Math Practice MCQs

Mathematics - Set Theory

1. $A - B = A$ iff

A.$A \subseteq B$B.$B \subseteq A$C.$A \equiv B$D.$A \cap B = \emptyset$

Mathematics - Quadratic Equations

2. If one root of the equation $(x - 1)(7 - x) = \lambda$ is three times the other, then $\lambda =$

A.3B.4C.5D.1

Mathematics - Definite Integrals

3. Value of $\int_{0}^{\pi/2} \frac{\sin x}{\sin x + \cos x} \, dx$ is

A.0B.$\frac{\pi}{4}$C.$\frac{\pi}{3}$D.$\frac{\pi}{2}$

Mathematics - Integral Calculus

4. The area bounded by $y = |x - 1|$ and $y = 1$ is

A.$2\text{ sq. unit}$B.$1\text{ sq. unit}$C.$\frac{1}{2}\text{ sq. unit}$D.$1\frac{1}{2}\text{ sq. unit}$

Mathematics - Complex Numbers

5. If $\frac{2z_1}{3z_2}$ is a purely imaginary number, then $\left|\frac{z_1 - z_2}{z_1 + z_2}\right|$ is

A.2/3B.3/2C.1D.4/9

Mathematics - Trigonometry

6. If $A, B, C$ are the angles of a $\triangle$, then the value of $\cot\frac{A}{2} + \cot\frac{B}{2} + \cot\frac{C}{2}$ is

A.$\frac{s}{R}$B.$\frac{R}{s}$C.$\frac{\Delta}{s^2}$D.$\frac{s^2}{\Delta}$

Mathematics - Algebra

7. If $x^2 + 6x - 27 > 0$ and $x^2 - 3x - 4 < 0$ then,

A.$x > 3$B.$x < 4$C.$3 < x < 4$D.$x = \frac{7}{2}$

Mathematics - Trigonometry

8. What is the value of $\cos 2100^\circ$?

A.1B.1/2C.-1/2D.0

Mathematics - Probability

9. An unbiased die is tossed until a number greater than 4 appears. The probability that an even number of tosses is needed is

A.1/2B.2/5C.1/5D.2/3

Mathematics - Number Theory

10. The digit at the unit place in number $19^{2005} + 11^{2005} - 9^{2005}$ is

A.2B.1C.0D.8

Mathematics - Arithmetic

11. A man sells two houses for Rs. 1955 each. On one of them he gains 15% and on the other he loses 15%. Find the total gain or loss.

A.90B.120C.85D.0

Mathematics - Coordinate Geometry

12. The slopes of the lines represented by $x^2 + 2hxy + 2y^2 = 0$ are in the ratio $1 : 2$, then $h$ equals

A.$\pm \frac{1}{2}$B.$\pm \frac{3}{2}$C.$\pm 1$D.$\pm 3$

Mathematics - Functions

13. The domain of function $f(x) = \log_2(\log_3(\log_4 x))$ is

A.$(-\infty, 4)$B.$(4, \infty)$C.$(0, 4)$D.$[1, 4)$

Mathematics - Conic Sections

14. The length of latus rectum of the conic $xy + 4 = 0$ is

A.$2\sqrt{2}$B.$4\sqrt{2}$C.$\sqrt{2}$D.4

Mathematics - Sequences and Series

15. 7 times the sum of $1^{\text{st}}$ $n$ natural numbers is equal to sum of their squares, then $n =$

A.9B.10C.11D.14

Mathematics - Sequences and Series

16. If $|x| < 1$, then $\frac{2}{3}x + \frac{3}{2}x^2 + \frac{4}{3}x^3 + \dots \text{ to } \infty$

A.$\frac{x}{1-x} + \log(1 - x)$B.$\frac{x}{1-x} + \log(1 + x)$C.$\frac{x}{1-x} - \log(1 - x)$D.$\frac{x}{1+x} - \log(1 - x)$

Mathematics - Binomial Theorem

17. ${}^{15}C_3 + {}^{15}C_5 + \dots + {}^{15}C_{15} =$

A.$2^{14}$B.$2^{14} - 15$C.$2^{14} + 15$D.$2^{14} - 1$

Mathematics - Matrices and Determinants

18. If $A$ and $B$ are square matrices of the same order and $AB = 3I$, then $A^{-1}$ is equal to

A.$3B$B.$\frac{1}{3}B$C.$3B^{-1}$D.$\frac{1}{3}B^{-1}$

Mathematics - Matrices and Determinants

19. If $\operatorname{adj} B = A$, $|P| = |Q| = 1$, then $\operatorname{adj}(Q^{-1}BP^{-1}) =$

A.$PQ$B.$QAP$C.$PAQ$D.$PA^{-1}Q$

Mathematics - Vectors

20. A force $\vec{F} = 3\hat{i} + 2\hat{j} - 4\hat{k}$ is applied at the point $(1, -1, 2)$. What is the moment of the force about the point $(2, -1, 3)$?

A.$\hat{i} + 4\hat{j} + 4\hat{k}$B.$2\hat{i} + \hat{j} + 2\hat{k}$C.$2\hat{i} - 7\hat{j} - 2\hat{k}$D.$2\hat{i} + 4\hat{j} - \hat{k}$

Mathematics - 3D Geometry

21. A line $L$ passes through the point $P(5, -6, 7)$ and is parallel to the planes $x + y + z = 1$ and $2x - y - 2z = 3$. What is the equation of the line $L$?

A.$\frac{x-5}{-1} = \frac{y+6}{4} = \frac{z-7}{-3}$B.$\frac{x+5}{-1} = \frac{y-6}{4} = \frac{z+7}{-3}$C.$\frac{x-5}{-1} = \frac{y+6}{-4} = \frac{z-7}{3}$D.$\frac{x-5}{-1} = \frac{y+6}{-4} = \frac{z-7}{-3}$

Mathematics - Statistics

22. What is the mean deviation from the mean of the numbers $10, 9, 21, 16, 24$?

A.5.2B.5.0C.4.5D.4.0

Mathematics - Trigonometry

23. If $\sin x = \frac{1}{\sqrt{5}}, \sin y = \frac{1}{\sqrt{10}}$, where $0 < x < \frac{\pi}{2}, 0 < y < \frac{\pi}{2}$, then what is $(x + y)$ equal to?

A.$\pi$B.$\frac{\pi}{2}$C.$\frac{\pi}{4}$D.0

Mathematics - Circles

24. A chord subtends an angle $120^\circ$ at the centre of a unit circle. What is the length of the chord?

A.$\sqrt{2} - 1\text{ units}$B.$\sqrt{3} - 1\text{ units}$C.$\sqrt{2}\text{ units}$D.$\sqrt{3}\text{ units}$

Mathematics - Conic Sections

25. For the hyperbola $\frac{x^2}{\cos^2\alpha} - \frac{y^2}{\sin^2\alpha} = 1$, which of the following remains fixed when $\alpha$ varies?

A.directrixB.verticesC.fociD.eccentricity

Mathematics - Calculus

26. The value of $\lim_{x \to 1} \frac{x + x^2 + \dots + x^n - n}{x - 1}$ is

A.$n$B.$\frac{n+1}{2}$C.$\frac{n(n+1)}{2}$D.$\frac{n(n-1)}{2}$

Mathematics - Calculus

27. What is the derivative of $e^{e^x}$ with respect to $e^x$?

A.$e^{e^x}$B.$e^x$C.$e^{e^x} e^x$D.$e e^x$

Mathematics - Calculus

28. Where does the tangent to the curve $y = e^x$ at the point $(0, 1)$ meet the x-axis?

A.$(1, 0)$B.$(-1, 0)$C.$(2, 0)$D.$\left(-\frac{1}{2}, 0\right)$

Mathematics - Integral Calculus

29. $\int (\ln x)^{-1} \, dx - \int (\ln x)^{-2} \, dx$ is equal to

A.$x(\ln x)^{-1} + C$B.$x(\ln x)^{-2} + C$C.$x(\ln x) + C$D.$x(\ln x)^2 + C$

Mathematics - Differential Equations

30. The order and degree of the following differential equation $\left[1 + \left(\frac{dy}{dx}\right)^2\right]^{5/2} = \frac{d^3y}{dx^3}$ are respectively

A.3, 2B.3, 10C.2, 3D.3, 5

1. $A - B = A$ iff

2. If one root of the equation $(x - 1)(7 - x) = \lambda$ is three times the other, then $\lambda =$

3. Value of $\int_{0}^{\pi/2} \frac{\sin x}{\sin x + \cos x} \, dx$ is

4. The area bounded by $y = |x - 1|$ and $y = 1$ is

5. If $\frac{2z_1}{3z_2}$ is a purely imaginary number, then $\left|\frac{z_1 - z_2}{z_1 + z_2}\right|$ is

6. If $A, B, C$ are the angles of a $\triangle$, then the value of $\cot\frac{A}{2} + \cot\frac{B}{2} + \cot\frac{C}{2}$ is

7. If $x^2 + 6x - 27 > 0$ and $x^2 - 3x - 4 < 0$ then,

8. What is the value of $\cos 2100^\circ$?

9. An unbiased die is tossed until a number greater than 4 appears. The probability that an even number of tosses is needed is

10. The digit at the unit place in number $19^{2005} + 11^{2005} - 9^{2005}$ is

11. A man sells two houses for Rs. 1955 each. On one of them he gains 15% and on the other he loses 15%. Find the total gain or loss.

12. The slopes of the lines represented by $x^2 + 2hxy + 2y^2 = 0$ are in the ratio $1 : 2$, then $h$ equals

13. The domain of function $f(x) = \log_2(\log_3(\log_4 x))$ is

14. The length of latus rectum of the conic $xy + 4 = 0$ is

15. 7 times the sum of $1^{\text{st}}$ $n$ natural numbers is equal to sum of their squares, then $n =$

16. If $|x| < 1$, then $\frac{2}{3}x + \frac{3}{2}x^2 + \frac{4}{3}x^3 + \dots \text{ to } \infty$

17. ${}^{15}C_3 + {}^{15}C_5 + \dots + {}^{15}C_{15} =$

18. If $A$ and $B$ are square matrices of the same order and $AB = 3I$, then $A^{-1}$ is equal to

19. If $\operatorname{adj} B = A$, $|P| = |Q| = 1$, then $\operatorname{adj}(Q^{-1}BP^{-1}) =$

20. A force $\vec{F} = 3\hat{i} + 2\hat{j} - 4\hat{k}$ is applied at the point $(1, -1, 2)$. What is the moment of the force about the point $(2, -1, 3)$?

21. A line $L$ passes through the point $P(5, -6, 7)$ and is parallel to the planes $x + y + z = 1$ and $2x - y - 2z = 3$. What is the equation of the line $L$?

22. What is the mean deviation from the mean of the numbers $10, 9, 21, 16, 24$?

23. If $\sin x = \frac{1}{\sqrt{5}}, \sin y = \frac{1}{\sqrt{10}}$, where $0 < x < \frac{\pi}{2}, 0 < y < \frac{\pi}{2}$, then what is $(x + y)$ equal to?

24. A chord subtends an angle $120^\circ$ at the centre of a unit circle. What is the length of the chord?

25. For the hyperbola $\frac{x^2}{\cos^2\alpha} - \frac{y^2}{\sin^2\alpha} = 1$, which of the following remains fixed when $\alpha$ varies?

26. The value of $\lim_{x \to 1} \frac{x + x^2 + \dots + x^n - n}{x - 1}$ is

27. What is the derivative of $e^{e^x}$ with respect to $e^x$?

28. Where does the tangent to the curve $y = e^x$ at the point $(0, 1)$ meet the x-axis?

29. $\int (\ln x)^{-1} \, dx - \int (\ln x)^{-2} \, dx$ is equal to

30. The order and degree of the following differential equation $\left[1 + \left(\frac{dy}{dx}\right)^2\right]^{5/2} = \frac{d^3y}{dx^3}$ are respectively

Test Result