MCQ 11 Mark
Directions: In the following questions, the Assertions $(A)$ and Reason(s) $(R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion $(A)$ Area of the triangle whose vertices are $(4, 4), (3, -2)$ and $(- 3, 16),$ is
Reason $(R)$ Area of triangle whose vertices are $(x_1, y_1), (x_2, y_2)$ and $(x_3, y_3),$ is $\frac{1}{2}\text{x}_1(\text{y}_2-\text{y}_3)+\text{x}_2(\text{y}_3-\text{y}_1)+\text{x}_3(\text{y}_1-\text{y}_2)$
- ✓
$A$ is true, $R$ is true; $R$ is acorrect explanation of $A$.
- B
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- C
$A$ is true; $R$ is false
- D
$A$ is false; $R$ is true.
AnswerCorrect option: A. $A$ is true, $R$ is true; $R$ is acorrect explanation of $A$.
Assertion Area of the triangle, whose vertices are $(4, 4), (3, -2)$ and $(-3, 16)$
$=\frac{1}{2}4(-2-16)+3(16-4)+(-3)(4+2)$
$=\frac{|72+36-18|}{2}=\frac{|-54|}{2}=\frac{54}{2}=27$
View full question & answer→MCQ 21 Mark
Directions: In the following questions, the Assertions $(A)$ and Reason(s) $(R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion $(A)$ The distance between the lines $4x + 3y = 11$ and $8x + 6y = 15$ is $\frac{7}{10}$
Reason $(R)$ The distance between lines the $ax + by = c_1 $ and $ax + by = c_2 $ is given by $\frac{\text{c}_1-\text{c}_2}{\sqrt{\text{a}^2+\text{b}^2}}$
- ✓
$A$ is true, $R$ is true; $R$ is acorrect explanation of $A.$
- B
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- C
$A$ is true; $R$ is false
- D
$A$ is false; $R$ is true.
AnswerCorrect option: A. $A$ is true, $R$ is true; $R$ is acorrect explanation of $A.$
Assertion Given lines are $4x + 3y = 11$ and $4x + 3y = \frac{15}{2}$
Distance between them
$=\Bigg|\frac{11-\frac{15}{2}}{\sqrt{16}+9}\Bigg|$
$=\Big|\frac{7}{2\times5}\Big|$
$=\frac{7}{10}$
View full question & answer→MCQ 31 Mark
Directions: In the following questions, the Assertions $(A)$ and Reason$(s) (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion $(A)$ Slope of line $3x - 4y +10 = 0$ is $\frac{3}{4}$
Reason $(R) x -$ intercept and $y-$intercept of $3x - 4y + 10 = 0$ respectively are $\frac{-10}{3} $ and $\frac{5}{2}.$
- A
$A$ is true, $R$ is true; $R$ is acorrect explanation of $A.$
- ✓
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- C
$A$ is true; $R$ is false
- D
$A$ is false; $R$ is true.
AnswerCorrect option: B. $A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
Assertion Given equation $3x - 4y + 10 = 0$ can be written as
$\text{y}=\frac{3}{4}\text{x}+\frac{5}{2}\ ...(\text{i}) $
Comparing Eq. $(i)$ with $y = mx + c,$ we have slope of the given line as $\text{m}=\frac{3}{4}.$
Reason Equation $3x - 4y + 10 = 0$ can be written as
$3\text{x}-4\text{y}=-10$ or $\frac{\text{x}}{-\frac{10}{3}}+\frac{\text{y}}{\frac{5}{2}}=1\ ...(\text{ii}) $
Comparing Eq. $(ii)$ with $\frac{\text{x}}{\text{a}}+\frac{\text{y}}{\text{b}}=1,$ we have
$x-$ intercept as $\text{a}=-\frac{10}{3}$ and $y-$intercept as $\text{b}=\frac{5}{2}.$
View full question & answer→MCQ 41 Mark
Directions: In the following questions, the Assertions $(A)$ and Reason$(s) (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion $(A)$ The slope of the line $x +7y = 0$ is $\frac{1}{7}$ and $y -$ intercept is $0.$
Reason $(R)$ The slope of the line ; $6x + 3y - 5 = 0$ is $- 2$ and $y -$ intercept is $\frac{5}{3}.$
- A
$A$ is true, $R$ is true; $R$ is acorrect explanation of $A.$
- B
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- C
$A$ is true; $R$ is false
- ✓
$A$ is false; $R$ is true.
AnswerCorrect option: D. $A$ is false; $R$ is true.
Assertion Given equation is $x + 7y = 0$
$\Rightarrow\text{y}=\frac{-\text{x}}{7}+0$
On comparing with $y = mx + c,$ we get
Slope $\text{(m)}=\frac{-1}{7}, y -$ intercept $= 0$
Reason Given equation is $6x + 3y - 5 = 0$
$\Rightarrow\text{y}=-2\text{x}+\frac{5}{3}$
On comparing with $y = mx + c,$ we get
Slope $(m) = - 2, y -$ intercept $\frac{5}{3}$
View full question & answer→MCQ 51 Mark
Directions: In the following questions, the Assertions $(A)$ and Reason$(s) (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion $(A)$ slope of line $3x - 4y + 10 = 0$ is $\frac{3}{4}.$
Reason $(R)\ x -$ intercept and $y-$ intercept of $3x - 4y + 10 = 0$ respectively are $\frac{-10}{3}$ and $\frac{5}{2}.$
- A
Both assertion and reason are true and reason is the correct explanation of assertion.
- ✓
Both assertion and reason are true but reason is not the correct explanation of assertion.
- C
Assertion is true but reason is false.
- D
Assertion is false but reason is true
AnswerCorrect option: B. Both assertion and reason are true but reason is not the correct explanation of assertion.
View full question & answer→MCQ 61 Mark
Directions: In the following questions, the Assertions $(A)$ and Reason$(s) (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion $(A)$ The point $(3, 0)$ is at $3$ units distance from the $Y-$axis measured along the positive $X -$ axis and has zero distance from the $X -$ axis.
Reason $(R)$ The point $(3, 0)$ is at $3$ units distance from the $X -$ axis measured along the positive $Y -$ axis and has zero distance from the $Y -$ axis
- A
$A$ is true, $R$ is true; $R$ is acorrect explanation of $A.$
- B
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- ✓
$A$ is true; $R$ is false
- D
$A$ is false; $R$ is true.
AnswerCorrect option: C. $A$ is true; $R$ is false
Assertion The point $(3, 0)$ is at $3$ units distance from the $Y-$axis measured along the positive $X-$axis and has zero distance from the $X-$axis.
View full question & answer→MCQ 71 Mark
Directions: In the following questions, the Assertions $(A)$ and Reason$(s) (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
if $A (- 2, - 1), B (4, 0), C (3, 3)$ and $D (- 3, 2)$ are the vertices of a parallelogram, then
Assertion $(A)$ Slope of $AB =$ Slope of $BC$ and Slope of $CD =$ Slope of $A D.$
Reason $(R)$ Mid $-$ point of $AC =$ Mid $-$ point of $BD.$
- A
$A$ is true, $R$ is true; $R$ is acorrect explanation of $A.$
- B
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- C
$A$ is true; $R$ is false
- ✓
$A$ is false; $R$ is true.
AnswerCorrect option: D. $A$ is false; $R$ is true.
$\because \text{ABCD}$ is a parallelogram.
$\therefore AB\| CD =$ Slope of $AB =$ Slope of $CD$
and $BC \| AD =$ Slope of $BC =$ Slope of $AD$
Reason mid $-$ point of
$\text{Ac}\Big(\frac{-2+3}{2},\frac{-1+3}{2}\Big)$
$=\Big(\frac{1}{2},\frac{2}{2}\Big)=\Big(\frac{1}{2},1\Big)$
and mid$-$point of $\text{BD}=\Big(\frac{4-3}{2},\frac{0+2}{2}\Big)$
$=\Big(\frac{1}{2},1\Big)$
$=$ Mid $-$ point of $AC =$ Mid$-$point of $BD$ View full question & answer→MCQ 81 Mark
Directions: In the following questions, the Assertions $(A)$ and Reason$(s) (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion $(A)$ The centroied cannot lie on the $Y -$ axis.
Reason $(R)$ The condition that the centroid may lie on the $x -$ axis is $a + b = 3.$
- A
Both assertion and reason are true and reason is the correct explanation of assertion.
- ✓
Both assertion and reason are true but reason is not the correct explanation of assertion.
- C
Assertion is true but reason is false.
- D
Assertion is false but reason is true
AnswerCorrect option: B. Both assertion and reason are true but reason is not the correct explanation of assertion.
View full question & answer→MCQ 91 Mark
Directions: In the following questions, the Assertions $(A)$ and Reason$(s) (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion $(A)$ Slope of $X-$ axis is zero and slope of $Y-$ axis is not defined
Reason $(R)$ Slope of $X-$ axis is not defined and slope of $Y-$ axis is zero.
- A
$A$ is true, $R$ is true; $R$ is acorrect explanation of $A.$
- B
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- ✓
$A$ is true; $R$ is false
- D
$A$ is false; $R$ is true.
AnswerCorrect option: C. $A$ is true; $R$ is false
Assertion Slope of $X-$ axis is zero and slope of $Y-$ axis is not defined.
View full question & answer→MCQ 101 Mark
Directions: In the following questions, the Assertions $(A)$ and Reason(s) $(R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion $(A)$ A point $P (h, k)$ lies on the straight line $x + y + 1 = 0$ and is at a distance $5$ units from the origin. If $k$ is negative, then h is equal to $- 3.$
Reason $(R)$ The distance formula is $\sqrt{\text{x}_2-\text{x}_1)^2+(\text{x}_2-\text{y}_1)^2}$
- A
$A$ is true, $R$ is true; $R$ is acorrect explanation of $A.$
- B
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- C
$A$ is true; $R$ is false
- ✓
$A$ is false; $R$ is true.
AnswerCorrect option: D. $A$ is false; $R$ is true.
Assertion Since, the point $(A, k)$ lies on $x + y + 1 = 0.$
$\Rightarrow h + k + 1 = 0$
and $h^2 + k^2 = 25$
$\Rightarrow (-1 - k)^2 + k^2 = 25$
$= 2k^2 + 2k - 24 = 0$
$k^2+ k - 12 = 0$
$\Rightarrow k = -4$ or $k =3$
$[k = 3$ rejected as $k < 0]$
$\therefore h = -1 - (-4) = 3$
View full question & answer→MCQ 111 Mark
Directions: In the following questions, the Assertions $(A)$ and Reason(s) $(R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
if the vertices of a triangle are $(1, a), (2, b)$ and $(c^2, - 3)$. Then,
Assertion $(A)$ The centroid cannot lie on the $Y-$axis.
Reason $(R)$ The condition that the centroid may lie on the $X -$ axis is $a + b = 3.$
- A
$A$ is true, $R$ is true; $R$ is acorrect explanation of $A.$
- ✓
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- C
$A$ is true; $R$ is false
- D
$A$ is false; $R$ is true.
AnswerCorrect option: B. $A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
Assertion Centroid of the triaigle is
$\text{G} \equiv\Big(\frac{1+2+\text{c}^2}{3},\frac{\text{a}+\text{b}-3}{2}\Big)$
$\text{i.e.}\Big(\frac{3+\text{c}^2}{3},\frac{\text{a}+\text{b}-3}{3}\Big)$
$\because$ $G$ will lie on $Y-$axis, then
$\frac{3+\text{c}^2}{3}=0\Rightarrow\text{c}^2=-3\text{ or}\text{ c}\equiv\pm\text{ i}\sqrt{3}$
$\because$ Both values of $c$ are imaginary.
Hence, $G$ cannot lie on $Y-$axis.
Reason $\because G$ will lies on $X-$axis, then
$\frac{\text{a}+\text{b}-3}{3}=0$
$\Rightarrow\text{a}+\text{b}-3=0$ or $\text{ a}+\text{b}=3$
View full question & answer→MCQ 121 Mark
Directions: In the following questions, the Assertions $(A)$ and Reason(s) $(R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion $(A)$ The angle between the lines $x + 2y - 3 = 0$ and $3x + y + 1 = 0$ is $\tan^{-1} (1).$
Reason $(R)$ Angle between two lines is given by $\tan^{-1}\Big[\pm\big(\frac{\text{m}_2-\text{m}_1}{1+\text{m}_1\text{m}_2}\Big)\Big]$
- ✓
$A$ is true, $R$ is true; $R$ is acorrect explanation of $A.$
- B
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- C
$A$ is true; $R$ is false
- D
$A$ is false; $R$ is true.
AnswerCorrect option: A. $A$ is true, $R$ is true; $R$ is acorrect explanation of $A.$
Assertion Let $m$, and $m$, be the slopes of the straight lines $x + 2y - 3 = 0$ and $3x + y + 1 = 0.$
Then, $\text{m}_1=-\frac{1}{2}$ and $m_2 = -3$
Let $\theta$ be the angle between the given lines.
Then, $\tan\theta=\pm\Big(\frac{\text{m}_2-\text{m}_1}{1+\text{m}_1\text{m}_2}\Big)$
$=\pm\Bigg(\frac{-3+\frac{1}{2}}{1+\frac{3}{2}}\Bigg)=\pm1$
$\Rightarrow\theta=\tan^1(1)$ or $\pi-\tan^{-1}(1)$
View full question & answer→MCQ 131 Mark
Directions: In the following questions, the Assertions $(A)$ and Reason$(s) (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
If the equation of line is $x - y = 4,$ then
Assertion $(A)$ The normal form of same equation is $\text{x}\cos\alpha+\text{y}\sin\alpha=\rho$ where $\alpha=315^\circ$ and $\rho=2\sqrt{2}.$
Reason $(R)$ The perpendicular distance of line from the origin is $3\sqrt{2}.$
- A
$A$ is true, $R$ is true; $R$ is acorrect explanation of $A.$
- B
$A$ is true, $R$ is true; $R$ is not a correct explanation of $A.$
- ✓
$A$ is true; $R$ is false
- D
$A$ is false; $R$ is true.
AnswerCorrect option: C. $A$ is true; $R$ is false
Assertion Given equation of line is $x - y = 4$
On dividing above equation by
$\sqrt{\text{(coefficient of x)}^2+\text{(conffcient of y})^2}$
$\text{i.e.} \sqrt{(1)^2+(-1)^2}=\sqrt{2},$ we get
$\frac{1}{\sqrt{2}}\text{x}-\frac{1}{\sqrt{2}}\text{y}=\frac{4}{\sqrt{2}}$
$\Rightarrow\cos45^\circ\text{x}-\sin45^\circ\text{ y}=2\sqrt{2}$
$[ \because \cos \text{x}$ is positive and $\sin \text{x}$ is negative, it is possible only in fourth quadrant$]$
$\Rightarrow\text{x}\cos(360^\circ-45^\circ)+\text{y}\sin(360^\circ-45^\circ)=2\sqrt{2}$
$\Big[\because\cos(360^\circ-\theta)=\cos\theta$ and $\sin(360^\circ-\theta)=-\sin\theta\Big]$
$\Rightarrow\text{x}=\cos315^\circ+\text{y }315^\circ=2\sqrt{2}$
On comparing with $\text{x}\cos\alpha+\text{y}\sin\alpha=\rho,$ we get
$\alpha=315^\circ$
and $\rho=2\sqrt{2}$
View full question & answer→