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Let $f$ be a non-constant continous function for all $x\ge 0$. Let $f$ satisfy the relation $f\left(x\right)f(a-x)=1$ for some $a\in R^{+}$. Then $I={\int }_0^a\frac{dx}{1+f\left(x\right)}$ is equal to

Find the
transpose of each of the following matrices:

(i) (ii) (iii)

Show
that the equation of the line passing through the origin and making
an angle θwith the line.

Prove the following by using the principle of mathematical induction for all n ∈ N:

Two distinct polynomial f(x) and g(x) are defined as follows:

$f\left(x\right)=x^2+ax+2;g\left(x\right)=x^2+2x+a$

If the equation f (x) = 0 and g(x) = 0 have a common root, then the sum of the roots of the equation f (x) + g(x) = 0 is

The length of the latus rectum of the curve traced by the point $(\frac{3}{2}[t+\frac{1}{t}],\frac{\sqrt{7}}{2}[t-\frac{1}{t}])$, where $t(\ne 0)$ is a parameter, is equal to

Verify Rolle’s Theorem for the
function

Sum of the binomial coefficients in the expansion of ${\left(a+b+c+d+e\right)}^n$ is

If a point $P=(\sqrt{3},0)$ on the line $y-\sqrt{3}x+3=0$ cuts the curve $y^2=x+2$ at $A$ and $B,$ then $|PA\cdot PB|=$

Seven athletes are participating in a race. If there are three prizes $Gold,Silver$ and $Bronze$, then the number of ways in which the prizes can be distributed to the athletes is

The value of
${\cos }^2{10}^{\circ }-\cos {10}^{\circ }\cos {50}^{\circ }+{\cos }^2{50}^{\circ }$ is:

Q18. Deepika spends 20 % of her income on rent, 40% of the remaining on groceries and 15 % of the remaining on rest of her needs. If she saves Rs 3,468 every month, what is her monthly salary?

दीपिका अपनी आय का 20 % किराए पर, शेष आय का 40 % किराने पर और शेष आय 15 % अपनी आवश्यकताओं की पूर्ति पर खर्च करती है। यदि वह हर महीने 3,468 रू. बचा लेती है, तो उसकी मासिक आय क्या है?

The image of the point $(1,3,4)$ in the plane $2x-y+z=-3$ is:

If the straight line y = mx + c touches the circle $x^2+y^2-4y=0,$ then the value of c will be

The value of $\frac{\cos {105}^{\circ }+\sin {105}^{\circ }}{\cos {105}^{\circ }-\sin {105}^{\circ }}$ is

The complex number(s) $z$ satisfying $Re\left(z^2\right)=0$ and $|z|=\sqrt{3}$ is (are)

If cos 2 B = $\frac{cos(A+C)}{cos(A-C)}$, then tan A, tan B, tan C are in

The solution of $\frac{dy}{dx}+\frac{y\cos x+\sin y+y}{\sin x+x\cos y+x}=0$ is

(where $C$ is an arbitrary constant)

If

Prove that (X ∩ Y) ∩ Z = X ∩ (Y ∩ Z)

Match $\text{List I}$ with $\text{List II}$ and select the correct answer using the code given below the lists :



$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{In a}△ABC,\text{if}{a}^2+b^2+c^2=ab+bc+ca,\text{then}& \text{(P)}& △ABC\text{is an equilateral triangle}\\ \text{(B)}& \text{In a}△ABC,\text{if}{a}^2+2b^2+c^2=2bc+2ab,\text{then}& \text{(Q)}& △ABC\text{is a right angled triangle}\\ \text{(C)}& \text{In a}△ABC,\text{if}{a}^2+b^2+c^2=\sqrt{2}a(b+c),\text{then}& \text{(R)}& △ABC\text{is a scalene triangle}\\ \text{(D)}& \text{In a}△ABC,\text{if}{a}^2+b^2+c^2=ca+\sqrt{3}ab,\text{then}& \text{(S)}& A={90}^{\circ },B={45}^{\circ },C={45}^{\circ }\end{array}$



Which of the following is the only CORRECT combination?

Probability that $A$ speaks truth is $\frac{3}{4}$. If a coin is tossed and $A$ reports that a head appears, then the probability that head actually appeared is

If one root of the equation $a{x}^2$ + bx + c = 0 be n times the other root, then

Let $a_1,a_2,a_3,\dots ,a_{49}$ be in $A.P.$ such that $\sum _{k=0}^{12}{a}_{4k+1}=416$ and $a_9+a_{43}=66.$ If $a_1^2+a_2^2+\dots +a_{17}^2=140m$, then $m$ is equal to:

2x squared + y squared + 8z squared - 2 root 2xy + 4 root 2yz - 8xz

$\underset{x\to 0}{lim}\frac{lo{g}_e(1+x)}{x}=$

If $|x-5|+|x-10|⩽20$, then x belongs to

$\text{If}\underset{x\to -\infty }{lim}(\sqrt{x^2-x+1}-ax-b)=0$ then the values of a and b are given by -

Given, $\left[\begin{array}{cc}2a+b& a-2b\\ 5c-d& 4c+3d\end{array}\right]=\left[\begin{array}{cc}4& -3\\ 11& 24\end{array}\right],$ then which of the following is/are not correct

Find the equation of the lines through the point (3, 2) which make an angle of ${45}^{\circ }$ with the line x - 2y = 3