Let X be a matrix of order 3 x 3, Y be a matrix of order 2 x 3 and Z be a matrix of order 3 × 2. Which of the following statements are correct?

I. (ZY)X is defined and is a square matrix of order 3.

II. Y(XZ) is defined and is a square matrix of order 2.

III. X(YZ) is not defined.

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Consider the following statements:

I. The set of all irrational numbers between $\sqrt{12}$ and $\sqrt{15}$ is an infinite set.

II. The set of all odd integers less than 1000 is a finite set.

Which of the statements given above is/are correct?

How many 4-digit numbers are there having all digits as odd? 

If ω ≠ 1 is a cube root of unity, then what is (1 + ω - ω²)¹⁰⁰ + (1 - ω + ω²)¹⁰⁰ equal to?  

Let A and B be two square matrices of same order. If AB is a null matrix, then which one of the following is correct? 

In the expansion of (1 + x)^p (1 + x)^q, if the coefficient of x³ is 35, then what is the value of (p + q)? 

If p times the pth term of an AP is equal to q times the qth term (p ≠ q), then what is the (p + q)th term equal to? 

Let p = ln(x), q = ln(x³) and r = ln(x⁵), where x > 1. Which of the following statements is/are correct? 

I. p, q and r are in AP.

Il. p, q and r can never be in GP.

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If $\rm z=\frac{1}{3}\begin{vmatrix}i&2i&1\\\ 2i&3i&2\\\ 3&1&3\end{vmatrix}=x+iy;i=\sqrt{-1}$ then what is modulus of Z equal to?  

What is the value of the sum $\rm \sum\limits_{n=1}^{20}(i^{n-1}+i^n+i^{n+1})$ where i = √-1 ?

Let x >1, y >1, z >1 be in GP. Then $\rm \frac{1}{1+ln x}, \frac{1}{1+lny}, \frac{1}{1+\ln z}$ are

If $\rm \omega=-\frac{1}{2}+i\frac{\sqrt3}{2}$ then what is $\rm \begin{vmatrix}1+\omega &1+\omega^2&\omega+\omega^2\\\ 1&\omega\ & \omega^2\\\ \frac{1}{\omega}&\frac{1}{\omega^2}&1\end{vmatrix}$ equal to? 

If the sum of the first n terms of a series is n(2n+1), then what is the nth term?

In how many ways can the letters of the word INDIA be permutated such that in each combination, vowels should occupy odd positions?

The letters of the word EQUATION are arranged in such a way that all vowels as well as consonants are together. How many such arrangements are there?

If n is a root of the equation x² + px + m = 0 and m is a root of the equation x² + px + n = 0, where m ≠ n, then what is the value of p + m + n?

In how many ways can a student choose (n - 2) courses out of n courses if 2 courses are compulsory (n > 4)? 

If $\rm D_n=\begin{vmatrix}n&20&30\\\ n^2&40&50\\\ n^3&60&70\end{vmatrix}$ then what is the value of $\rm \sum\limits_{n=1}^4D_n?$

Consider the following in respect of the matrices  $\rm P=\begin{bmatrix}0&c&-b\\\ -c&0&a\\\ b&-a&0\end{bmatrix}\ and\ \rm Q=\begin{bmatrix}a^2&ab&ac\\\ ab&b^2&bc\\\ ac&bc&c^2\end{bmatrix}$

I. PQ is a null matrix. 

II. QP is an identity matrix of order 3. 

III. PQ = QP

Which of the above is/are correct? 

If P is a skew-symmetric matrix of order 3, then what is det(P) equal to? 

If 4 sin^-1 x + cos^-1 x = π, then what is sin^-1 x + 4 cos^-1 x equal to?

What is cot²(sec^-1 2) + tan² (cosec^-1 3) equal to ?

In a triangle ABC $\rm \frac{a}{\cos A}=\frac{b}{\cos B}=\frac{c}{\cos C}$ What is the area of the triangle if a = 6 cm?

The roots of the equation 7x² - 6x + 1 = 0 are tan α and tan β, where 2α and 2β are the angles of a triangle. Which one of the following is correct?

In a triangle ABC, ∠A = 75° and ∠B = 45°. What is 2a - b equal to?

What is the number of solutions of the equation cot2x cot3x = 1 for 0 < x

What is the general solution of cos¹⁰⁰ x - sin¹⁰⁰ x = 1?

In a triangle ABC tan A + tan B + tan C = k  What is the value of cot A cot B cot C? 

What is sin12° sin48° equal to? 

What is $\rm \frac{\cos 17^\circ-\sin 17^\circ}{\cos 17^\circ+\sin ^\circ}$ equal to

Consider the following numbers : 

I. tan 22.5° 

II. cot 22.5° 

III. tan 22.5° - cot 22.5° 

How many of the above are irrational numbers?

If $\rm \frac{x}{\cos \theta}=\frac{y}{\cos \left(\frac{2\pi}{3}-\theta\right)}=\frac{z}{\cos\left(\frac{2\pi}{3}+\theta\right)}$ then what is x + y + z equal to? 

If p tan (θ - 30°) = q tan (θ + 120°), then what is (p + q) / (p - q) equal to?

Let P and Q be two non-void relations on a set A. Which of the following statements are correct?

I. P and Q are reflexive ⇒ P ∩ Q is reflexive.

II. P and Q are symmetric ⇒ P ∪ Q is symmetric.

III. P and Q are transitive ⇒ P ∩ Q is transitive.

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If A and B are two non-empty sets having 10 elements in common, then how many elements do A x B and B x A have in common?

What is the remainder when 7ⁿ - 6n is divided by 36 for n = 100?

What is the maximum number of possible points of intersection of four straight lines and a circle (intersection is between lines as well as circle and lines)? 

In an AP, the ratio of the sum of the first p terms to the sum of the first q terms is p² : q². Which one of the following is correct?

What is the number of real roots of the equation (x - 1)² + (x - 3)² + (x - 5)² = 0?

In a class of 240 students, 180 passed in English, 130 passed in Hindi and 150 passed in Sanskrit. Further, 60 passed in only one subject, 110 passed in only two subjects and 10 passed in none of the subjects. How many passed in all three subjects?

What is the common difference?

What is the sum of all five terms? 

Which one of the following statements is correct? 

Which one of the following is a root of the equation? 

What is A(adj A) equal to?  

What is A^-1 equal to?  

What is 3α + 2β equal to if (2î + 6ĵ + 27k̂) × (î + αĵ + βk̂) is a null vector?

For what value of the angle between the vectors $\rm \vec a\ and \ \vec b$ is the quantity $\rm |\vec a\times \vec b|+\sqrt3|\vec a.\vec b|$ maximum? 

Let θ be the angle between two unit vectors $\rm \vec a\ and\ \vec b. \ if\ \vec a+2\vec b$ is perpendicular to $\rm 5\vec a-4\vec b$ then what is cos θ + cos 2θ equal to?

Let ABCDEF be a regular hexagon. If $\rm \vec{AD}=m \vec {BC}\ and\ \vec {CF}=n\vec {AB}$ then what is mn equal to 

The vectors $\rm \vec a, \vec b\ and\ \vec c$ are of the same length. If taken pairwise they form equal angles. If $\rm \vec a=̂ i+̂ j\ and \ \vec b=̂ j+̂ k,$ then what can $\vec c$ be equal to? 

I. î + k̂ 

II. $\rm \frac{-\hat i+4\hat j-\hat k}{3}$

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The diagonals of a quadrilateral ABCD are along the lines x - 2y = 1 and 4x + 2y = 3. The quadrilateral ABCD may be a 

The foci of the ellipse 4x² + 9y² = 1 are at Q and R. If P(x, y) is any point on the ellipse, then what is PQ + PR equal to?

If P(2, 4), Q(8, 12), R(10, 14) and S(x, y) are vertices of a parallelogram, then what is (x + y) equal to?  

The equation of a circle is 

(x² - 4x + 3) + (y² - 6y + 8) = 0

Which of the following statements are correct?

I. The end points of a diameter of the circle are at (1, 2) and (3, 4).

II. The end points of a diameter of the circle are at (1, 4) and (3, 2).

III. The end points of a diameter of the circle are at (2, 4) and (4, 2).

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Consider the points P(4k, 4k) and Q(4k, -4k) lying on the parabola y² = 4kx. If the vertex is A, then what is ∠PAQ equal to?

What is BAC equal to?

What are the coordinates of A? 

What are the coordinates of vertex D?

What is the point of intersection of the diagonals of the trapezium?

What is the diameter of the sphere? 

The centre of the sphere lies on the plane 

Which of the following are the direction ratios of S? 

If 〈l, m, n〉 are direction cosines of S, then what is the value. of 43 (I² - m² - n²)?  

What are the direction ratios of the line?  

What is the point of intersection of L and P? 

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 f o g(x) = g o f(x), then what is the value of k?  

What is the minimum value of the function f(x) = log₁₀(x² + 2x + 11)? 

Which one of the following is correct regarding $\rm \lim\limits_{x\rightarrow 3}\frac{|x-3|}{x-3}$?

What is the maximum value of a cos x + b sin x + c? 

If f(2x) = 4x² + 1, then for how many real values of x will f(2x) be the GM of f(x) and f(4x)?  

lf f(x) = [x]² - 30[x] + 221 = 0, where [x] is the greatest integer function, then what is the sum of all integer solutions? 

f f(x) = 9x - 8√x such that g(x) = f(x) - 1, then which one of the following is correct?

What is $\rm \lim\limits_{x\rightarrow \frac{\pi}{2}}(\sec \theta-\tan \theta)$ equal to? 

Let f(x) f(y) = f(xy) for all real x, y. If f(2) = 4, then what is the value of f(1/2)?

Which one of the following is f(x)? 

Which one of the following is g(x)? 

What is f(0.999) + f(1.001) equal to

Consider the following statements : 

I. f(x) is continuous at x = 0. 

Il. f(x) is continuous at x = 1 

Which of the statements given above is/are correct? 

What is the greatest value of f(x)? 

What is the least value of f(x)? 

What is the value of k? 

What is the area of the parabola bounded by the latus rectum? 

What are the order and degree respectively of the differential equation? 

What is the solution of the differential equation? 

What is $\rm \int_0^2f(x)dx$ equal to

What is $\rm \int_h^3f(x)dx$ equal to

Consider the following statements :

I. f(t) is an odd function.

Il. g(t) is an odd function.

Which of the statements given above is/are correct?  

What is $\rm \int_{-\pi}^\pi g(t)dt$ equal to

What is h’(0) equal to? 

What is g’(0) equal to? 

What is $\rm \int_0^{\pi/2}\frac{d(x)}{g(x)}$ equal to

What is I equal to?

What is |U²(x) - V²(x)| equal to?

What is U(x) V(x) equal to? 

Let x - 3y + 4 = 0 and 2x - 7y + 8 = 0 be two lines of regression computed from some bivariate data. If b_yx and b_xy are regression coefficients of lines of regression of y on x and x on y respectively, then what is the value of b_xy + 7b_yx?

The mean of n observations

1, 4, 9, 16 ...., n² is 130. What is the value of n?

Three distinct natural numbers chosen at random from 1 to 10. What is the probability that they are consecutive?

A, B, C are three mutually exclusive and exhaustive events associated with a random experiment. If 3P(B) = 4P(A) and 3P(C) = 2P(B), then what is P(A) equal to?

A die has two faces with number 4, three faces with number 5 and one face with number 6. If the die is rolled once, then what is the probability of getting 4 or 5?

A box contains 2 black, 4 yellow and 6 white balls. Three balls are drawn in succession with replacement. What is the probability that all three are of the same colour?

A can hit a target 5 times in 6 shots, B can hit 4 times in 5 shots and C can hit 3 times in 4 shots. What is the probability that A and C may hit but B may lose?

The letters of the word ZOOLOGY are arranged in all possible ways. What is the probability that the consonants and vowels occur alternatively?

A natural number x is chosen at random from the first 100 natural numbers. What is the probability that x² + x > 50?

What is the mean deviation of the first 10 natural numbers?

Let $\rm \Sigma_{i=1}^9x_i^2=885$ If M is the mean and σ is the standard deviation of  x₁, x₂, x₃.....x₉ then what is the value of M² + σ²?

The mean of the series x₁, x₂...x_n is x̅ If x_n is replaced by k, then what is the new mean?

A fair coin is tossed till two heads occur in succession. What is the probability that the number of tosses required is less than 6?  

Urn A contains 2 white and 2 black balls while urn B contains 3 white and 2 black balls. One ball is transferred from urn A to urn B and then a ball is drawn out of urn B. What is the probability that the ball is white?

For two events A and  B, P(A) = P(A|B) = 0.25 and P(BIA) = 0.5. Which of the following are correct?

I. A and B are independent.

II. P(A^c ∪ B^c) = 0.875

III. P(A^c ∩ B^c) = 0.375

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Two perfect dice are thrown. What C is the probability that the sum of the numbers on the faces is neither 9 nor 10?

The occurrence of a disease in an industry is such that the workers have 20% chance of suffering from it. What is the probability that out of 6 workers chosen at random, 4 or more will suffer from the disease?

Three perfect dice are rolled. Under the condition that no two show the same face, what is the probability that one of the faces shown is an ace (one)?

Three perfect dice D₁, D₂ and D₃ and are rolled. Let x, represent the numbers on D₁, D₂ and D₃ respectively. What is the number of possible outcomes such that x < y < z?

In a binomial distribution, if the mean is 6 and the standard deviation is √2, then what are the values of the parameters n and p respectively?