NDA Mathematics 21 April 2024

Q1. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow: $x_i$123... $n$$f_i$1$2^{-1}$$2^{-2}$...$2^{-(n-1)}$ --- What is the mean of the distribution?

Q2. mcq single +2.5 / 0.83

An edible oil is sold at the rates 150, 200, 250, 300 rupees per litre in four consecutive years. Assuming that an equal amount of money is spent on oil by a family in every year during these years, what is the average price of oil in rupees (approximately) per litre?

Q3. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow: The marks obtained by 10 students in a Statistics test are 24, 47, 18, 32, 19, 15, 21, 35, 50 and 41. --- What is the mean deviation of the largest five observations?

Q4. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow: $x_i$123... $n$$f_i$1$2^{-1}$$2^{-2}$...$2^{-(n-1)}$ --- $$ \text { What is } \sum_i^n x_i f_i \text { equal to? } $$

Q5. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow: The marks obtained by 10 students in a Statistics test are 24, 47, 18, 32, 19, 15, 21, 35, 50 and 41. --- What is the variance of the largest five observations?

Q6. mcq single +2.5 / 0.83

From data (-4, 1), (-1, 2), (2, 7) and (3, 1), the regression line of y on x is obtained as $y = a + bx$, then what is the value of $2a + 15b$?

Q7. mcq single +2.5 / 0.83

If two random variables $X$ and $Y$ are connected by relation $\frac{2 X-3 Y}{5 X+4 Y}=4$ and $X$ follows Binomial distribution with parameters $n=10$ and $p=\frac{1}{2}$, then what is the variance of $Y$ ?

Q8. mcq single +2.5 / 0.83

Let $R$ be a relation on the open interval $(-1, 1)$ and is given by $R = \{(x, y) : |x + y| < 2\}$. Then which one of the following is correct?

Q9. mcq single +2.5 / 0.83

Let $A = \{1, 2, 3, 4, 5\}$ and $B = \{6, 7\}$. What is the number of onto functions from $A$ to $B$?

Q10. mcq single +2.5 / 0.83

Let $A = \{x \in \mathbb{R} : -1 <x <1\}$. Which of the following is/are bijective functions from A to itself? 1. $f(x) = x|x|$ 2. $g(x) = \cos(\pi x)$ Select the correct answer using the code given below:

Q11. mcq single +2.5 / 0.83

For any three non-empty sets $A, B, C$, what is $$(A \cup B - \{(A - B) \cup (B - A) \cup (A \cap B)\})$$ equal to?

Q12. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow: Let $f(x) = \frac{x}{\ln x}; (x > 1)$ --- Consider the following statements : 1. $f(x)$ is increasing in the interval $(e, \infty)$ 2. $f(x)$ is decreasing in the interval $(1, e)$ 3. $9 \ln 7 > 7 \ln 9$ Which of the statements given above are correct ?

Q13. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow: Let $f(x) = |x| + 1$ and $g(x) = [x] - 1$, where [.] is the greatest integer function. Let $h(x) = \frac{f(x)}{g(x)}$. --- What is $\lim\limits_{x \to 0-} h(x) + \lim\limits_{x \to 0+} h(x)$ equal to?

Q14. mcq single +2.5 / 0.83

What is the remainder when $2^{120}$ is divided by 7?

Q15. mcq single +2.5 / 0.83

If $a, b, c$ are in HP, then what is $\frac{1}{b-a} + \frac{1}{b-c}$ equal to? 1. $\frac{2}{b}$ 2. $\frac{1}{a} + \frac{1}{c}$ 3. $\frac{1}{2} \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right)$ Select the correct answer using the code given below:

Q16. mcq single +2.5 / 0.83

If $a, b, c$ are in AP; $b, c, d$ are in GP; $c, d, e$ are in HP, then which of the following is/are correct? 1. $a, c,$ and $e$ are in GP 2. $\frac{1}{a}, \frac{1}{c}, \frac{1}{e}$ are in GP Select the correct answer using the code given below:

Q17. mcq single +2.5 / 0.83

If a vector of magnitude 2 units makes an angle $\frac{\pi}{3}$ with $2\hat{i}$, $\frac{\pi}{4}$ with $3\hat{j}$ and an acute angle $\theta$ with $4\hat{k}$, then what are the components of the vector?

Q18. mcq single +2.5 / 0.83

For any vector $\vec{r}$, what is $\left(\vec{r}\cdot\hat{i}\right)\left(\vec{r}\times\hat{i}\right) + \left(\vec{r}\cdot\hat{j}\right)\left(\vec{r}\times\hat{j}\right) + \left(\vec{r}\cdot\hat{k}\right)\left(\vec{r}\times\hat{k}\right)$ equal to?

Q19. mcq single +2.5 / 0.83

Consider the following in respect of moment of a force: 1. The moment of force about a point is independent of point of application of force. 2. The moment of a force about a line is a vector quantity. Which of the statements given above is/are correct?

Q20. mcq single +2.5 / 0.83

Let $\vec{a} = \hat{i} - \hat{j} + \hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} - \hat{k}$. If $\vec{a} \times (\vec{b} \times \vec{a}) = \alpha \hat{i} - \beta \hat{j} + \gamma \hat{k}$, then what is the value of $\alpha + \beta + \gamma$?

Q21. mcq single +2.5 / 0.83

If the direction cosines <l, m, n> of a line are connected by relation $l + 2m + n = 0, 2l - 2m + 3n = 0$, then what is the value of $l^{2} + m^{2} - n^{2}$?

Q22. mcq single +2.5 / 0.83

Let $\vec{a}$ and $\vec{b}$ be two vectors of magnitude 4 inclined at an angle $\frac{\pi}{3}$, then what is the angle between $\vec{a}$ and $\vec{a} - \vec{b}$?

Q23. mcq single +2.5 / 0.83

What is $\tan^{-1} \left( \frac{a}{b} \right) - \tan^{-1} \left( \frac{a - b}{a + b} \right)$ equal to?

Q24. mcq single +2.5 / 0.83

What is $\sqrt{15 + \cot^2 \left( \frac \pi 4 - 2 \cot^{-1} 3 \right)}$ equal to?

Q25. mcq single +2.5 / 0.83

If $\cos^{-1} x = \sin^{-1} x$, then which one of the following is correct?

Q26. mcq single +2.5 / 0.83

Three different numbers are selected at random from the first 15 natural numbers. What is the probability that the product of two of the numbers is equal to third number?

Q27. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow: $A$, $B$ and $C$ are three events such that $P(A) = 0.6$, $P(B) = 0.4$, $P(C) = 0.5$, $P(A \cup B) = 0.8$, $P(A \cap C) = 0.3$ and $P(A \cap B \cap C) = 0.2$ and $P(A \cup B \cup C) \geq 0.85$. --- What is the minimum value of $P(B \cap C)$?

Q28. mcq single +2.5 / 0.83

Let $m = 77^n$. The index $n$ is given a positive integral value at random. What is the probability that the value of $m$ will have $1$ in the units place?

Q29. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow: An unbiased coin is tossed $n$ times. The probability of getting at least one tail is $p$ and the probability of at least two tails is $q$ and $p - q = \frac{5}{32}$. --- What is the value of $n$?

Q30. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow: Let $A$ and $B$ be two events such that $P(A \cup B) \geq 0.75$ and $0.125 \leq P(A \cap B) \leq 0.375$. --- What is the minimum value of $P(A) + P(B)$?

Q31. mcq single +2.5 / 0.83

If a random variable $(x)$ follows binomial distribution with mean 5 and variance 4 , and $5^{23} P(X=3)=\lambda 4^\lambda$, then what is the value of $\lambda$ ?

Q32. mcq single +2.5 / 0.83

A bag contains 5 black and 4 white balls. A man selects two balls at random. What is the probability that both of these are of the same colour?

Q33. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow: An unbiased coin is tossed $n$ times. The probability of getting at least one tail is $p$ and the probability of at least two tails is $q$ and $p - q = \frac{5}{32}$. --- What is the value of $p + q$?

Q34. mcq single +2.5 / 0.83

**Passage:** Let $A$ and $B$ be two events such that $P(A \cup B) \geq 0.75$ and $0.125 \leq P(A \cap B) \leq 0.375$. --- What is the maximum value of $P(A) + P(B)$?

Q35. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow: $A$, $B$ and $C$ are three events such that $P(A) = 0.6$, $P(B) = 0.4$, $P(C) = 0.5$, $P(A \cup B) = 0.8$, $P(A \cap C) = 0.3$ and $P(A \cap B \cap C) = 0.2$ and $P(A \cup B \cup C) \geq 0.85$. --- What is the maximum value of $P(B \cap C)$?

Q36. mcq single +2.5 / 0.83

If the letters of the word 'TIRUPATI' are written down at random, then what is the probability that both Ts are always consecutive?

Q37. mcq single +2.5 / 0.83

ABC is an acute angled isosceles triangle. Two equal sides AB and AC lie on the lines 7x - y - 3 = 0 and x + y - 5 = 0. If θ is one of the equal angles, then what is cotθ equal to?

Q38. mcq single +2.5 / 0.83

ABC is a triangle with A(3, 5). The mid-points of sides AB, AC are at (-1, 2), (6, 4) respectively. What are the coordinates of centroid of the triangle ABC?

Q39. mcq single +2.5 / 0.83

Let $x+2 y+1=0$ and $2 x+3 y+4=0$ are two lines of regression computed from some bivariate data. If $\theta$ is the acute angle between them, then what is the value of $488 \tan 3 \theta$ ?

Q40. mcq single +2.5 / 0.83

Two points $P$ and $Q$ lie on line $y = 2x + 3$. These two points $P$ and $Q$ are at a distance 2 units from another point $R(1, 5)$. What are the coordinates of the points $P$ and $Q$?

Q41. mcq single +2.5 / 0.83

The number of points represented by the equation $x = 5$ on the $xy$-plane is

Q42. mcq single +2.5 / 0.83

If two sides of a square lie on the lines $2x + y - 3 = 0$ and $4x + 2y + 5 = 0$, then what is the area of the square in square units?

Q43. mcq single +2.5 / 0.83

What is the equation to the straight line passing through the point $(-sin\theta, cos\theta)$ and perpendicular to the line $xcos\theta + ysin\theta = 9$?

Q44. mcq single +2.5 / 0.83

If a variable line passes through the point of intersection of the lines $x + 2y - 1 = 0$ and $2x - y - 1 = 0$ and meets the coordinate axes in $A$ and $B$, then what is the locus of the mid-point of $AB$?

Q45. mcq single +2.5 / 0.83

What is $\sin 9^\circ - \cos 9^\circ$ equal to?

Q46. mcq single +2.5 / 0.83

$$ \text { What is } \frac{\sqrt{3} \cos 10^{\circ}-\sin 10^{\circ}}{\sin 25^{\circ} \cos 25^{\circ}} \text { equal to ? } $$

Q47. mcq single +2.5 / 0.83

What is the value of $\tan 65^\circ +2 \tan 45^\circ-2 \tan 40^\circ-\tan 25^\circ$?

Q48. mcq single +2.5 / 0.83

What is the number of solutions of $(\sin \theta - \cos \theta)^2 = 2$ where $-\pi < \theta < \pi$?

Q49. mcq single +2.5 / 0.83

If $f(\theta)=\frac{1}{1+\tan \theta}$ and $\alpha+\beta=\frac{5\pi}{4}$, then what is the value of $f(\alpha) f(\beta)$?

Q50. mcq single +2.5 / 0.83

If $\tan \alpha$ and $\tan \beta$ are the roots of the equation $x^2-6x+8=0$, then what is the value of $\cos(2 \alpha+2 \beta)$?

Q51. mcq single +2.5 / 0.83

What is the value of $\sin10^{\circ} \cdot \sin50^{\circ} + \sin50^{\circ} \cdot \sin250^{\circ} + \sin250^{\circ} \cdot \sin10^{\circ}$ equal to?

Q52. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow: Let $\varphi(a) = \int_{a}^{a + 100 \pi} |\sin x| dx$ --- What is $\varphi(a)$ equal to?

Q53. mcq single +2.5 / 0.83

The non-negative values of $b$ for which the function $\frac{16x^3}{3} - 4bx^2 + x$ has neither maximum nor minimum in the range $x>0$ is

Q54. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow: Let $\varphi(a) = \int_{a}^{a + 100 \pi} |\sin x| dx$ --- What is $\varphi'(a)$ equal to?

Q55. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow: A differentiable function $f(x)$ has a local maximum at $x = 0$. Let $y = 2f(x) + ax - b$. --- Which of the following is/are correct? 1. $f'(0) = 0$ 2. $f''(0) < 0$ Select the correct answer using the code given below:

Q56. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow: A differentiable function $f(x)$ has a local maximum at $x = 0$. Let $y = 2f(x) + ax - b$. --- The function $y$ has a relative maxima at $x = 0$ for

Q57. mcq single +2.5 / 0.83

If (1, −1, 2) and (2, 1, −1) are the end points of a diameter of a sphere $x^2 + y^2 + z^2 + 2ux + 2vy + 2wz − 1 = 0$, then what is $u + v + w$ equal to?

Q58. mcq single +2.5 / 0.83

A line through $(1, −1, 2)$ with direction ratios $\langle 3, 2, 2 \rangle$ meets the plane $x + 2y + 3z = 18$. What is the point of intersection of line and plane?

Q59. mcq single +2.5 / 0.83

If $p$ is the perpendicular distance from origin to the plane passing through $(1, 0, 0)$, $(0, 1, 0)$ and $(0, 0, 1)$, then what is $3p^2$ equal to?

Q60. mcq single +2.5 / 0.83

If $\langle l, m, n \rangle$ are the direction cosines of a normal to the plane $2x − 3y + 6z + 4 = 0$, then what is the value of $49(7l^2 + m^2 − n^2)$?

Q61. mcq single +2.5 / 0.83

In the parabola $y^2 = 8x$, the focal distance of a point P lying on it is 8 units. Which of the following statements is/are correct? 1. The coordinates of $P$ can be $\left(6, 4\sqrt{3}\right)$. 2. The perpendicular distance of $P$ from the directrix of parabola is 8 units. Select the correct answer using the code given below:

Q62. mcq single +2.5 / 0.83

What is the coefficient of $x^{10}$ in the expansion of $(1-x^2)^{20}\left(2-x^2-\frac{1}{x^2}\right)^{-5}$?

Q63. mcq single +2.5 / 0.83

If the 4th term in the expansion of $\left(mx + \frac{1}{x}\right)^n$ is $\frac{5}{2}$, then what is the value of $mn$?

Q64. mcq single +2.5 / 0.83

In a binomial expansion of $(x+y)^{2 n+1}(x-y)^{2 n+1}$, the sum of middle terms is zero. What is the value of $\left(\frac{x^2}{y^2}\right)$ ?

Q65. mcq single +2.5 / 0.83

For $x \geq y > 1$, let $\log_x\left( \frac{x}{y} \right) + \log_y\left(\frac{y}{x}\right) = k$, then the value of $k$ can never be equal to :

Q66. mcq single +2.5 / 0.83

What is the number of solutions of $\log_4(x - 1) = \log_2(x - 3)$?

Q67. mcq single +2.5 / 0.83

If $\log _b a=p, \log _d c=2 p$ and $\log _f e=3 p$, then what is $(a c e)^{\frac{1}{p}}$ equal to ?

Q68. mcq single +2.5 / 0.83

Let $z_1$ and $z_2$ be two complex numbers such that $\left|\frac{z_1 + z_2}{z_1 - z_2}\right| = 1$, then what is $\operatorname{Re} \left(\frac{z_1}{z_2}\right) + 1$ equal to?

Q69. mcq single +2.5 / 0.83

If $z$ is any complex number and $i z^3+z^2-z+i=0$, where $i=\sqrt{-1}$, then what is the value of $(|z|+1)^2$ ?

Q70. mcq single +2.5 / 0.83

If $\omega \neq 1$ is a cube root of unity, then what are the solutions of $(z-100)^3 + 1000 = 0$?

Q71. mcq single +2.5 / 0.83

What is $(1 + i)^4 + (1 - i)^4$ equal to, where $i=\sqrt{-1}$?

Q72. mcq single +2.5 / 0.83

If $x, y$ and $z$ are the cube roots of unity, then what is the value of $xy + yz + zx$?

Q73. mcq single +2.5 / 0.83

If $f(x)=ax-b$ and $g(x)=cx+d$ are such that $f(g(x))=g(f(x))$, then which one of the following holds?

Q74. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow : Let $3f(x) + f\left(\frac{1}{x}\right) = \frac{1}{x} + 1$ --- What is $f(x)$ equal to?

Q75. mcq single +2.5 / 0.83

Which one of the following is correct in respect of $f(x) = \frac{1}{\sqrt{|x| - x}}$ and $g(x) = \frac{1}{\sqrt{x - |x|}}$?

Q76. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow: Let $f(x)$ and $g(x)$ be two functions such that $g(x) = x - \frac{1}{x}$ and $f \circ g(x) = x^3 - \frac{1}{x^3}$. --- What is $g[f(x) - 3x]$ equal to?

Q77. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow: Let $f(x) = |x| + 1$ and $g(x) = [x] - 1$, where [.] is the greatest integer function. Let $h(x) = \frac{f(x)}{g(x)}$. --- Consider the following statements: 1. $f(x)$ is differentiable for all $x < 0$ 2. $g(x)$ is continuous at $x = 0.0001$ 3. The derivative of $g(x)$ at $x = 2.5$ is 1 Which of the statements given above are correct?

Q78. mcq single +2.5 / 0.83

Let $y_1(x)$ and $y_2(x)$ be two solutions of the differential equation $\frac{dy}{dx} = x$. If $y_1(0) = 0$ and $y_2(0) = 4$, then what is the number of points of intersection of the curves $y_1(x)$ and $y_2(x)$?

Q79. mcq single +2.5 / 0.83

If $\frac{dy}{dx} = 2e^xy^3$, $y(0)= \frac{1}{2}$ then what is $4y^2(2-e^x)$ equal to?

Q80. mcq single +2.5 / 0.83

The differential equation, representing the curve $y = e^{x}(a\cos{x} + b\sin{x})$ where $a$ and $b$ are arbitrary constants, is

Q81. mcq single +2.5 / 0.83

What are the order and degree respectively of the differential equation $$ \left\{2-\left(\frac{d y}{d x}\right)^2\right\}^{0-6}=\frac{d^2 y}{d x^2} ? $$

Q82. mcq single +2.5 / 0.83

If (a, b) is the centre and c is the radius of the circle $x^2 + y^2 + 2x + 6y + 1 = 0$, then what is the value of $a^2 + b^2 + c^2$?

Q83. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow: The circle $x^2 + y^2 - 2x = 0$ is partitioned by line $y = x$ in two segments. Let $A_1, A_2$ be the areas of major and minor segments respectively. --- What is the value of $A_1$?

Q84. mcq single +2.5 / 0.83

**Passage:** The circle $x^2 + y^2 - 2x = 0$ is partitioned by line $y = x$ in two segments. Let $A_1, A_2$ be the areas of major and minor segments respectively. --- What is the value of $\frac{2(A_1 + A_2)}{ A_1 - 3A_2}$?

Q85. mcq single +2.5 / 0.83

If $x^2 + mx + n$ is an integer for all integral values of $x$, then which of the following is/are correct? 1. $m$ must be an integer 2. $n$ must be an integer Select the correct answer using the code given below:

Q86. mcq single +2.5 / 0.83

If $a$, $b$, and $c$ $(a > 0, c > 0)$ are in GP, then consider the following in respect of the equation $ax^2 + bx + c = 0$: 1. The equation has imaginary roots. 2. The ratio of the roots of the equation is $1 : \omega$ where $\omega$ is a cube root of unity. 3. The product of roots of the equation is $\left(\frac{b^2}{a^2}\right)$. Which of the statements given above are correct?

Q87. mcq single +2.5 / 0.83

Under which one of the following conditions does the equation $\left(\cos \beta-1\right)x^2+(\cos \beta)x+\sin \beta=0$ in $x$ have a real root for $\beta \in [0, \pi]$?

Q88. mcq single +2.5 / 0.83

If $\sqrt{2}$ and $\sqrt{3}$ are roots of the equation $a_0 + a_1 x + a_2 x^2 + a_3 x^3 + x^4 = 0$ where $a_0, a_1, a_2, a_3$ are integers, then which one of the following is correct?

Q89. mcq single +2.5 / 0.83

What is the eccentricity of the ellipse if the angle between the straight lines joining the foci to an extremity of the minor axis is 90°?

Q90. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow: Let $f(x) = \frac{x}{\ln x}; (x > 1)$ --- Consider the following statements : 1. $f''(e) = \frac{1}{e}$ 2. $f(x)$ attains local minimum value at $x = e$ 3. A local minimum value of $f(x)$ is $e$ Which of the statements given above are correct ?

Q91. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow: Let $f(x)$ and $g(x)$ be two functions such that $g(x) = x - \frac{1}{x}$ and $f \circ g(x) = x^3 - \frac{1}{x^3}$. --- What is $f''(x)$ equal to?

Q92. mcq single +2.5 / 0.83

How many four-digit natural numbers are there such that all of the digits are even?

Q93. mcq single +2.5 / 0.83

What is the number of different matrices, each having 4 entries that can be formed using 1, 2, 3, 4 (repetition is allowed)?

Q94. mcq single +2.5 / 0.83

A triangle $PQR$ is such that 3 points lie on the side $PQ$, 4 points on $QR$ and 5 points on $RP$ respectively. Triangles are constructed using these points as vertices. What is the number of triangles so formed?

Q95. mcq single +2.5 / 0.83

What is the sum of all four-digit numbers formed by using all digits 0, 1, 4, 5 without repetition of digits?

Q96. mcq single +2.5 / 0.83

A man has 7 relatives (4 women and 3 men). His wife also has 7 relatives (3 women and 4 men). In how many ways can they invite 3 women and 3 men so that 3 of them are man's relatives and 3 of them are his wife's relatives?

Q97. mcq single +2.5 / 0.83

If $26! = n8^k$, where $k$ and $n$ are positive integers, then what is the maximum value of $k$?

Q98. mcq single +2.5 / 0.83

Four digit numbers are formed by using the digits 1, 2, 3, 5 without repetition of digits. How many of them are divisible by 4?

Q99. mcq single +2.5 / 0.83

**Passage:** Let $f(x)=|x-1|, g(x)=[x]$ and $h(x)=f(x) g(x)$ where [.] is greatest integer function. --- What is $\int_{0}^{2} h(x) dx$ equal to?

Q100. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow: Let $3f(x) + f\left(\frac{1}{x}\right) = \frac{1}{x} + 1$ --- What is $8\int_1^2 f(x)dx$ equal to?

Q101. mcq single +2.5 / 0.83

What is $\int^{\pi/2}_0 \frac{a+\sin x}{2a+\sin x+\cos x} dx$ equal to?

Q102. mcq single +2.5 / 0.83

**Passage:** Let $f(x)=|x-1|, g(x)=[x]$ and $h(x)=f(x) g(x)$ where [.] is greatest integer function. --- What is $\int\limits_{-1}^{0} h(x) dx$ equal to?

Q103. mcq single +2.5 / 0.83

Let $p=\int_a^b f(x) d x$ and $q=\int_a^b|f(x)| d x$. If $f(x)=e^{-x}$, then which one of the following is correct ?

Q104. mcq single +2.5 / 0.83

What is $\int^{1}_{-1}(3\sin x-\sin 3x)\cos^2 xdx$ equal to?

Q105. mcq single +2.5 / 0.83

If $A=\left[\begin{array}{ccc}\sin 2 \theta & 2 \sin ^2 \theta-1 & 0 \\ \cos 2 \theta & 2 \sin \theta \cos \theta & 0 \\ 0 & 0 & 1\end{array}\right]$, then which of the following statements is/are correct? 1. $A^{-1}=\operatorname{adj} A$ 2. A is skew-symmetric matrix 3. $A^{-1}=A^T$ Select the correct answer using the code given below :

Q106. mcq single +2.5 / 0.83

Let $A$ and $B$ be matrices of order $3 \times 3$. If $|A| = \frac{1}{2 \sqrt{2}}$ and $|B| = \frac{1}{729}$, then what is the value of $|2B(adj(3A))|$?

Q107. mcq single +2.5 / 0.83

Consider the following statements in respect of a non-singular matrix $A$ of order $n$: 1. $A(\text{adj}A^T) = A(\text{adj}A)^T$ 2. If $A^2 = A$, then $A$ is identity matrix of order $n$ 3. If $A^3 = A$, then $A$ is identity matrix of order $n$ Which of the statements given above are correct?

Q108. mcq single +2.5 / 0.83

Consider the following statements in respect of a skew-symmetric matrix $A$ of order $3$: 1. All diagonal elements are zero. 2. The sum of all the diagonal elements of the matrix is zero. 3. $A$ is orthogonal matrix. Which of the statements given above are correct?

Q109. mcq single +2.5 / 0.83

Consider the following statements in respect of two non-singular matrices $A$ and $B$ of the same order $n$: - 1.$adj(AB) = (adjA)(adjB)$ - 2. $adj(AB) = adj(BA)$ - 3. $(AB) adj(AB) - |AB| I_n$ is a null matrix of order $n$ How many of the above statements are correct?

Q110. mcq single +2.5 / 0.83

Consider the following statements: 1. In a triangle $ABC$, if $\cot A \cdot \cot B \cdot \cot C>0$, then the triangle is an acute-angled triangle. 2. In a triangle $ABC$, if $\tan A \cdot \tan B \cdot \tan C > 0$, then the triangle is an obtuse-angled triangle. Which of the statements given above is/are correct?

Q111. mcq single +2.5 / 0.83

In a triangle $ABC$, $AB=16 \text{ cm}, BC=63 \text{ cm}$ and $AC=65 \text{ cm}$. What is the value of $\cos 2A+\cos 2B+\cos 2C$?

Q112. mcq single +2.5 / 0.83

$ABC$ is a triangle such that angle $C = 60^{\circ}$, then what is $\frac{\cos A + \cos B}{\cos \left(\frac{A - B}{2}\right)}$ equal to?

Q113. mcq single +2.5 / 0.83

If $a, b, c$ are the sides of a triangle $ABC$, then what is $$ \begin{vmatrix} a^2 & b \sin A & c \sin A \\ b \sin A & 1 & \cos A \\ c \sin A & \cos A & 1 \end{vmatrix}$$ equal to?

Q114. mcq single +2.5 / 0.83

If *ABC* is a triangle, then what is the value of the determinant $$ \left|\begin{array}{ccc} \cos C & \sin B & 0 \\ \tan A & 0 & \sin B \\ 0 & \tan (B+C) & \cos C \end{array}\right| ? $$

Q115. mcq single +2.5 / 0.83

For what value of $n$ is the determinant $$ \left|\begin{array}{ccc} C(9,4) & C(9,3) & C(10, n-2) \\ C(11,6) & C(11,5) & C(12, n) \\ C(m, 7) & C(m, 6) & C(m+1, n+1) \end{array}\right|=0 $$ for every $m>n$ ?

Q116. mcq single +2.5 / 0.83

If in a triangle $ABC$, $\sin^3A + \sin^3B + \sin^3C = 3\sin A \sin B \sin C$, then what is the value of the determinant $\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}$, where $a$, $b$, $c$ are sides of the triangle?

Q117. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow: Given that $\int \frac{3 \cos x + 4 \sin x}{2 \cos x + 5 \sin x} dx = \frac{\alpha x}{29} + \frac{\beta}{29} \ln |2 \cos x + 5 \sin x| + c$ --- What is the value of $\alpha$ ?

Q118. mcq single +2.5 / 0.83

What is the value of $\alpha$?

Q119. mcq single +2.5 / 0.83

**Passage:** Consider the following for the next two (02) items that follow: Given that $\int \frac{3 \cos x + 4 \sin x}{2 \cos x + 5 \sin x} dx = \frac{\alpha x}{29} + \frac{\beta}{29} \ln |2 \cos x + 5 \sin x| + c$ --- What is the value of $\beta$ ?

Q120. mcq single +2.5 / 0.83

**Passage:** Let $\int \frac{d x}{\sqrt{x+1}-\sqrt{x-1}}=\alpha(x+1)^{\frac{3}{2}}+$ $$ \beta(x-1)^{\frac{3}{2}}+c $$ --- What is the value of $\beta$?

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