The number of 4-digit numbers
Question: The number of 4-digit numbers which are neither multiple of 7 nor multiple of 3 is_______. Solution: $\mathrm{A}=4$ - digit numbers divisible by 3 $A=1002,1005, \ldots, 9999$ $9999=1002+(n-1) 3$ $\Rightarrow(\mathrm{n}-1) 3=8997 \Rightarrow \mathrm{n}=3000$ $\mathrm{B}=4$-digit numbers divisible by 7 $\mathrm{B}=1001,1008, \ldots, 9996$ $\Rightarrow 9996=1001+(\mathrm{n}-1) 7$ $\Rightarrow \mathrm{n}=1286$ $A \cap B=1008,1029, \ldots, 9996$ $9996=1008+(n-1) 21$ $\Rightarrow \mathrm{n}=...
Read More →Let (x) denote the
Question: Let $[\mathrm{x}]$ denote the greatest integer $\leq \mathrm{x}$, where $x \in \mathbf{R}$. If the domain of the real valued function $f(x)=\sqrt{\frac{\mid x] \mid-2}{|[x]|-3}}$ is $(-\infty, a) \cup[b, c) \cup[4, \infty), abc$, then the value of $a+b+c$ is:81-2-3Correct Option: , 3 Solution: For domain, $\frac{|[x]|-2}{|[x]|-3} \geq 0$ Case I : When $|[x]|-2 \geq 0$ and $|[x]|-30$ $\therefore x \in(-\infty,-3) \cup[4, \infty)$.....(1) Case II : When $|[\mathrm{x}]|-2 \leq 0$ and $|[x...
Read More →Suppose the line x-2/α = y-2/-5 = z+2/2 lies on
Question: Suppose the line $\frac{x-2}{\alpha}=\frac{y-2}{-5}=\frac{z+2}{2}$ lies on the plane $x+3 y-2 z+\beta=0$. Then $(\alpha+\beta)$ is equal to_____. Solution: Point $(2,2,-2)$ also lies on given plane So $2+3 \times 2-2(-2)+\beta=0$ $\Rightarrow 2+6+4+\beta=0 \Rightarrow \beta=-12$ Also $\alpha \times 1-5 \times 3+2 \times-2=0$ $\Rightarrow \alpha-15-4=0 \Rightarrow \alpha=19$ $\therefore \alpha+\beta=19-12=7$...
Read More →Solve this
Question: Let $a, b \in \mathbf{R}, b \neq 0$, Define a function $f(x)= \begin{cases}\operatorname{asin} \frac{\pi}{2}(x-1), \text { for } x \leq 0 \\ \frac{\tan 2 x-\sin 2 x}{b x^{3}}, \text { for } x0\end{cases}$ If $\mathrm{f}$ is continuous at $\mathrm{x}=0$, then $10-\mathrm{ab}$ is equal to_____________ Solution: $f(x)= \begin{cases}a \sin \frac{\pi}{2}(x-1), x \leq 0 \\ \frac{\tan 2 x-\sin 2 x}{b x^{3}}, x0\end{cases}$ For continuity at ' 0 ' $\lim _{x \rightarrow 0^{+}} f(x)=f(0)$ $\Righ...
Read More →If in a triangle
Question: If in a triangle $\mathrm{ABC}, \mathrm{AB}=5$ units, $\angle \mathrm{B}=\cos ^{-1}\left(\frac{3}{5}\right)$ and radius of circumcircle of $\triangle \mathrm{ABC}$ is 5 units, then the area (in sq. units) of $\triangle \mathrm{ABC}$ is :$10+6 \sqrt{2}$$8+2 \sqrt{2}$$6+8 \sqrt{3}$$4+2 \sqrt{3}$Correct Option: , 3 Solution: As, $\operatorname{cosB}=\frac{3}{5} \Rightarrow \mathrm{B}=53^{\circ}$ As, $R=5 \Rightarrow \frac{c}{\sin c}=2 R$ $\Rightarrow \frac{5}{10}=\sin \mathrm{c} \Rightarr...
Read More →If the coefficient of a^7 b^8 in the expansion
Question: If the coefficient of $a^{7} b^{8}$ in the expansion of $(a+2 b+4 a b)^{10}$ is $K \cdot 2^{16}$, then $K$ is equal to_________. Solution: $\frac{10 !}{\alpha ! \beta ! \gamma !} \mathrm{a}^{\alpha}(2 \mathrm{~b})^{\beta} \cdot(4 \mathrm{ab})^{\gamma}$ $\frac{10 !}{\alpha ! \beta ! \gamma !} \mathrm{a}^{\alpha+\gamma} \cdot \mathrm{b}^{\beta+\gamma} \cdot 2^{\beta} \cdot 4^{\gamma}$ $\alpha+\beta+\gamma=10$ ............(1) $\alpha+\gamma=7$ .............(2) $\beta+\gamma=8$ ..............
Read More →The mean and variance of 7 observations are 8 and 16 respectively.
Question: The mean and variance of 7 observations are 8 and 16 respectively. If two observations are 6 and 8 , then the variance of the remaining 5 observations is:$\frac{92}{5}$$\frac{134}{5}$$\frac{536}{25}$$\frac{112}{5}$Correct Option: , 3 Solution: Let $8,16, x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$ be the observations. Now $\frac{x_{1}+x_{2}+\ldots+x_{5}+14}{7}=8$ $\Rightarrow \sum_{i=1}^{5} x_{i}=42$ ..........(1) Also $\frac{x_{1}^{2}+x_{2}^{2}+\ldots x_{5}^{2}+8^{2}+6^{2}}{7}-64=16$ $\Rightar...
Read More →If z and
Question: If $z$ and $\omega$ are two complex numbers such that $|z \omega|=1$ and $\arg (z)-\arg (\omega)=\frac{3 \pi}{2}$, then $\arg \left(\frac{1-2 \bar{z} \omega}{1+3 \bar{z} \omega}\right)$ is : (Here $\arg (z)$ denotes the principal argument of complex number $z$ )$\frac{\pi}{4}$$-\frac{3 \pi}{4}$$-\frac{\pi}{4}$$\frac{3 \pi}{4}$Correct Option: , 2 Solution: As $|z \omega|=1$ $\Rightarrow \operatorname{If}|z|=r$, then $|\omega|=\frac{1}{r}$ Let $\arg (\mathrm{z})=\theta$ $\therefore \arg ...
Read More →The number of three-digit even numbers, formed by the digits 0,1,3,4,6,7 if the repetition of digits is not allowed, is
Question: The number of three-digit even numbers, formed by the digits $0,1,3,4,6,7$ if the repetition of digits is not allowed, is___________ Solution: (i) When ' 0 ' is at unit place Number of numbers $=20$ (ii) When 4 or 6 are at unit place Number of numbers $=32$ So number of numbers $=52$...
Read More →If [x] is the greatest integer
Question: If $[x]$ is the greatest integer $\leq x$, then $\pi^{2} \int_{0}^{2}\left(\sin \frac{\pi x}{2}\right)(x-[x])^{[x]} d x$ is equal to:$2(\pi-1)$$4(\pi-1)$$4(\pi+1)$$2(\pi+1)$Correct Option: , 2 Solution: $\pi^{2}\left[\int_{0}^{1} \sin \frac{\pi x}{2} d x+\int_{1}^{2} \sin \frac{\pi x}{2}(x-1) d x\right]$ $=\pi^{2}\left[-\frac{2}{\pi}\left(\cos \frac{\pi x}{2}\right)+\left((x-1)\left(-\frac{2}{\pi} \cos \frac{\pi x}{2}\right)\right)_{1}^{2}-\int_{1}^{2}-\frac{2}{\pi} \cos \frac{\pi x}{2...
Read More →If y=y(x) is an implicit function of x such that
Question: If $y=y(x)$ is an implicit function of $x$ such that $\log _{c}(x+y)=4 x y$, then $\frac{d^{2} y}{d x^{2}}$ at $x=0$ is equal to________________ Solution: $\ln (x+y)=4 x y$ $($ At $x=0, y=1)$ $x+y=e^{4 x y}$ $\Rightarrow 1+\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{e}^{4 \mathrm{xy}}\left(4 \mathrm{x} \frac{\mathrm{dy}}{\mathrm{dx}}+4 \mathrm{y}\right)$ At $x=0$ $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\mathrm{e}^{4 \mathrm{xy}}\left(4 \mathrm{x} \frac{\mathrm{dy}}{\mathrm{dx}}+...
Read More →Let f be any continuous function on [0,2]
Question: Let $f$ be any continuous function on $[0,2]$ and twice differentiable on $(0,2)$. If $f(0)=0, f(1)=1$ and $\mathrm{f}(2)=2$, then$\mathrm{f}^{\prime \prime}(\mathrm{x})=0$ for all $\mathrm{x} \in(0,2)$$f^{\prime \prime}(x)=0$ for some $x \in(0,2)$$f^{\prime}(x)=0$ for some $x \in[0,2]$$\mathrm{f}^{\prime \prime}(\mathrm{x})0$ for all $\mathrm{x} \in(0,2)$Correct Option: , 2 Solution: $f(0)=0 \quad f(1)=1$ and $f(2)=2$ Let $\mathrm{h}(\mathrm{x})=f(\mathrm{x})-\mathrm{x}$ has three roo...
Read More →Solve the Following Questions
Question: Let $A=\left[\begin{array}{ll}2 3 \\ a 0\end{array}\right], a \in \mathbf{R}$ be written as $P+Q$ where $P$ is a symmetric matrix and $Q$ is skew symmetric matrix. If $\operatorname{det}(Q)=9$, then the modulus of the sum of all possible values of determinant of $P$ is equal to:36244518Correct Option: 1 Solution: $A=\left[\begin{array}{ll}2 3 \\ a 0\end{array}\right], a \in R$ and $P=\frac{A+A^{T}}{2}=\left[\begin{array}{cc}2 \frac{3+a}{2} \\ \frac{a+3}{2} 0\end{array}\right]$ and $Q=\...
Read More →The number of solutions of the equation
Question: The number of solutions of the equation $32^{\tan ^{2} x}+32^{\sec ^{2} x}=81,0 \leq x \leq \frac{\pi}{4}$ is :3102Correct Option: , 2 Solution: $(32)^{\tan ^{2} x}+(32)^{\sec ^{2} x}=81$ $\Rightarrow(32)^{\tan ^{2} x}+(32)^{1+\tan ^{2} x}=81$ $\Rightarrow(32)^{\tan ^{2} x}=\frac{81}{33}$ In interval $\left[0, \frac{\pi}{4}\right]$ only one solution...
Read More →The locus of a point, which moves such that the sum of squares of its distances from the points (0,0),(1,0),(0,1)(1,1) is 18 units,
Question: The locus of a point, which moves such that the sum of squares of its distances from the points $(0,0),(1,0),(0,1)(1,1)$ is 18 units, is a circle of diameter $\mathrm{d}$. Then $\mathrm{d}^{2}$ is equal to Solution: Let $\mathrm{P}(\mathrm{x}, \mathrm{y})$ $x^{2}+y^{2}+x^{2}+(y-1)^{2}+(x-1)^{2}+y^{2}+(x-1)^{2}+(y-1)^{2}$ $\Rightarrow 4\left(x^{2}+y^{2}\right)-4 y-4 x=14$ $\Rightarrow x^{2}+y^{2}-x-y-\frac{7}{2}=0$ $\mathrm{d}=2 \sqrt{\frac{1}{4}+\frac{1}{4}+\frac{7}{2}}$ $\Rightarrow \...
Read More →Solve the Following Questions
Question: If $\alpha$ and $\beta$ are the distinct roots of the equation $x^{2}+(3)^{1 / 4} x+3^{1 / 2}=0$, then the value of $\alpha^{96}\left(\alpha^{12}-1\right)+\beta^{96}\left(\beta^{12}-1\right)$ is equal to :$56 \times 3^{25}$$56 \times 3^{24}$$52 \times 3^{24}$$28 \times 3^{25}$Correct Option: , 3 Solution: As, $\left(\alpha^{2}+\sqrt{3}\right)=-(3)^{1 / 4} \cdot \alpha$ $\Rightarrow\left(\alpha^{4}+2 \sqrt{3} \alpha^{2}+3\right)=\sqrt{3} \alpha^{2}$ (On squaring) $\therefore\left(\alpha...
Read More →Let A be the set of all points
Question: Let $A$ be the set of all points $(\alpha, \beta)$ such that the area of triangle formed by the points $(5,6),(3,2)$ and $(\alpha, \beta)$ is 12 square units. Then the least possible length of a line segment joining the origin to a point in A, is:$\frac{4}{\sqrt{5}}$$\frac{16}{\sqrt{5}}$$\frac{8}{\sqrt{5}}$$\frac{12}{\sqrt{5}}$Correct Option: , 3 Solution: $4 \alpha-2 \beta=\pm 24+8$ $\Rightarrow 4 \alpha-2 \beta=+24+8 \Rightarrow 2 \alpha-\beta=16$ $2 x-y-16=0$ $\ldots(1)$ $\Rightarro...
Read More →The area of the region
Question: The area of the region $S=\left\{(x, y): 3 x^{2} \leq 4 y \leq 6 x+24\right\}$ is Solution: For A \ B $3 x^{2}=6 x+24 \Rightarrow x^{2}-2 x-8=0$ $\Rightarrow x=-2,4$ Area $=\int_{-2}^{4}\left(\frac{3}{2} x+6-\frac{3}{4} x^{2}\right) d x$ $=\left[\frac{3 x^{2}}{4}+6 x-\frac{x^{3}}{4}\right]_{-2}^{4}=27$...
Read More →The value of the integral
Question: The value of the integral $\int_{-1}^{1} \log _{e}(\sqrt{1-x}+\sqrt{1+x}) d x$ is equal to :$\frac{1}{2} \log _{\mathrm{e}} 2+\frac{\pi}{4}-\frac{3}{2}$$2 \log _{e} 2+\frac{\pi}{4}-1$$\log _{e} 2+\frac{\pi}{2}-1$$2 \log _{e} 2+\frac{\pi}{2}-\frac{1}{2}$Correct Option: , 3 Solution: Let $I=2 \int_{0}^{1} \underbrace{\ln (\sqrt{1-x}+\sqrt{1+x})}_{\text {(I) }} 1 \mathrm{dx}$ (I.B.P.) $\therefore \mathrm{I}=2\left[(\mathrm{x} \cdot \ln (\sqrt{1-\mathrm{x}}+\sqrt{1-\mathrm{x}}))_{0}^{1}\ri...
Read More →A wire of length 36m is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of
Question: A wire of length $36 \mathrm{~m}$ is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is $\mathrm{k}$ (meter), then $\left(\frac{4}{\pi}+1\right) \mathrm{k}$ is equal to____________ Solution: Let $x+y=36$ $x$ is perimeter of square and $y$ is perimeter of circle side of square $=\mathrm{x} / 4$ side of square $=x / 4$ radius of circle $=\frac...
Read More →Let a1, a2, a3,.......... be a A. P. if
Question: Let $\mathrm{a}_{1}, \mathrm{a}_{2}, \mathrm{a}_{3}, \ldots$ be an A.P. If $\frac{\mathrm{a}_{1}+\mathrm{a}_{2}+\ldots+\mathrm{a}_{10}}{\mathrm{a}_{1}+\mathrm{a}_{2}+\ldots+\mathrm{a}_{\mathrm{p}}}=\frac{100}{\mathrm{p}^{2}}, \mathrm{p} \neq 10$, then $\frac{\mathrm{a}_{11}}{\mathrm{a}_{10}}$ is equal to :$\frac{19}{21}$$\frac{100}{121}$$\frac{21}{19}$$\frac{121}{100}$Correct Option: 3 Solution: $\frac{\frac{10}{2}\left(2 \mathrm{a}_{1}+9 \mathrm{~d}\right)}{\frac{\mathrm{p}}{2}\left(2...
Read More →The mean of 6 distinct observations
Question: The mean of 6 distinct observations is $6.5$ and their variance is $10.25$. If 4 out of 6 observations are 2 , 4,5 and 7 , then the remaining two observations are:10, 113, 188, 131, 20Correct Option: 1 Solution: Let other two numbers be $\mathrm{a},(21-\mathrm{a})$ Now, $10.25=\frac{\left(4+16+25+49+a^{2}+(21-a)^{2}\right)}{6}-(6.5)^{2}$ (Using formula for variance) $\Rightarrow 6(10.25)+6(6.5)^{2}=94+a^{2}+(21-a)^{2}$ $\Rightarrow a^{2}+(21-a)^{2}=221$ $\therefore a=10$ and $(21-a)=21...
Read More →Let a be a positive real number
Question: Let a be a positive real number such that $\int_{0}^{\mathrm{a}} \mathrm{e}^{\mathrm{x}-[\mathrm{x}]} \mathrm{dx}=10 \mathrm{e}-9$ where $[\mathrm{x}]$ is the greatest integer less than or equal to $x$. Then a is equal to :$10-\log _{e}(1+e)$$10+\log _{e} 2$$10+\log _{\mathrm{e}} 3$$10+\log _{\mathrm{e}}(1+\mathrm{e})$Correct Option: , 2 Solution: $a0$ Let $n \leq an+1, n \in W$ Here $[a]=n$ Now, $\int_{0}^{a} e^{x-[x]} d x=10 e-9$ $\Rightarrow \int_{0}^{n} e^{\{x\}} d x+\int_{n}^{a} e...
Read More →If z is a complex number such that
Question: If $z$ is a complex number such that $\frac{z-i}{z-1}$ is purely imaginary, then the minimum value of $|z-(3+3 i)|$ is:$2 \sqrt{2}-1$$3 \sqrt{2}$$6 \sqrt{2}$$2 \sqrt{2}$Correct Option: , 4 Solution: $\frac{z-i}{z-1}$ is purely Imaginary number Let $z=x+i y$ $\therefore \frac{x+i(y-1)}{(x-1)+i(y)} \times \frac{(x-1)-i y}{(x-1)-i y}$ $\Rightarrow \frac{x(x-1)+y(y-1)+i(-y-x+1)}{(x-1)^{2}+y^{2}}$is purely Imaginary number $\Rightarrow x(x-1)+y(y-1)=0$ $\Rightarrow\left(x-\frac{1}{2}\right)...
Read More →Solve this
Question: If ${ }^{1} \mathrm{P}_{1}+2 \cdot{ }^{2} \mathrm{P}_{2}+3 \cdot{ }^{3} \mathrm{P}_{3}+\ldots+15 \cdot{ }^{15} \mathrm{P}_{15}={ }^{9} \mathrm{P}-\mathrm{s}, 0 \leq \mathrm{s} \leq 1$ then ${ }^{4+5} C_{x-s}$ is equal to_________ Solution: ${ }^{1} P_{1}+2 \cdot{ }^{2} P_{2}+3 \cdot{ }^{3} P_{3}+\ldots+15 \cdot{ }^{15} P_{15}$ $=1 !+2.2 !+3.3 !+\ldots .15 \times 15 !$ $=\sum_{r=1}^{15}(r+1-1) r !$ $=\sum_{r=1}^{15}(r+1) !-(r) !$ $=16 !-1$ $={ }^{16} P_{16}-1$ $\Rightarrow \mathrm{q}=\m...
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