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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 \(2,2^{2}, 2^{3}, 2^{4}, \ldots, 2^{\mathrm{n}}\) 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 \( (\mathrm{y}+\mathrm{z}-\mathrm{x}) \) 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 \( \mathrm{P}+\mathrm{N} \)?

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 \( \frac{5 \sqrt{3}}{2} \) 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 \(32\ \mathrm{cm}\). 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 \( \sin2\alpha + \cos2\alpha \) equal to?

Consider the following : I. \(1-\sin^{6} \alpha = \cos^{2} \alpha(\cos^{4} \alpha - 3\cos^{2} \alpha + 3)\) II. \(\cos^{8} \alpha - \sin^{8} \alpha = 2\sin^{2} \alpha(1 - \cos^{4} \alpha + \sin^{2} \alpha \cos^{2} \alpha)\) 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. \( (\cosec \alpha - \sec \alpha ) \) 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?