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MCQOPTIONS
This section includes 8666 Mcqs, each offering curated multiple-choice questions to sharpen your Joint Entrance Exam - Main (JEE Main) knowledge and support exam preparation. Choose a topic below to get started.
| 1151. |
The differential equations of all conies whose axes coincide with the co-ordinate axis |
| A. | \[xy\frac{{{d}^{2}}y}{d{{x}^{2}}}+x{{\left( \frac{dy}{dx} \right)}^{2}}+y\frac{dy}{dx}=0\] |
| B. | \[xy\frac{{{d}^{2}}y}{d{{x}^{2}}}+x{{\left( \frac{dy}{dx} \right)}^{2}}+x\frac{dy}{dx}=0\] |
| C. | \[xy\frac{{{d}^{2}}y}{d{{x}^{2}}}+x{{\left( \frac{dy}{dx} \right)}^{2}}-y\frac{dy}{dx}=0\] |
| D. | \[xy\frac{{{d}^{2}}y}{d{{x}^{2}}}-x{{\left( \frac{dy}{dx} \right)}^{2}}+y\frac{dy}{dx}=0\] |
| Answer» D. \[xy\frac{{{d}^{2}}y}{d{{x}^{2}}}-x{{\left( \frac{dy}{dx} \right)}^{2}}+y\frac{dy}{dx}=0\] | |
| 1152. |
A differential equation associated with the primitive \[y=a+b{{e}^{5x}}+c{{e}^{-~7x}}\] is |
| A. | \[{{y}_{3}}+2{{y}_{2}}-{{y}_{1}}=0\] |
| B. | \[{{y}_{3}}+2{{y}_{2}}-35{{y}_{1}}=0\] |
| C. | \[4{{y}_{3}}+5{{y}_{2}}-20{{y}_{1}}=0\] |
| D. | None of these |
| Answer» C. \[4{{y}_{3}}+5{{y}_{2}}-20{{y}_{1}}=0\] | |
| 1153. |
The particular solution of the differential equation \[{{\sin }^{-1}}\left( \frac{{{d}^{2}}y}{d{{x}^{2}}}-1 \right)=x\], where\[y=\frac{dy}{dx}=0\] when\[x=0\], is |
| A. | \[y={{x}^{2}}+x-\sin x\] |
| B. | \[y=\frac{{{x}^{2}}}{2}+x-\sin x\] |
| C. | \[y=\frac{{{x}^{2}}}{2}+\frac{x}{2}-\sin x\] |
| D. | \[2y={{x}^{2}}+x-\sin x\] |
| Answer» C. \[y=\frac{{{x}^{2}}}{2}+\frac{x}{2}-\sin x\] | |
| 1154. |
The degree of differential equation satisfying the relation\[\sqrt{1+{{x}^{2}}}+\sqrt{1+{{y}^{2}}}=\lambda (x\sqrt{1+{{y}^{2}}}-y\sqrt{1+{{x}^{2}}})\] is |
| A. | 1 |
| B. | 2 |
| C. | 3 |
| D. | 4 |
| Answer» B. 2 | |
| 1155. |
The differential equation\[(1+{{y}^{2}})xdx-(1+{{x}^{2}})ydy=0\] Represents a family of: |
| A. | Ellipses of constant eccentricity |
| B. | Ellipses of variable eccentricity- |
| C. | Hyperbolas of constant eccentricity |
| D. | Hyperbolas of variable eccentricity |
| Answer» E. | |
| 1156. |
Which of the following does not represent the orthogonal trajectory of the system of curves \[{{\left( \frac{dy}{dx} \right)}^{2}}=\frac{a}{x}\] |
| A. | \[9a{{(y+c)}^{2}}=4{{x}^{3}}\] |
| B. | \[y+c=\frac{-2}{3\sqrt{a}}{{x}^{3/2}}\] |
| C. | \[y+c=\frac{2}{3\sqrt{a}}{{x}^{3/2}}\] |
| D. | All are orthogonal trajectories |
| Answer» E. | |
| 1157. |
The general solution of \[(x+1)\frac{dy}{dx}+1=2{{e}^{-y}}\] is |
| A. | \[{{e}^{y}}(x+1)=x+C\] |
| B. | \[{{e}^{-y}}=2x+C\] |
| C. | \[{{e}^{y}}(x+1)=2x+C\] |
| D. | \[{{e}^{y}}(x+1)=C\] |
| Answer» D. \[{{e}^{y}}(x+1)=C\] | |
| 1158. |
The differential equation of the family of circles with fixed radius 5 units and centre on the line \[y=2\] is |
| A. | \[(y-2)y{{'}^{2}}=25-{{(y-2)}^{2}}\] |
| B. | \[{{(y-2)}^{2}}y{{'}^{2}}=25-{{(y-2)}^{2}}\] |
| C. | \[{{(x-2)}^{2}}y{{'}^{2}}=25-{{(y-2)}^{2}}\] |
| D. | \[(x-2)y{{'}^{2}}=25-{{(y-2)}^{2}}\] |
| Answer» C. \[{{(x-2)}^{2}}y{{'}^{2}}=25-{{(y-2)}^{2}}\] | |
| 1159. |
What is the solution of \[\frac{dy}{dx}+2y=1\] satisfying\[y(0)=0\]? |
| A. | \[y=\frac{1-{{e}^{-2x}}}{2}\] |
| B. | \[y=\frac{1+{{e}^{-2x}}}{2}\] |
| C. | \[y=1+{{e}^{x}}\] |
| D. | \[y=\frac{1+{{e}^{x}}}{2}\] |
| Answer» B. \[y=\frac{1+{{e}^{-2x}}}{2}\] | |
| 1160. |
Which one of the following differential equations represents the family of straight lines which are at unit distance from the origin? |
| A. | \[{{\left( y-x\frac{dy}{dx} \right)}^{2}}=1-{{\left( \frac{dy}{dx} \right)}^{2}}\] |
| B. | \[{{\left( y+x\frac{dy}{dx} \right)}^{2}}=1+{{\left( \frac{dy}{dx} \right)}^{2}}\] |
| C. | \[{{\left( y-x\frac{dy}{dx} \right)}^{2}}=1+{{\left( \frac{dy}{dx} \right)}^{2}}\] |
| D. | \[{{\left( y+x\frac{dy}{dx} \right)}^{2}}=1-{{\left( \frac{dy}{dx} \right)}^{2}}\] |
| Answer» D. \[{{\left( y+x\frac{dy}{dx} \right)}^{2}}=1-{{\left( \frac{dy}{dx} \right)}^{2}}\] | |
| 1161. |
The solution of \[\frac{dy}{dx}=\sqrt{1-{{x}^{2}}-{{y}^{2}}+{{x}^{2}}{{y}^{2}}}\] is |
| A. | \[si{{n}^{-1}}y=si{{n}^{-1}}x+c\] |
| B. | \[2si{{n}^{-1}}y=\sqrt{1-{{x}^{2}}}+si{{n}^{-1}}x+c\] |
| C. | \[2si{{n}^{-1}}y=x\sqrt{1-{{x}^{2}}}+si{{n}^{-1}}x+c\] |
| D. | \[2si{{n}^{-1}}y=x\sqrt{1-{{x}^{2}}}+{{\cos }^{-1}}x+c\] |
| Answer» E. | |
| 1162. |
The general solution of the differential equation \[\frac{dy}{dx}-\frac{\tan \,\,y}{1+x}=(1+x){{e}^{x}}\sec \,\,y\] is |
| A. | \[\sin (1+x)=y({{e}^{x}}+c)\] |
| B. | \[y\sin (1+x)=c{{e}^{x}}\] |
| C. | \[(1+x)sin\,\,y={{e}^{x}}+c\] |
| D. | \[\sin \,\,y=(1+x)({{e}^{x}}+c)\] |
| Answer» E. | |
| 1163. |
The curve satisfying the equation \[\frac{dy}{dx}=\frac{y(x+{{y}^{3}})}{x({{y}^{3}}-x)}\]and passing through the point (4, -2) is |
| A. | \[{{y}^{2}}=-2x\] |
| B. | \[y=-2x\] |
| C. | \[{{y}^{3}}=-2x\] |
| D. | None of these |
| Answer» D. None of these | |
| 1164. |
The solution to of the differential equation \[(x+1)\frac{dy}{dx}-y={{e}^{3x}}{{(x+1)}^{2}}\] is |
| A. | \[y=(x+1){{e}^{3x}}+c\] |
| B. | \[3y=(x+1)+{{e}^{3x}}+c\] |
| C. | \[\frac{3y}{x+1}={{e}^{3x}}+c\] |
| D. | \[y{{e}^{-3x}}=3(x+1)+c\] |
| Answer» D. \[y{{e}^{-3x}}=3(x+1)+c\] | |
| 1165. |
A function \[y=f(x)\] satisfies the condition \[f'(x)\sin x+f(x)\cos x=1\] being bounded when\[x\to 0\]. If\[l=\int_{0}^{\pi /2}{f(x)dx}\], then |
| A. | \[\frac{\pi }{2}<l<\frac{{{\pi }^{2}}}{4}\] |
| B. | \[\frac{\pi }{4}<l<\frac{{{\pi }^{2}}}{2}\] |
| C. | \[1<l<\frac{\pi }{2}\] |
| D. | \[0<l<1\] |
| Answer» B. \[\frac{\pi }{4}<l<\frac{{{\pi }^{2}}}{2}\] | |
| 1166. |
Solution of differential equation \[{{x}^{2}}=1+{{\left( \frac{x}{y} \right)}^{-1}}\frac{dy}{dx}+\frac{{{\left( \frac{x}{y} \right)}^{-2}}{{\left( \frac{dy}{dx} \right)}^{2}}}{2!}\]\[+\frac{{{\left( \frac{x}{y} \right)}^{-3}}{{\left( \frac{dy}{dx} \right)}^{3}}}{3!}+.........\] is |
| A. | \[{{y}^{2}}={{x}^{2}}(ln\,\,{{x}^{2}}-1)+C\] |
| B. | \[y={{x}^{2}}(ln\,\,x-1)+C\] |
| C. | \[{{y}^{2}}=x(ln\,\,x-1)+C\] |
| D. | \[y={{x}^{2}}{{e}^{{{x}^{2}}}}+C\] |
| Answer» B. \[y={{x}^{2}}(ln\,\,x-1)+C\] | |
| 1167. |
The solution of the differential equation\[\frac{dy}{dx}=\frac{1-3y-3x}{1+x+y}\] is |
| A. | \[x+y-\ell n\left| x+y \right|=c\] |
| B. | \[3x+y+2\ell n\left| 1-x-y \right|=c\] |
| C. | \[x+3y-2\ell n\left| 1-x-y \right|=c\] |
| D. | None of these |
| Answer» C. \[x+3y-2\ell n\left| 1-x-y \right|=c\] | |
| 1168. |
The solution to the differential equation\[\frac{dy}{dx}=\frac{yf'(x)-{{y}^{2}}}{f(x)}\] Where f(x) is a given function is |
| A. | \[f(x)=y(x+c)\] |
| B. | \[f(x)=cxy\] |
| C. | \[f(x)=c(x+y)\] |
| D. | \[yf(x)=cx\] |
| Answer» B. \[f(x)=cxy\] | |
| 1169. |
The differential equation which represents the three parameter family of circles\[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\] is |
| A. | \[y'''=\frac{3y'y'{{'}^{2}}}{1+y{{'}^{2}}}\] |
| B. | \[y'''=\frac{3y'{{'}^{2}}}{1+y{{'}^{2}}}\] |
| C. | \[y'''=\frac{3y'}{1+y{{'}^{2}}}\] |
| D. | \[y'''=\frac{3y'}{1-y{{'}^{2}}}\] |
| Answer» B. \[y'''=\frac{3y'{{'}^{2}}}{1+y{{'}^{2}}}\] | |
| 1170. |
Given \[a=x/(y-z),\]\[b=y/(z-x)\]and \[c=z/(x-y),\] where x, y and z are not all zero, Then the value of \[ab+bc+ca\] |
| A. | \[0\] |
| B. | \[1\] |
| C. | \[-1\] |
| D. | None of these |
| Answer» D. None of these | |
| 1171. |
\[A=\left| \begin{matrix} 2a & 3r & x \\ 4b & 6s & 2y \\ -2c & -3t & -z \\ \end{matrix} \right|=\lambda \left| \begin{matrix} a & r & x \\ b & s & y \\ c & t & z \\ \end{matrix} \right|,\] then what is the value of \[\lambda \]? |
| A. | \[12\] |
| B. | \[-12\] |
| C. | \[7\] |
| D. | \[-7\] |
| Answer» C. \[7\] | |
| 1172. |
If \[A=\left[ \begin{matrix} 3 & 2 \\ 1 & 4 \\ \end{matrix} \right],\] then what is A (adj A) equal to? |
| A. | \[\left[ \begin{matrix} 0 & 10 \\ 10 & 0 \\ \end{matrix} \right]\] |
| B. | \[\left[ \begin{matrix} 10 & 0 \\ 0 & 10 \\ \end{matrix} \right]\] |
| C. | \[\left[ \begin{matrix} 1 & 10 \\ 10 & 1 \\ \end{matrix} \right]\] |
| D. | \[\left[ \begin{matrix} 10 & 1 \\ 1 & 10 \\ \end{matrix} \right]\] |
| Answer» C. \[\left[ \begin{matrix} 1 & 10 \\ 10 & 1 \\ \end{matrix} \right]\] | |
| 1173. |
If A, B, and C are the angles of a triangle and \[\left| \begin{matrix} 1 & 1 & 1 \\ 1+\sin A & 1+\sin B & 1+\sin C \\ \sin A+{{\sin }^{2}}A & \sin B+{{\sin }^{2}}B & \sin C+{{\sin }^{2}}C \\ \end{matrix} \right|=0,\]then the triangle must be |
| A. | Isosceles |
| B. | Equilateral |
| C. | Right-angled |
| D. | None of these |
| Answer» B. Equilateral | |
| 1174. |
If in a triangle ABC, \[\left| \begin{matrix} 1 & \sin A & {{\sin }^{2}}A \\ 1 & \sin B & {{\sin }^{2}}B \\ 1 & \sin C & {{\sin }^{2}}C \\ \end{matrix} \right|=0\] then the triangle is |
| A. | Equilateral or isosceles |
| B. | Equilateral or right-angled |
| C. | Right angled or isosceles |
| D. | None of these |
| Answer» B. Equilateral or right-angled | |
| 1175. |
If \[f(x)=\left| \begin{matrix} {{2}^{-x}} & {{e}^{x{{\log }_{e}}2}} & {{x}^{2}} \\ {{2}^{-3x}} & {{e}^{3x{{\log }_{e}}2}} & {{x}^{4}} \\ {{2}^{-5x}} & {{e}^{5x{{\log }_{e}}2}} & 1 \\ \end{matrix} \right|,\] then |
| A. | \[f(x)+f(-x)=0\] |
| B. | \[f(x)-f(-x)=0\] |
| C. | \[f(x)+f(-x)=2\] |
| D. | None of these |
| Answer» B. \[f(x)-f(-x)=0\] | |
| 1176. |
If \[{{A}_{1}}{{B}_{1}}{{C}_{1}},\] \[{{A}_{2}}{{B}_{2}}{{C}_{2}}\] and \[{{A}_{3}}{{B}_{3}}{{C}_{3}}\]are three digit numbers, each of which is divisible by k, then \[\Delta =\left| \begin{matrix} {{A}_{1}} & {{B}_{1}} & {{C}_{1}} \\ {{A}_{2}} & {{B}_{2}} & {{C}_{2}} \\ {{A}_{3}} & {{B}_{3}} & {{C}_{3}} \\ \end{matrix} \right|\] is |
| A. | Divisible by k |
| B. | Divisible by \[{{k}^{2}}\] |
| C. | Divisible by \[{{k}^{3}}\] |
| D. | None of these |
| Answer» B. Divisible by \[{{k}^{2}}\] | |
| 1177. |
If a, b, c are in GP, then what is the value of \[\left| \begin{matrix} a & b & a+b \\ b & c & b+c \\ a+b & b+c & 0 \\ \end{matrix} \right|?\] |
| A. | \[0\] |
| B. | \[1\] |
| C. | \[-1\] |
| D. | None of these |
| Answer» B. \[1\] | |
| 1178. |
The determinant \[\left| \begin{matrix} a+b+c & a+b & a \\ 4a+3b+2c & 3a+2b & 2a \\ 10a+6b+3c & 6a+3b & 3a \\ \end{matrix} \right|\] is independent of which one of the following? |
| A. | a and b |
| B. | b and c |
| C. | a and c |
| D. | All of these |
| Answer» C. a and c | |
| 1179. |
The rank of the matrix \[\left[ \begin{matrix} -1 & 2 & 5 \\ 2 & -4 & a-4 \\ 1 & -2 & a+1 \\ \end{matrix} \right]\] is |
| A. | 1 if \[a=6\] |
| B. | 2 if \[a=1\] |
| C. | 3 if \[a=2\] |
| D. | 1 if\[a=4\] |
| Answer» C. 3 if \[a=2\] | |
| 1180. |
If \[\left| \begin{matrix} a & \cot A/2 & \lambda \\ b & \cot B/2 & \mu \\ c & \operatorname{cotC}/2 & \gamma \\ \end{matrix} \right|=0,\] where a, b, c, A, B, and C are elements of a triangle ABC with usual meaning. Then, the value of a \[(\mu -\gamma )+b(\gamma -\lambda )+c(\lambda -\mu )=0\] is |
| A. | \[0\] |
| B. | \[abc\] |
| C. | \[ab+bc+ca\] |
| D. | \[2abc\] |
| Answer» B. \[abc\] | |
| 1181. |
If \[{{a}_{r}}={{(\cos 2r\pi +i\sin 2r\pi )}^{\frac{1}{9}}},\]then the value of \[\left| \begin{matrix} {{a}_{1}} & {{a}_{2}} & {{a}_{3}} \\ {{a}_{4}} & {{a}_{5}} & {{a}_{6}} \\ {{a}_{7}} & {{a}_{8}} & {{a}_{9}} \\ \end{matrix} \right|\] is |
| A. | \[1\] |
| B. | \[-1\] |
| C. | \[0\] |
| D. | None of these |
| Answer» D. None of these | |
| 1182. |
If \[\omega \] is a complex cube root of unity, then value of\[\Delta =\left| \begin{matrix} {{a}_{1}}+{{b}_{1}}\omega & {{a}_{1}}{{\omega }^{2}}+{{b}_{1}} & {{c}_{1}}+{{b}_{1}}\bar{\omega } \\ {{a}_{2}}+{{b}_{2}}\omega & {{a}_{2}}{{\omega }^{2}}+{{b}_{2}} & {{c}_{2}}+{{b}_{2}}\bar{\omega } \\ {{a}_{3}}+{{b}_{3}}\omega & {{a}_{3}}{{\omega }^{2}}+{{b}_{3}} & {{c}_{3}}+{{b}_{3}}\bar{\omega } \\ \end{matrix} \right|\] is |
| A. | \[0\] |
| B. | \[-1\] |
| C. | \[2\] |
| D. | None of these |
| Answer» B. \[-1\] | |
| 1183. |
If \[\text{l}_{r}^{2}+m_{r}^{2}+n_{r}^{2}=\text{1};\] \[r=1,2,3\] and \[{{\text{l}}_{r}}{{\text{l}}_{s}}+{{m}_{r}}{{m}_{s}}+{{n}_{r}}{{n}_{s}}=0;\]\[r\ne s,\]\[r=1,2,3;\] \[s=1,2,3,\]then the value of \[\left| \begin{matrix} {{\text{l}}_{1}} & {{m}_{1}} & {{n}_{1}} \\ {{\text{l}}_{2}} & {{m}_{2}} & {{n}_{2}} \\ {{\text{l}}_{3}} & {{m}_{3}} & {{n}_{3}} \\ \end{matrix} \right|\] is |
| A. | \[0\] |
| B. | \[\pm 1\] |
| C. | \[2\] |
| D. | None of these |
| Answer» C. \[2\] | |
| 1184. |
If \[{{a}_{1}},{{a}_{2}},{{a}_{3}},............\]are positive numbers in G.P. then the value of \[\left| \begin{matrix} \log {{a}_{n}} & \log {{a}_{n+1}} & \log {{a}_{n+2}} \\ \log {{a}_{n+1}} & \log {{a}_{n+2}} & {{\operatorname{loga}}_{n+3}} \\ \log {{a}_{n+2}} & \log {{a}_{n+3}} & \log {{a}_{n+4}} \\ \end{matrix} \right|\] |
| A. | \[1\] |
| B. | \[4\] |
| C. | \[3\] |
| D. | \[0\] |
| Answer» E. | |
| 1185. |
If \[\left| A \right|=8,\] where.4 is square matrix of order 3, then what is \[\left| adj\,\,A \right|\] equal to? |
| A. | \[16\] |
| B. | \[24\] |
| C. | \[64\] |
| D. | \[512\] |
| Answer» D. \[512\] | |
| 1186. |
Suppose the system of equations \[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}z={{d}_{1}}\] \[{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}z={{d}_{2}}\] \[{{a}_{3}}x+{{b}_{3}}y+{{c}_{3}}z={{d}_{3}}\] has a unique solution \[({{x}_{0}},{{y}_{0}},{{z}_{0}})\]. If \[{{x}_{0}}=0,\] then which one of the following is correct? |
| A. | \[\left| \begin{matrix} {{a}_{1}} & {{b}_{1}} & {{c}_{1}} \\ {{a}_{2}} & {{b}_{2}} & {{c}_{2}} \\ {{a}_{3}} & {{b}_{3}} & {{c}_{3}} \\ \end{matrix} \right|=0\] |
| B. | \[\left| \begin{matrix} {{d}_{1}} & {{b}_{1}} & {{c}_{1}} \\ {{d}_{2}} & {{b}_{2}} & {{c}_{2}} \\ {{d}_{3}} & {{b}_{3}} & {{c}_{3}} \\ \end{matrix} \right|=0\] |
| C. | \[\left| \begin{matrix} {{d}_{1}} & {{a}_{1}} & {{c}_{1}} \\ {{d}_{2}} & {{a}_{2}} & {{c}_{2}} \\ {{d}_{3}} & {{a}_{3}} & {{c}_{3}} \\ \end{matrix} \right|=0\] |
| D. | \[\left| \begin{matrix} {{d}_{1}} & {{a}_{1}} & {{b}_{1}} \\ {{d}_{2}} & {{a}_{2}} & {{b}_{2}} \\ {{d}_{3}} & {{a}_{3}} & {{b}_{3}} \\ \end{matrix} \right|=0\] |
| Answer» C. \[\left| \begin{matrix} {{d}_{1}} & {{a}_{1}} & {{c}_{1}} \\ {{d}_{2}} & {{a}_{2}} & {{c}_{2}} \\ {{d}_{3}} & {{a}_{3}} & {{c}_{3}} \\ \end{matrix} \right|=0\] | |
| 1187. |
Let A and B be two matrices of order\[n\times n\]. Let A be non-singular and B be singular. Consider the following: 1. AB is singular 2. AB is non-singular 3. \[{{A}^{-1}}B\] is singular 4.\[{{A}^{-1}}B\] is non singular Which of the above is/ are correct? |
| A. | 1 and 3 |
| B. | 2 and 4 only |
| C. | 1 only |
| D. | 3 only |
| Answer» C. 1 only | |
| 1188. |
If x, y, z are complex numbers, and\[\Delta =\left| \begin{matrix} 0 & -y & -z \\ {\bar{y}} & 0 & -x \\ {\bar{z}} & {\bar{x}} & 0 \\ \end{matrix} \right|\] then \[\Delta \] is |
| A. | Purely real |
| B. | Purely imaginary |
| C. | Complex |
| D. | 0 |
| Answer» C. Complex | |
| 1189. |
Let \[\Delta =\left| \begin{matrix} 1+{{x}_{1}}{{y}_{1}} & 1+{{x}_{1}}{{y}_{2}} & 1+{{x}_{1}}{{y}_{3}} \\ 1+{{x}_{2}}{{y}_{1}} & 1+{{x}_{2}}{{y}_{2}} & \,1+{{x}_{2}}{{y}_{3}} \\ 1+{{x}_{3}}{{y}_{1}} & 1+{{x}_{3}}{{y}_{2}} & 1+{{x}_{3}}{{y}_{3}} \\ \end{matrix} \right|\] then value of \[\Delta \] is |
| A. | \[{{x}_{1}}{{x}_{2}}{{x}_{3}}+{{y}_{1}}{{y}_{2}}{{y}_{3}}\] |
| B. | \[{{x}_{1}}{{x}_{2}}{{x}_{3}}{{y}_{1}}{{y}_{2}}{{y}_{3}}\] |
| C. | \[{{x}_{2}}{{x}_{3}}{{y}_{2}}{{y}_{3}}+{{x}_{3}}{{x}_{1}}{{y}_{3}}{{y}_{1}}+{{x}_{1}}{{x}_{2}}{{y}_{1}}{{y}_{2}}\] |
| D. | 0 |
| Answer» E. | |
| 1190. |
If \[a>0,b>0,c>0\] are respectively the pth, qth,rth terms of GP, then the value of the determinant \[\left| \begin{matrix} \log a & p & 1 \\ \log b & q & 1 \\ \log c & r & 1 \\ \end{matrix} \right|\] is |
| A. | \[0\] |
| B. | \[1\] |
| C. | \[-1\] |
| D. | None of these |
| Answer» B. \[1\] | |
| 1191. |
The value of \[\left| \begin{matrix} ^{10}{{C}_{4}} & ^{10}{{C}_{5}} & ^{11}{{C}_{m}} \\ ^{11}{{C}_{6}} & ^{11}{{C}_{7}} & ^{12}{{C}_{m+2}} \\ ^{12}{{C}_{8}} & ^{12}{{C}_{9}} & ^{13}{{C}_{m+4}} \\ \end{matrix} \right|=0,\] when m is equal to |
| A. | \[6\] |
| B. | \[5\] |
| C. | \[4\] |
| D. | \[1\] |
| Answer» C. \[4\] | |
| 1192. |
If \[A\left[ \begin{matrix} 1 & 2 \\ 3 & 5 \\ \end{matrix} \right],\] then the value of the determinant \[|{{A}^{2009}}-5{{A}^{2008}}|\] is |
| A. | \[-6\] |
| B. | \[-5\] |
| C. | \[-4\] |
| D. | \[4\] |
| Answer» B. \[-5\] | |
| 1193. |
If \[a,b,c,d>0,x\text{ }\in \text{R}\] and \[({{a}^{2}}+{{b}^{2}}+{{c}^{2}}){{x}^{2}}-2(ab+bc+cd)x+{{b}^{2}}+{{c}^{2}}+{{d}^{2}}\le 0.\]Then, \[\left| \begin{matrix} 33 & 14 & \log a \\ 65 & 27 & \log b \\ 97 & 40 & \log c \\ \end{matrix} \right|\] is equal to |
| A. | \[1\] |
| B. | \[-1\] |
| C. | \[2\] |
| D. | \[0\] |
| Answer» E. | |
| 1194. |
If \[g(x)=\left| \begin{matrix} {{a}^{-x}} & {{e}^{x{{\log }_{e}}a}} & {{x}^{2}} \\ {{a}^{-3x}} & {{e}^{3x{{\log }_{e}}a}} & {{x}^{4}} \\ {{a}^{-5x}} & {{e}^{5x{{\log }_{e}}a}} & 1 \\ \end{matrix} \right|,\] then |
| A. | \[g(x)+g(-x)=0\] |
| B. | \[g(x)-g(-x)=0\] |
| C. | \[g(x)\times g(-x)=0\] |
| D. | None of these |
| Answer» B. \[g(x)-g(-x)=0\] | |
| 1195. |
Let A be an \[n\times n\] matrix. If \[\det \,(\lambda A)={{\lambda }^{s}}\det \,(A),\] what is the value of s? |
| A. | \[0\] |
| B. | \[1\] |
| C. | \[-1\] |
| D. | \[n\] |
| Answer» E. | |
| 1196. |
If \[|{{A}_{n\times n}}|=3\] and \[|adj\,\,A|=243,\] what is the value of n? |
| A. | \[4\] |
| B. | \[5\] |
| C. | \[6\] |
| D. | \[7\] |
| Answer» D. \[7\] | |
| 1197. |
Consider the following statements: 1. If det \[A=0,\]then det \[(adj\,A)=0\] 2. If A is non- singular, then \[\det \,({{A}^{-1}})={{(\det \,A)}^{-1}}\] |
| A. | 1 only |
| B. | 2 only |
| C. | Both 1 and 2 |
| D. | Neither 1 nor 2 |
| Answer» D. Neither 1 nor 2 | |
| 1198. |
Let \[{{S}_{k}}={{\alpha }^{k}}+{{\beta }^{k}}+{{\gamma }^{k}},\] then \[\Delta =\left| \begin{matrix} {{S}_{0}} & {{S}_{1}} & {{S}_{2}} \\ {{S}_{1}} & {{S}_{2}} & {{S}_{3}} \\ {{S}_{2}} & {{S}_{3}} & {{S}_{4}} \\ \end{matrix} \right|\] is equal to |
| A. | \[{{S}_{6}}\] |
| B. | \[{{S}_{5}}-{{S}_{3}}\] |
| C. | \[{{S}_{6}}-{{S}_{4}}\] |
| D. | None |
| Answer» E. | |
| 1199. |
If \[f(x)=\left| \begin{matrix} 1+{{\sin }^{2}}x & {{\cos }^{2}}x & 4\sin 2x \\ {{\sin }^{2}}x & 1+{{\cos }^{2}}x & 4\sin 2x \\ {{\sin }^{2}}x & {{\cos }^{2}}x & 1+4\sin 2x \\ \end{matrix} \right|\]What is the maximum value of \[f(x)\]? |
| A. | 2 |
| B. | 4 |
| C. | 6 |
| D. | 8 |
| Answer» D. 8 | |
| 1200. |
If \[A=\left[ \begin{matrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \\ \end{matrix} \right],\] then the value of \[|adj\,\,A|\] is |
| A. | \[{{a}^{27}}\] |
| B. | \[{{a}^{9}}\] |
| C. | \[{{a}^{6}}\] |
| D. | \[{{a}^{2}}\] |
| Answer» D. \[{{a}^{2}}\] | |