Math Practice MCQs

Mathematics - Algebra

1. The number of real roots of the equation $x^2 - 3|x| + 2 = 0$ is

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

Mathematics - Functions

2. If $f(x) = \ln(x + \sqrt{1 + x^2})$, then which one of the following is correct?

A.$f(x) + f(-x) = 0$B.$f(x) - f(-x) = 0$C.$2f(x) = f(-x)$D.$f(x) = 2f(-x)$

Mathematics - Logarithms

3. The value of $\frac{1}{\log_3 e} + \frac{1}{\log_3 e^2} + \frac{1}{\log_3 e^4} + \dots$ up to infinite terms is

A.$\log_3 9$B.0C.1D.$\log_e 3$

Mathematics - Binomial Theorem

4. Consider the expansion of $(1 + x)^{2n+1}$. If the coefficients of $x^r$ and $x^{r+1}$ are equal in the expansion, then $r$ is equal to

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

Mathematics - Permutations and Combinations

5. Rank of the word "GOOGLE" in the dictionary of all words formed by using the letters of the word "GOOGLE".

A.86B.87C.88D.89

Mathematics - Complex Numbers

6. What is $\sqrt{\frac{1+\omega^2}{1+\omega}}$ equal to, where $\omega$ is the cube root of unity?

A.1B.$\omega$C.$\omega^2$D.$i\omega$, where $i = \sqrt{-1}$

Mathematics - Set Theory

7. If $A = \{x : x \text{ is a multiple of } 2\}$, $B = \{x : x \text{ is a multiple of } 5\}$ and $C = \{x : x \text{ is a multiple of } 10\}$, then $A \cap (B \cap C)$ is equal to

A.$A$B.$B$C.$C$D.$\{x : x \text{ is a multiple of } 100\}$

Mathematics - Vectors

8. If $|a| = 2$ and $|b| = 3$, then $|a \times b|^2 + |a \cdot b|^2$ is equal to

A.72B.64C.48D.36

Mathematics - Vectors

9. What is the area of the parallelogram having diagonals $3\hat{i} + \hat{j} - 2\hat{k}$ and $\hat{i} - 3\hat{j} + 4\hat{k}$?

A.$5\sqrt{5} \text{ sq. units}$B.$4\sqrt{5} \text{ sq. units}$C.$5\sqrt{3} \text{ sq. units}$D.$15\sqrt{2} \text{ sq. units}$

Mathematics - Matrices and Determinants

10. If $A = \begin{pmatrix} 1 & 2 \\ 2 & 3 \end{pmatrix}$ and $A^2 - kA - I_2 = 0$, where $I_2$ is the $2 \times 2$ identity matrix, then what is the value of $k$?

A.4B.-4C.8D.-8

Mathematics - Trigonometry

11. Let $f_k(x) = \frac{1}{k}(\sin^k x + \cos^k x)$ where $x \in \mathbb{R}$ and $k \ge 1$. Then $f_4(x) - f_6(x)$ equals

A.1/4B.1/6C.1/3D.1/12

Mathematics - Inverse Trigonometry

12. If $\cos^{-1}\left(\frac{x}{10}\right) + \cos^{-1}\left(\frac{x-2}{10}\right) = \frac{\pi}{2}$, then $x$ is equal to

A.6B.8C.4D.2

Mathematics - Calculus

13. Consider the function $f(x) = \left(\frac{1}{x}\right)^{2x^2}$, where $x > 0$. At what value of $x$ does the function attain maximum value?

A.$e$B.$\sqrt{e}$C.$\frac{1}{\sqrt{e}}$D.$\frac{1}{e}$

Mathematics - Calculus

14. What is the minimum value of the function $f(x) = \log_{10}(x^2 + 2x + 11)$?

A.0B.1C.2D.10

Mathematics - 3D Geometry

15. A plane $P$ passes through the line of intersection of the planes $2x - y + 3z = 2$, $x + y - z = 1$ and the point $(1, 0, 1)$. What is the equation of the plane $P$?

A.$2x + 5y - 2 = 0$B.$5x + 2y - 5 = 0$C.$x + z - 2 = 0$D.$2x - y - 2z = 0$

Mathematics - Coordinate Geometry

16. The area of the figure formed by the lines $ax + by + c = 0$, $ax - by + c = 0$, $ax + by - c = 0$ and $ax - by - c = 0$ is

A.$\frac{c^2}{ab}$B.$\frac{2c^2}{ab}$C.$\frac{c^2}{2ab}$D.$\frac{c^2}{4ab}$

Mathematics - Circles

17. The circle $x^2 + y^2 + 4x - 7y + 12 = 0$ cuts an intercept on the Y-axis equal to

A.1B.3C.4D.7

Mathematics - Conic Sections

18. What is the eccentricity of rectangular hyperbola?

A.$\sqrt{2}$B.$\sqrt{3}$C.$\sqrt{5}$D.$\sqrt{6}$

Mathematics - Calculus

19. What is $\lim_{x \to 0} \frac{e^x - (1+x)}{x^2}$ equal to?

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

Mathematics - Calculus

20. The derivative of $\ln(x + \sin x)$ with respect to $(x + \cos x)$ is

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

Mathematics - Calculus

21. If $x^a y^b = (x - y)^{a+b}$, then the value of $\frac{dy}{dx} - \frac{y}{x}$ is equal to

A.$\frac{a}{b}$B.$\frac{b}{a}$C.1D.0

Mathematics - Calculus

22. Derivative of $y = \sqrt{e^x + \sqrt{e^x + \sqrt{e^x + \dots}}}$

A.$\frac{e^x}{2e^x - 1}$B.$\frac{-e^x}{2e^x - 1}$C.$\frac{-e^x}{2y - 1}$D.$\frac{e^x}{2y - 1}$

Mathematics - Integral Calculus

23. $\int 2x^3 e^{x^2} \, dx =$

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

Mathematics - Definite Integrals

24. $\int_{0}^{\pi} \sin 5x \cos 4x \, dx =$

A.10/9B.0C.5/4D.2/3

Mathematics - Definite Integrals

25. $\int_{3}^{5} \frac{dx}{\sqrt{(x-3)(5-x)}} =$

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

Mathematics - Differential Equations

26. The differential equation of the family of circles passing through the origin and having centres on the X-axis is

A.$2xy\frac{dy}{dx} = x^2 - y^2$B.$2xy\frac{dy}{dx} = y^2 - x^2$C.$2xy\frac{dy}{dx} = x^2 + y^2$D.$2xy\frac{dy}{dx} + x^2 + y^2 = 0$

Mathematics - Integral Calculus

27. Find the area enclosed by the curves $x = \sqrt{y}$ and $y = \sqrt{x}$.

A.1/3B.2/3C.$\frac{2\sqrt{3}}{5}$D.$\sqrt{3}$

Mathematics - Statistics

28. The mean and the standard deviation (s.d.) of 10 observations are 20 and 2 respectively. Each of these 10 observations is multiplied by $p$ and then reduced by $q$. If the new mean and new s.d. become half of their original values, then $q$ is equal to ($p \ne 0 \ \& \ q \ne 0$):

A.-5B.-20C.-10D.10

Mathematics - Probability

29. Two dice are thrown at the same time. Find the probability that the sum of the two numbers appearing on the top of the dice is 8:

A.1/36B.5/36C.7/36D.3/18

Mathematics - Trigonometry

30. The top of a hill when observed from the top and bottom of a building of height $h$ are at angles of elevation $p$ and $q$ respectively. What is the height of the hill?

A.$\frac{h\cot q}{\cot q - \cot p}$B.$\frac{h\cot p}{\cot p - \cot q}$C.$\frac{2h\tan p}{\tan p - \tan q}$D.$\frac{2h\tan p}{\tan q - \tan p}$

1. The number of real roots of the equation $x^2 - 3|x| + 2 = 0$ is

2. If $f(x) = \ln(x + \sqrt{1 + x^2})$, then which one of the following is correct?

3. The value of $\frac{1}{\log_3 e} + \frac{1}{\log_3 e^2} + \frac{1}{\log_3 e^4} + \dots$ up to infinite terms is

4. Consider the expansion of $(1 + x)^{2n+1}$. If the coefficients of $x^r$ and $x^{r+1}$ are equal in the expansion, then $r$ is equal to

5. Rank of the word "GOOGLE" in the dictionary of all words formed by using the letters of the word "GOOGLE".

6. What is $\sqrt{\frac{1+\omega^2}{1+\omega}}$ equal to, where $\omega$ is the cube root of unity?

7. If $A = \{x : x \text{ is a multiple of } 2\}$, $B = \{x : x \text{ is a multiple of } 5\}$ and $C = \{x : x \text{ is a multiple of } 10\}$, then $A \cap (B \cap C)$ is equal to

8. If $|a| = 2$ and $|b| = 3$, then $|a \times b|^2 + |a \cdot b|^2$ is equal to

9. What is the area of the parallelogram having diagonals $3\hat{i} + \hat{j} - 2\hat{k}$ and $\hat{i} - 3\hat{j} + 4\hat{k}$?

10. If $A = \begin{pmatrix} 1 & 2 \\ 2 & 3 \end{pmatrix}$ and $A^2 - kA - I_2 = 0$, where $I_2$ is the $2 \times 2$ identity matrix, then what is the value of $k$?

11. Let $f_k(x) = \frac{1}{k}(\sin^k x + \cos^k x)$ where $x \in \mathbb{R}$ and $k \ge 1$. Then $f_4(x) - f_6(x)$ equals

12. If $\cos^{-1}\left(\frac{x}{10}\right) + \cos^{-1}\left(\frac{x-2}{10}\right) = \frac{\pi}{2}$, then $x$ is equal to

13. Consider the function $f(x) = \left(\frac{1}{x}\right)^{2x^2}$, where $x > 0$. At what value of $x$ does the function attain maximum value?

14. What is the minimum value of the function $f(x) = \log_{10}(x^2 + 2x + 11)$?

15. A plane $P$ passes through the line of intersection of the planes $2x - y + 3z = 2$, $x + y - z = 1$ and the point $(1, 0, 1)$. What is the equation of the plane $P$?

16. The area of the figure formed by the lines $ax + by + c = 0$, $ax - by + c = 0$, $ax + by - c = 0$ and $ax - by - c = 0$ is

17. The circle $x^2 + y^2 + 4x - 7y + 12 = 0$ cuts an intercept on the Y-axis equal to

18. What is the eccentricity of rectangular hyperbola?

19. What is $\lim_{x \to 0} \frac{e^x - (1+x)}{x^2}$ equal to?

20. The derivative of $\ln(x + \sin x)$ with respect to $(x + \cos x)$ is

21. If $x^a y^b = (x - y)^{a+b}$, then the value of $\frac{dy}{dx} - \frac{y}{x}$ is equal to

22. Derivative of $y = \sqrt{e^x + \sqrt{e^x + \sqrt{e^x + \dots}}}$

23. $\int 2x^3 e^{x^2} \, dx =$

24. $\int_{0}^{\pi} \sin 5x \cos 4x \, dx =$

25. $\int_{3}^{5} \frac{dx}{\sqrt{(x-3)(5-x)}} =$

26. The differential equation of the family of circles passing through the origin and having centres on the X-axis is

27. Find the area enclosed by the curves $x = \sqrt{y}$ and $y = \sqrt{x}$.

28. The mean and the standard deviation (s.d.) of 10 observations are 20 and 2 respectively. Each of these 10 observations is multiplied by $p$ and then reduced by $q$. If the new mean and new s.d. become half of their original values, then $q$ is equal to ($p \ne 0 \ \& \ q \ne 0$):

29. Two dice are thrown at the same time. Find the probability that the sum of the two numbers appearing on the top of the dice is 8:

30. The top of a hill when observed from the top and bottom of a building of height $h$ are at angles of elevation $p$ and $q$ respectively. What is the height of the hill?

Test Result