Jee Math Test Series Quiz-1

In a class of 140 students numbered 1 to 140, all even-numbered students opted Mathematics course, those whose number is divisible by 3 opted Physics course and those whose number is divisible by 5 opted Chemistry course. Then the number of students who did not opt for any of the three courses is:

\(
\text { If } f(x)=\log _e\left(\frac{1-x}{1+x}\right),|x|<1 \text {, then } f\left(\frac{2 x}{1+x^2}\right) \text { is equal to : }
\)

For any \(\theta \in\left(\frac{\pi}{4}, \frac{\pi}{2}\right)\) the expression \(3(\sin \theta-\cos \theta)^4+6(\sin \theta+\cos \theta)^2+4 \sin ^6 \theta\) equals:

If \(\operatorname{cos}(\alpha+\beta)=\frac{3}{5}, \sin (\alpha-\beta)=\frac{5}{13}\) and \(0<\alpha, \beta<\frac{\pi}{4}\), then \(\tan (2 \alpha)\) is equal to:

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

Let \(S=\left\{\theta \in[-2 \pi, 2 \pi]: 2 \cos ^2 \theta+3 \sin \theta=0\right\}\). Then the sum of the elements of S is:

Let \(S=\{x \in R: x \geq 0\) and \(2|\sqrt{\mathrm{x}}-3|+\sqrt{\mathrm{x}}(\sqrt{\mathrm{x}}-6)+6=0\). Then \(\mathrm{S}:\)

If sum of all the solutions of the equation \(8 \cos x \cdot\left(\cos \left(\frac{\pi}{6}+x\right) \cdot \cos \left(\frac{\pi}{6}-x\right)-\frac{1}{2}\right)-1\) in \([0, \pi]\) is \(k \pi\), then \(k\) is equal to :

The number of solutions of \(\sin 3 x=\cos 2 x\), in the interval \(\left(\frac{\pi}{2}, \pi\right)\) is

Let \(\mathrm{N}\) denote the set of all natural numbers. Define two binary relations on \(N\) as \(R_1=\{(x, y) \in N \times N: 2 x+y=10\}\) and \(R_2=\{(x, y) \in N \times N: x+2 y=10\}\). Then

Consider the following two binary relations on the set \(A=\{a, b, c\}: R_1=\{(\mathrm{c}, a)(b, b),(\mathrm{a}, c),(c, c),(b, c),(a, a)\}\) and \(\mathrm{R}_2=\{(\mathrm{a}, \mathrm{b}),(\mathrm{b}, \mathrm{a}),(\mathrm{c}, \mathrm{c}),(\mathrm{c}, \mathrm{a}),(\mathrm{a}, \mathrm{a}),(\mathrm{b}, \mathrm{b}),(\mathrm{a}, \mathrm{c})\). Then

Let \(f: \mathrm{A} \rightarrow \mathrm{B}\) be a function defined as \(f(x)=\frac{x-1}{x-2}\), where \(A\) \(=R-\{2\}\) and \(B=R-\{1\}\). Then \(f\) is

If \(f(x)+2 f\left(\frac{1}{x}\right)=3 x, x \neq 0\) and \(\mathrm{S}=\{\mathrm{x} \in \mathrm{R}: \mathrm{f}(\mathrm{x})=\mathrm{f}(-\mathrm{x})\}\); then \(\mathrm{S}\):

Let \(P=\{\theta: \sin \theta-\cos \theta=\sqrt{2} \cos \theta\}\) and \(Q=\{\theta: \sin \theta+\cos \theta\) \(=\sqrt{2} \sin \theta\}\) be two sets. Then:

A relation on the set \(\mathrm{A}=\{\mathrm{x}:|\mathrm{x}|<3, \mathrm{x} \in \mathrm{Z}\}\), where \(Z\) is the set of integers is defined by \(\mathrm{R}=\{(\mathrm{x}, \mathrm{y}): \mathrm{y}=|\mathrm{x}|, \mathrm{x} \neq-1\}\). Then the number of elements in the power set of \(R\) is:

Let \(X=\{1,2,3,4,5\}\). The number of different ordered pairs \((\mathrm{Y}, \mathrm{Z}\) ) that can formed such that \(Y \subseteq X, \mathrm{Z} \subseteq \mathrm{X}\) and \(\mathrm{Y} \cap \mathrm{Z}\) is empty is :

If \(A, B\) and \(C\) are three sets such that \(A \cap B=A \cap C\) and \(A \cup B=A \cup C\), then

In a certain town, \(25 \%\) of the families own a phone and \(15 \%\) own a car; \(65 \%\) families own neither a phone nor a car and 2,000 families own both a car and a phone. Consider the following three statements :
(A) \(5 \%\) families own both a car and a phone
(B) \(35 \%\) families own either a car or a phone
(C) 40,000 families live in the town
Then,

The range of the function \(f(x)=\frac{x}{1+|x|}, x \in R\), is

The domain of the function \(f(x)=\frac{1}{\sqrt{|x|-x}}\) is

Domain of definition of the function \(f(x)=\frac{3}{4-x^2}+\log _{10}\left(x^3-x\right)\), is

Let \(f(n)=\left[\frac{1}{3}+\frac{3 n}{100}\right] n\), where \([\mathrm{n}]\) denotes the greatest integer less than or equal to \(n\). Then \(\sum_{n=1}^{56} f(n)\) is equal to:

Let \(f\) be an odd function defined on the set of real numbers such that for \(x \geq 0\), \(f(x)=3 \sin x+4 \cos x\)
Then \(f(x)\) at \(x=-\frac{11 \pi}{6}\) is equal to:

A real-valued function \(f(x)\) satisfies the functional equation
\(
f(x-y)=f(x) f(y)-f(a-x) f(a+y)
\)
where a is a given constant and \(f(0)=1\), \(f(2 a-x)\) is equal to

The graph of the function \(y=f(x)\) is symmetrical about the line \(x=2\), then

If \(f: R \rightarrow R\) satisfies \(f(x+y)=f(x)+f(y)\), for all \(x\), \(y \in R\) and \(f(1)=7\), then \(\sum_{r=1}^n f(r)\) is

Let \(f_k(x)=\frac{1}{k}\left(\sin ^k x+\cos ^k x\right)\) where \(x \in R\) and \(k \geq 1\). Then \(f_4(x)-f_6(x)\) equals

If \(2 \cos \theta+\sin \theta=1\left(\theta \neq \frac{\pi}{2}\right)\), then \(7 \cos \theta+6 \sin \theta\) is equal to:

The expression \(\frac{\tan A}{1-\cot A}+\frac{\cot A}{1-\tan A}\) can be written as :

The value of \(\cos 255^{\circ}+\sin 195^{\circ}\) is

Let \(f(x)=\sin x, \mathrm{~g}(x)=x\).
Statement 1: \(f(x) \leq g(x)\) for \(x\) in \((0, \infty)\)
Statement 2: \(f(x) \leq 1\) for \(x\) in \((0, \infty)\) but \(g(x) \rightarrow \infty\) as \(x \rightarrow \infty\).

A triangular park is enclosed on two sides by a fence and on the third side by a straight river bank. The two sides having fence are of same length \(x\). The maximum area enclosed by the park is

\(
\text { If } 5\left(\tan ^2 x-\cos ^2 x\right)=2 \cos 2 x+9 \text {, then the value of } \cos 4 x \text { is: }
\)

If \(\mathrm{m}\) and \(\mathrm{M}\) are the minimum and the maximum values of \(4+\frac{1}{2} \sin ^2 2 x-2 \cos ^4 x, x \in R\), then \(M-m\) is equal to :

If \(\cos \alpha+\cos \beta=\frac{3}{2}\) and \(\sin \alpha+\sin \beta \quad \frac{1}{2}\) and \(\theta\) is the the arithmetic mean of \(\alpha\) and \(\beta\), then \(\sin 2 \theta+\cos 2 \theta\) is equal to :

If \(\operatorname{cosec} \theta=\frac{p+q}{p-q}(p \neq q \neq 0)\), then \(\left|\cot \left(\frac{\pi}{4}+\frac{\theta}{2}\right)\right|\) is equal to:

If \(A=\sin ^2 x+\cos ^4 x\), then for all real \(x\) :

Let \(\cos (\alpha+\beta)=\frac{4}{5}\) and \(\sin (\alpha-\beta)=\frac{5}{13}\), where \(0 \leq \alpha, \beta \leq \frac{\pi}{4}\). Then \(\tan 2 \alpha=\)

Let A and B denote the statements
\(
\begin{aligned}
& \text { A: } \cos \alpha+\cos \beta+\cos \gamma=0 \\
& \text { B : } \sin \alpha+\sin \beta+\sin \gamma=0
\end{aligned}
\)
\(
\text { If } \cos (\beta-\gamma)+\cos (\gamma-\alpha)+\cos (\alpha-\beta)=-\frac{3}{2} \text {, then : }
\)

If \(p\) and \(q\) are positive real numbers such that \(p^2+q^2=1\), then the maximum value of \((p+q)\) is

If \(0<x<\pi\) and \(\cos x+\sin x=\frac{1}{2}\), then \(\tan \mathrm{x}\) is

If \(u=\sqrt{a^2 \cos ^2 \theta+b^2 \sin ^2 \theta}+\sqrt{a^2 \sin ^2 \theta+b^2 \cos ^2 \theta}\) then the difference between the maximum and minimum values of \(u^2\) is given by

Let \(\alpha, \beta\) be such that \(\pi<\alpha-\beta<3 \pi\).
If \(\sin \alpha+\sin \beta=-\frac{21}{65}\) and \(\cos \alpha+\cos \beta=-\frac{27}{65}\), then the value of \(\cos \frac{\alpha-\beta}{2}\)

The function \(f(x)=\log \left(x+\sqrt{x^2+1}\right)\), is

The period of \(\sin ^2 \theta\) is

Which one is not periodic?

If \(0 \leq x<2 \pi\), then the number of real values of \(x\), which satisfy the equation
\(\cos x+\cos 2 x+\cos 3 x+\cos 4 x=0\) is:

The number of \(x \in[0,2 \pi]\) for which \(\left|\sqrt{2 \sin ^4 x+18 \cos ^2 x}-\sqrt{2 \cos ^4 x+18 \sin ^2 x}\right|=1\) is

If \(\mathrm{X}\) and \(\mathrm{Y}\) are two sets such that \(\mathrm{X} \cup \mathrm{Y}\) has 50 elements, \(\mathrm{X}\) has 28 elements and \(\mathrm{Y}\) has 32 elements, how many elements does \(\mathrm{X} \cap \mathrm{Y}\) have ?

In a class of 35 students, 24 like to play cricket and 16 like to play football. Also, each student likes to play at least one of the two games. How many students like to play both cricket and football?