Assertion (A) & Reason (B) MCQ

MCQ 11 Mark

Statement-1 (A): For $0<\theta \leq 90^{\circ}, \operatorname{cosec} \theta-\cot \theta$ and $\operatorname{cosec} \theta+\cot \theta$ are reciprocal of each other.
Statement-2 (R): $\cot ^2 \theta-\operatorname{cosec}^2 \theta=1$

A
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
B
Statement-1 is true, Statement- 2 is true; Statement- 2 is not a correct explanation for Statement-1.
✓
Statement-1 is true, Statement-2 is false.
D
Statement-1 is false, Statement-2 is true.

Answer

Correct option: C.

Statement-1 is true, Statement-2 is false.

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MCQ 21 Mark

Statement-1 (A): If $\tan \theta+\cot \theta=2$, then $\tan ^2 \theta+\cot ^2 \theta=4$.
Statement-2 (R): If $\operatorname{cosec} A=\sqrt{2}$, then $\frac{2 \sin ^2 A+3 \cot ^2 A}{4 \tan ^2 A-2 \cos ^2 A}=\frac{4}{3}$.

A
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
B
Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
C
Statement-1 is True, Statement-2 is False.
✓
Statement-1 is False, Statement-2 is True.

Answer

Correct option: D.

Statement-1 is False, Statement-2 is True.

(D)Statement-1 is False, Statement-2 is True.
We have, $\tan \theta+\cot \theta=2$
$
\begin{array}{ll}
\Rightarrow & (\tan \theta+\cot \theta)^2=2^2 \\
\Rightarrow & \tan ^2 \theta+\cot ^2 \theta+2 \tan \theta \cot \theta=4 \Rightarrow \tan ^2 \theta+\cot ^2 \theta+2=4 \Rightarrow \tan ^2 \theta+\cot ^2 \theta=2
\end{array}
$
So, statement-1 is not true.
We have, $\operatorname{cosec} A=\sqrt{2} \Rightarrow \sin A=\frac{1}{\sqrt{2}} \Rightarrow \sin A=\sin 45^{\circ} \Rightarrow A=45^{\circ}$
$\begin{array}{l}\therefore \quad \frac{2 \sin ^2 A+3 \cot ^2 A}{4 \tan ^2 A-2 \cos ^2 A}=\frac{2 \sin ^2 45+3 \cot ^2 45^{\circ}}{4 \tan ^2 45^{\circ}-2 \cos ^2 45^{\circ}}=\frac{2\left(\frac{1}{2}\right)+3(1)}{4 \times 1-2\left(\frac{1}{2}\right)}=\frac{4}{3} \\ \text { So, statement-2 is true. }\end{array}$

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MCQ 31 Mark

Statement-1 (A): If $\sin \theta+\sin ^2 0=1$, then $\cos ^2 0+\cos ^4 0=1$
Statement-2 (R): $1-\sin ^2 0=\cos ^2 0$.

✓
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
B
Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
C
Statement-1 is True, Statement-2 is False.
D
Statement-1 is False, Statement-2 is True.

Answer

Correct option: A.

Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.

(A)Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
Clearly, statement-2 is true as $\sin ^2 \theta+\cos ^2 \theta=1$.
$
\begin{array}{ll}
\text { Now, } & \sin \theta+\sin ^2 \theta=1 \\
\Rightarrow & \sin \theta=1-\sin ^2 \theta
\end{array}
$
$
\Rightarrow \quad \sin \theta=\cos ^2 \theta \qquad
$[Using statement-2]
$
\Rightarrow \quad \sin ^2 \theta=\cos ^4 \theta \qquad
$[Squaring both sides]
$
\Rightarrow \quad 1-\cos ^2 \theta=\cos ^4 \theta \qquad
$[Using statement-2]
$
\Rightarrow \quad \cos ^2 \theta+\cos ^4 \theta=1
$
So, statement-1 is also correct and statement-2 is a correct explanation for statement-1.
Hence, option (a) is correct.

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MCQ 41 Mark

Statement-1 (A): Let $a, b$ be non-zero real numbers. Then, $\sec ^2 \theta=\frac{4 a b}{(a+b)^2}$ is true if and only if $a=b$.
Statement-2 (R): $\sec ^2 \theta \geq 1$ for $0 \leq \theta<90^{\circ}$.

✓
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
B
Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
C
Statement-1 is True, Statement-2 is False.
D
Statement-1 is False, Statement-2 is True.

Answer

Correct option: A.

Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.

(A)Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
In a right triangle if $\theta$ is one of the acute angles, then
$\sec \theta=\frac{\text { Hypotenuse }}{\text { Base }} \geq 1 \Rightarrow \sec ^2 \theta \geq 1 \quad[\because$ Hypotenuse $\geq$ Base $]$
So, statement-2 is true.
Now, $\quad \sec ^2 \theta=\frac{4 a b}{(a+b)^2}$
$\Rightarrow \quad \frac{4 a b}{(a+b)^2} \geq 1 \quad\left[\because \sec ^2 \theta \geq 1\right]$
$\begin{array}{ll}\Rightarrow & 1-\frac{4 a b}{(a+b)^2} \geq 0 \\ \Rightarrow & \frac{(a+b)^2-4 a b}{(a+b)^2} \leq 0\end{array}$
$\Rightarrow \quad \frac{(a-b)^2}{(a+b)^2} \leq 0 \Rightarrow\left(\frac{a-b}{a+b}\right)^2 \leq 0 \Rightarrow\left(\frac{a-b}{a+b}\right)^2=0 \quad\left[\because\left(\frac{a-b}{a+b}\right)^2\right.$ cannot be negative $]$
$
\Rightarrow \quad \frac{a-b}{a+b}=0 \Rightarrow a-b=0 \Rightarrow a=b .
$
So, statement-1 is true and slatement-2 is a correct explanation for statement-1.

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MCQ 51 Mark

Statement-1 (A) : For $0 < \theta < 90^{\circ}, \sec \theta+\tan \theta$ and $\sec \theta-\tan \theta$ are reciprocal of each other.
Statement-2 (R): $\tan ^2 \theta-\sec ^2 \theta=1$

A
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
B
Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
✓
Statement-1 is True, Statement-2 is False.
D
Statement-1 is False, Statement-2 is True.

Answer

Correct option: C.

Statement-1 is True, Statement-2 is False.

(C)Statement-1 is True, Statement-2 is False.
Clearly, statement-2 is not true. We know that
$
\sec ^2 \theta-\tan ^2 \theta=1
$
$
\Rightarrow(\sec \theta-\tan \theta)(\sec \theta+\tan \theta)=1 \Rightarrow \sec \theta+\tan \theta=\frac{1}{\sec \theta-\tan \theta} \text { and, } \sec \theta-\tan \theta=\frac{1}{\sec \theta+\tan \theta}
$
i.e. $\sec \theta+\tan \theta$ and $\sec \theta-\tan \theta$ are reciprocal of each other. Thus, statement -1 is true.

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MCQ 61 Mark

Statement-1 (A) : The value of the product $P_1=\cos 1^{\circ} \cos 2^{\circ} \cos 3^{\circ} \ldots \cos 179^{\circ} \cos 180^{\circ}$ is zero.
Statement-2 (R) : The value of the product $P_2=\tan 1^{\circ} \tan 2^{\circ} \ldots \tan 89^{\circ}$ is 1 .

A
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
B
Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
C
Statement-1 is True, Statement-2 is False.
D
Statement-1 is False, Statement-2 is True.

Answer

(B)Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.

We find that in the product
$P_1=\cos 1^{\circ} \cos 2^{\circ} \cos 3^{\circ} \ldots \cos 179^{\circ} \cos 180^{\circ}$
one of the terms is $\cos 90^{\circ}$
which is equal to zero. Hence, product
$P_1=0$.
So, statement- 1 is true.

The product $P_2$ can be written as
$P_2=\left(\tan 1^{\circ} \tan 89^{\circ}\right)\left(\tan 2^{\circ} \tan 88^{\circ}\right) \ldots\left(\tan 44^{\circ} \tan 46^{\circ}\right) \tan 45^{\circ}$
$\Rightarrow \quad P_2=\left(\tan 1^{\circ} \cot 1^{\circ}\right)\left(\tan 2^{\circ} \cot 2^{\circ}\right) \ldots\left(\tan 44^{\circ} \cot 44^{\circ}\right) \tan 45^{\circ}$
$=1\left[\begin{array}{l}\text { Using }: \tan \left(90^{\circ}-\theta\right)=\cot \theta \\ \text { for } \theta=1^{\circ}, 2^{\circ}, \ldots, 44^{\circ}\end{array}\right]$
So, statement $P_2$ is also true.
it is not a correct explanation for $P_1 .$

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MCQ 71 Mark

Statement-1 (A): For $0 \leq \theta < 90^{\circ}, \sec ^2 \theta+\cos ^2 \theta \geq 2$.
Stalement-2 (R) : For $x > 0, x+\frac{1}{x} \geq 2$

✓
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
B
Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
C
Statement-1 is True, Statement-2 is False.
D
Statement-1 is False, Statement-2 is True.

Answer

Correct option: A.

Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.

(A)Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
For any $x > 0$, we know that
$\left(\sqrt{x}-\frac{1}{\sqrt{x}}\right)^2 \geq 0 \Rightarrow x+\frac{1}{x}-2 \geq 0 \Rightarrow x+\frac{1}{x} \geq 2$.
So, statement- 2 is true.
For $0 \leq \theta<90^{\circ}$, we find that $\sec ^2 \theta$ and $\cos ^2 \theta$ are positive real numbers such that $\sec ^2 \theta=\frac{1}{\cos ^2 \theta}$.
Using statement-2, we obtain
$\cos ^2 \theta+\frac{1}{\cos ^2 \theta} \geq 2 \Rightarrow \sec ^2 \theta+\cos ^2 \theta \geq 2$
Thus, statement- 1 is true and statement- 2 is a correct explanation for statement- 2 . Hence, option (a) is correct.

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MCQ 81 Mark

Statement-1 (A): If $\sin A=\frac{1}{3}\left(0^{\circ}<A<90^{\circ}\right)$, then the value of $\cos A$ is $\frac{2 \sqrt{2}}{3}$.
Statement-2 (R): For every angle $\theta, \sin ^2 \theta+\cos ^2 \theta=1$.

✓
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
B
Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
C
Statement-1 is True, Statement-2 is False.
D
Statement-1 is False, Statement-2 is True.

Answer

Correct option: A.

Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.

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MCQ 91 Mark

Statement-1 (A): For $0<\theta \leq 90^{\circ}, \operatorname{cosec}^2 \theta+\sin ^2 \theta \geq 2$.
Statement-2 (R): For any $x>0, x+\frac{1}{x} \geq 2$.

✓
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
B
Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
C
Statement-1 is True, Statement-2 is False.
D
Statement-1 is False, Statement-2 is True.

Answer

Correct option: A.

Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.

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MCQ 101 Mark

Statement-1 (A): If $x=a \cos \theta$ and $y=b \sin \theta$, then $b^2 x^2+a^2 y^2=a^2 b^2$.
Statement-2 (R): $\quad \cos ^2 \theta+\sin ^2 \theta=1$.

✓
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
B
Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
C
Statement-1 is True, Statement-2 is False.
D
Statement-1 is False, Statement-2 is True.

Answer

Correct option: A.

Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.

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MCQ 111 Mark

Statement-1 (A): For $0 \leq \theta<90^{\circ}, \sec \theta+\tan \theta$ and $\sec \theta-\tan \theta$ are reciprocal of each other.
Statement-2 (R): $\operatorname{cosec}^2 \theta-\cot ^2 \theta=1$

A
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
B
Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
✓
Statement-1 is True, Statement-2 is False.
D
Statement-1 is False, Statement-2 is True.

Answer

Correct option: C.

Statement-1 is True, Statement-2 is False.

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MCQ 121 Mark

Statement-1 (A): The value of the product of $P=\cos 1^{\circ} \cos 2^{\circ} \ldots \ldots \cos 179^{\circ}$ as $180^{\circ}$ is zero.
Statement-2 (R) : The value of $\cos 90^{\circ}$ is zero.

✓
Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.
B
Statement-1 and Statement-2 are True; Statement-2 is not a correct explanation for Statement-1.
C
Statement-1 is True, Statement-2 is False.
D
Statement-1 is False, Statement-2 is True.

Answer

Correct option: A.

Statement-1 and Statement-2 are True; Statement-2 is a correct explanation for Statement-1.

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MCQ 131 Mark

Statement-1 (A): The value of the product $P=\tan 1^{\circ} \tan 2^{\circ} \tan 3^{\circ}, \ldots \ldots, \tan 89^{\circ}$ is 1 .
Statement-2 (R): For $0 < \theta \leq 90^{\circ}, \tan \left(90^{\circ}-\theta\right)=\cot \theta$ and $\tan 45^{\circ}=1$.

✓
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
B
Statement-1 is true, Statement- 2 is true; Statement- 2 is not a correct explanation for Statement-1.
C
Statement-1 is true, Statement-2 is false.
D
Statement-1 is false, Statement-2 is true.

Answer

Correct option: A.

Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

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Maths Assertion (A) & Reason (B) MCQ

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