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Question 1:

A real number \( x \) is such that the sum of the number and four times its square is the least. What is that number?

-0.625

-0.125

0.125

1

Question 2:

The difference of the square of two natural numbers \(m\) and \(n\) (\(m>n\)) is 72. How many pairs of natural numbers will satisfy?

3

4

5

6

Question 3:

Let \( N \) be a 5-digit number. When \( N \) is divided by \( 6, 12, 15, 24 \) it leaves respectively \( 2, 8, 11, 20 \) as remainders. What is the greatest value of \( N \)?

99960

99956

99950

99946

Question 4:

What is the remainder when \(111^{222} + 222^{333} + 333^{444}\) is divided by 5 ?

N/A

Question 5:

What are the last three digits in the multiplication of \(4321012345 \times 98766789\)?

1,0,5

2,0,5

2,1,5

3,0,5

Question 6:

\( p \) varies directly as \( x^{2}+y^{2}+z^{2} \). When \( x=1, y=2, z=3 \), then \( p=70 \). What is the value of \( p \) when \( x=-1, y=1 \), \( z=5 \)?

100

125

135

140

Question 7:

Let \( N \) be the least positive multiple of 11 that leaves a remainder of 5 when divided by \( 6,12,15,18 \). Which one of the following is correct?

\(900

\(1000

\(1100

\(1200

Question 8:

What is \( \frac{1}{\sqrt{10}+\sqrt{9}}+\frac{1}{\sqrt{11}+\sqrt{10}}+\frac{1}{\sqrt{12}+\sqrt{11}}+\ldots+\frac{1}{\sqrt{196}+\sqrt{195}} \) equal to?

17

14

11

10

Question 9:

Train \( X \) crosses a man standing on the platform in 24 seconds and train \( Y \) crosses a man standing on the platform in 18 seconds. They cross each other while running in opposite directions in 20 seconds. What is the ratio of speed of \( X \) to speed of \( Y \)?

\( 1: 2 \)

\( 2: 3 \)

\( 1: 3 \)

\( 3: 4 \)

Question 10:

Let \( p, q \) be the roots of the equation \( x^{2} + m x - n = 0 \) and \( m, n \) be the roots of the equation \( x^{2} + p x - q = 0 \) (m, n, p, q are non-zero numbers). Which of the following statements is/are correct?
I. \( m(m+n) = -1 \)
II. \( p+q = 1 \)

I only

II only

Both I and II

Neither I nor II

Question 11:

What is the maximum value of \( 8 \sin \theta-4 \sin ^{2} \theta \) ?

3

4

8

12

Question 12:

What is \( (1+\tan \alpha \tan \beta)^{2}+(\tan \alpha-\tan \beta)^{2} \) equal to?

\( \tan^2 \alpha \tan^2 \beta \)

\( \sec^2 \alpha \sec^2 \beta \)

\( \tan^2 \alpha \cot^2 \beta \)

\( \sec^2 \alpha \tan^2 \beta \)

Question 13:

Consider the following statements:
I. \( \tan 50^{\circ}-\cot 50^{\circ} \) is positive
II. \( \cot 25^{\circ}-\tan 25^{\circ} \) is negative
Which of the statements is/are correct?

I only

II only

Both I and II

Neither I nor II

Question 14:

यदि \( 0 \leqslant(\alpha-\beta) \leqslant(\alpha+\beta) \leqslant \frac{\pi}{2} \), \( \tan (\alpha+\beta)=\sqrt{3} \) और \( \tan (\alpha-\beta)=\frac{1}{\sqrt{3}} \), तो \( \tan \alpha \cdot \cot 2 \beta \) किसके बराबर है ?

1

\( \sqrt{2} \)

\( \sqrt{3} \)

\( \frac{1}{\sqrt{3}} \)

Question 15:

If \(0 \leqslant (\alpha-\beta) \leqslant (\alpha+\beta) \leqslant \frac{\pi}{2}\), \(\tan(\alpha+\beta)=\sqrt{3}\) and \(\tan(\alpha-\beta)=\frac{1}{\sqrt{3}}\), then what is \(\tan \alpha \cdot \cot 2\beta\) equal to?

1

\(\sqrt{2}\)

\(\sqrt{3}\)

\(\frac{1}{\sqrt{3}}\)

Question 16:

\( \sin^{2} \theta \cos^{2} \theta\bigl(\sec^{2} \theta + \cosec^{2} \theta\bigr) \) का मान किसके बराबर है ?

0

1

2

4

Question 17:

What is the value of \( \sin^{2} \theta \cos^{2} \theta (\sec^{2} \theta + \operatorname{cosec}^{2} \theta) \) equal to?

0

1

2

4

Question 18:

यदि \(64^{\sin^{2} \theta}+64^{\cos^{2} \theta}=16\) जहां \(0 \leqslant \theta \leqslant \frac{\pi}{2}\), तो \(\tan \theta+\cot \theta\) का मान क्या है?

1

2

3

4

Question 19:

If \(64^{\sin^{2}\theta}+64^{\cos^{2}\theta}=16\) where \(0\leqslant \theta\leqslant \frac{\pi}{2}\), then what is the value of \(\tan\theta+\cot\theta\)?

1

2

3

4

Question 20:

यदि \( \operatorname{cosec} \theta - \cot \theta = m \) और \( \sec \theta - \tan \theta = n \), तो \( \operatorname{cosec} \theta + \sec \theta \) किसके बराबर है ?

\( \tfrac{1}{2}\bigl(m+n+\tfrac{1}{m}+\tfrac{1}{n}\bigr) \)

\( m+n+\tfrac{1}{m}+\tfrac{1}{n} \)

\( \tfrac{1}{2}\bigl(m+n-\tfrac{1}{m}-\tfrac{1}{n}\bigr) \)

\( m+n-\tfrac{1}{m}-\tfrac{1}{n} \)

Question 21:

If \( \operatorname{cosec} \theta-\cot \theta=m \) and \( \sec \theta-\tan \theta=n \), then what is \( \operatorname{cosec} \theta+\sec \theta \) equal to?

\( \frac{1}{2}\left(m+n+\frac{1}{m}+\frac{1}{n}\right) \)

\( \left(m+n+\frac{1}{m}+\frac{1}{n}\right) \)

\( \frac{1}{2}\left(m+n-\frac{1}{m}-\frac{1}{n}\right) \)

\( \left(m+n-\frac{1}{m}-\frac{1}{n}\right) \)

Question 22:

एक नदी के पुल पर एक बिंदु \( X \) से, नदी के विपरीत किनारे पर दो बिंदुओं \( P \) और \( Q \) के अवनमन कोण क्रमशः \( \alpha \) और \( \beta \) हैं। यदि बिंदु \( X \) नदी की सतह से \( h \) ऊंचाई पर है, तो नदी की चौड़ाई क्या है यदि \( \alpha \) और \( \beta \) पूरक हैं ?

\( 2 h(\tan \alpha+\cot \alpha) \)

\( H \tan \alpha \cdot \tan \beta \)

\( h \cot \alpha \cdot \cot \beta \)

\( h \sec \alpha \cdot \operatorname{cosec} \alpha \)

Question 23:

From a point \(X\) on a bridge across a river, the angles of depression of two points \(P\) and \(Q\) on the banks on opposite side of the river are \(\alpha\) and \(\beta\) respectively. If the point \(X\) is at a height \(h\) above the surface of the river, what is the width of the river if \(\alpha\) and \(\beta\) are complementary?

\(2h(\tan \alpha+\cot \alpha)\)

\(H\tan \alpha \cdot \tan \beta\)

\(h\cot \alpha \cdot \cot \beta\)

\(h\sec \alpha \cdot \operatorname{cosec} \alpha\)

Question 24:

त्रिभुज \( A B C \) में, \( \angle A B C=60^{\circ} \) और \( A D \) ऊँचाई है। यदि \( A B=6\ \mathrm{cm} \) और \( B C=8\ \mathrm{cm} \), तो त्रिभुज का क्षेत्रफल क्या है?

12 square cm

\( 12 \sqrt{3} \) square cm

24 square cm

\( 24 \sqrt{3} \) square cm

Question 25:

In a triangle \( A B C, \angle A B C=60^{\circ} \) and \( A D \) is the altitude. If \( A B=6 \mathrm{~cm} \) and \( B C=8 \mathrm{~cm} \), then what is the area of the triangle?