If the curves, x squer/a + y squer/b = 1 and
Question: If the curves, $\frac{x^{2}}{a}+\frac{y^{2}}{b}=1$ and $\frac{x^{2}}{c}+\frac{y^{2}}{d}=1$ intersect each other at an angle of $90^{\circ}$, then which of the following relations is TRUE?$a+b=c+d$$a-b=c-d$$a-c=b+d$$a b=\frac{c+d}{a+b}$Correct Option: , 2 Solution: For orthogonal curves $a-c=b-d$ $\Rightarrow a-b=c-d$...
Read More →The function
Question: The function $f(x)=\frac{4 x^{3}-3 x^{2}}{6}-2 \sin x+(2 x-1) \cos x:$ increases in $\left[\frac{1}{2}, \infty\right)$increases in $\left(-\infty, \frac{1}{2}\right]$decreases in $\left[\frac{1}{2}, \infty\right)$decreases in $\left(-\infty, \frac{1}{2}\right]$Correct Option: 1 Solution: $f(x)=\frac{4 x^{3}-3 x^{2}}{6}-2 \sin x+(2 x-1) \cos x$ $f^{\prime}(x)=\left(2 x^{2}-x\right)-2 \cos x+2 \cos x-\sin x(2 x-1)$ $=(2 x-1)(x-\sin x)$ for $x0, x-\sin x0$ $x0, x-\sin x0$ for $x \in(-\inf...
Read More →The image of the point (3,5)
Question: The image of the point $(3,5)$ in the line $x-y+1=0$, lies on :$(x-2)^{2}+(y-2)^{2}=12$$(x-4)^{2}+(y+2)^{2}=16$$(x-4)^{2}+(y-4)^{2}=8$$(x-2)^{2}+(y-4)^{2}=4$Correct Option: , 4 Solution: $\frac{x-3}{1}=\frac{y-5}{-1}=-2\left(\frac{3-5+1}{1+1}\right)$ $\mathrm{So}, x=4, y=4$ Hence, $(x-2)^{2}+(y-4)^{2}=4$...
Read More →The value of
Question: The value of $\left|\begin{array}{lll}(a+1)(a+2) a+2 1 \\ (a+2)(a+3) a+3 1 \\ (a+3)(a+4) a+4 1\end{array}\right|$ is$(a+2)(a+3)(a+4)$$-2$$(a+1)(a+2)(a+3)$0Correct Option: , 2 Solution: $\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{1}$ and $\mathrm{R}_{3} \rightarrow \mathrm{R}_{3}-\mathrm{R}_{1}$ $\Delta=\left|\begin{array}{ccc}(a+1)(a+2) a+2 1 \\ (a+2)(a+3-a-1) 1 0 \\ a^{2}+7 a+12-a^{2}-3 a-2 2 0\end{array}\right|$ $=\left|\begin{array}{ccc}a^{2}+3 a+2 a+2 1 \\ 2(a+2) 1 0 \\ ...
Read More →Consider the three planes
Question: Consider the three planes $P_{1}: 3 x+15 y+21 z=9$ $P_{2}: x-3 y-z=5$, and $P_{3}: 2 x+10 y+14 z=5$ Then, which one of the following is true ?$\mathrm{P}_{1}$ and $\mathrm{P}_{2}$ are parallel$P_{1}$ and $P_{3}$ are parallel$P_{2}$ and $P_{3}$ are parallel$\mathrm{P}_{1}, \mathrm{P}_{2}$ and $\mathrm{P}_{3}$ all are parallelCorrect Option: , 2 Solution: $P_{1}: x+5 y+7 z=3$ $P_{2}: x-3 y-z=5$ $P_{3}: x+5 y+7 z=\frac{5}{2}$ so $\mathrm{P}_{1}$ and $\mathrm{P}_{3}$ are parallel....
Read More →Solve this following
Question: The value of $-{ }^{15} C_{1}+2 \cdot{ }^{15} C_{2}-3 \cdot{ }^{15} C_{3}+\ldots \ldots$ $-15 .{ }^{15} \mathrm{C}_{15}+{ }^{14} \mathrm{C}_{1}+{ }^{14} \mathrm{C}_{3}+{ }^{14} \mathrm{C}_{5}+\ldots .+{ }^{14} \mathrm{C}_{11}$ is : $2^{16}-1$$2^{13}-14$$2^{14}$$2^{13}-13$Correct Option: , 2 Solution: $\left(-{ }^{15} C_{1}+2 .{ }^{15} C_{2}-3 .{ }^{15} C_{3}+\ldots \ldots-15 .{ }^{15} C_{15}\right)$ $+\left({ }^{14} C_{1}+{ }^{14} C_{3}+\ldots .+{ }^{14} C_{11}\right)$ $=\sum_{r=1}^{15...
Read More →The intersection of three lines
Question: The intersection of three lines $x-y=0, x+2 y=3$ and $2 x+y=6$ is aRight angled triangleEquilateral triangleIsosceles triangleNone of the aboveCorrect Option: , 3 Solution: $L_{1}: x-y=0$ $L_{2}: x+2 y=3$ $\mathrm{L}_{3}: \mathrm{x}+\mathrm{y}=6$ on solving $\mathrm{L}_{1}$ and $\mathrm{L}_{2}$ : $\mathrm{y}=\mathrm{L}$ and $\mathrm{x}=1$ $\mathrm{L}_{1}$ and $\mathrm{L}_{3}$ : $x=2$ $y=2$ $\mathrm{L}_{2}$ and $\mathrm{L}_{3}:$ $x+y=3$ $2 x+y=6$ $x=3$ $y=0$ $\mathrm{AC}=\sqrt{4+1}=\sqr...
Read More →Solve this following
Question: If $\int \frac{\cos x-\sin x}{\sqrt{8-\sin 2 x}} d x=a \sin ^{-1}\left(\frac{\sin x+\cos x}{b}\right)+c$ where $c$ is a constant of integration, then the ordered pair $(a, b)$ is equal to : $(-1,3)$$(3,1)$$(1,3)$$(1,-3)$Correct Option: 3, Solution: $\int \frac{\cos x-\sin x}{\sqrt{8-\sin 2 x}} d x$ $=\int \frac{\cos x-\sin x}{\sqrt{9-(\sin x+\cos x)^{2}}} d x$ Let $\sin x+\cos x=t$ $\int \frac{\mathrm{dt}}{\sqrt{9-\mathrm{t}^{2}}}=\sin ^{-1} \frac{\mathrm{t}}{3}+\mathrm{c}$ $=\sin ^{-1...
Read More →The maximum slope of the curve
Question: The maximum slope of the curve $y=\frac{1}{2} x^{4}-5 x^{3}+18 x^{2}-19 x$ occurs at the point$(2,2)$$(0,0)$$(2,9)$$\left(3, \frac{21}{2}\right)$Correct Option: 1 Solution: $\frac{\mathrm{dy}}{\mathrm{dx}}=2 \mathrm{x}^{3}-15 \mathrm{x}^{2}+36 \mathrm{x}-19$ Since, slope is maximum so, $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=6 \mathrm{x}^{2}-30 \mathrm{x}+36=0$ $y=\frac{1}{2} \times 16-5 \times 8+18 \times 4-19 \times 2$ $=8-40+72-38=80-78=2$ point $(2,2)$...
Read More →Let the line (2 - i )z = (2 + i)
Question: Let the lines $(2-\mathrm{i}) \mathrm{z}=(2+\mathrm{i}) \overline{\mathrm{z}}$ and $(2+\mathrm{i}) \mathrm{z}+(\mathrm{i}-2) \overline{\mathrm{z}}-4 \mathrm{i}=0$, (here $\mathrm{i}^{2}=-1$ ) be normal to a circle $C$. If the line $i z+\bar{z}+1+i=0$ is tangent to this circle $\mathrm{C}$, then its radius is:$\frac{3}{\sqrt{2}}$$\frac{1}{2 \sqrt{2}}$$3 \sqrt{2}$$\frac{3}{2 \sqrt{2}}$Correct Option: , 4 Solution: (i) $(2-i) z=(2+i) \bar{z}$ $y=\frac{x}{2}$ (ii) $(2+i) z+(i-2) \bar{z}-4 ...
Read More →Solve this following
Question: If the tangent to the curve $\mathrm{y}=\mathrm{x}^{3}$ at the point $\mathrm{P}\left(\mathrm{t}, \mathrm{t}^{3}\right)$ meets the curve again at $\mathrm{Q}$, then the ordinate of the point which divides PQ internally in the ratio $1: 2$ is: $-2 \mathrm{t}^{3}$0$-\mathrm{t}^{3}$$2 \mathrm{t}^{3}$Correct Option: 1 Solution: Slope of tangent at $\left.P\left(t, t^{3}\right)=\frac{d y}{d x}\right]_{\left(t, t^{3}\right)}$ $=\left(3 x^{2}\right)_{x=t}=3 t^{2}$ So equation tangent at $\mat...
Read More →Solve the Following Questions
Question: Let $f$ be any function defined on $\mathrm{R}$ and let it satisfy the condition : $|f(\mathrm{x})-f(\mathrm{y})| \leq\left|(\mathrm{x}-\mathrm{y})^{2}\right|, \forall(\mathrm{x}, \mathrm{y}) \in \mathrm{R}$ If $f(0)=1$, then :$f(x)$ can take any value in $R$$f(\mathrm{x})0, \forall \mathrm{x} \in \mathrm{R}$$f(\mathrm{x})=0, \forall \mathrm{x} \in \mathrm{R}$$f(\mathrm{x})0, \forall \mathrm{x} \in \mathrm{R}$Correct Option: , 4 Solution: $\left|\frac{f(\mathrm{x})-f(\mathrm{y})}{(\mat...
Read More →All possible values of θ ∈ [0, 2π] for
Question: All possible values of $\theta \in[0,2 \pi]$ for which $\sin 2 \theta+\tan 2 \theta0$ lie in :$\left(0, \frac{\pi}{2}\right) \cup\left(\pi, \frac{3 \pi}{2}\right)$$\left(0, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \frac{3 \pi}{4}\right) \cup\left(\pi, \frac{7 \pi}{6}\right)$$\left(0, \frac{\pi}{4}\right) \cup\left(\frac{\pi}{2}, \frac{3 \pi}{4}\right) \cup\left(\frac{3 \pi}{2}, \frac{11 \pi}{6}\right)$$\left(0, \frac{\pi}{4}\right) \cup\left(\frac{\pi}{2}, \frac{3 \pi}{4}\right) \...
Read More →The distance of the point (1,1,9) from the point
Question: The distance of the point $(1,1,9)$ from the point of intersection of the line $\frac{x-3}{1}=\frac{y-4}{2}=\frac{z-5}{2}$ and the plane $x+y+z=17$ is: $2 \sqrt{19}$$19 \sqrt{2}$38$\sqrt{38}$Correct Option: , 4 Solution: Let $\frac{x-3}{1}=\frac{y-4}{2}=\frac{z-5}{2}=t$ $\Rightarrow \quad x=3+t, y=2 t+4, z=2 t+5$ for point of intersection with $x+y+z=17$ $3+t+2 t+4+2 t+5=17$ $\Rightarrow 5 \mathrm{t}=5 \Rightarrow \mathrm{t}=1$ $\Rightarrow$ point of intersection is $(4,6,7)$ distance ...
Read More →The number of seven digit
Question: The number of seven digit integers with sum of the digits equal to 10 and formed by using the digits 1,2 and 3 only is42827735Correct Option: , 3 Solution: (I) First possiblity is $1,1,1,1,1,2,3$ required number $=\frac{7 !}{5 !}=7 \times 6=42$. (II) Second possiblity is $1,1,1,1,2,2,2$ required number $=\frac{7 !}{4 ! 3 !}=\frac{7 \times 6 \times 5}{6}=35$ Total $=42+35=77$...
Read More →A tangent is drawn to the parabola
Question: A tangent is drawn to the parabola $y^{2}=6 x$ which is perpendicular to the line $2 x+y=1$. Which of the following points does NOT lie on it?$(-6,0)$$(4,5)$$(5,4)$$(0,3)$Correct Option: , 3 Solution: Slope of tangent $=\mathrm{m}_{\mathrm{T}}=\mathrm{m}$ So, $\mathrm{m}(-2)=-1 \Rightarrow \mathrm{m}=\frac{1}{2}$ Equation : $y=m x+\frac{a}{m}$ $\Rightarrow \mathrm{y}=\frac{1}{2} \mathrm{x}+\frac{3}{2 \times \frac{1}{2}}\left(\mathrm{a}=\frac{6}{4}=\frac{3}{2}\right)$ $\Rightarrow y=\fr...
Read More →Solve this following
Question: If $f: \mathrm{R} \rightarrow \mathrm{R}$ is a function defined by $f(x)=[x-1] \cos \left(\frac{2 x-1}{2}\right) \pi$, where [.] denotes the greatest integer function, then $f$ is : discontinuous at all integral values of $x$ except at $x=1$continuous only at $x=1$continuous for every real $x$discontinuous only at $x=1$Correct Option: , 3 Solution: For $x=n, n \in Z$ $\mathrm{LHL}=\lim _{x \rightarrow \mathrm{n}^{-}} f(\mathrm{x})=\lim _{\mathrm{x} \rightarrow \mathrm{n}^{-}}[\mathrm{x...
Read More →Solve the Following Questions
Question: If $\frac{\sin ^{-1} x}{a}=\frac{\cos ^{-1} x}{b}=\frac{\tan ^{-1} y}{c} ; 0x1$, then the value of $\cos \left(\frac{\pi c}{a+b}\right)$ is$\frac{1-y^{2}}{y \sqrt{y}}$$1-y^{2}$$\frac{1-y^{2}}{1+y^{2}}$$\frac{1-y^{2}}{2 y}$Correct Option: , 3 Solution: $\frac{\sin ^{-1} x}{r}=a, \frac{\cos ^{-1} x}{r}=b, \frac{\tan ^{-1} y}{r}=c$ So, $a+b=\frac{\pi}{2 r}$ $\cos \left(\frac{\pi c}{a+b}\right)=\cos \left(\frac{\pi \tan ^{-1} y}{\frac{\pi}{2 r} r}\right)$ $=\cos \left(2 \tan ^{-1} y\right)...
Read More →A man is observing, from the top of a tower,
Question: A man is observing, from the top of a tower, a boat speeding towards the tower from a certain point $A$, with uniform speed. At that point, angle of depression of the boat with the man's eye is $30^{\circ}$ (Ignore man's height). After sailing for 20 seconds, towards the base of the tower (which is at the level of water), the boat has reached a point $\mathrm{B}$, where the angle of depression is $45^{\circ}$. Then the time taken (in seconds) by the boat from $B$ to reach the base of t...
Read More →Solve the Following Questions
Question: If $(1,5,35),(7,5,5),(1, \lambda, 7)$ and $(2 \lambda, 1,2)$ are coplanar, then the sum of all possible values of $\lambda$ is$\frac{39}{5}$$-\frac{39}{5}$$\frac{44}{5}$$-\frac{44}{5}$Correct Option: , 3 Solution: $\mathrm{A}(1,5,35), \mathrm{B}(7,5,5), \mathrm{C}(1, \lambda, 7), \mathrm{D}(2 \lambda, 1,2)$ $\overline{\mathrm{AB}}=6 \hat{\mathrm{i}}-30 \hat{\mathrm{k}}, \overline{\mathrm{BC}}=-6 \hat{\mathrm{i}}(\lambda-5) \hat{\mathrm{j}}+2 \hat{\mathrm{k}}$ $\overrightarrow{\mathrm{C...
Read More →The value of
Question: The value of $\int_{-1}^{1} x^{2} e^{\left[x^{3}\right]} d x$, where $[t]$ denotes the greatest integer $\leq t$, is :$\frac{\mathrm{e}-1}{3 \mathrm{e}}$$\frac{\mathrm{e}+1}{3}$$\frac{\mathrm{e}+1}{3 \mathrm{e}}$$\frac{1}{3 \mathrm{e}}$Correct Option: , 3 Solution: $I=\int_{-1}^{1} x^{2} e^{\left[x^{3}\right]} d x$ $=\int_{-1}^{0} x^{2} e^{\left[x^{3}\right]} d x+\int_{0}^{1} x^{2} e^{\left[x^{3}\right]} d x$ $=\int_{-1}^{0} x^{2} e^{-1} d x+\int_{0}^{1} x^{2} e^{0} d x$ $=\frac{1}{e} ...
Read More →The system of linear equations
Question: The system of linear equations $3 x-2 y-k z=10$ $2 x-4 y-2 z=6$ $x+2 y-z=5 m$ is inconsistent if: $\mathrm{k}=3, \mathrm{~m}=\frac{4}{5}$$\mathrm{k} \neq 3, \mathrm{~m} \in \mathrm{R}$$\mathrm{k} \neq 3, \mathrm{~m} \neq \frac{4}{5}$$\mathrm{k}=3, \mathrm{~m} \neq \frac{4}{5}$Correct Option: , 4 Solution: $\Delta=\left|\begin{array}{ccc}3 -2 -k \\ 2 -4 -2 \\ 1 2 -1\end{array}\right|=0$ $\Rightarrow 24-2(0)-\mathrm{k}(8)=0 \Rightarrow \mathrm{k}=3$ $\Delta_{\mathrm{x}}=\left|\begin{arra...
Read More →The rate of growth of bacteria in a culture
Question: The rate of growth of bacteria in a culture is proportional to the number of bacteris present and the bacteria count is 1000 at initial time $\mathrm{t}=0$. The number of bacteria is increased by $20 \%$ in 2 hours. If the population of bacteria is 2000 after $\frac{\mathrm{k}}{\log _{\mathrm{e}}\left(\frac{6}{5}\right)}$ hours, then $\left(\frac{\mathrm{k}}{\log _{\mathrm{e}} 2}\right)^{2}$ is equal to48216Correct Option: 1 Solution: $\frac{\mathrm{dB}}{\mathrm{dt}}=\lambda \mathrm{B}...
Read More →The value of the integral
Question: The value of the integral $\int \frac{\sin \theta \cdot \sin 2 \theta\left(\sin ^{6} \theta+\sin ^{4} \theta+\sin ^{2} \theta\right) \sqrt{2 \sin ^{4} \theta+3 \sin ^{2} \theta+6}}{1-\cos 2 \theta} \mathrm{d} \theta$ is: (where $c$ is a constant of integration)$\frac{1}{18}\left[11-18 \sin ^{2} \theta+9 \sin ^{4} \theta-2 \sin ^{6} \theta\right]^{\frac{3}{2}}+c$$\frac{1}{18}\left[9-2 \cos ^{6} \theta-3 \cos ^{4} \theta-6 \cos ^{2} \theta\right]^{\frac{3}{2}}+c$$\frac{1}{18}\left[9-2 \s...
Read More →The maximum value of the term independent
Question: The maximum value of the term independent of $\mathrm{t}^{\prime}$ in the expansion of $\left(\mathrm{tx}^{\frac{1}{5}}+\frac{(1-\mathrm{x})^{\frac{1}{10}}}{\mathrm{t}}\right)^{10}$ where $x \in(0,1)$ is$\frac{10 !}{\sqrt{3}(5 !)^{2}}$$\frac{2.10 !}{3 \sqrt{3}(5 !)^{2}}$$\frac{2.10 !}{3(5 !)^{2}}$$\frac{10 !}{3(5 !)^{2}}$Correct Option: , 2 Solution: Term independent of t will be the middle term due to exect same magnitude but opposite sign powers of $t$ in the binomial expression give...
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