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A real number M is squared to give the value N. What is the minimum value of \( (\mathrm{M}+\mathrm{N}) \)?

If \( \alpha \) and \( \beta \) are the roots of the equation \( x+a+b=\frac{a b x}{a b+a x+b x} \), then what is \( (\alpha \beta+\alpha+\beta) \) equal to ?

N is the smallest 5-digit number which when divided by \(2,2^{2},2^{3},2^{4},\ldots,2^{n}\) leaves a remainder 1. What is the value of \(n\)?

What is the sum of all 3-digit numbers that give a remainder of 5 when they are divided by 50?

What is the minimum value of \( p \) for which \( \frac{1}{532900}+\frac{\mathrm{p}^{2}}{266450}+\frac{\mathrm{p}^{4}}{523900} \) is an integer?

If the average of \(64,69,72,75,x\) lies between 62 and 76 (excluding 62 and 76), then what is the number of possible integer values of \(x\)?

Let \(x, y, z\) be variables such that \((x+y+z)=k\), where \(k\) is a constant. If \((x+z-y)\times(x-z+y)\) is proportional to yz, then \((y+z-x)\) is proportional to:

Let \( p \) be the remainder when \( 7^{84} \) is divided by 342 and \( q \) be the remainder when \( 7^{84} \) is divided by 344. What is \( (p-q) \) equal to?

Consider a 2-digit number N. Let P be the product of the digits of the number. If \(P\) is added to the square of the digit in the tens place of N, we get 84. If \(P\) is added to the square of the digit in the unit place of N, we get 60. What is the value of \(P+N\)?

A mixture of 100 L contains kerosene and turpentine oil in the ratio \( 3: 2 \). What is the minimum quantity of kerosene in litres (whole number) that should be mixed in the mixture so that the resulting mixture has \( 20 \% \) of kerosene?

A lamp is kept on a vertical pole. The height of the top of the lamp above the ground is \( \frac{5 \sqrt{3}}{2} \mathrm{~m} \). The perpendicular distances of the bottom of the pole from two adjacent walls meeting perpendicularly are 0.7 m and 2.4 m. What is the distance of the top of the lamp from the corner point of the walls on the ground?

C is the centre of a circle of radius \(20\ \mathrm{cm}\). AB is a chord of length 32 cm. E is a point on AB such that \(CE = 13\ \mathrm{cm}\). What is \(AE \times EB\) equal to?

The inside of a bowl is part of a sphere. When water is put into the bowl to a depth d, the water surface becomes a circle of radius 2d. What is the radius of the sphere?

In a triangle \( \mathrm{ABC}, \mathrm{AB}=2 \mathrm{~cm}, \mathrm{BC}=4 \mathrm{~cm} \) and \( \mathrm{AC}=3 \mathrm{~cm} \). The bisector of angle A meets BC at D and the bisector of angle B meets AD at E. What is AE : ED equal to ?

In a triangle ABC, the bisector of angle A cuts BC at D. If \( \mathrm{AB}+\mathrm{AC}=10 \mathrm{~cm} \) and \( \mathrm{BD}:\mathrm{DC}=3:1 \), then what is the length of AC?

In a triangle \( \mathrm{ABC}, \mathrm{AB}+\mathrm{BC}=7 \cdot 1\,\mathrm{cm},\; \mathrm{BC}+\mathrm{CA}=12 \cdot 1\,\mathrm{cm}\) and \( \mathrm{CA}+\mathrm{AB}=7 \cdot 2\,\mathrm{cm}\). What is the area of the triangle?

The adjacent sides of a parallelogram are 10 cm and 8 cm and the angle between them is \(150^{\circ}\). What is the area of the parallelogram?

The measure of an angle formed by the bisectors of the angles A and C of the triangle ABC is \( 130^{\circ} \). What is the measure of the angle B ?

What is \(\log_{10} 2000 + \log_{10} 400 + 4\log_{10} 25 + 5\log_{10} 20\) equal to?

If \( \frac{\log_{10}\bigl(100001-4^{x}\bigr)}{5-x}=1 \), then what is \( x \) equal to?

If \(2\sin^4\alpha + 2\cos^4\alpha - 1 = 0\), where \(0 \leq \alpha < \pi/2\), then what is \(\sin 2\alpha + \cos 2\alpha\) equal to?

Consider the following :
I. \(1 - \sin^{6} \alpha = \cos^{2} \alpha\bigl(\cos^{4} \alpha - 3\cos^{2} \alpha + 3\bigr)\)
II. \(\cos^{8} \alpha - \sin^{8} \alpha = 2\sin^{2} \alpha\bigl(1 - \cos^{4} \alpha + \sin^{2} \alpha\cos^{2} \alpha\bigr)\)
Which of the above is/are identities?

If \( p=\frac{1}{\operatorname{cosec} \theta+\cot \theta} \) and \( q=\operatorname{cosec} \theta \), then what is \( p^{2}-2 p q \) equal to?

Consider the following statements:
I. \( (\operatorname{cosec} \alpha - \sec \alpha) \) is always positive in the first quadrant.
II. \( \tan \alpha - \cot \alpha \) is always negative in the first quadrant.
Which of the statements given above is/are correct?

A tower subtends an angle \(60^{\circ}\) at a point A on the same level as the foot of the tower. B is a point vertically above A and \(AB = h\). The angle of depression of the foot of the tower, measured from B is \(30^{\circ}\). What is the height of the tower?

What is \( \frac{\sin \theta}{1-\cot \theta}+\frac{\cos \theta}{1-\tan \theta}(\theta \neq \pi / 4) \) equal to?

The length of an arc of a circle of radius 4 cm is \( \pi \mathrm{cm}\). What is the magnitude of the angle subtended by the arc at the centre?