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

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

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

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 \( \mathrm{P}+\mathrm{N} \)?

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 \( \frac{5 \sqrt{3}}{2} \mathrm{~m} \). 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 \(32 \mathrm{~cm}\). E is a point on AB such that \(CE=13 \mathrm{~cm}\). What is \(AE \times EB\) equal to?

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 \(\mathrm{AE}:\mathrm{ED}\) equal to?

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

In a triangle \(ABC\), \(AB+BC=7\cdot1\,\mathrm{cm}\), \(BC+CA=12\cdot1\,\mathrm{cm}\) and \(CA+AB=7\cdot2\,\mathrm{cm}\). 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 ?

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. \(1-\sin^{6}\alpha=\cos^{2}\alpha(\cos^{4}\alpha-3\cos^{2}\alpha+3)\)
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. (\( \operatorname{cosec} \alpha - \sec \alpha \)) 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?

If \(\cot^{2}\theta - 3\sqrt{3}\cot\theta + 6 = 0\), where \(\frac{\pi}{6} \le \theta < \frac{\pi}{2}\), then what is a value of \(\sin\theta + \cos 2\theta\)?

Which of the following equations is/are possible? I. \( \sin^{2}\theta = \frac{(x+y)^{2}}{4xy} \), where \( x, y \) are positive unequal real quantities. II. \( \sin\theta + \cos\theta = x + \frac{1}{x} \), where \( x \) is a positive real quantity.

If \(m^{2}(\sin\theta - 1) + n^{2}(\sin\theta + 1) = 0\), where \(0 < \theta < \tfrac{\pi}{2}\), then what is \((m^{2} + n^{2})\cos\theta - (m^{2} - n^{2})\cot\theta\) equal to?

If \( \sin \alpha+\cos \alpha=\sqrt{2} \), where \( 0<\alpha<\frac{\pi}{2} \), then what is \( \sin^{3} \alpha-\cos^{3} \alpha \) equal to?

What is \( (1+\cot \alpha-\operatorname{cosec} \alpha)(1+\tan \alpha+\sec \alpha) \) equal to?

If \(\tan \theta=\frac{\sin \alpha-\cos \alpha}{\sin \alpha+\cos \alpha}\), where \(\theta\) and \(\alpha\) (\(\alpha\neq\frac{\pi}{4}\)) are acute angles, then what is \(\sqrt{2}\sin\theta\) equal to?

For how many values of \( \alpha \) does the expression \( (\sin \alpha+2)(\sin \alpha+4)(\sin \alpha-2)(\sin \alpha-4) \) become zero?

What is the value of x, where \(0 \leq \mathrm{x}<30^{\circ}\), satisfying \(\tan 3 \mathrm{x}\tan 6 \mathrm{x}=1\)?