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$$AB$$ is a straight road of length 400 metres. From $$A$$, Samrud runs at a speed of $$6 \text{ m/s}$$ towards $$B$$ and at the same time Saket starts from $$B$$ and runs towards $$A$$ at a speed of $$5 \text{ m/s}$$. After reaching their destinations, they return with the same speeds. They repeat it again and again. How many times do they meet each other in 15 minutes?
In 15 minutes, that is 900 seconds, the two runners together cover $$(6 + 5) \times 900 = 9900$$ metres. Along a road of length 400 metres they come together once for every 400 metres of combined running. Since $$\frac{9900}{400} = 24.75$$, exactly 24 such meetings are completed, so they meet 24 times.
In the adjoining figure, the measure of the angle $$x$$ is
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The value of $$x$$ which satisfies $$\frac{1}{x+a} + \frac{1}{x+b} = \frac{1}{x+a+b} + \frac{1}{x}$$ is
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Two sides of an isosceles triangle are 23 cm and 17 cm respectively. The perimeter of the triangle (in cm ) is
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$$ABCDE$$ is a pentagon with $$\angle B = 90^\circ$$ and $$\angle E = 150^\circ$$. If $$\angle C + \angle D = 180^\circ$$ and $$\angle A + \angle D = 180^\circ$$, then the external angle $$\angle D$$ is
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The unit's digit of the product $$3^{2025} \times 7^{2024}$$ is
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The smallest positive integer $$n$$ for which $$18900 \times n$$ is a perfect cube is
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Two numbers $$a$$ and $$b$$ are respectively $$20\%$$ and $$50\%$$ more of a third number $$c$$. The percentage of $$a$$ to $$b$$ is
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If $$a + b = 2$$, $$\frac{1}{a} + \frac{1}{b} = 18$$, then $$a^3 + b^3$$ lies between
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If $$\sqrt{12 + \sqrt[3]{x}} = \frac{7}{2}$$ and $$x = \frac{p}{q}$$, $$p, q$$ are natural numbers with G.C.D. $$(p, q) = 1$$, then $$p + q$$ is
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The smallest number of 4-digits leaving a remainder of 1 when divided by 2 or 3 or 4 or 6 has
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If $$a \colon b = 2 \colon 3$$, $$b \colon c = 4 \colon 5$$ and $$a + c = 736$$, then the value of $$b$$ is
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In the given figure, $$\angle B = 110^\circ$$; $$\angle C = 80^\circ$$; $$\angle F = 120^\circ$$; $$\angle ADC = 30^\circ$$ $$2\angle DGF = \angle DEF$$. The measure of $$\angle BHF$$ is
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If $$\frac{1}{b+c} + \frac{1}{c+a} = \frac{2}{a+b}$$, then the value of $$\frac{a^2 + b^2}{c^2}$$ is
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If 3 men or 4 women can do a job in 43 days, the number of days the same job is done by 7 men and 5 women is
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The expression $$49(a + b)^2 - 46(a - b)^2$$ is factorized into $$(la + mb)(na + pb)$$, then the numerical value of $$(l + m + n + p)$$ is
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The integer part of the solution of the equation in $$x$$, $$\frac{1}{3}(x - 3) - \frac{1}{4}(x - 8) = \frac{1}{5}(x - 5)$$ is
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In the adjoining figure, $$ABC$$ is a triangle in which $$\angle BAC = 100^\circ$$, $$\angle ACB = 30^\circ$$. An equilateral triangle, a square and a regular hexagon are drawn as shown in the figure. The measure (in degrees) of $$(x + y + z)$$ is
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The mean of 5 numbers is 105 . The first number is $$\frac{2}{5}$$ times the sum of the other 4 numbers. The first number is
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$$PQRS$$ is a square. The sides $$PQ$$ and $$RS$$ are increased by 30 % each and the sides $$QR$$ and $$PS$$ are increased by 20 % each. The area of the quadrilateral thus obtained exceeds the area of the square by _________ %.
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If $$x^2 + (2 + \sqrt{3})x - 1 = 0$$ and $$x^2 + \frac{1}{x^2} = a + b\sqrt{c}$$, then $$(a + b + c)$$ is
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In the given figure, $$ABCD$$ is a rectangle. The measure of angle $$x$$ is _______________ degrees.
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The sum of all positive integers $$m, n$$ which satisfy $$m^2 + 2mn + n = 44$$ is
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Given $$a = 2025$$, $$b = 2024$$, the numerical value of $$\left(a + b - \frac{4ab}{a+b}\right) \div \left(\frac{a}{a+b} - \frac{b}{b-a} + \frac{2ab}{b^2 - a^2}\right)$$ is
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In the sequence $$0, 7, 26, 63, 124, \ldots$$ the $$6^{\text{th}}$$ term is
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If $$A = \sqrt{281 + \sqrt{53 + \sqrt{112 + \sqrt{81}}}}$$, $$B = \sqrt{92 + \sqrt{55 + \sqrt{75 + \sqrt{36}}}}$$ then $$A - B$$ is
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The average of the numbers $$a, b, c, d$$ is $$(b + 4)$$. The average of pairs $$(a, b)$$, $$(b, c)$$ and $$(c, a)$$ are respectively 16, 26 and 25 . Then the average of $$d$$ and 67 is
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$$ABC$$ is a quadrant of a circle of radius 10 cm . Two semicircles are drawn as in the figure.
The area of the shaded portion is $$k\pi$$, where $$k$$ is a positive integer. The value of $$k$$ is
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In the figure, $$ABC$$ and $$PQR$$ are two triangles such that $$\angle A \colon \angle B \colon \angle C = 5 \colon 6 \colon 7$$ and $$\angle PRQ = \angle B$$. $$PS$$ makes an angle $$\frac{\angle P}{3}$$ with $$PQ$$ and $$RS$$ makes an angle $$\frac{\angle SRT}{5}$$ with $$RQ$$. Then the measure of $$\angle S$$ is
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In a two-digit positive integer, the units digit is one less than the tens digit. The product of one less than the units digit and one more than the tens digit is 40. The number of such two-digit integers is
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