Bihar Board 12th Maths Objective Answers Chapter 8 Application of Integrals

Bihar Board 12th Maths Objective Questions and Answers

Bihar Board 12th Maths Objective Answers Chapter 8 Application of Integrals

Question 1.
If y = 2 sin x + sin 2x for 0 ≤ x ≤ 2π, then the area enclosed by the curve and x-axis is
(a) \(\frac{9}{2}\) sq. units
(b) 8 sq. units
(c) 12 sq. units
(d) 4 sq. units
Answer:
(c) 12 sq. units

Question 2.
The area bounded by the curve y = x² + 4x + 5, the axes of coordinates and minimum ordinate is
(a) \(3 \frac{2}{3}\) sq. units
(b) \(4 \frac{2}{3}\) sq. units
(c) \(5 \frac{2}{3}\) sq. units
(d) None of these
Answer:
(b) \(4 \frac{2}{3}\) sq. units

Question 3.
The area of the ellipse \(\frac{x^{2}}{4^{2}}+\frac{y^{2}}{9^{2}}=1\) is
(a) 6π sq. units
(b) \(\frac{\pi\left(a^{2}+b^{2}\right)}{4}\) sq. units
(c) p(a + b) sq. units
(d) none of these
Answer:
(d) none of these

Question 4.
The area bounded by the curve 2x² + y² = 2 is
(a) π sq. units
(b) √2π sq. units
(c) \(\frac{\pi}{2}\) sq. units
(d) 2π sq. units
Answer:
(b) √2π sq. units

Question 5.
Area of the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) is
(a) 4πab sq.units
(b) 2πab sq.units
(c) πab sq.units
(d) \(\frac{\pi a b}{2}\) sq.units
Answer:
(c) πab sq.units

Question 6.
Determine the area under the curve \(y=\sqrt{a^{2}-x^{2}}\) included between the lines x = 0 and x = a.
(a) \(\frac{\pi a^{a}}{4}\)
(b) \(\frac{\pi a^{3}}{4}\)
(c) \(\frac{\pi a^{2}}{8}\)
(d) None of these
Answer:
(a) \(\frac{\pi a^{a}}{4}\)

Question 7.
The area enclosed by curve \(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\) is
(a) 10π sq. units
(b) 20π sq. units
(c) 5π sq. units
(d) 4π sq. units
Answer:
(b) 20π sq. units

Question 8.
The area bounded by the curve y = x² – 1 and the straight line x + y = 3 is
(a) \(\frac{9}{2}\) sq. units
(b) 4 sq. units
(c) \(\frac{7 \sqrt{17}}{2}\) sq. units
(d) \(\frac{17 \sqrt{17}}{6}\) sq. units
Answer:
(d) \(\frac{17 \sqrt{17}}{6}\) sq. units

Question 9.
The area of the region R = ((x, y) : |x| ≤ |y| and x² + y² ≤ 1) is
(a) \(\frac{3 \pi}{8}\) sq. units
(b) \(\frac{5 \pi}{8}\) sq. units
(c) \(\frac{\pi}{2}\) sq. units
(d) \(\frac{\pi}{8}\) sq. units
Answer:
(c) \(\frac{\pi}{2}\) sq. units

Question 10.
The area enclosed between the curve y² = 4x and the line y = x is
(a) \(\frac{8}{3}\) sq. units
(b) \(\frac{4}{3}\) sq. units
(c) \(\frac{2}{3}\) sq. units
(d) \(\frac{1}{2}\) sq. units
Answer:
(a) \(\frac{8}{3}\) sq. units

Question 11.
The area bounded by the lines y = |x – 2|, x = 1, x = 3 and the x-axis is
(a) 1 sq. units
(b) 2 sq. units
(c) 3 sq. units
(d) 4 sq. units
Answer:
(b) 2 sq. units

Question 12.
Area of the region bounded by the curve y = x² and the line y = 4 is
(a) \(\frac{11}{3}\) sq. units
(b) \(\frac{32}{3}\) sq. units
(c) \(\frac{43}{3}\) sq. units
(d) \(\frac{47}{3}\) sq. units
Answer:
(b) \(\frac{32}{3}\) sq. units

Question 13.
Area of the smaller region bounded by x² + y² = 9 and the line x = 1 is
(a) (2 – 3 sec^-1 3) sq. units
(b) (√8 – 3sec^-1 3) sq.units
(c) (9sec^-1 3 – √8) sq. units
(d) (sec^-1 3 – 3√8) sq.units
Answer:
(c) (9sec^-1 3 – √8) sq. units

Question 14.
The area bounded by the curve y² = x, line y = 4 and y-axis is
(a) \(\frac{16}{3}\) sq. units
(b) \(\frac{64}{3}\) sq. units
(c) 7√2 sq. units
(d) none of these
Answer:
(b) \(\frac{64}{3}\) sq. units

Question 15.
The area bounded by the curve x = 3y² – 9 and the line x = 0, y = 0 and y = 1 is
(a) 8 sq. units
(b) \(\frac{8}{3}\) sq. units
(c) \(\frac{3}{8}\) sq. units
(d) 3 sq. units
Answer:
(a) 8 sq. units

Question 16.
Area bounded by the curve y² = 16x and line y = mx is \(\frac{2}{3}\) then m is equal to
(a) 3
(b) 4
(c) 1
(d) 2
Answer:
(b) 4

Question 17.
Find the area enclosed by parabola y² = x and the line y + x = 2 and the x-axis.
(a) \(\frac{5}{6}\) sq. units
(b) \(\frac{7}{6}\) sq. units
(c) \(\frac{6}{7}\) sq. units
(d) \(\frac{4}{7}\) sq. units
Answer:
(b) \(\frac{7}{6}\) sq. units

Question 18.
The area bounded by the curve x² + y² = 1 and 1st quadrant is
(a) \(\frac{\pi}{4}\) sq.units
(b) \(\frac{\pi}{2}\) sq. units
(c) \(\frac{\pi}{3}\) sq.units
(d) \(\frac{\pi}{6}\) sq.units
Answer:
(a) \(\frac{\pi}{4}\) sq.units

Question 19.
Area bounded by the curve y = cos x between x = 0 and x = \(\frac{3 \pi}{2}\) is
(a) 1 sq. units
(b) 2 sq. units
(c) 3 sq. units
(d) 4 sq. units
Answer:
(c) 3 sq. units

Question 20.
The area of the region bounded by the curve \(y=\sqrt{4-x^{2}}\) and x-axis is
(a) 8π sq. units
(b) 2π sq. units
(c) 16π sq. units
(d) 6π sq. units
Answer:
(b) 2π sq. units

Question 21.
The area enclosed by the curve y = √x and x = -√y , the circle x² + y² = 2 above the x-axis is
(a) \(\frac{\pi}{4}\) sq. units
(b) \(\frac{3 \pi}{2}\) sq. units
(c) π sq. units
(d) \(\frac{\pi}{2}\) sq. units
Answer:
(d) \(\frac{\pi}{2}\) sq. units

Question 22.
The ratio in which the x-axis divides the area of the region bounded by the curves y = x² – 4x and y = 2x – x²
(a) 4 : 23
(b) 4 : 27
(c) 4 : 19
(d) none of these
Answer:
(a) 4 : 23

Question 23.
Area bounded by the lines y = |x| and y = 1 – |x – 1| is equal to
(a) 4 sq. units
(b) 6 sq. units
(c) 2 sq. units
(d) 8 sq. units
Answer:
(a) 4 sq. units

Question 24.
The area bounded by the lines y = |x – 1| and y = 3 – |x| is
(a) 2 sq. units
(b) 3 sq. units
(c) 4 sq. units
(d) 6 sq. units
Answer:
(c) 4 sq. units

Question 25.
The area bounded by the line y = 2x – 2, y = -x and x-axis is given by
(a) \(\frac{9}{2}\) sq. units
(b) \(\frac{43}{6}\) sq. units
(c) \(\frac{35}{6}\) sq. units
(d) None of these
Answer:
(d) None of these

Question 26.
The area of smaller portion bounded by |y| = -x + 1 and y² = 4x is
(a) 1 sq. units
(b) 2 sq. units
(c) 3 sq. units
(d) none of these
Answer:
(d) none of these

Question 27.
The area lying above x-axis and included between the circle x² + y² = 8x and inside of parabola y² = 4x is
(a) \(\frac{1}{3}\) (2 + 3π) sq. units
(b) \(\frac{2}{3}\) (4 + 3π) sq. units
(c) (6 + 3π) sq. units
(d) \(\frac{4}{3}\) (8 + 3π) sq. units
Answer:
(d) \(\frac{4}{3}\) (8 + 3π) sq. units

Question 28.
Find the area enclosed by the parabola 4y = 3x² and the line 2y = 3x + 12.
(a) 27 sq. units
(b) 28 sq. units
(c) 54 sq. units
(d) 30 sq. units
Answer:
(a) 27 sq. units

Question 29.
The area included between the curves x² = 4by and y² = 4ax
(a) 16ab sq. units
(b) \(\frac{16 a b}{3}\) sq. units
(c) 4ab sq. units
(d) 16πab sq. units
Answer:
(b) \(\frac{16 a b}{3}\) sq. units

Question 30.
Area of the region between the curves x² + y² = π², y = sin x and y-axis in first quadrant is
(a) \(\left(\frac{\pi^{3}-8}{4}\right)\) sq. units
(b) \(\left(\frac{\pi^{3}-4}{8}\right)\) sq. units
(c) \(\left(\frac{\pi^{2}-8}{4}\right)\) sq. units
(d) \(\left(\frac{\pi^{2}-4}{8}\right)\) sq. units
Answer:
(a) \(\left(\frac{\pi^{3}-8}{4}\right)\) sq. units

Question 31.
The area bounded by the curves x² + y² = 9 and y² = 8x is
(a) 0 sq. units
(b) \(\left(\frac{2 \sqrt{2}}{3}+\frac{9 \pi}{2}-9 \sin ^{-1} \frac{1}{3}\right)\) sq. units
(c) 16π sq. units
(d) None of these
Answer:
(b) \(\left(\frac{2 \sqrt{2}}{3}+\frac{9 \pi}{2}-9 \sin ^{-1} \frac{1}{3}\right)\) sq. units

Question 32.
The area bounded by the curves y = sin x, y = cos x and x = 0 is
(a) (√2 – 1) sq. units
(b) 1 sq. units
(c) √2 sq. units
(d) (1 + √2) sq. units
Answer:
(a) (√2 – 1) sq. units

Question 33.
The area common to the circle x² + y² = 16a² and the parabola y² = 6ax is
(a) \(\frac{4 a^{2}}{3}(4 \pi-\sqrt{3})\) sq. units
(b) \(\frac{4 a^{2}}{3}(8 \pi-3) \text { sq. units }\) sq. units
(c) \(\frac{4 a^{2}}{3}(4 \pi+\sqrt{3})\) sq. units
(d) None of these
Answer:
(c) \(\frac{4 a^{2}}{3}(4 \pi+\sqrt{3})\) sq. units

Question 34.
The area included between curves y = x² – 3x + 2 and y = -x² + 3x – 2 is
(a) \(\frac{1}{6}\) sq. units
(b) \(\frac{1}{2}\) sq. units
(c) 1 sq. units
(d) \(\frac{1}{3}\) sq. units
Answer:
(d) \(\frac{1}{3}\) sq. units

Question 35.
The area bounded by x = – 4y² and x – 1 = -5y² is
(a) 1 sq. unit
(b) \(\frac{2}{3}\) sq. units
(c) \(\frac{2}{3}\) sq. units
(d) 2 sq. units
Answer:
(c) \(\frac{2}{3}\) sq. units

Question 36.
The area bounded by the curves \(y=-\sqrt{4-x^{2}}\), x² = -√2y and x = y is
(a) \(\left(\pi+\frac{1}{3}\right)\) sq. units
(b) \(\left(\pi-\frac{1}{3}\right)\) sq. units
(c) \(\left(\pi+\frac{2}{3}\right)\) sq. units
(d) \(\left(\pi-\frac{2}{3}\right)\) sq. units
Answer:
(a) \(\left(\pi+\frac{1}{3}\right)\) sq. units

Question 37.
The area common to the ellipses \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) and \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1\), 0 < b < a is
(a) \((a+b)^{2} \tan ^{-1} \frac{b}{a}\)
(b) \((a+b)^{2} \tan ^{-1} \frac{a}{b}\)
(c) \(4 a b \tan ^{-1} \frac{b}{a}\)
(d) \(4 a b \tan ^{-1} \frac{a}{b}\)
Answer:
(c) \(4 a b \tan ^{-1} \frac{b}{a}\)

Question 38.
The area enclosed by the parabola y² = 2x and tangents through the point (-2, 0) is
(a) 3 sq. units
(b) 4 sq. units
(c) \(\frac{4}{3}\) sq. units
(d) \(\frac{8}{3}\) sq. units
Answer:
(d) \(\frac{8}{3}\) sq. units

Question 39.
The area bounded by the lines y = 4x + 5, y = 5 – x and 4y = x + 5 is
(a) \(\frac{15}{2}\) sq. units
(b) \(\frac{9}{2}\) sq. units
(c) \(\frac{13}{2}\) sq. units
(d) None of these
Answer:
(a) \(\frac{15}{2}\) sq. units

Question 40.
The area bounded by the curves x + 2y² = 0 and x + 3y² = 1 is
(a) 1 sq. units
(b) \(\frac{1}{3}\) sq. units
(c) \(\frac{2}{3}\) sq. units
(d) \(\frac{4}{3}\) sq. units
Answer:
(d) \(\frac{4}{3}\) sq. units

Question 41.
The area bounded by \(y=(2 x)^{1 / 2}\) and \(x=(2 y)^{1 / 2}\) is
(a) \(\frac{4}{3}\) sq. units
(b) \(\frac{13}{2}\) sq. units
(c) \(\frac{12}{5}\) sq. units
(d) \(\frac{4}{25}\) sq. units
Answer:
(a) \(\frac{4}{3}\) sq. units

Question 42.
The area of the region {(x, y) : y² = x, x² + y² = 2} is
(a) \(\left(\frac{\pi}{4}-\frac{1}{3}\right)\) sq. units
(b) \(\left(\frac{\pi}{4}+\frac{1}{3}\right)\) sq. units
(c) \(\left(\frac{\pi}{4}-\frac{1}{6}\right)\) sq. units
(d) \(\left(\frac{\pi}{2}+\frac{1}{3}\right)\) sq. units
Answer:
(d) \(\left(\frac{\pi}{2}+\frac{1}{3}\right)\) sq. units

Question 43.
The area of the circle 4x² + 4y² = 9 which is interior to the parabola x² = 4y is
(a) \(\frac{\sqrt{2}}{6}+\frac{9}{4} \sin ^{-1}\left(\frac{2 \sqrt{2}}{3}\right)\) sq. units
(b) \(\frac{\sqrt{2}}{6}-\frac{1}{4} \sin ^{-1}\left(\frac{2 \sqrt{2}}{3}\right)\) sq. units
(c) \(\frac{3}{2}\) sq. units
(d) \(\frac{7}{2}\) sq. units
Answer:
(a) \(\frac{\sqrt{2}}{6}+\frac{9}{4} \sin ^{-1}\left(\frac{2 \sqrt{2}}{3}\right)\) sq. units

Question 44.
The area bounded by the curve x² = 4y = 4y + 4 and line 3x + 4y = 0 is
(a) \(\frac{25}{4}\) sq. units
(b) \(\frac{125}{8}\) sq. units
(c) \(\frac{125}{16}\) sq. units
(d) \(\frac{125}{24}\) sq. units
Answer:
(d) \(\frac{125}{24}\) sq. units

Question 45.
The area enclosed between the graph of y = x³ and the lines x = 0, y = 1, y = 8 is
(a) \(\frac{45}{4}\)
(b) 14
(c) 7
(d) none of these
Answer:
(a) \(\frac{45}{4}\)