Prove the following
Question: If $g(x)=x^{2}+x-1$ and $(g \circ f)(x)=4 x^{2}-10 x+5$, then $f\left(\frac{5}{4}\right)$ is equal to$\frac{3}{2}$$-\frac{1}{2}$$-\frac{3}{2}$$\frac{1}{2}$Correct Option: , 2 Solution: $g(x)=x^{2}+x-1$ $g(f(x))=4 x^{2}-10 x+5$ $=(2 x-2)^{2}+(2-2 x)-1$ $=(2-2 x)^{2}+(2-2 x)-1$ $\Rightarrow f(x)=2-2 x$ $f\left(\frac{5}{4}\right)=\frac{-1}{2}$...
Read More →If the sum and product of the first three term in an A.P
Question: If the sum and product of the first three term in an A.P. are 33 and 1155 , respectively, then a value of its $11^{\text {th }}$ term is :-$-25$25$-36$$-35$Correct Option: 1 Solution:...
Read More →Solve this following
Question: The common tangent to the circles $x^{2}+y^{2}=4$ and $x^{2}+y^{2}+6 x+8 y-24=0$ also passes through the point :-$(-4,6)$$(6,-2)$$(-6,4)$$(4,-2)$Correct Option: , 2 Solution:...
Read More →If the system of equations
Question: If the system of equations $2 x+3 y-z=0, x+$ $\mathrm{ky}-2 \mathrm{z}=0$ and $2 \mathrm{x}-\mathrm{y}+\mathrm{z}=0$ has a non-trival solution $(\mathrm{x}, \mathrm{y}, \mathrm{z})$, then $\frac{\mathrm{x}}{\mathrm{y}}+\frac{\mathrm{y}}{\mathrm{z}}+\frac{\mathrm{z}}{\mathrm{x}}+\mathrm{k}$ is equal to:- $\frac{3}{4}$$-4$$\frac{1}{2}$$-\frac{1}{4}$Correct Option: , 3 Solution:...
Read More →Solve this following
Question: If $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ is a differentiable function and $f(2)=6$, then $\lim _{x \rightarrow 2} \int_{6}^{f(x)} \frac{2 t d t}{(x-2)}$ is :-0$2 f^{\prime}(2)$$12 f^{\prime}(2)$$24 f^{\prime}(2)$Correct Option: , 3 Solution:...
Read More →Some identical balls are arranged in rows to form an equilateral triangle.
Question: Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If 99 more identical balls are addded to the total number of balls used in forming the equilaterial triangle, then all these balls can be arranged in a square whose each side contains exactly 2 balls less than the number of balls each side of the triangle contains. Then the number of balls used to form the equilateral triangle is...
Read More →If the tangent to the parabola
Question: If the tangent to the parabola $\mathrm{y}^{2}=\mathrm{x}$ at a point $(\alpha, \beta),(\beta0)$ is also a tangent to the ellipse, $x^{2}+2 y^{2}=1$, then $\alpha$ is equal to :$2 \sqrt{2}+1$$\sqrt{2}-1$$\sqrt{2}+1$$2 \sqrt{2}-1$Correct Option: , 3 Solution:...
Read More →An ellipse, with foci at (0,2) and (0,-2)
Question: An ellipse, with foci at $(0,2)$ and $(0,-2)$ and minor axis of length 4 , passes through which of the following points ?$(1,2 \sqrt{2})$$(2, \sqrt{2})$$(2,2 \sqrt{2})$$(\sqrt{2}, 2)$Correct Option: , 4 Solution:...
Read More →If the area (in sq. units) bounded by the parabola
Question: If the area (in sq. units) bounded by the parabola $y^{2}=4 \lambda x$ and the line $y=\lambda x, \lambda0$, is $\frac{1}{9}$, then $\lambda$ is equal to:2448$4 \sqrt{3}$$2 \sqrt{6}$Correct Option: 1 Solution:...
Read More →A plane which bisects the angle between the two
Question: A plane which bisects the angle between the two given planes $2 \mathrm{x}-\mathrm{y}+2 \mathrm{z}-4=0$ and $x+2 y+2 z-2=0$, passes through the point:$(2,4,1)$$(2,-4,1)$$(1,4,-1)$$(1,-4,1)$Correct Option: , 2 Solution:...
Read More →The Boolean expression
Question: The Boolean expression $\sim(\mathrm{p} \Rightarrow(\sim \mathrm{q}))$ is equivalent to :$(\sim p) \Rightarrow q$$\mathrm{p} \vee \mathrm{q}$$\mathrm{q} \Rightarrow \sim \mathrm{p}$$p^{\wedge} \mathrm{q}$Correct Option: , 4 Solution: $\sim(\mathrm{p} \rightarrow(\sim \mathrm{q}))=\sim(\sim \mathrm{p} \vee \sim \mathrm{q})$ $=\mathrm{p} \wedge \mathrm{q}$...
Read More →Solve this following
Question: If the function $f: \mathrm{R}-\{1,-1\} \rightarrow \mathrm{A}$ defined by $f(\mathrm{x})=\frac{\mathrm{x}^{2}}{1-\mathrm{x}^{2}}$, is surjective, then $\mathrm{A}$ is equal to$\mathrm{R}-[-1,0)$$\mathrm{R}-(-1,0)$$R-\{-1\}$$[0, \infty)$Correct Option: 1 Solution:...
Read More →The angle of elevation of the top of vertical tower
Question: The angle of elevation of the top of vertical tower standing on a horizontal plane is observed to be $45^{\circ}$ from a point $\mathrm{A}$ on the plane. Let $\mathrm{B}$ be the point 30 $m$ vertically above the point $A$. If the angle of elevation of the top of the tower from $B$ be $30^{\circ}$, then the distance (in m) of the foot of the tower from the point $A$ is:$15(3-\sqrt{3})$$15(3+\sqrt{3})$$15(1+\sqrt{3})$$15(5-\sqrt{3})$Correct Option: , 2 Solution:...
Read More →If one end of a focal chord of the parabola,
Question: If one end of a focal chord of the parabola, $y^{2}=16 x$ is at $(1,4)$, then the length of this focal chord is25242022Correct Option: 1 Solution:...
Read More →Prove the following identities.
Question: Let $f(x)=5-|x-2|$ and $g(x)=|x+1|, x \in R$. If $f(x)$ attains maximum value at $\alpha$ and $g(x)$ attains minimum value at $\beta$, then $\lim _{x \rightarrow-\alpha \beta} \frac{(x-1)\left(x^{2}-5 x+6\right)}{x^{2}-6 x+8}$ is equal to :$1 / 2$$-3 / 2$$3 / 2$$-1 / 2$Correct Option: 1 Solution:...
Read More →Solve this following
Question: If $\left[\begin{array}{ll}1 1 \\ 0 1\end{array}\right] \cdot\left[\begin{array}{ll}1 2 \\ 0 1\end{array}\right] \cdot\left[\begin{array}{ll}1 3 \\ 0 1\end{array}\right] \ldots . .\left[\begin{array}{cc}1 \mathrm{n}-1 \\ 0 1\end{array}\right]=\left[\begin{array}{cc}1 78 \\ 0 1\end{array}\right]$, then the inverse of $\left[\begin{array}{ll}1 \mathrm{n} \\ 0 1\end{array}\right]$ is$\left[\begin{array}{cc}1 -13 \\ 0 1\end{array}\right]$$\left[\begin{array}{cc}1 0 \\ 12 1\end{array}\right...
Read More →A person throws two fair dice.
Question: A person throws two fair dice. He wins Rs. 15 for throwing a doublet (same numbers on the two dice), wins Rs. 12 when the throw results in the sum of 9 , and loses Rs. 6 for any other outcome on the throw. Then the expected gain/loss (in Rs.) of the person is :2 gain$\frac{1}{2}$ loss$\frac{1}{4}$ loss$\frac{1}{2}$ gainCorrect Option: , 2 Solution: win Rs.15 $\rightarrow$ number of cases $=6$ win Rs.12 $\rightarrow$ number of cases $=4$ loss Rs. $6 \rightarrow$ number of cases $=26$ $\...
Read More →Solve this following
Question: If the function $f$ defined on $\left(\frac{\pi}{6}, \frac{\pi}{3}\right)$ by $f(x)=\left\{\begin{array}{cc}\frac{\sqrt{2} \cos x-1}{\cot x-1}, x \neq \frac{\pi}{4} \\ \mathrm{k}, \mathrm{x}=\frac{\pi}{4}\end{array}\right.$ then $\mathrm{k}$ is equal to $\frac{1}{2}$1$\frac{1}{\sqrt{2}}$2Correct Option: 1, Solution:...
Read More →The general solution of the differential equation
Question: The general solution of the differential equation $\left(y^{2}-x^{3}\right) d x-x y d y=0(x \neq 0)$ is : (where $\mathrm{c}$ is a constant of integration)$y^{2}+2 x^{3}+c x^{2}=0$$\mathrm{y}^{2}+2 \mathrm{x}^{2}+\mathrm{cx} 3=0$$y^{2}-2 x^{3}+c x^{2}=0$$\mathrm{y}^{2}-2 \mathrm{x}^{2}+c \mathrm{x}^{3}=0$Correct Option: 1 Solution:...
Read More →Solve the following systems of equations:
Question: Let $z \in C$ with $\operatorname{Im}(z)=10$ and it satisfies $\frac{2 z-n}{2 z+n}=2 i-1$ for some natural number $n$. Then:$\mathrm{n}=20$ and $\operatorname{Re}(\mathrm{z})=-10$$\mathrm{n}=20$ and $\operatorname{Re}(\mathrm{z})=10$$\mathrm{n}=40$ and $\operatorname{Re}(\mathrm{z})=-10$$\mathrm{n}=40$ and $\operatorname{Re}(\mathrm{z})=10$Correct Option: , 3 Solution:...
Read More →The equation of a common tangent to the curves,
Question: The equation of a common tangent to the curves, $y^{2}=16 x$ and $x y=-4$ is :$x+y+4=0$$x-2 y+16=0$$2 x-y+2=0$$x-y+4=0$Correct Option: , 4 Solution:...
Read More →If a tangent to the circle
Question: If a tangent to the circle $x^{2}+y^{2}=1$ intersects the coordinate axes at distinct points $\mathrm{P}$ and $\mathrm{Q}$, then the locus of the mid-point of $P Q$ is$x^{2}+y^{2}-2 x y=0$$x^{2}+y^{2}-16 x^{2} y^{2}=0$$x^{2}+y^{2}-4 x^{2} y^{2}=0$$x^{2}+y^{2}-2 x^{2} y^{2}=0$Correct Option: , 3 Solution:...
Read More →A group of students comprises of 5 boys and
Question: A group of students comprises of 5 boys and $\mathrm{n}$ girls. If the number of ways, in which a team of 3 students can randomly be selected from this group such that there is at least one boy and at least one girl in each team, is 1750 , then $n$ is equal to:25282724Correct Option: 1 Solution:...
Read More →Solve this following
Question: The value of $\cos ^{2} 10^{\circ}-\cos 10^{\circ} \cos 50^{\circ}+\cos ^{2} 50^{\circ}$ is$\frac{3}{2}\left(1+\cos 20^{\circ}\right)$$\frac{3}{4}$$\frac{3}{4}+\cos 20^{\circ}$$\frac{3}{2}$Correct Option: , 2 Solution:...
Read More →The tangents to the curve
Question: The tangents to the curve $y=(x-2)^{2}-1$ at its points of intersection with the line $x-y=3$, intersect at the point :$\left(-\frac{5}{2},-1\right)$$\left(-\frac{5}{2}, 1\right)$$\left(\frac{5}{2},-1\right)$$\left(\frac{5}{2}, 1\right)$Correct Option: , 3 Solution: Put $\mathrm{x}-2=\mathrm{X} \ \mathrm{y}+1=\mathrm{Y}$ $\therefore$ given curve becomes $\mathrm{Y}=\mathrm{X}^{2}$ and $\mathrm{Y}=\mathrm{X}$...
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