 & C & 1 \\ 6 & 1 & C \\ \end{matrix} \right]\] is

If \[A=\left[ \begin{matrix} 1 & 2 & -1 \\ -1 & 1 & 2 \\ 2 & -1 & 1 \\ \end{matrix} \right],\] then det(adj (adj A)) is

If \[{{A}^{-1}}=\left[ \begin{matrix} 1 & -2 \\ -2 & 2 \\ \end{matrix} \right],\] what is det(A)?

The determinant \[\left| \begin{matrix} x & \sin \theta & \cos \theta \\ -\sin \theta & -x & 1 \\ \cos \theta & 1 & x \\ \end{matrix} \right|\] is independent of

Let \[f(x)=\left| \begin{matrix} n & n+1 & n+2 \\ ^{n}{{P}_{n}} & ^{n+1}{{P}_{n+1}} & ^{n+2}{{P}_{n+2}} \\ ^{n}{{C}_{n}} & ^{n+1}{{C}_{n+1}} & ^{n+2}{{C}_{n+2}} \\ \end{matrix} \right|,\] where the symbols have their usual meanings. The \[f(x)\] is divisible by

If [ ] denotes the greatest integer less than or equal to the real number under consideration and \[-1\le x

If \[[a]\] denotes the integral part of a and \[x={{a}_{3}}y+{{a}_{2}}z,\] \[y={{a}_{1}}z+{{a}_{3}}z\] and \[z={{a}_{2}}x+{{a}_{1}}y,\]where x, y, z are not all zero. If \[{{a}_{1}}=m-[m],\] m being a non-integral constant, then \[{{a}_{1}}{{a}_{2}}{{a}_{3}}\] is

Let \[x

For what values of k, does the system of linear equation \[x+y+z=2,\] \[2x+y-z=3,\] \[3x+2y+kz=4\] have a unique solution?

Let \[\lambda \] and \[\alpha \] be real. The set of all values of x for which the system of linear equations \[\lambda x+(\sin \alpha )y+(cos\alpha )z=0\] \[x+(cos\alpha )y+(sin\alpha )z=0\] \[-x+(\sin \alpha )-(\cos \alpha )z=0\] has a non-trivial solution, is

If A is an orthogonal matrix of order 3 and \[B=\left[ \begin{matrix} 1 & 2 & 3 \\ -3 & 0 & 2 \\ 2 & 5 & 0 \\ \end{matrix} \right],\] then which of the following is/are correct? 1. \[|AB|=\pm 47\] 2. \[AB=BA\] Select the correct answer using the code given below:

If \[\left| \begin{matrix} {{x}^{2}}+x & 3x-1 & -x+3 \\ 2x+1 & 2+{{x}^{2}} & {{x}^{3}}-3 \\ x-3 & {{x}^{2}}+4 & 3x \\ \end{matrix} \right|\]\[={{a}_{0}}+{{a}_{1}}x+{{a}_{2}}{{x}^{2}}+....+{{a}_{7}}{{x}^{7}},\] then the value of \[{{a}_{0}}\] is

If \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}\ne -2\] and\[f(x)=\left| \begin{matrix} (1+{{a}^{2}})x & (1+{{b}^{2}})x & (1+{{c}^{2}})x \\ (1+{{a}^{2}})x & (1+{{b}^{2}})x & (1+{{c}^{2}})x \\ (1+{{a}^{2}})x & (1+{{b}^{2}})x & (1+{{c}^{2}})x \\ \end{matrix} \right|\]then \[f(x)\] is a polynomial of degree

For what value of p, is the system of equations : \[{{p}^{3}}x+{{(p+1)}^{3}}y={{(p+2)}^{3}}\] \[px+(p+1)y=p+2\] \[x+y=1\] consistent?

If \[\left. \begin{align} & f(x)=sin\text{ }x,when\,\,x\,\,is\,\,rational \\ & \,\,\,\,\,\,\,\,\,\,=\,\cos x,when\,\,x\,\,is\,\,irrational \\ \end{align} \right\}\]

If \[{{I}_{n}}=\frac{{{d}^{n}}}{d{{x}^{n}}}({{x}^{n}}\log \,x)\], then \[{{I}_{n}}-n{{I}_{n-1}}=\]

Suppose\[f(x)={{e}^{ax}}+{{e}^{bx}}\], where\[a\ne b\], and that \[f{{\,}^{n}}(x)-2f'(x)-15f(x)=0\] for all x. Then the product ab is

Let \[3f(x)-2f(1/x)=x,\] then \[f'(2)\] is equal to

What is the value of k for which the following function f(x) is continuous for all x? \[f(x)=\left\{ \begin{align} & \frac{{{x}^{3}}-3x+2}{{{(x-1)}^{2}}},for\,\,x\ne 1 \\ & k\,\,\,\,\,\,\,\,,for\,\,x=1 \\ \end{align} \right.\]

If \[u=f({{x}^{3}}),v=g({{x}^{2}}),f'(x)=\cos x\] and \[g'(x)=\sin x,\] then \[\frac{du}{dv}=\]

If the functions \[f(x)\] and \[g(x)\] are continuous in [a, b] and differentiable in (a, b), then the equation \[\left| \begin{matrix} f(a) & f(b) \\ g(a) & g(b) \\ \end{matrix} \right|=(b-a)\left| \begin{matrix} f(a) & f'(x) \\ g(a) & g'(x) \\ \end{matrix} \right|\] has in the interval [a, b]