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There is a 6-digit number in which the first and the fourth digit from the first are the same, the second and the fifth digit from the first are the same and the third and the sixth digit from the first are the same. Then the number is always
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There is a 6-digit number in which the first and the fourth digit from the first are the same, the second and the fifth digit from the first are the same and the third and the sixth digit from the first are the same. Then the number is always
Let the number be $$abcabc$$. Its value is $$100000a+10000b+1000c+100a+10b+c=1001(100a+10b+c)$$. Since $$1001=7\times 11\times 13$$, the number is always divisible by $$11$$.
Starting from the number $$1$$, Ritu generates a series of numbers as $$1,3,6,11,18,29,42,\ldots$$ such that the differences of the consecutive numbers from the beginning give consecutive primes. In this series she came across a perfect square for the first time. The square root of this perfect square is
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The expression $$\frac{x\left(\frac{\sqrt{x}+\sqrt{y}}{2y\sqrt{x}}\right)^{-1}+y\left(\frac{\sqrt{x}+\sqrt{y}}{2x\sqrt{y}}\right)^{-1}}{\left(\frac{x+\sqrt{xy}}{2xy}\right)^{-1}+\left(\frac{y+\sqrt{xy}}{2xy}\right)^{-1}}$$ reduces to
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The sum of the digits of a two-digit number is multiplied by $$8$$ and the result is found to be $$13$$ more than the number. Then the two-digit number is
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A water tank is fitted with four different taps as outlets. If the tank is full, it takes $$1\ \text{hour}$$ to empty the tank when the first tap alone is opened, it takes $$2\ \text{hours}$$ to empty the tank when the second tap alone is opened, it takes $$3\ \text{hours}$$ when the third tap alone is opened and $$4\ \text{hours}$$ to empty the tank when the fourth tap alone is opened. . When all the taps are opened simultaneously, the full tank will be emptied in
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Two primes $$p,q$$ are such that $$p+q$$ is odd and $$q-10p=23$$. Then $$q-20p$$ equals
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Which one of the following is a false statement?
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Soham has his $$23^{\text{rd}}$$ birthday on $$1^{\text{st}}$$ January $$2024$$ and he noticed that $$2024$$ is divisible by $$23$$. If he lives till $$100$$ years of age, how many times other than the above, his age would be a divisor of the then year?
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Consider the two figures shown here. $$AB=16\ \text{cm}$$ in both the figures. Points $$P,Q,R$$ divide $$AB$$ into equal lengths in figure $$1$$. Similarly, $$P,Q,R,S,T,L,M$$ divide $$AB$$ into equal lengths in figure $$2$$. All the curves are semicircles. If $$[a]$$ and $$[b]$$ are the areas of the shaded figures respectively in figure $$1$$ and figure $$2$$, then
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The sum of $$11$$ consecutive natural numbers is $$121$$. The sum of the next three numbers is
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A big ship wrecked and $$1000$$ people landed on a remote island. The food material was available for them for $$60$$ days. After $$16$$ days another small ship, which had no food stock, wrecked and $$100$$ people landed on the same island. The number of days the food material for all of them is available is
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Two numbers are respectively $$28\%$$ and $$70\%$$ of a third number. The percentage of the first number to the second is
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The sum of two natural numbers is $$150$$. Their HCF is $$15$$. The number of pairs of such numbers is
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$$ABC$$ and $$ADE$$ are isosceles triangles. If $$\angle BFD=156^\circ$$, then $$\angle A$$ equals
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Some students are made to stand in rows of equal number, one behind the other. Saket is in the $$3^{\text{rd}}$$ row from the front and $$5^{\text{th}}$$ row from the back. He is $$4^{\text{th}}$$ from the left and $$6^{\text{th}}$$ from the right. The total number of students is
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In the adjoining figure, $$ABCD$$ is a rectangle. Then $$\angle EBD$$ is ______ degrees.
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An infinite sequence of positive numbers $$x_1,x_2,x_3,\ldots,x_n,x_{n+1},\ldots$$ satisfies $$x_n^2=(3n+7)+(n-3)x_{n+1}$$, where $$x_n$$ is the $$n^{\text{th}}$$ term of the sequence. Then the numerical value of $$x_1$$ is ______.
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For $$n\ge 2$$ and $$n\in\mathbb{Z}$$, the smallest positive integer $$n$$ for which none of the fractions $$\frac{17}{n+17},\frac{18}{n+18},\frac{19}{n+19},\ldots,\frac{100}{n+100}$$ can be simplified is ______.
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In triangle $$ABC$$, $$AB=15\ \text{cm}$$, $$BC=20\ \text{cm}$$ and $$CA=25\ \text{cm}$$. Then the length of the shortest altitude of the triangle in $$\text{cm}$$ is ______.
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The units digit of $$19^{2025}+999^{2023}$$ is ______.
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$$N$$ is a two-digit number. When $$6$$ is added to the tens digit and $$2$$ is subtracted from the units digit, we get a two-digit number which is equal to $$3N$$. Then $$N$$ is ______.
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$$ABCD$$ is a quadrilateral. $$AB$$ is parallel to $$CD$$ and $$AB>CD$$. If $$AD=AB=BC$$ and $$\angle ADC=140^\circ$$, then the measure of $$\angle CAB$$ is ______ degrees.
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The product of two positive numbers $$x$$ and $$y$$ is $$4$$ times their sum and the same product is $$8$$ times their difference. If $$x\ge y$$, then $$x$$ is ______.
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In the adjoining figure, $$ABCDEFGH$$ is a regular octagon. The measure of $$\angle ADG$$ in degrees is ______.
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If $$2^{3a+2}=4^{b+7}$$ and $$3^{a+10}=27^{2b+10}$$, then the value of $$a^2+b^2$$ is ______.
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$$ABCD$$ is a rectangle. $$AB=6$$ and $$AD=10$$. $$E$$ is a point on $$BC$$ such that $$AE=10$$. Then the area of triangle $$\triangle ADE$$ in square units is ______.
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The numbers $$1,4,7,10$$ and $$13$$ are placed in each box of the figure, such that the sums of the numbers in the horizontal or vertical boxes are the same. The largest possible value of the horizontal or vertical sum is ______.
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The number of integer pairs $$(m,n)$$ such that $$m(n^2+1)=48$$ is ______.
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In the adjoining figure, triangles $$\triangle ABD$$ and $$\triangle BCE$$ are equilateral triangles. The measure of $$\angle AFC$$ is ______ degrees.
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The value of $$\frac{\sqrt[4]{27\cdot\sqrt[3]{9}}}{\sqrt[6]{9\cdot 3^3\cdot\sqrt{3}}}$$ is ______.
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