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APPLICATION OF INTEGRALS MATHS - APPLICATION OF INTEGRALS

Rajasthan Board (RBSE) - STD 12 Science - MATHS (Rajasthan - English Medium). 386 questions in this chapter.

Question types

APPLICATION OF INTEGRALS question types

386 questions across 8 question groups — pick any mix to generate a MATHS paper with step-by-step answer keys.

386
Questions
8
Question groups
5
Question types
Sample Questions

APPLICATION OF INTEGRALS questions

One sample from each question group in this chapter. Select any group above to see the full set with answer keys.

Q 1M.C.Q (1 Marks)1 Mark
The area bounded by the curvey $=\sqrt{\text{x}}$ the line 2y + 3 = x and the x - axis in the first quadrant is:
  1. $9$
  2. $\frac{27}{4}$
  3. $36$
  4. $18$
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Q 6Assertion (A) & Reason (B) MCQ1 Mark
Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices:
Assertion: The area of the smaller region bounded by the ellipse $\frac{\text{x}^2}{9}+\frac{\text{y}^2}{4}=1$ the line $\frac{\text{x}}{3}+\frac{\text{y}}{2}=1$ is $\frac{3}{2}(\pi-2)\text{ sq.units}$
Reason: Formula to calculate the area of the smaller region bounded by the ellipse $\frac{\text{x}^2}{\text{a}^2}+\frac{\text{y}^2}{\text{b}^2}=1$ and the line $\frac{\text{x}}{\text{a}}+\frac{\text{y}}{\text{b}}=1$ is $\frac{\text{ab}}{4}(\pi-2) \text{ sq.units}$ 
  1. Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
  2. Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
  3. Assertion is correct statement but Reason is wrong statement.
  4. Assertion is wrong statement but Reason is correct statement.
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Q 7Assertion (A) & Reason (B) MCQ1 Mark
Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices:
Assertion: The area of the region bounded by the curve $y^2 = 4x$ and the line $x = 3$ is $8\sqrt{3} \text{ sq.units}$
Reason: The area of the region bounded by the curve $x^2 = 4y$ and the line $x = 4y - 2$ is $\frac{9}{8}\text{ sq.units}$
  • A
    Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
  • B
    Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
  • C
    Assertion is correct statement but Reason is wrong statement.
  • ✓
    Assertion is wrong statement but Reason is correct statement.

Answer: D.

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Q 8Assertion (A) & Reason (B) MCQ1 Mark
Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices:
Asseration: Thearea bounded bythe parabola $y^{2 }= 4ax$ and the line $x = a$ and $x = 4a$ is $\frac{56\text{a}^2}{3}\text{ sq.units}$
Reason: The area bounded by the curves $y = 3x$ and $y = x^{2 }$ is $9.5\text{ sq.units}$
  • A
    Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
  • B
    Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
  • C
    Assertion is correct statement but Reason is wrong statement.
  • D
    Assertion is wrong statement but Reason is correct statement.
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Q 9Assertion (A) & Reason (B) MCQ1 Mark
Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices:
Assertion: The area bounded by the curves $\text{y}^2 = 4\text{a}^2(\text{x} — 1) $ and lines $\text{x}=1$ and $\text{y}=4$ a is $\frac{8\text{a}}{3}\text{sq.units}$
Reason: The area enclosed between the parabola $\text{y}=\text{x}^2-\text{x}+2$ and the line $\text{y}=\text{x+2}$ is $\frac{4}{3}\text{ sq.units}$
  • A
    Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
  • B
    Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
  • C
    Assertion is correct statement but Reason is wrong statement.
  • ✓
    Assertion is wrong statement but Reason is correct statement.

Answer: D.

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Q 101 Marks Question1 Mark
The area of the region bounded by the circle $x^2 + y^2 = 1$ is$:$
  • A
    $2\pi\text{ sq.}\text{ units}$
  • ✓
    $\pi\text{ sq.}\text{ units}$
  • C
    $3\pi\text{ sq.}\text{ units}$
  • D
    $4\pi\text{ sq.}\text{ units}$

Answer: B.

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Q 121 Marks Question1 Mark
Sketch the graph of $\text{y}=\sqrt{\text{x}+1}$ in [0, 4] and determine the area of the region enclosed by the curve, the x-axis and the lines x = 0, x = 4.
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Q 132 Marks Questions2 Marks
If $\text{A}=\begin{bmatrix}2&3\\5&7\end{bmatrix},\text{ B}=\begin{bmatrix}-1&0&2\\3&4&1\end{bmatrix},\text{C}=\begin{bmatrix}-1&2&3\\2&1&0\end{bmatrix},$ find
A + B and B + C
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Q 163 Marks Question3 Marks
Draw a rough sketch of the curve $\text{y}=\sqrt{\text{x}-1}$ in the interval [1, 5]. Find the area under the curve and between the lines x = 1 and x = 5.
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Q 183 Marks Question3 Marks
Find the intervals in which the function f given by $\text{f}\text{(x)}=\text{x}^3+\frac{1}{\text{x}^3},\text{x}\neq0\text{ is}$ (i) increasing (ii) decreasing.
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Q 195 Marks Questions5 Marks
Prove that the curves $\text{y}^{2} = 4x \text{ and } x^{2} = 4y $ divided the area of square bounded by $x = 0, x = 4,y=4 \text{ and } y = 0$ into three equal parts.
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Q 205 Marks Questions5 Marks
Tangent to the circle $\text{x}^{2} + \text{y}^{2} = 4$ at any point on it in the first quadrant makes intercepts OA and OB on x and y axes respectively, O being the centre of the circle. Find the minimum value of (OA + OB).
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Q 215 Marks Questions5 Marks
If the area bounded by the parabola $\text{y}^{2} = 16\text{ax}$ and the line $\text{y = 4 mx}$ is  $\frac{\text{a}^{2}}{12}$ sq. units, then using integration, find the value of m.
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Q 235 Marks Questions5 Marks
Prove that the curves $\text{y}^{2} = 4x \text{ and } x^{2} = 4y $ divided the area of square bounded by $x = 0, x = 4,y=4 \text{ and } y = 0$ into three equal parts.
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Q 24Case study (4 Marks)4 Marks
Consider the following equations of curves $x^2 = y$ and $y = x.$ On the basis of above information, answer the following questions.
  1. The point$(s)$ of intersection of both the curves is $($are$).$
  1. $(0, 0)(2, 2)$
  2. $(0, 0)(1, 1)$
  3. $(0, 0)(-1, -1)$
  4. $(0, 0)(-2, -2)$
  1. Area bounded by the curves is represented by which of the following graph?
  1. The value of the integral $\int\limits_{1}^{0}\text{x}\ \text{dx}$ is.
  1. $\frac{1}{4}$
  2. $\frac{1}{3}$
  3. $\frac{1}{2}$
  4. $1$
  1. The value of the integral $\int\limits_{0}^{1}\text{x}^2\ \text{dx}$ is.
  1. $\frac{1}{4}$
  2. $\frac{1}{3}$
  3. $\frac{1}{2}$
  4. $1$
  1. The value of area bounded by the curves $x^2 = y$ and $x = y$ is.
  1. $\frac{1}{6}\text{ sq}.\text{unit}$
  2. $\frac{1}{3}\text{ sq}.\text{unit}$
  3. $\frac{1}{2}\text{ sq}.\text{unit}$
  4. ${1}\text{ sq}.\text{unit}$
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Q 25Case study (4 Marks)4 Marks
Ajay cut two circular pieces of cardboard and placed one upon other as shown in figure. One of the circle represents the equation $(x - 1)^2 + y^2 = 1,$ while other circle represents the equation $x^2 + y^2 = 1.$

Based on the above information, answer the following questions.
  1. Both the circular pieces of cardboard meet each other at
  1. $\text{x}=1$
  2. $\text{x}=\frac{1}{2}$
  3. $\text{x}=\frac{1}{3}$
  4. $\text{x}=\frac{1}{4}$
  1. Graph of given two curves can be drawn as.
  4. None of these
  1. Value of $\int\limits_{0}^{\frac{1}{2}}\sqrt{1-(\text{x}-1)^2}\text{dx}$ is.
  1. $\frac{\pi}{6}-\frac{\sqrt{3}}{8}$
  2. $\frac{\pi}{6}+\frac{\sqrt{3}}{8}$
  3. $\frac{\pi}{2}+\frac{\sqrt{3}}{4}$
  4. $\frac{\pi}{2}-\frac{\sqrt{3}}{4}$
  1. Value of $\int\limits_{\frac{1}{2}}^{1}\sqrt{1-\text{x}^2}\text{dx}$ is.
  1. $\frac{\pi}{6}+\frac{\sqrt{3}}{4}$
  2. $\frac{\pi}{6}+\frac{\sqrt{3}}{8}$
  3. $\frac{\pi}{6}-\frac{\sqrt{3}}{8}$
  4. $\frac{\pi}{2}-\frac{\sqrt{3}}{4}$
  1. Area of hidden portion of lower circle is.
  1. $\bigg(\frac{2\pi}{3}+\frac{\sqrt{3}}{2}\bigg)\text{ sq.units}$
  2. $\bigg(\frac{\pi}{3}-\frac{\sqrt{3}}{8}\bigg)\text{ sq.units}$
  3. $\bigg(\frac{\pi}{3}+\frac{\sqrt{3}}{8}\bigg)\text{ sq.units}$
  4. $\bigg(\frac{2\pi}{3}-\frac{\sqrt{3}}{2}\bigg)\text{ sq.units}$
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Q 26Case study (4 Marks)4 Marks
Consider the following equation of curve $y^2 = 4x$ and straight line $x + y = 3.$
Based on the above information, answer the following questions.
  1. The line $x + y = 3$ cuts the $x-$axis and $y-$axis respectively at.
  1. $(0, 2), (2, 0)$
  2. $(3, 3), (0, 0)$
  3. $(0, 3), (3, 0)$
  4. $(3, 0), (0, 3)$
  1. Point(s) of intersection of two given curves is $($are$).$
  1. $(1, -2), (-9, 6)$
  2. $(2, 1), (-6, 9)$
  3. $(1, 2), (9, -6)$
  4. None of these.
  1. Which of the following shaded portion re present the area bounded by given curves?
    4. None of these
  1. Value of the integral $\int\limits_{-6}^{2}(3-\text{y})\text{ dy}$ is
  1. $10$
  2. $20$
  3. $30$
  4. $40$
  1. Value of area bounded by given curves is.
  1. $56\text{ sq.units}$
  2. $\frac{63}{5}\text{ sq. units}$
  3. $\frac{64}{3}\text{ sq. units}$
  4. $31\text{ sq.units}$
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Q 27Case study (4 Marks)4 Marks
A mirror in the shape of an ellipse represented by $\frac{\text{x}^2}{9}+-\frac{\text{y}^2}{4}=1$ was hanging on the wall. Arun and his sister were playing with ball inside the house, even their mother refused to do so. All of sudden, ball hit the mirror and got a scratch in the shape of line represented by $\frac{\text{x}}{3}+\frac{\text{y}}{2}=1$ Based on the above information, answer the following questions.
  1. Point(s) of intersection of ellipse and scratch (straight line) is (are).
  1. (0, 2), (3, 0)
  2. (2, 0), (3, 0)
  3. (2, 3), (0, 0)
  4. (0, 3), (3, 0)
  1. Area of smaller region bounded by the ellipse and line is represented by.
  1. The value of $\frac{2}{3}\int\limits_{0}^{3}\sqrt{9-\text{x}^2}\text{dx}$ is.
    1. $\frac{\pi}{2}$
    2. $\pi$
    3. $\frac{3\pi}{2}$
    4. $\frac{\pi}{4}$
  1. The value of  $2\int\limits_{0}^{3}\bigg(1-\frac{\text{x}}{3}\bigg)\text{dx}$ is.
    1. 0
    2. 1
    3. 2
    4. 3
  1. Area of the smaller region bounded by the mirror and scratch is.
  1. $3\Big(\frac{\pi}{2}+1\Big)\text{ sq.units}$
  2. $\Big(\frac{\pi}{2}+1\Big)\text{ sq.units}$
  3. $\Big(\frac{\pi}{2}-1\Big)\text{ sq.units}$
  4. $3\Big(\frac{\pi}{2}-1\Big)\text{ sq.units}$
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Q 28Case study (4 Marks)4 Marks
Graphs of two function $\text{f}(\text{x})=\text{sin}\text{ x}$ and $\text{(g)}\text{x}=\text{cos}\text{ x}$ is given below:

Based on the above information, answer the following questions.
  1. In $(0, \pi)$, the curves $\text{f}(\text{x})=\text{sin}\text{ x}$ and $\text{g}\text{ (x)}=\text{cos}\text{ x}$ at $\text{x}=$ 
    1. $\frac{\pi}{2}$
    2. $\frac{\pi}{3}$
    3. $\frac{\pi}{4}$
    4. ${\pi}$
  2. Value of $\int\limits_{0}^{\frac{\pi}{4}}\text{sin}\text{ x}\text{ dx}$ is.
    1. $1-\frac{1}{\sqrt{2}}$
    2. $1+\frac{1}{\sqrt{2}}$
    3. $2-\frac{1}{\sqrt{2}}$
    4. $2+\frac{1}{\sqrt{2}}$
  1. Value of $\int\limits_\frac{\pi}{4}^{\frac{\pi}{2}}\text{cos}\text{ x}\text{ dx}$ is.
    1. $1+\frac{1}{\sqrt{2}}$
    2. $1-\frac{1}{\sqrt{2}}$
    3. $2-\sqrt{2}$
    4. $2+\sqrt{2}$
  2. Value of $\int\limits_{0}^{\pi}\text{sin}\text{ x}\text{ dx}$ is.
  1. 0
  2. 1
  3. 2
  4. -2
  1. Value of $\int\limits_{0}^\frac{\pi}{2}\text{sin}\text{ x}\text{ dx}$ is.
  1. 0
  2. 1
  3. 3
  4. 4
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Q 29Fill In The Blanks[1 Marks ]1 Mark
A mirror in the shape of an ellipse represented by $\frac{\text{x}^2}{9}+-\frac{\text{y}^2}{4}=1$ was hanging on the wall. Arun and his sister were playing with ball inside the house, even their mother refused to do so. All of sudden, ball hit the mirror and got a scratch in the shape of line represented by $\frac{\text{x}}{3}+\frac{\text{y}}{2}=1$ Based on the above information, answer the following questions.
  1. Point(s) of intersection of ellipse and scratch (straight line) is (are).
  1. (0, 2), (3, 0)
  2. (2, 0), (3, 0)
  3. (2, 3), (0, 0)
  4. (0, 3), (3, 0)
  1. Area of smaller region bounded by the ellipse and line is represented by.
  1. The value of $\frac{2}{3}\int\limits_{0}^{3}\sqrt{9-\text{x}^2}\text{dx}$ is.
    1. $\frac{\pi}{2}$
    2. $\pi$
    3. $\frac{3\pi}{2}$
    4. $\frac{\pi}{4}$
  1. The value of  $2\int\limits_{0}^{3}\bigg(1-\frac{\text{x}}{3}\bigg)\text{dx}$ is.
    1. 0
    2. 1
    3. 2
    4. 3
  1. Area of the smaller region bounded by the mirror and scratch is.
  1. $3\Big(\frac{\pi}{2}+1\Big)\text{ sq.units}$
  2. $\Big(\frac{\pi}{2}+1\Big)\text{ sq.units}$
  3. $\Big(\frac{\pi}{2}-1\Big)\text{ sq.units}$
  4. $3\Big(\frac{\pi}{2}-1\Big)\text{ sq.units}$
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Q 30Fill In The Blanks[1 Marks ]1 Mark
A child cut a pizza with a knife. Pizza is circular in shape which is represented by $x^2 + y^2 = 4$ and sharp edge of knife represents a straight line given by $\text{x}=\sqrt{3\text{y}}$ Based on the above information, answer the following questions.
  1. The point$(s)$ of intersection of the edge of knife $($line$)$ and pizza shown in the figure is $($are$)$.
  1. $(1, \sqrt{3}),(-1,-\sqrt{3})$
  2. $(\sqrt{3},1),(-\sqrt{3,}-1)$
  3. $(\sqrt{2,}0),(0,\sqrt{3})$
  4. $(-\sqrt{3,}),(1,-\sqrt{3})$
  1. Which of the following shaded portion represent the smaller area bounded by pizza and edge of knife in first quadrant?
  1. Value of area of the region bounded by circular pizza and edge of knife in first quadrant is.
  1. $\frac{\pi}{2}\text{ sq.units}$
  2. $\frac{\pi}{3}\text{ sq.units}$
  3. $\frac{\pi}{5}\text{ sq.units}$
  4. $\pi\text{ sq.units}$
  1. Area of each slice of pizza when child cut the pizza into $4$ equal pieces is.
  1. $\pi\text{ sq.units}$
  2. $\frac{\pi}{2}\text{ sq.units}$
  3. $3\pi\text{ sq.units}$
  4. $2\pi\text{ sq.units}$
  1. Area of whole pizza is.
  1. $3\pi\text{ sq.units}$
  2. $2\pi\text{ sq.units}$
  3. $5\pi\text{ sq.units}$
  4. $4\pi\text{ sq.units}$
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Q 31Fill In The Blanks[1 Marks ]1 Mark
Graphs of two function $\text{f}(\text{x})=\text{sin}\text{ x}$ and $\text{(g)}\text{x}=\text{cos}\text{ x}$ is given below:

Based on the above information, answer the following questions.
  1. In $(0, \pi)$, the curves $\text{f}(\text{x})=\text{sin}\text{ x}$ and $\text{g}\text{ (x)}=\text{cos}\text{ x}$ at $\text{x}=$ 
    1. $\frac{\pi}{2}$
    2. $\frac{\pi}{3}$
    3. $\frac{\pi}{4}$
    4. ${\pi}$
  2. Value of $\int\limits_{0}^{\frac{\pi}{4}}\text{sin}\text{ x}\text{ dx}$ is.
    1. $1-\frac{1}{\sqrt{2}}$
    2. $1+\frac{1}{\sqrt{2}}$
    3. $2-\frac{1}{\sqrt{2}}$
    4. $2+\frac{1}{\sqrt{2}}$
  1. Value of $\int\limits_\frac{\pi}{4}^{\frac{\pi}{2}}\text{cos}\text{ x}\text{ dx}$ is.
    1. $1+\frac{1}{\sqrt{2}}$
    2. $1-\frac{1}{\sqrt{2}}$
    3. $2-\sqrt{2}$
    4. $2+\sqrt{2}$
  2. Value of $\int\limits_{0}^{\pi}\text{sin}\text{ x}\text{ dx}$ is.
  1. 0
  2. 1
  3. 2
  4. -2
  1. Value of $\int\limits_{0}^\frac{\pi}{2}\text{sin}\text{ x}\text{ dx}$ is.
  1. 0
  2. 1
  3. 3
  4. 4
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Q 32Fill In The Blanks[1 Marks ]1 Mark
Consider the following equations of curves $x^2 = y$ and $y = x. $ On the basis of above information, answer the following questions.
  1. The point$(s)$ of intersection of both the curves is $($are$).$
  1. $(0, 0)(2, 2)$
  2. $(0, 0)(1, 1$)
  3. $(0, 0)(-1, -1)$
  4. $(0, 0)(-2, -2)$
  1. Area bounded by the curves is represented by which of the following graph?
  1. The value of the integral $\int\limits_{1}^{0}\text{x}\ \text{dx}$ is.
  1. $\frac{1}{4}$
  2. $\frac{1}{3}$
  3. $\frac{1}{2}$
  4. $1$
  1. The value of the integral $\int\limits_{0}^{1}\text{x}^2\ \text{dx}$ is.
  1. $\frac{1}{4}$
  2. $\frac{1}{3}$
  3. $\frac{1}{2}$
  4. $1$
  1. The value of area bounded by the curves $x^2 = y$ and $x = y$ is.
  1. $\frac{1}{6}\text{ sq}.\text{unit}$
  2. $\frac{1}{3}\text{ sq}.\text{unit}$
  3. $\frac{1}{2}\text{ sq}.\text{unit}$
  4. ${1}\text{ sq}.\text{unit}$
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Q 33Fill In The Blanks[1 Marks ]1 Mark
Location of three houses of a society is represented by the points A(-1, 0), B(1, 3) and C(3, 2) as shown in figure.

Based on the above information, answer the following questions
  1. Equation of line AB is.
    1. $\text{y}=\frac{3}{2}(\text{x}+1)$
    2. $\text{y}=\frac{3}{2}(\text{x}-1)$
    3. $\text{y}=\frac{1}{2}(\text{x}+1)$
    4. $\text{y}=\frac{1}{2}(\text{x}-1)$
  2. Equation of line BC is.
    1. $\text{y}=\frac{1}{2}\text{x}-\frac{7}{2}$
    2. $\text{y}=\frac{3}{2}\text{x}-\frac{7}{2}$
    3. $\text{y}=\frac{-1}{2}\text{x}+\frac{7}{2}$
    4. $\text{y}=\frac{3}{2}\text{x}+\frac{7}{2}$
  3. Area of region ABCD is.
  1. 2 sq. units
  2. 4 sq. units
  3. 6 sq. units
  4. 8 sq. units
  1. Area of $\triangle\text{ADC}$ is,
  1. 4 sq. units
  2. 8 sq. units
  3. 16 sq. units
  4. 32 sq. units
  1. Area of $\triangle\text{ABC}$ is.
  1. 3 sq. units
  2. 4 sq. units
  3. 5 sq. units
  4. 6 sq. units
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