GATE Mathematics Mock Test - 1 - Mathematics MCQ

This is a case of mixed series

3 + 3 = 6

6 x 4 = 24

24 + 5 = 29

29 x 6 = 174

174 + 7 = 181

∴ The missing number is 181

Therefore, original gross receipts = ps. Let new price = 0.80p, and new sales = 1.15s.

Therefore, new gross receipts = 0.92ps. Gross receipts are only 92% of what they were.

The sum of series 1, 2, 3, …..n is n(n+1)/2

So, according to problem statement

n2 + n – 2k = 30n

n2 – 29n = 2k

n (n – 29) = 2k

So, at n ≤ 29, the above expression is invalid. Because at n ≤ 29, the value of k become either zero or negative.

So, least value of n is 30.

At n = 30, we have

30(30 – 29) = 2k

k = 15

3 x 5 x 4 = 60

5 x 7 x 2 = 70

8 x 6 x ? = 144 = 144/48 = 3, which is the answer.

2014 starts with a Wednesday

1234567

WTFSSM T

So if Wednesday is day n = 1 then Sundays will come on days n = 5,12,19,…96, total 15 Sundays.

Similarly if we start with-

Thursday - 14 sundays 

Friday - 14 sundays 

Saturday - 15 sundays 

Sunday - 15 sundays 

Monday - 14 sundays 

Tuesday - 14 sundays

So, probability 

⇒ P = 2/7

Let Y = (y_n) be the sequence of real numbers given by

Clearly, Y is not a monotone sequence. However, if m > n, then

Since , it follows that if m > n, then

Therefore, it follows that (y_n) is a Cauchy sequence. Hence it converges to a
limit y. At the present moment we cannot evaluate y directly; however, passing
to the limit (with respect to m) in the above inequality, we obtain

Hence we can calculate y to any desired accuracy by calculating the terms yn
for sufficiently large n. The reader should to this and show that y is approxi-
mately equal to 0.632 120559.

The LCM of 4,5,6 and 7 = 420

so, the required number = 420k + 3

which is exactly divisible by 9 for some value of k

Now, 420K + 3 = 46 x 9k + 6k + 3

When k = 1, 6k + 3 = 9, which is divisible by 9

So, the required number = 420 x 1 + 3 = 423

Condition 1: 

(It is Homogeneous)

Condition 2: 

Equation (1) can be written as 

It is not a linear form.

It is in linear form.

Condition 3: 

So, it is an exact equation.

We know that for maximum or minimum value of 

Therefore f(x) has maximum value at .

An elastic collision is a collision in which there is no net loss in kinetic energy in the system as a result of the collision. Both momentum and kinetic energy are conserved quantities in elastic collisions. Suppose two similar trolleys are traveling toward each other with equal speed.

For solenoidal vector, 

The given equation is,

Differentiating equation (1) with respect to x, we get

Now, again differentiating equation (1) with respect to y, we get

Hence, the correct answer is 0.

The matrix form of the given system of equations is,

The given system of equations will have a unique solution if and only if the coefficient matrix is non-singular.

Performing , we get,

Performing , we get

Therefore the coefficient matrix will be non-singular if and only if,

Thus the given system will have a unique solution if 

Showing that given equations are inconsistent in this case. Thus if  

= -3.5 no solution exists.

Hence, the correct answer is −3.5

In the question, y : x to infinity x : 0 to infinity.
Now changing the order of integration:
y = x
y tends to infinity
y : 0 to infinity x : 0 to y

Hence, the correct answer is 1.

According to the property of the Eigen values, the sum of the Eigen values of a matrix is its trace that is the sum of the elements of the principal diagonal.

Therefore, the sum of the Eigen values = 3 + 4 + 1 = 8.

Let n be the number of children in each family 1/12 = 

We want extreme values of f(x, y) = xy subject to the constraints

To do so, we first find the value of x,y and λ, for which

the gradient eq. in eq (1) gives

from which we find

and
so y = 0 or λ = ±2. we now consider these Wvo cases
case I if
y = O then x = y = 0 But (0, 0) is not on the ellipse therefore y + 0
case II If y ≠ 0 then λ = ± 2 and x = substituting this in the eq. g(x,y) = 0
given

Since here a_n = p(n)

If let p(x) = 

Radius of convergence of power series is given by

According to L'Hospital's Rules,

Adding (i) and (ii), we get

Integrating 

According to the property of Eigen values, the product of the Eigen values of a given matrix is equal to the determinant of the matrix |A| = 1(12-0) – 2(0) + 4(8)

= -60.

Note that a matrix is singular if and only if its determinant is zero.

So we compute the determinant of the matrix A as follows.

Thus the determinant of A is zero if

det(A) = −2x² + 4x − 2=0,

equivalently,

x² − 2x + 1 = (x − 1)² = 0.

Thus, the determinant of the matrix A is zero if and only if x = 1.

Hence the matrix A is singular if and only if x = 1.