The sum of the digits at the 10^th place of all numbers formed with the help of 2, 4, 5, 7 taken all at a time is

(a)

432

(b)

108

(c)

36

(d)

18

In a plane there are 10 points are there out of which 4 points are collinear, then the number of triangles formed is

(a)

110

(b)

¹⁰C₃

(c)

120

(d)

116

In ²ⁿC₃ : ⁿC₃ = 11 : 1 then n is

(a)

5

(b)

6

(c)

11

(d)

7

The product of r consecutive positive integers is divisible by _________

(a)

r!

(b)

r!+1

(c)

(r+1)

(d)

none of these

There are 10 points in a plane and 4 of them are collinear. The number of straight lines joining any two of them is _________

(a)

45

(b)

40

(c)

39

(d)

38

is _________

(a)

\(\lfloor{n}(n+2)\)

(b)

(c)

(d)

none of these

If ¹⁰⁰C_r = ¹⁰⁰C_3r then r is _________

(a)

24

(b)

25

(c)

20

(d)

50

How many words can be formed using all the letters of the word ANAND _________

(a)

30

(b)

35

(c)

40

(d)

45

There are 10 lamps in a hall. Each one of them can be switched on independently. The number of ways in which the hall can be illuminated is  _________

(a)

10²

(b)

10²³

(c)

2¹⁰

(d)

10!

The number of positive integral solution of \(x\times y\times z=30\) is  _________

(a)

3

(b)

1

(c)

9

(d)

27

There are 15 points in a plane of which exactly 8 are collinear. The number of straight lines obtained by joining these points is  _________

(a)

105

(b)

28

(c)

77

(d)

78

If ⁿC₁₀ = ⁿC₆, then ⁿC₂ =  _________

(a)

16

(b)

4

(c)

120

(d)

240

The unit vector parallel to the resultant of the vectors \(\hat{i}+\hat{j}-\hat{k}\) and \(\hat{i}-2\hat{j}+\hat{k}\) is

(a)

\({\hat{i}-\hat{j}+\hat{k}\over\sqrt{5}}\)

(b)

\({2\hat{i}+\hat{j}\over\sqrt{5}}\)

(c)

\({2\hat{i}-\hat{j}+\hat{k}\over\sqrt{5}}\)

(d)

\({2\hat{i}-\hat{j}\over\sqrt{5}}\)

If ABCD is a parallelogram, then \(\overrightarrow{AB}+\overrightarrow{AD}+\overrightarrow{CB}+\overrightarrow{CD}\) is equal to

(a)

\(2(\overrightarrow{AB}+\overrightarrow{AD})\)

(b)

\(4\overrightarrow{AC}\)

(c)

\(4\overrightarrow{BD}\)

(d)

\(\overrightarrow{0}\)

Two vertices of a triangle have position vectors \(3\hat{i}+4\hat{j}-4\hat{k}\) and \(2\hat{i}+3\hat{j}+4\hat{k}\) . If the position vector of the centroid is \(\hat{i}+2\hat{j}+3\hat{k}\), then the position vector of the third vertex is

(a)

\(-2\hat{i}-\hat{j}+9\hat{k}\)

(b)

\(-2\hat{i}-\hat{j}-6\hat{k}\)

(c)

\(2\hat{i}-\hat{j}+6\hat{k}\)

(d)

\(-2\hat{i}+\hat{j}+6\hat{k}\)

If \(\overrightarrow{a}\)  and \(\overrightarrow{b}\) having same magnitude and angle between them is 60° and their scalar product is \({1\over2}\) then \(|\overrightarrow{a}|\) is 

(a)

2

(b)

3

(c)

7

(d)

1

Vectors \(\overrightarrow{a}\) and \(\overrightarrow{b}\) are inclined at an angle \(\theta =120^o\). If \(|\overrightarrow{a}|=1,|\overrightarrow{b}|=2,\) then \([(\overrightarrow{a}+3\overrightarrow{b})\times (3\overrightarrow{a}-\overrightarrow{b})]^2\) is equal to

(a)

225

(b)

275

(c)

325

(d)

300

If   \(\overrightarrow{a}\)  and   \(\overrightarrow{b}\) are two vectors of magnitude 2 and inclined at an angle 60°, then the angle between   \(\overrightarrow{a}\)  and \(\overrightarrow{a}+\overrightarrow{b}\) is

(a)

30°

(b)

60°

(c)

45°

(d)

90°

If the points whose position vectors \(10\hat{i}+3\hat{j},12\hat{i}-5\hat{j}\) and \(a\hat{i}+11\hat{j}\) are collinear then a is equal to

(a)

6

(b)

3

(c)

5

(d)

8

If \(y={1\over a-z}\) , then \({dz\over dy}\) is

(a)

\((a-z)^2\)

(b)

-(z - a)²

(c)

(z + a)²

(d)

-(z + a)²

If y = mx + c and f(0) =\(f '(0)=1\), then f(2) is

(a)

1

(b)

2

(c)

3

(d)

-3

If x = a sin \(\theta\) and y = b cos \(\theta\), then \({d^2y\over dx^2}\)is

(a)

\({a \over b^2}sec^2 \theta\)

(b)

\(-{b \over a}sec^2 \theta\)

(c)

\(-{b \over a^2}sec^3 \theta\)

(d)

\(-{b^2\over a^2}sec^3 \theta\)

If f(x) = x + 2, then f '(f(x)) at x = 4 is

(a)

8

(b)

1

(c)

4

(d)

5

If g(x) = (x² + 2x + 3) f(x) and f(0) = 5 and \(lim_{x \rightarrow 0}{f(x)-5\over x}=4\), then g'(0) is

(a)

20

(b)

14

(c)

18

(d)

12

The number of points in R in which the function \(f(x)=|x-1|+|x-3|+sin \ x\) is not differentiable, is

(a)

3

(b)

2

(c)

1

(d)

4