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

If different permutations of the letters of the word 'MATHEMATICS' are listed as in a dictionary, how many words (with or without meaning) are there in the list before the first word that starts with C ?

Consider the following statements :

1. If f is the subset of Z × Z defined by f = {(xy, x − y); x, y ∈ Z}, then f is a function from Z to Z.

2. If f is the subset of N × N defined by f = {(xy, x + y); x, y ∈ N}, then f is a function from N to N.

Which of the statements given above is/are correct?

Consider the determinant

Δ = $\left|\begin{array}{lll}\text{a}_{11} & \text{a}_{12} & \text{a}_{13} \\ \text{a}_{21} & \text{a}_{22} & \text{a}_{23} \\ \text{a}_{31} & \text{a}_{32} & \text{a}_{33}\end{array}\right|$

If a13 = yz, a23 = zx, a33 = xy and the minors of a13, a23, a33 are respectively (z − y), (z − x), (y − x) then what is the value of Δ ?

If A = $\left(\begin{array}{ccc}1 & 0 & 0 \\ 0 & \cos \theta & \sin \theta \\ 0 & \sin \theta & −\cos \theta\end{array}\right)$, then which of the following are correct?

1. A + adj A is a null matrix

2. A−1 + adj A is a null matrix

3. A − A−1 is a null matrix

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If X is a matrix of order 3 × 3, Y is a matrix of order 2 × 3 and Z is a matrix of order 3 × 2, then which of the following are correct?

1. (ZY)X is a square matrix having 9 entries.

2. Y(XZ) is a square matrix having 4 entries.

3. X(YZ) is not defined.

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For how many quadratic equations, the sum of roots is equal to the product of roots?

Consider the following statements :

1. The set of all irrational numbers between $\sqrt{2}$ and $\sqrt{5}$ is an infinite set.

2. The set of all odd integers less than 100 is a finite set.

Which of the statements given above is/are correct?

Consider the following statements :

1. 2 + 4 + 6 + ........ + 2n = n2 + n

2. The expression n2 + n + 41 always gives a prime number for every natural number n

Which of the above statements is/are correct ?

What is the diameter of a circle inscribed in a regular polygon of 12 sides, each of length 1 cm ?

Let A = {7, 8, 9, 10, 11, 12, 13; 14, 15, 16} and let f ∶ A → N be defined by f(x) = the highest prime factor of x.

How many elements are there in the range of f?

Let R be a relation from N to N defined by R = {(x, y): x, y ∈ N and x2 = y³}. Which of the following are not correct?

1. (x, x) ∈ R for all x ∈ N

2. (x, y) ∈ R ⇒ (y, x) ∈ R

3. (x, y) ∈ R and (y, z) ∈ R ⇒ (x, z) ∈ R

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Consider the following :

1. A ∩ B = A ∩ C ⇒ B = C

2. A ∪ B = A ∪ C ⇒ B = C

Which of the above is/are correct ?

What is the modulus of z?

What is angle θ such that z is purely real ?

where n is an integer

What is angle θ such that z is purely imaginary ?

where n is an integer

What is the value of q if the coefficients of x3 and x6 are equal ?

What is the ratio of the coefficients of middle terms in the expansion (when expanded in ascending powers of x)?

Under what condition the coefficients of x2 and x4 are equal ?

How many 4-letter words each of two vowels and two consonants with or without meaning, can be formed ?

How many 8-letter words with or without meaning, can be formed such that consonants and vowels occupy alternate positions?

How many 8-letter words with or without meaning, can be formed so that all consonants are together?

What is the value of a11C11 + a12C12 + a13C13 ?

What is the value of $a_{21}C_{11}+a_{22}C_{12}+a_{23}C_{13}$?

What is the value of $\left|\begin{array}{lll}\text{a}_{21} & \text{a}_{31} & \text{a}_{11} \\ \text{a}_{23} & \text{a}_{33} & \text{a}_{13} \\ \text{a}_{22} & \text{a}_{32} & \text{a}_{12}\end{array}\right|$ ?

If $\displaystyle\sum_{x=2}^n$​f(x) = 2044, then what is the value of n ?

What is $\displaystyle\sum_{x=1}^5$f(2x − 1) equal to ?

What is $\displaystyle\sum_{x=1}^6$2^x f(x) equal to ?

How many received medals in exactly two of the three sports ?

How many received medals in at least two of three sports ?

How many received medals in exactly one of three sports ?

What is P equal to ?

What is Q equal to ?

What is the minimum value of determinant of A ?

What is the height of the lamp post ?

What is $\frac{\text{AB}}{\sin \text{C}}$ equal to ?

What is cos A + cos B + cos C equal to ?

At what height is the top of the tower above the ground level ?

If θ is the inclination of the tower to the horizontal, then what is cot θ equal to ?

What is the length of the tower ?

What is the value of cosec$\left(−\frac{73 \pi}{3}\right)$ ?

What is the value of $\cos \left(\frac{5 \pi}{17}\right)+\cos \left(\frac{7 \pi}{17}\right)+2 \cos \left(\frac{11 \pi}{17}\right) \cos \left(\frac{\pi}{17}\right)$ ?

In a triangle ABC, a = 4, b = 3, c = 2. What is cos 3C equal to ?

What is cos 36° − cos 72° equal to ?

If sec x = $\frac{25}{24}$ and x lies in the fourth quadrant, then what is the value of tan x + sin x ?

What is the value of $\sin\left(2\text{n}\pi+\frac{5\pi}{6}\right)\sin\left(2\text{n}\pi−\frac{5\pi}{6}\right)$ where n ∈ Z ?

If 1 + 2(sin x + cos x)(sin x − cos x) = 0 where 0 < x < 360°, then how many values does x take ?

Consider the following statements in respect of the line passing through origin and inclining at an angle of 75° with the positive direction of x-axis :

1. The line passes through the point $\left(1, \frac{1}{2−\sqrt{3}}\right)$.

2. The line entirely lies in first and third quadrants.

Which of the statements given above is/are correct ?

If P(3, 4) is the mid-point of a line segment between the axes, then what is the equation of the line?

The base AB of an equilateral triangle ABC with side 8 cm lies along the y-axis such that the mid-point of AB is at the origin and B lies above the origin. What is the equation of line passing through (8, 0) and parallel to the side AC ?

The centre of the circle passing through the origin and making positive intercepts 4 and 6 on the coordinate axes, lies on the line

The centre of an ellipse is at (0, 0), major axis is on the y-axis. If the ellipse passes through (3, 2) and (1, 6), then what is its eccentricity ?

Consider the points A(2, 4, 6), B(−2, −4, −2), C(4, 6, 4), and D(8, 14, 12). Which of the following statements is/are correct?

1. The points are the vertices of a rectangle ABCD.

2. The mid-point of A C is the same as that of BD.

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Consider the equation of a sphere x2 + y2 + z2 − 4x − 6y − 8z − 16 = 0.

Which of the following statements is/are correct ?

1. z-axis is tangent to the sphere.

2. The centre of the sphere lies on the plane x + y + z − 9 = 0.

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A plane cuts intercepts 2, 2, 1 on the coordinate axes. What are the direction cosines of the normal to the plane?

Consider the following statements :

1. The direction ratios of y-axis can be <0, 4, 0>

2. The direction ratios of a line perpendicular to z-axis can be <5, 6, 0>

Which of the statements given above is/are correct?

Let $\vec{\text{a}}$ and $\vec{\text{b}}$ are two unit vectors such that $\vec{\text{a}}+2 \vec{\text{b}}$ and $5\vec{\text{a}}−4\vec{\text{b}}$ are perpendicular. What is the angle between $\vec{\text{a}}$ and $\vec{\text{b}}$ ?

Let $\vec{\text{a}}$, $\vec{\text{b}}$ and $\vec{\text{c}}$ be unit vectors lying on the same plane. What is $\{(3\vec{\text{a}} + 2\vec{\text{b}}) × (5\vec{\text{a}} − 4\vec{\text{c}})\}⋅(\vec{\text{b}} + 2\vec{\text{c}})$ equal to ?

Let z = [y] and y = [x] − x, where [.] is the greatest integer function. If x is not an integer but positive, then what is the value of z ?

​If f(x) = 4x + 1 and g(x) = kx + 2 such that fog(x) = gof(x), then what is the value of k ?

If y = $\frac{x \sqrt{x^2−16}}{2} − 8 \ln\left|x + \sqrt{x^2−16}\right|$, then what is $\frac{\text{dy}}{\text{dx}}$ equal to ?

What is the area of the region (in the first quadrant) bounded by y = $\sqrt{1−\text{x}^2}$, y = x and y = 0 ?

What is the area of the region bounded by x − |y| = 0 and x − 2 = 0 ?

If f(α) = $\sqrt{\sec^2\alpha−1}$, then what is $\frac{f(\alpha)+f(\beta)}{1−f(\alpha) f(\beta)}$ equal to ?

What is $\displaystyle\lim_{x \rightarrow 0} \frac{x}{\sqrt{1−\cos 4x}}$ equal to ?

Let f(x) be a function such that f'(x) = g(x) and f''(x) = −f(x). Let h(x) = {f(x)}² + {g(x)}². Then consider the following statements :

1. h'(3) = 0

2. h(1) = h(2)

Which of the statements given above is/are correct ?

Consider the following statements in respect of the function f(x) = $\left\{\begin{array}{rc}|x|+1, & 0<|x| \leqslant 3 \\ 1, & x = 0\end{array}\right.$

1. The function attains maximum value only at x = 3

2. The function attains local minimum only at x = 0

Which of the statements given above is/are correct ?

What is $\displaystyle\int_0^1 \ln \left(\frac{1}{x}−1\right)$dx equal to ?

If $\displaystyle\int_0^{\pi/2}$(sin⁴ x + cos⁴ x)dx = k, then what is the value of $\displaystyle\int_0^{20 \pi}$(sin⁴ x + cos⁴ x)dx ?

Consider the following statements:

1. The degree of the differential equation $\frac{\text{dy}}{\text{dx}} + \cos \left(\frac{\text{dy}}{\text{dx}}\right)$ = 0 is 1.

2. The order of the differential equation $\left(\frac{\text{d}^2\text{y}}{\text{dx}^2}\right)^3 + \cos \left(\frac{\text{dy}}{\text{dx}}\right) $ = 0 is 2.

Which of the statements above is/are correct?

What is the differential equation of the family of parabolas having a vertex at origin and axis along positive y-axis?

What is the solution of the differential equation (dy − dx) + cos x(dy + dx) = 0 ?

What is the probability of getting a composite number in the list of natural numbers from 1 to 50?

If n > 7, then what is the probability that C(n, 7) is a multiple of 7?

Two numbers x and y are chosen at random from a set of the first 10 natural numbers. What is the probability that (x + y) is divisible by 4 ?

Three fair dice are tossed once. What is the probability that they show different numbers that are in AP?

If P(A) = 0.5, P(B) = 0.7 and P(A ∩ B) = 0.3, then what is the value of P(A' ∩ B') + P(A' ∩ B) + P(A ∩ B') ?

Five coins are tossed once. What is the probability of getting at most four tails ?

Three fair dice are thrown. What is the probability of getting a total greater than or equal to 15 ?

The probability that a person hits a target is 0.5. What is the probability of at least one hit in 4 shots ?

A box contains 2 white balls, 3 black balls, and 4 red balls. What is the number of ways of drawing 3 balls from the box with at least one black ball?

During war, one ship out of 5 was sunk on an average in making a certain voyage. What is the probability that exactly 3 out of 5 ships would arrive safely?

A card is drawn from a pack of 52 cards. A gambler bets that it is either a spade or an ace. The odds against his winning are

The coefficient of correlation between ages of husband and wife at the time of marriage for a given set of 100 couples was noted to be 0.7. Assume that all these couples survive to celebrate the silver jubilee of their marriage. The coefficient of correlation at that point of time will be

The completion of a construction job may be delayed due to strike. The probability of strike is 0.6. The probability that the construction job gets completed on time if there is no strike is 0.85 and the probability that the construction job gets completed on time if there is a strike is 0.35. What is the probability that the construction job will not be completed on time ?

Which one of the following subjects shows highest variability of marks ?

What is the coefficient of variation of marks in Mathematics ?

What is the median of the distribution ?

What is mean deviation about the median ?

What is the mean deviation about the mean ?