Mathematics Mock Test - 11

If $2f(x)+f\left(\frac{1}{x}\right)=\log x$, for all $x>0$, then $f(e^x)$ is:

Find the principal value of ${\cot }^{-1}\left(-\frac{1}{\sqrt{3}}\right)$.

If $\Delta =\left|\begin{array}{ccc}0& a-b& a-c\\ b-a& 0& b-c\\ c-a& c-b& 0\end{array}\right|$, then $\Delta$ equals:

Let $f(x)=\frac{\log (1-x+x^2)+\log (1+x+x^2)}{\sec x-\cos x}$ for $x\ne 0$. Then the value of $f(0)$ so that $f$ is continuous at $x=0$ is:

The equation of tangent to the curve $x^{2/3}+y^{2/3}=a^{2/3}$ at $(a,0)$ passes through which of the following points?

Let $y=f\left(\frac{x+e^x}{e^x}\right)$ satisfy $f^{\prime }(1)=2$. Then the value of $\frac{dy}{dx}$ at $x=0$ equals:

Match List I with List II:List I (Integrals)List II (Values)A. ${\int }_0^{\pi /2}\frac{\sin 2x}{{\sin }^{4}x+{\cos }^{4}x}dx$ I. $\pi /4$ B. ${\int }_0^{\pi /2}\frac{\sin x}{\sin x+\cos x}dx$ II. $\pi$ C. ${\int }_0^{2\pi }\frac{1}{1+e^{\sin x}}dx$ III. $\pi /12$ D. ${\int }_0^{\pi /2}\frac{1}{9{\sin }^{2}x+4{\cos }^{2}x}dx$ IV. $\pi /2$ Choose the correct answer from the options given below:

If $x=a(\cos \theta +\theta \sin \theta )$ and $y=a(\sin \theta -\theta \cos \theta )$, then at $\theta =\frac{\pi }{4}$, which of the following is true?

If $f(x+y)=f(x)\cdot f(y)$ for all $x,y\in \mathbb{R}$ such that $f(5)=2$ and $f^{\prime }(0)=3$, then the value of $f^{\prime }(5)$ is:

If $A=\{1,2,3\}$ and $R$ is a relation defined on $A$ such that $R=\{(1,1),(1,2),(2,3),(2,2),(3,3),(3,1)\}$. Then the relation $R$ is:

Ashok bought tables and chairs for a marriage hall. The cost of a table is $₹2500$ and a chair is $₹500$. He has a budget of $₹60000$ and space for at most $60$ items. If the profit is $₹75$ per table and $₹25$ per chair, find the maximum profit.

The system of linear equations$x+y+z=4$,$x+2y+3z=7$,$x+4y+\lambda z=\mu$ has a unique solution if:

If the difference between the greatest and the least value of the function $f(x)={\sin }^{2}x+\cos x$ on $[0,\pi ]$ is $\frac{a-4}{4}$, then $a$ equals:

Consider $f(x)=2x^3-9x^2+12x+6$. Then:

Let $m\in \mathbb{Z}$ and consider the relation $R_m$ defined by $a{R}_{m}b$ if and only if $a\equiv b(\mathrm{mod}m)$. Then $R_m$ is:

For the function $f(x)=\sin x-\sqrt{3}\cos x-x$ on the interval $[0,\pi ]$, the stationary points will be:

Which of the following is one of the feasible solutions for a Linear Programming Problem with constraints: $x+2y\ge 2$, $5x+4y\le 20$, $2x-y\le 4$, $x\ge 0$, $y\ge 0$?

Consider a system of linear equations:$4x-2y=3$$6x+\alpha y=5$ For what values of constant $\alpha$ does the system have a unique solution?

Consider a plane passing through the points $A(2,2,1)$, $B(3,4,2)$ and $C(7,0,6)$. Which one of the following points lies on this plane?

Consider the plane passing through the points $A(2,2,1)$, $B(3,4,2)$ and $C(7,0,6)$. What are the direction ratios of the normal to the plane?

Maximize $Z=4x+6y$ subject to $3x+2y\le 12$, $x+y\ge 2$, $x,y\ge 0$.

The number of straight lines that are equally inclined to the three-dimensional coordinate axes is:

Match List I with List II regarding the domain of functions:List I (Function)List II (Domain)A. $f(x)=\frac{1}{\sqrt{x-2}}+\frac{1}{\sqrt{3-x}}$ I. $[2,3]$ B. $f(x)=\sqrt{\frac{x-2}{3-x}}$ II. $(2,3]$ C. $f(x)=\sqrt{\frac{3-x}{x-2}}$ III. $[2,3)$ D. $f(x)=\sqrt{(x-2)(3-x)}$ IV. $(2,3)$ Choose the correct answer:

The solution of the differential equation $\cos (x+y)dy=dx$ is:

Let $\int \frac{\sqrt{x}}{\sqrt{1-x^3}}dx=\frac{2}{3}(g\circ f)(x)+c$. Then:

Let $L$ be a line that is perpendicular to the plane $x-3y+2z=k$ and passes through the point $(1,2,-2)$. The point on the line $L$ that is nearest to the $y$ -axis is:

Let $P$ be the plane passing through the intersection of planes $P_1:x+y+z-6=0$ and $P_2:2x+3y+4z+5=0$. If the plane $P$ contains the origin, then the equation of the plane $P$ is:

The reflection of the origin in the plane $P_2:2x+3y+4z+5=0$ is:

The unit vector perpendicular to line $L$ (DRs: $1,-3,2$) and the normal to the plane $P_1:x+y+z-6=0$ is:

If the probability of $A$ to fail in an examination is $0.2$ and that for $B$ is $0.3$, then the probability that either $A$ or $B$ fails (meaning exactly one of them fails) is:

A line joining the points $(1,2,0)$ and $(4,13,5)$ is perpendicular to a plane. Then the coefficients of $x,y$ and $z$ in the equation of the plane are respectively:

The area of the region bounded by $Y$ -axis, $y=\cos x$ and $y=\sin x$ for $0\le x\le \frac{\pi }{4}$ is:

For an objective function $Z=x+y$ with constraints $2x+2y\le 5$ and $4x-3y\ge 3$ ($x,y\ge 0$), at what point(s) will $Z$ achieve its maximum value?

The present value of a perpetuity is $₹50,000$ when the payment of $₹1,000$ is made at the end of each period. Find the present value of the perpetuity if the payments are made at the beginning of each period.

The point that lies in the region bounded by the lines $7x+y\ge 40$ and $2x+3y\le 25$ is:

If the coordinates of $A$ and $B$ are $(1,2,3)$ and $(7,8,7)$, then the projections of the line segment $AB$ on the coordinate axes are:

A coin is tossed $3$ times. If $X$ is a random variable representing the number of tails, then the mathematical expectation of $X$ is:

Nisha needs $₹2,00,000$ (after 50% subsidy) in 4 years for her daughter's education. If the interest rate is 8% p.a., how much does she need to save annually? (Assume $\frac{1.08^4-1}{0.08}=4.5$)

An objective function $Z=ax+by$ is maximum at points $(9,3)$ and $(5,7)$. If $a,b\ge 0$ and $ab=16$, then the maximum value of the function is:

A company has a variable cost $V(x)=3x^2+3500$ and a fixed cost of $₹2000$. The marginal cost of producing $50$ units is:

Maximize $Z=30x-18y$ subject to $3x+4y\le 60$ and $5x-3y\le 20$ ($x,y\ge 0$). In the feasible region, the maximum value occurs at:

Find the area between the curves $y=16x^2$ and $y=9$.

A study investigates if the average sunlight in February was greater than the annual average of $5$ units/m${}^2$. What is the null hypothesis ($H_0$) for this problem?

If $A=\left[\begin{array}{cc}6& -3\\ 4& -2\end{array}\right]$ and $B=\left[\begin{array}{cc}2& y\\ x& 7\end{array}\right]$, such that $AB=O$ (Zero matrix), find the values of $x$ and $y$.

To maximize profit, what condition must be met regarding Marginal Cost (MC) and Marginal Revenue (MR)?

In a five-year plan with a total allocation of $₹36,000$ crore, the angle for Employment is $120^{\circ }$. Which head is allocated maximum funds if the angles are Agriculture ($90^{\circ }$), Industry ($45^{\circ }$), Education ($30^{\circ }$), and Miscellaneous ($75^{\circ }$)?

In the same five-year plan (Total $₹36,000$ crore), how much money is allocated to Education if its angle is $30^{\circ }$?

In the same plan, how much money is allocated to both Agriculture (Angle: $90^{\circ }$) and Employment (Angle: $120^{\circ }$)?

How much excess money is allocated to Miscellaneous (Angle: $75^{\circ }$) over Education (Angle: $30^{\circ }$)?

Three vertices of a parallelogram $ABCD$ are $A(1,2,3)$, $B(-1,-2,-1)$ and $C(2,3,2)$. The coordinates of the fourth vertex $D$ are: