Which one of the following statements is not a tautology ?
Question: Which one of the following statements is not a tautology ?$(p \wedge q) \rightarrow p$$(\mathrm{p} \wedge \mathrm{q}) \rightarrow(\sim \mathrm{p}) \vee \mathrm{q}$$\mathrm{p} \rightarrow(\mathrm{p} \vee \mathrm{q})$$(\mathrm{p} \vee \mathrm{q}) \rightarrow(\mathrm{p} \vee(\sim \mathrm{q}))$Correct Option: , 4 Solution:...
Read More →Consider the following three statements:
Question: Consider the following three statements: $P: 5$ is a prime number. Q : 7 is a factor of 192 . R : L.C.M. of 5 and 7 is 35 . Then the truth value of which one of the following statements is true?$(P \wedge Q) \vee(\sim R)$$(\sim \mathrm{P}) \wedge(\sim \mathrm{Q} \wedge \mathrm{R})$$(\sim P) \vee\left(Q^{\wedge} R\right)$$\mathrm{P} \vee\left(\sim \mathrm{Q}^{\wedge} \mathrm{R}\right)$Correct Option: , 4 Solution: It is obvious $\therefore$ Option (4)...
Read More →The plane which bisects the line segment joining the points
Question: The plane which bisects the line segment joining the points $(-3,-3,4)$ and $(3,7,6)$ at right angles, passes through which one of the following points ?$(4,-1,7)$$(4,1,-2)$$(-2,3,5)$$(2,1,3)$Correct Option: , 2 Solution: $p: 3(x-0)+5(y-2)+1(z-5)=0$ $3 x+5 y+z=15$ $\therefore$ Option (2)...
Read More →Solve this following
Question: With the usual notation, in $\triangle A B C$, if $\angle \mathrm{A}+\angle \mathrm{B}=120^{\circ}, \mathrm{a}=\sqrt{3}+1$ and $\mathrm{b}=\sqrt{3}-1$ then the ratio $\angle \mathrm{A}: \angle \mathrm{B}$, is :$7: 1$$5: 3$$9: 7$$3: 1$Correct Option: 1 Solution: $\mathrm{A}+\mathrm{B}=120^{\circ}$ $\tan \frac{\mathrm{A}-\mathrm{B}}{2}=\frac{\mathrm{a}-\mathrm{b}}{\mathrm{a}+\mathrm{b}} \cot \left(\frac{\mathrm{C}}{2}\right)$ $=\frac{\sqrt{3}+1-\sqrt{3}+1}{2(\sqrt{3})} \cot \left(30^{\ci...
Read More →The area (in sq. units) of the region bounded
Question: The area (in sq. units) of the region bounded by the parabola, $y=x^{2}+2$ and the lines, $y=x+1, x=0$ and $x=3$, is :$\frac{15}{4}$$\frac{15}{2}$$\frac{21}{2}$$\frac{17}{4}$Correct Option: , 2 Solution: Req. area $=\int_{0}^{3}\left(\mathrm{x}^{2}+2\right) \mathrm{dx}-\frac{1}{2} \cdot 5 \cdot 3=9+6-\frac{15}{2}=\frac{15}{2}$...
Read More →The vector equation of the plane through the line of intersection of the planes
Question: The vector equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y+ 4z = 5 which is perpendicular to the plane x y + z = 0 is :$\overrightarrow{\mathrm{r}} \times(\hat{\mathrm{i}}+\hat{\mathrm{k}})+2=0$$\overrightarrow{\mathrm{r}} \cdot(\hat{\mathrm{i}}-\hat{\mathrm{k}})-2=0$$\overrightarrow{\mathrm{r}} \cdot(\hat{\mathrm{i}}-\hat{\mathrm{k}})+2=0$$\overrightarrow{\mathrm{r}} \times(\hat{\mathrm{i}}-\hat{\mathrm{k}})+2=0$Correct Option: , 3 Soluti...
Read More →An ordered pair ( α , β ) for which the system of linear equations
Question: An ordered $\operatorname{pair}(\alpha, \beta)$ for which the system of linear equations $(1+\alpha) x+\beta y+z=2$ $\alpha x+(1+\beta) y+z=3$ $\alpha x+\beta y+2 z=2$ has a unique solution is$(1,-3)$$(-3,1)$$(2,4)$$(-4,2)$Correct Option: , 3 Solution: For unique solution $\Delta \neq 0 \Rightarrow\left|\begin{array}{ccc}1+\alpha \beta 1 \\ \alpha 1+\beta 1 \\ \alpha \beta 2\end{array}\right| \neq 0$ $\left|\begin{array}{ccc}1 -1 0 \\ 0 1 -1 \\ \alpha \beta 2\end{array}\right| \neq 0 \...
Read More →The value of
Question: The value of $\cot \left(\sum_{n=1}^{19} \cot ^{-1}\left(1+\sum_{p=1}^{n} 2 p\right)\right)$ is : $\frac{22}{23}$$\frac{23}{22}$$\frac{21}{19}$$\frac{19}{21}$Correct Option: , 3 Solution: $\cot \left(\sum_{n=1}^{19} \cot ^{-1}(1+n(n+1))\right.$ $\cot \left(\sum_{n=1}^{19} \cot ^{-1}\left(n^{2}+n+1\right)\right)=\cot \left(\sum_{n=1}^{19} \tan ^{-1} \frac{1}{1+n(n+1)}\right)$ $\sum_{n=1}^{19}\left(\tan ^{-1}(n+1)-\tan ^{-1} n\right)$ $\cot \left(\tan ^{-1} 20-\tan ^{-1} 1\right)=\frac{\...
Read More →Given that the slope of the tangent to a curve
Question: Given that the slope of the tangent to a curve $\mathrm{y}=\mathrm{y}(\mathrm{x})$ at any point $(\mathrm{x}, \mathrm{y})$ is $\frac{2 \mathrm{y}}{\mathrm{x}^{2}}$. If the curve passes through the centre of the circle $x^{2}+y^{2}-2 x-2 y=0$, then its equation is$x \log _{e}|y|=2(x-1)$$x \log _{e}|y|=x-1$$x^{2} \log _{e}|y|=-2(x-1)$$x \log _{e}|y|=-2(x-1)$Correct Option: 1, Solution: $\operatorname{given} \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{2 \mathrm{y}}{\mathrm{x}^{2}}$ $\Rightarrow...
Read More →Solve this following
Question: Let $\vec{\alpha}=(\lambda-2) \vec{a}+\vec{b}$ and $\vec{\beta}=(4 \lambda-2) \vec{a}+3 \vec{b} \quad$ be two given vectors where vectors $\vec{a}$ and $\vec{b}$ are non-collinear. The value of $\lambda$ for which vectors $\vec{\alpha}$ and $\vec{\beta}$ are collinear, is : $-3$43$-4$Correct Option: , 4 Solution: $\vec{\alpha}=(\lambda-2) \vec{\alpha}+\vec{b}$ $\vec{\beta}=(4 \lambda-2) \vec{\alpha}+3 \vec{b}$ $\frac{\lambda-2}{4 \lambda-2}=\frac{1}{3}$ $3 \lambda-6=4 \lambda-2$ $\lamb...
Read More →If the solve the problem
Question: Let $S(\alpha)=\left\{(\mathrm{x}, \mathrm{y}): \mathrm{y}^{2} \leq \mathrm{x}, 0 \leq \mathrm{x} \leq \alpha\right\}$ and $\mathrm{A}(\alpha)$ is area of the region $S(\alpha)$. If for a $\lambda, 0\lambda4$, $\mathrm{A}(\lambda): \mathrm{A}(4)=2: 5$, then $\lambda$ equals$2\left(\frac{4}{25}\right)^{\frac{1}{3}}$$4\left(\frac{4}{25}\right)^{\frac{1}{3}}$$2\left(\frac{2}{5}\right)^{\frac{1}{3}}$$4\left(\frac{2}{5}\right)^{\frac{1}{3}}$Correct Option: Solution: $\mathrm{S}(\alpha)=\lef...
Read More →Considering only the principal values of inverse functions,
Question: Considering only the principal values of inverse functions, the set $A=\left\{x \geq 0: \tan ^{-1}(2 x)+\tan ^{-1}(3 x)=\frac{\pi}{4}\right\}$is an empty setContains more than two elementsContains two elementsis a singletonCorrect Option: , 4 Solution: $\tan ^{-1}(2 x)+\tan ^{-1}(3 x)=\pi / 4$ $\Rightarrow \frac{5 x}{1-6 x^{2}}=1$ $\Rightarrow 6 x^{2}+5 x-1=0$ $x=-1$ or $x=\frac{1}{6}$ $x=\frac{1}{6} \quad \because x0$...
Read More →If the fourth term in the binomial expansion of
Question: If the fourth term in the binomial expansion of $\left(\sqrt{\frac{1}{x^{1+\log _{10} x}}}+x^{\frac{1}{12}}\right)^{6}$ is equal to 200 , and $x1$, then the value of x is :$10^{3}$100$10^{4}$10Correct Option: , 4 Solution: $200={ }^{6} \mathrm{C}_{3}\left(\mathrm{x}^{\frac{1}{x+\log _{10} x}}\right)^{\frac{3}{2}} \times \mathrm{x}^{\frac{1}{4}}$ $\Rightarrow 10=x^{\frac{3}{2\left(1+\log _{10} x\right)}+\frac{1}{4}}$ $\Rightarrow 1=\left(\frac{3}{2(1+t)}+\frac{1}{4}\right) t$ where $t=\...
Read More →Solve this following
Question: Two sides of a parallelogram are along the lines, $x+y=3$ and $x-y+3=0$. If its diagonals intersect at $(2,4)$, then one of its vertex is : $(2,6)$$(2,1)$$(3,5)$$(3,6)$Correct Option: , 4 Solution: $\frac{x_{1}+0}{2}=2 ; \quad x_{i}=4 \quad$ similarly $\quad y_{1}=5$ $\mathrm{C} \Rightarrow(4,5)$ Now equation of $B C$ is $x-y=-1$ and equation of CD is $x+y=9$ Solving $x+y=9$ and $x-y=-3$ Point $\mathrm{D}$ is $(3,6)$ Option (4)...
Read More →Prove the following
Question: $\lim _{x \rightarrow \pi / 4} \frac{\cot ^{3} x-\tan x}{\cos (x+\pi / 4)}$ is :4$8 \sqrt{2}$8$4 \sqrt{2}$Correct Option: , 3 Solution: $\lim _{x \rightarrow \pi / 4} \frac{\cot ^{3} x-\tan x}{\cos \left(x+\frac{\pi}{4}\right)}$ $\lim _{x \rightarrow \pi / 4} \frac{\left(1-\tan ^{4} x\right)}{\cos (x+\pi / 4)}$ $2 \lim _{x \rightarrow \pi / 4} \frac{\left(1-\tan ^{2} x\right)}{\cos (x+\pi / 4)}$ $R \lim _{x \rightarrow \pi / 4} \frac{\cos ^{2} x-\sin ^{2} x}{\frac{\cos x-\sin x}{\sqrt{...
Read More →The Boolean expression
Question: The Boolean expression $((\mathrm{p} \wedge \mathrm{q}) \vee(\mathrm{p} \vee \sim \mathrm{q})) \wedge(\sim \mathrm{p} \wedge \sim \mathrm{q})$ is equivalent to: $\mathrm{p} \wedge(\sim \mathrm{q})$$p \vee(\sim q)$$(\sim p) \wedge(\sim q)$$\mathrm{p} \wedge \mathrm{q}$Correct Option: , 3 Solution: (3) $(\sim p) \wedge(\sim q)$...
Read More →The maximum area (in sq. units) of a rectangle having
Question: The maximum area (in sq. units) of a rectangle having its base on the $\mathrm{x}$-axis and its other two vertices on the parabola, $y=12-x^{2}$ such that the rectangle lies inside the parabola, is :-$20 \sqrt{2}$$18 \sqrt{3}$3236Correct Option: , 3 Solution: $f(a)=2 a(12-a)^{2}$ $f^{\prime}(a)=2\left(12-3 a^{2}\right)$ maximum at $\mathrm{a}=2$ maximum area $=\mathrm{f}(2)=32$...
Read More →Two vertices of a triangle are
Question: Two vertices of a triangle are $(0,2)$ and $(4,3)$. If its orthocentre is at the origin, then its third vertex lies in which quadrant ?FourthSecondThirdFirstCorrect Option: , 2 Solution: $\mathrm{m}_{\mathrm{BD}} \times \mathrm{m}_{\mathrm{AD}}=-1 \Rightarrow\left(\frac{3-2}{4-0}\right) \times\left(\frac{\mathrm{b}-0}{\mathrm{a}-0}\right)=-1$ $\Rightarrow b+4 a=0$ ...............(I) $\mathrm{m}_{\mathrm{AB}} \times \mathrm{m}_{\mathrm{CF}}=-1 \Rightarrow\left(\frac{(\mathrm{b}-2)}{\mat...
Read More →Let f and g be continuous functions on
Question: Let $f$ and $g$ be continuous functions on $[0$, a such that $f(x)=f(a-x)$ and $g(x)+g(a-x)=4$, then $\int_{0}^{a} f(x) g(x) d x$ is equal to :-$4 \int_{0}^{a} f(x) d x$$2 \int_{0}^{a} f(x) d x$$-3 \int_{0}^{a} f(x) d x$$\int_{0}^{a} f(x) d x$Correct Option: , 2 Solution: $I=\int_{0}^{a} f(x) g(x) d x$ $I=\int_{0}^{a} f(a-x) g(a-x) d x$ $I=\int_{0}^{a} f(x)(4-g(x) d x$ $I=4 \int_{0}^{a} f(x) d x-I$ $\Rightarrow I=2 \int_{0}^{a} f(x) d x$...
Read More →Solve this following
Question: The values of $\lambda$ such that sum of the squares of the roots of the quadratic equation, $x^{2}+(3-\lambda) x+2=\lambda$ has the least value is : 2$\frac{4}{9}$$\frac{15}{8}$1Correct Option: 1 Solution: $\alpha+\beta=\lambda-3$ $\alpha \beta=2-\lambda$ $\alpha^{2}+\beta^{2}=(\alpha+\beta)^{2}-2 \alpha \beta=(\lambda-3)^{2}-2(2-\lambda)$ $=\lambda^{2}+9-6 \lambda-4+2 \lambda$ $=\lambda^{2}-4 \lambda+5$ $=(\lambda-2)^{2}+1$ $\therefore \lambda=2$ Option (1)...
Read More →If the straight line, 2 x-3 y+17=0 is perpendicular to the line passing
Question: If the straight line, $2 x-3 y+17=0$ is perpendicular to the line passing through the points $(7,17)$ and $(15, \beta)$, then $\beta$ equals :-$-5$$-\frac{35}{3}$$\frac{35}{3}$5Correct Option: , 4 Solution: $\frac{17-\beta}{-8} \times \frac{2}{3}=-1$ $\beta=5$...
Read More →If the solve the problem
Question: Let $f(\mathrm{x})=\mathrm{a}^{\mathrm{x}}(\mathrm{a}0)$ be written as $f(\mathrm{x})=f_{1}(\mathrm{x})+f_{2}(\mathrm{x})$, where $f_{1}(\mathrm{x})$ is an even function of $f_{2}(\mathrm{x})$ is an odd function. Then $f_{1}(\mathrm{x}+\mathrm{y})+f_{1}(\mathrm{x}-\mathrm{y})$ equals$2 f_{1}(\mathrm{x}) f_{1}(\mathrm{y})$$2 f_{1}(\mathrm{x}) f_{2}(\mathrm{y})$$2 f_{1}(\mathrm{x}+\mathrm{y}) f_{2}(\mathrm{x}-\mathrm{y})$$2 f_{1}(\mathrm{x}+\mathrm{y}) f_{1}(\mathrm{x}-\mathrm{y})$Correc...
Read More →Solve this following
Question: Let $\mathrm{N}$ be the set of natural numbers and two functions $f$ and $g$ be defined as $f, g: N \rightarrow N$ such that $: f(n)=\left(\begin{array}{ll}\frac{n+1}{2} \text { if } n \text { is odd } \\ \frac{n}{2} \text { if } n \text { is even }\end{array}\right.$ and $g(n)=n-(-1)^{n}$. The fog is : Both one-one and ontoOne-one but not ontoNeither one-one nor ontoonto but not one-oneCorrect Option: , 4 Solution: $f(x)= \begin{cases}\frac{n+1}{2} n \text { is odd } \\ n / 2 n \text ...
Read More →In a random experiment,
Question: In a random experiment, a fair die is rolled until two fours are obtained in succession. The probability that the experiment will end in the fifth throw of the die is equal to:$\frac{150}{6^{5}}$$\frac{175}{6^{5}}$$\frac{200}{6^{5}}$$\frac{225}{6^{5}}$Correct Option: , 2 Solution: $\frac{1}{6^{2}}\left(\frac{5^{3}}{6^{3}}+\frac{2 C_{1} \cdot 5^{2}}{6^{3}}\right)=\frac{175}{6^{5}}$...
Read More →Let C1 and C2 be the centres of the circles
Question: let $C_{1}$ and $C_{2}$ be the centres of the circles $x^{2}+y^{2}-2 x-2 y-2=0$ and $x^{2}+y^{2}-6 x-6 y+14=0$ respectively. If $P$ and $Q$ are the points of intersection of these circles, then the area (in sq. units) of the quadrilateral $\mathrm{PC}_{1} \mathrm{QC}_{2}$ is :8694Correct Option: , 4 Solution: Area $=2 \times \frac{1}{2} \cdot 4=2$...
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